+-----------------------+-----------------------+----------------------+
| **Date**              | **Author**            | **Subject**          |
+-----------------------+-----------------------+----------------------+
| April 22, 2007        | Kalb, Kris            | [(no                 |
|                       |                       | subject)](#KrisKalb) |
+-----------------------+-----------------------+----------------------+
| March 19, 2007        | Alsop, Linda          | [reaction to         |
|                       |                       | Preliminary Math     |
|                       |                       | Advisory Document,   |
|                       |                       | Ja                   |
|                       |                       | n.\'07](#LindaAlsop) |
+-----------------------+-----------------------+----------------------+
| March 12, 2007        | Garelick, Barry       | [Comment on NMP      |
|                       |                       | Preliminary          |
|                       |                       | Rep                  |
|                       |                       | ort](#BarryGarelick) |
+-----------------------+-----------------------+----------------------+
| February 28, 2007     | Migdal, Paul          | [Math                |
|                       |                       | Panel](#PaulMigdal)  |
+-----------------------+-----------------------+----------------------+
| February 24, 2007     | Forman, Frank         | [Eureka: Human brain |
|                       |                       | region functions     |
|                       |                       | like digital         |
|                       |                       | compu                |
|                       |                       | ter,](#FrankForman3) |
|                       |                       |                      |
|                       |                       | [says CU-Boulder     |
|                       |                       | profe                |
|                       |                       | ssor](#FrankForman3) |
+-----------------------+-----------------------+----------------------+
| February 24, 2007     | Forman, Frank         | [Eureka: Why do      |
|                       |                       | black and Latino     |
|                       |                       | boys lag behind in   |
|                       |                       | m                    |
|                       |                       | ath?](#FrankForman2) |
+-----------------------+-----------------------+----------------------+
| February 24, 2007     | Forman, Frank         | [Eureka: Studies     |
|                       |                       | yield insight into   |
|                       |                       | the numerical        |
|                       |                       | brain](#FrankForman) |
+-----------------------+-----------------------+----------------------+
| December 23, 2006     | Forman, Frank         | [Frank H. Heppner,   |
|                       |                       | Karen R. Kouttab,    |
|                       |                       | William Croasdale:   |
|                       |                       | Inquiry: Does it     |
|                       |                       | Favor the Prepared   |
|                       |                       | Mind? (with reply    |
|                       |                       | and                  |
|                       |                       | respo                |
|                       |                       | nse)](#FrankForman2) |
+-----------------------+-----------------------+----------------------+
| December 18, 2006     | Cotter, Joan          | [Comments to         |
|                       |                       | NMP](#JoanCotter2)   |
+-----------------------+-----------------------+----------------------+
| November 22, 2006     | Johnston, Ron and     | [The Role of English |
|                       | Linda                 | Instruction in       |
|                       |                       | Mathematics](#       |
|                       |                       | RonAndLindaJohnston) |
+-----------------------+-----------------------+----------------------+
| November 5, 2006      | Shacter, John         | [How to Make MAJOR   |
|                       |                       | (not just Marginal)  |
|                       |                       | Educational          |
|                       |                       | Improvem             |
|                       |                       | ents](#JohnShacter4) |
+-----------------------+-----------------------+----------------------+
| October 30, 2006      | Moschkovich, Judit N. | [Statement for       |
|                       |                       | National Math        |
|                       |                       | Panel                |
|                       |                       | ](#JuditMoschkovich) |
+-----------------------+-----------------------+----------------------+
| October 27, 2006      | Treloar, TJ           | [Nat\'l Math         |
|                       |                       | Panel](#TJTreloar)   |
+-----------------------+-----------------------+----------------------+
| October 22, 2006      | Carthel, Chris        | [Comments for the    |
|                       |                       | National Math        |
|                       |                       | P                    |
|                       |                       | anel](#ChrisCarthel) |
+-----------------------+-----------------------+----------------------+
| October 17, 2006      | Shacter, John         | [Re: Ideas for the   |
|                       |                       | National Math        |
|                       |                       | P                    |
|                       |                       | anel](#JohnShacter2) |
+-----------------------+-----------------------+----------------------+
| October 16, 2006      | Marshall, John        | [RE: Some views of   |
|                       |                       | US                   |
|                       |                       | math](#JohnMarshall) |
+-----------------------+-----------------------+----------------------+
| October 13, 2006      | Grebow, David         | [Math Panel          |
|                       |                       | Co                   |
|                       |                       | mment](#DavidGrebow) |
+-----------------------+-----------------------+----------------------+
| October 13, 2006      | Berger, Norman        | [Public Comments on  |
|                       |                       | Math                 |
|                       |                       | Educa                |
|                       |                       | tion](#NormanBerger) |
+-----------------------+-----------------------+----------------------+
| September 08, 2006    | Fretwell, Jack        | [RE: June 29th       |
|                       |                       | Public               |
|                       |                       | Com                  |
|                       |                       | ment](#JackFretwell) |
+-----------------------+-----------------------+----------------------+
| September 04, 2006    | Marshall, John        | [Some Views of US    |
|                       |                       | Math](#JohnMarshall) |
+-----------------------+-----------------------+----------------------+
| September 01, 2006    | Marain, Dave          | [Statement to the    |
|                       |                       | National Math        |
|                       |                       | Panel](#DaveMarain3) |
+-----------------------+-----------------------+----------------------+
| September 01, 2006    | Beck, Bert            | [REVISED Paper       |
|                       |                       | Submitted for        |
|                       |                       | National Mathematics |
|                       |                       | Advisory             |
|                       |                       | Panel](#BertBeck)    |
+-----------------------+-----------------------+----------------------+
| August 03, 2006       | Hu, Arthur            | [Parent For Real     |
|                       |                       | Math](#ArthurHu)     |
+-----------------------+-----------------------+----------------------+
| August 03, 2006       | Marain, Dave          | [Re: Questions to    |
|                       |                       | the                  |
|                       |                       | Panel](#DaveMarain4) |
+-----------------------+-----------------------+----------------------+
| July 28, 2006         | Shacter, John         | [Teaching Math with  |
|                       |                       | the \"KISS\"         |
|                       |                       | Approach             |
|                       |                       | (r                   |
|                       |                       | ev.)](#JohnShacter2) |
+-----------------------+-----------------------+----------------------+
| July 13, 2006         | Shacter, John         | [Teaching Math with  |
|                       |                       | the \"KISS\"         |
|                       |                       | App                  |
|                       |                       | roach](#JohnShacter) |
+-----------------------+-----------------------+----------------------+
| June 12, 2006         | Johnson, Doug         | [Math                |
|                       |                       | Ed](#DougJohnson)    |
+-----------------------+-----------------------+----------------------+
| June 11, 2006         | Avent, Kirk           | [High School         |
|                       |                       | Math](#KirkAvent)    |
+-----------------------+-----------------------+----------------------+
| June 09, 2006         | Hirakawa, Diane       | [Suggestions for     |
|                       |                       | Elem.                |
|                       |                       | M                    |
|                       |                       | ath](#DianeHirakawa) |
+-----------------------+-----------------------+----------------------+
| June 06, 2006         | Good, Pamela          | [Math Skills         |
|                       |                       | Comm                 |
|                       |                       | entary](#PamelaGood) |
+-----------------------+-----------------------+----------------------+
| May 30, 2006          | Berger, Norman        | [Some                |
|                       |                       | ba                   |
|                       |                       | sics](#NormanBerger) |
+-----------------------+-----------------------+----------------------+
| May 26, 2006          | Morris, Larry         | [No                  |
|                       |                       | Su                   |
|                       |                       | bject](#LarryMorris) |
+-----------------------+-----------------------+----------------------+
| May 25, 2006          | Raeth, John S.        | [Algebra             |
|                       |                       | Reform](#JohnRaeth)  |
+-----------------------+-----------------------+----------------------+
| May 21, 2006          | Gray, Marta           | [RE: Curriculum      |
|                       |                       | Dir                  |
|                       |                       | ectors](#MartaGray1) |
+-----------------------+-----------------------+----------------------+
| May 18, 2006          | Gray, Marta           | [Curriculum          |
|                       |                       | Dir                  |
|                       |                       | ectors](#MartaGray2) |
+-----------------------+-----------------------+----------------------+
| May 17, 2006          | Cotter, Joan          | [First Grade         |
|                       |                       | Mathematics          |
|                       |                       | Re                   |
|                       |                       | search](#JoanCotter) |
+-----------------------+-----------------------+----------------------+
| May 15, 2006          | Johnson, Lisa         | [Math Panel          |
|                       |                       | In                   |
|                       |                       | quiry](#LisaJohnson) |
+-----------------------+-----------------------+----------------------+

[]{#KrisKalb .anchor}

\-\-\-\--Original Message\-\-\-\--

From: Kris Kalb \[mailto:kriskalb@patmedia.net\]

Sent: Sunday, April 22, 2007 9:10 PM

To: National Math Panel

Subject: (no subject)

I am an elementary math teacher at Barley Sheaf School in Flemington,
NJ. I have taught elementary mathematics for the past 15 years.

I wish to offer a response to the following paragraph from the National
Mathematics Advisory Panel's Preliminary Report January 2007: "The
discussion about math skills has persisted for many decades. One aspect
of the debate is over how explicitly children must be taught skills
based on formulas or algorithms (fixed step by step procedures for
solving math problems) versus a more inquiry-based approach in which
students are exposed to real-world problems that help them develop
fluency in number sense, reasoning, and problem-solving skills. In this
latter approach, computational skills and correct answers are not the
primary goals of instruction."

It is my sincere hope that the distinguished members of the Advisory
Panel will consult with some of the authors and resource books I quote
below and come to the realization that skill instruction versus inquiry
based instruction in mathematics is not a choice between two opposite
and conflicting ways of thinking, but rather two instructional
approaches that are interconnected and dependent upon each other. Both
instructional approaches have as their main goal correct answers.

John A. Van De Walle wrote in his book /\_Elementary and Middle School
Mathematics: Teaching Developmentally\_/: "In spite of our beliefs that
understanding and skills can and should develop together, we must make
it clear that we assume the primary goal of mathematics instructions is
conceptual understanding. But we must also make it clear that setting
conceptual understanding, as the primary goal does not mean ignoring
computation skills. In fact, we have found that instruction for
understanding can help students construct skills that can be recalled
when needed and be adjusted to fit new situations. Most, if not all,
important mathematics concepts and procedures can best be taught through
problem solving."(p 36)

Marilyn Burns writes in her book /\_About Teaching Mathematics: A K-8
Resource Book\_/: "If teaching computation skills by starting with
numerical symbols is a backward approach, the traditional sequence of
teaching computation first and then applying those skills to problem
situations is just as backward. It doesn't make sense to teach
arithmetic skills in isolation from situations for which those skills
are needed. As supported by research findings, this pedagogical version
of putting the cart before the horse doesn't work." (p12)

Karen C. Fuson writes in Chapter 6\_: Developing Mathematical Power in
Whole Number Operations/ /\_from /\_A Research Companion to Principles
and Standards for School Mathematics:\_/ "For many years, researchers
have contrasted conceptual and procedural aspects of learning
mathematics debating which aspect should come first. Recent research,
however, portrays a more complex relationship between these conceptual
and procedural aspects, concluding that they are continually intertwined
and potentially facilitate each other. These conceptual and procedural
interconnections are forged in individual ways, and attempts to
distinguish between them may not even be useful, because doing and
understanding are always intertwined in complex ways." (p. 68)

Dolores D. Pesek and David Kirshner write in their article
/\_Interference of Instrumental Instruction in Subsequent Relational
Learning\_/ (JRME 2000); "Initial rote learning of a concept can create
interference to later meaningful learning."

Other educators who have written extensively about constructivism in
teaching mathematics are

Cathy Fosnot: /\_Young Mathematicians at Work\_/

Constance Kamii: /\_Young Children Reinvent Arithmetic\_/

Susan Jo Russell: /\_ Beyond Arithmetic: Changing Mathematics in the

Elementary Classroom\_/

In her article "Teaching Fractions: Fostering children's Own Reasoning"
Kamii discusses Piaget's fundamental distinctions among three kinds of
knowledge: physical, social (or conventional) and logicomathematical
knowledge. Since logicomathematical knowledge develops out of children's
own mental actions, they need opportunities to struggle to invent
solutions because it is this thinking that helps them construct new
relationships. Kamii has presented data proving that algorithms are
harmful to the development of children's numerical thinking.

Social knowledge or conventions need to be introduced (told by the
teacher) such as the meaning of symbols, and vocabulary such as geometry
terminology. When planning lessons, teachers need to ask not only what
the content of the curriculum is, but also: where does this knowledge
come from? If the purpose is to introduce a lot of vocabulary (social
knowledge) use the word wall, teach by telling. If the purpose of the
lesson is to help students understand that addition is the total sum of
the parts, they need to have experiences that will let them construct
this understanding inside their own brains.

The questions is not whether to teach skills or inquiry based lessons,
but rather what kind of thinking will the students be expected to use,
and plan lessons accordingly. The correct answer is always the goal, no
matter if the lesson focuses on skills and algorithms or problem
solving, reasoning and sense making.

Thank you,

Kristina S. Kalb

\-\-\-\--Original Message\-\-\-\--\
**From:** []{#LindaAlsop .anchor}Linda Alsop
\[mailto:lindaalsop@hotmail.com\]\
**Sent:** Monday, March 19, 2007 7:50 PM\
**To:** National Math Panel\
**Subject:** reaction to Preliminary Math Advisory Document, Jan.\'07

Dear Tyrell,

I have just finished reading the Preliminary Math Advisory Panel Report
and would like to share my reactions.

Thank you. Linda Alsop

March 19, 2007

Dear Tyrell,

I have just finished reading the Preliminary Math Advisory Panel Report.
I have some comments to make and would very much appreciate it if you
would send a copy of these comments to the panel members for their
review.

I was a speaker at the Chapel Hill meeting in June of 2006 and I can be
reached at

<lindaalsop@hotmail.com> If you or the members of the panel have any
comments or suggestions concerning these viewpoints, please do not
hesitate to contact me.

Thank you, in advance, for any attention to these matters.

As an elementary math specialist and teacher, I am pleased that the
panel is beginning to put their research together to make some
significant suggestions for how we can improve mathematics teaching in
our country. I am thankful that so much work is being done and look
forward to the completed document next winter. However, after reading
the preliminary report from the National Math Advisory Panel, I feel
compelled to write some reaction to the part of the report that reads;
"One aspect of the debate is over how explicitly children must be taught
skills based on formulas or algorithms (fixed, step-by step procedures
for solving math problems) versus a more inquiry-based approach in which
students are exposed to real-world problems that help them develop
fluency in number sense, reasoning and problem-solving skills. In this
latter approach, computational skills and correct answers are not the
primary goals of instruction."

To say that computational skills and correct answers are not an
important or primary goal in an inquiry based approach is absurd.
Obviously, attaining the correct answer and developing computational
skills are an essential part of a balanced mathematics program, and the
above comment from the National Math Advisory Panel is a misleading
representation of inquiry-based methods.

Furthermore, as a title one support math teacher with many years of
teaching experience in grades K-5 mathematics, I have had ample
experiences in teaching traditional, reform, and currently a balanced
approach to children of all abilities, backgrounds, and cultures. I
remember using several traditional math series and teaching memorization
of math skills only to repeat the procedure when so many children forgot
over the summer and needed a good two to three months reviewing concepts
that should have been learned from the previous year in math class. I
remember when NCTM began their work on the Standards to address these
issues. Too many of our youngsters did not have the basics in concepts
or skills to be able to adequately compute or problem solve in their
prospective grades. The reform movement grew out of the need to help
children internalize their learning so that they could remember, apply,
and move on to more efficient strategies and rigorous concepts. The
inquiry method of teaching was a way to enable

children to take charge of their own learning, developing deep
understandings where they could articulate, and justify their problem
solving strategies. The emphasis is and was totally centered

around getting the right answer -- for math is not a subject of opinion,
it is centered in correct answers. The inquiry approach enables students
to place their discoveries in their long-term memories, ready for easy
retrieval as new skills are addressed. A balanced program is designed
for the teacher to carefully use inquiry and through analyzing
assessments and communication of strategies, to know when and how skills
need to be practiced so that the concepts are firmly embedded.
Computational skills and correct answers have always been a focus for a
good math teacher.

Last year all of my student support title I students graduated from
partial proficient into proficient level on their state testing, with
half of them moving from partially proficient to advanced proficient in
one year. My job, along with the classroom teacher, was to motivate
these students to work diligently, not just on memorizing their facts,
but in risk taking initiatives to problem solve using invented
strategies and then revising those strategies to become as efficient as
possible so that the least amount of energy was spent on trivial math
and more could be spent on deeper thinking. Both of us were in a math
master's program. I have since graduated. I can tell you that we are on
to something as we employ similar techniques to this year's student
support population as well as the advanced thinkers in our fourth grade
class.

I would hate to see us take a step back in our mathematics pedagogy --
this we cannot afford to do given the circumstances that so many of our
students are mathematically ignorant.

I am sure that your research includes successful math practices that
Asian countries utilize as they educate their students. Presently, I use
many Singapore and Japanese parts to whole strategies and diagramming
techniques in my own teaching. However, there is a renewed interest in
many of these countries to investigate the advantage of including
inquiry-oriented mathematics so that their students not only perform
well on a math test, but they are creative mathematical thinkers. I
refer you to an article written in 2005 by Kwon, Oh Nam from Korea
entitled. "Towards Inquiry-Oriented Mathematics Instruction in the
University." It is my sincerest hope that your research will not
overlook some of the outstanding benefits from the inquiry process and
that we can come to a logical balance so that future teachers utilize
the best methods for long-term mathematical retention and motivation. We
need a future of students who are so excited and competent in math that
they choose courses and occupations that allow them to rise to their
mathematical potentials.

Sincerely,

Linda Alsop

Student Support Math Specialist

Francis A. Desmares School

<lalsop@frsd.k12.nj.us>

\-\-\-\--Original Message\-\-\-\--\
**From:**[]{#BarryGarelick .anchor} Barry Garelick
\[mailto:barryg99@yahoo.com\]\
**Sent:** Monday, March 12, 2007 3:56 PM\
**To:** Flawn, Tyrrell\
**Subject:** Comment on NMP Preliminary Report

Ms Flawn:

I have just read through the preliminary report of the National Math
Advisory Panel.  I wish to express a concern I have regarding a
discussion that appears on pp 1 and 2 of the report.  The two paragraphs
of concern are:

 

\"The discussion about math skills has persisted for many decades.  One
aspect of the debate is over how explicitly children must be taught
skills based on formulas or algorithms (fixed, step-by-step procedures
for solving math problems) versus a more inquiry-based approach in which
students are exposed to real-world problems that help them develop
fluency in number sense, reasoning, and problem-solving skills.  In this
latter approach, computational skills and correct answers are not the
primary goals of instruction.

 

\"Those who disagree with the inquiry-based philosophy maintain that
students must first develop computational skills before they can
understand concepts of mathematics.  These skills should be memorized
and practiced until they become automatic.  In this view, estimating
answers is insufficient and, in fact, is considered to be dependent on
strong foundational skills.  Learning abstract concepts of mathematics
is perceived to depend on a solid base of knowledge of the tools of the
subject.  Of course, teaching in very few classrooms would be
characterized by the extremes of these philosophies.  In reality, there
is a mixing of approaches to instruction in the classroom, perhaps with
one predominating.\"

 

 I am concerned with the last two sentences of the second paragraph. 
The statements that extremes of either type of these philosophies are
not used exclusively in classrooms and that actually both types are
mixed implies that there is no problem.  To suggest that the
inquiry-based philosophy has had no effect because it has not been used
in its pure form, or because it is mixed with direct instruction is a
specious argument and conveniently sidesteps an extremely significant
issue.

 

The problem is more complex than characterized by these last two
sentences.  First of all, there are degrees of discovery or
inquiry-based learning.  There is general agreement within the
psychological community that knowledge is ultimately constructed by the
learner in order to be absorbed.  But such construction can occur with
passive type learning (i.e., direct instruction) just as it can with
hands-on activities (discovery learning).  Thus all types of learning is
discovery oriented, and one has to look at the gradations of discovery
learning.  Some types have minimal guidance, and other types rely on
structured guidance such as that found in textbooks such as Singapore,
Saxon, or Dolciani.

 

There are a host of math programs being used, however, that are informed
by constructivist theory of the minimal guidance variety, such as
Investigations in Number, Data and Space; Everyday Math, Connected Math,
IMP, Core Plus, and Math Trailblazers.  Some of these programs such as
Investigations, Trailblazers and Everyday Math, do not have textbooks. 
Teachers who must teach from such programs are unwittingly conducted
discovery-based classes by virtue of how the program is put together. 
Students are often not given enough prior information before being
presented with a problem that they must solve in group work, leading to
inefficient solutions.

 

Furthermore such programs typically do not teach to mastery
since students will be exposed again next year to the same topic through
\"spiraling.\"   The \"spiraling\" concept is picked up by other texts
and programs, which then engenders the use of discovery in classrooms,
since mastery is no longer as pertinent as it once was.  The last two
sentences would seem to ignore the highjacking of math programs going on
because of the increasing pervasiveness of the inquiry-based
philosophy. 

 

I would hope that consideration is given to better characterizing the
discussion of inquiry-based learning versus direct instruction.

 

Barry Garelick

\-\-\-\--Original Message\-\-\-\--\
**From:** []{#PaulMigdal .anchor}Palmig \[mailto:palmig@earthlink.net\]\
**Sent:** Wednesday, February 28, 2007 9:11 PM\
**To:** Graban, Jennifer\
**Subject:** Math Panel

Dear Ms. Graban: I am a retired attorney functioning pro bono as a non
profit consultant with a particular interest in the learning gaps among
income groups in elementary school and the impact on those gaps of
learning transfer from and to mathematics with other subjects especially
reading and music. To that end I have been following the transcripts of
hearings of the Panel. The September 13,2006 transcript at pages 9-10
contains Dr. Clements reference to research showing that "early
mathematics is predictive of . . .later achievement in literacy and
reading." ; "gaps between income groups and between nations . . . ." can
be closed " . . .using research-based developmental trajectories of
mathematical concepts and skills . . . ." ; and " . . .early geometry
work leads to higher mathematics achievement in second grade, . . .
higher literacy achievement and also higher IQ scores." At page 25 Dr.
Boykin requested references to that report. Can you provide me with
those references or with copies of them if not otherwise available. I
assumed Dr. Boykin requested references to a report that covered all of
the above quoted comments by Dr. Clemnents, including the transfer to
reading and IQ enhancement since they are all a part of gap closing --
am I correct? Thank you very much and I apologize for the intrusion.
Paul Migdal

 \-\-\-\--Original Message\-\-\-\--

From: []{#FrankForman3 .anchor}Premise Checker
\[mailto:checker@panix.com\]

Sent: Saturday, February 24, 2007 6:02 AM

To: National Math Panel

Subject: Eureka: Human brain region functions like digital computer,

says CU-Boulder professor

Human brain region functions like digital computer, says CU-Boulder
professor

www.eurekalert.org/pub_releases/2006-10/uoca-hbr100406.php

Public release date: 5-Oct-2006

Contact: Randall O\'Reilly

randy.oreilly@colorado.edu

720-938-0523

University of Colorado at Boulder

A region of the human brain that scientists believe is critical to

human intellectual abilities surprisingly functions much like a

digital computer, according to psychology Professor Randall

O\'Reilly of the University of Colorado at Boulder.

The finding could help researchers better understand the

functioning of human intelligence.

In a review of biological computer models of the brain appearing in

the Oct. 6 edition of the journal Science, O\'Reilly contends that

the prefrontal cortex and basal ganglia operate much like a digital

computer system.

\"Many researchers who create these models shun the computer

metaphor,\" O\'Reilly said. \"My work comes out of a tradition that

says people\'s brains are nothing like computers, and now all of a

sudden as we look at them, in fact, in a certain respect they are

like computers.\"

Digital computers operate by turning electrical signals into binary

\"on and off states\" and flexibly manipulating these states by using

switches. O\'Reilly found the same operating principles in the

brain.

\"The neurons in the prefrontal cortex are binary \-- they have two

states, either active or inactive \-- and the basal ganglia is

essentially a big switch that allows you to dynamically turn on and

off different parts of the prefrontal cortex,\" O\'Reilly said.

The brain as a whole operates more like a social network than a

digital computer, with neurons communicating to allow learning and

the creation of memory, according to O\'Reilly.

However, the computer-like features of the prefrontal cortex

broaden the social networks, helping the brain become more flexible

in processing novel and symbolic information, O\'Reilly said.

The prefrontal cortex is the executive center of the brain and

supports \"higher level\" cognition, including decision making and

problem solving. Researchers believe that the prefrontal cortex is

critical to human intellectual ability, and better understanding it

is crucial to understanding more about human intelligence,

according to O\'Reilly.

If researchers can gain a better understanding of this synthesis of

the prefrontal cortex and the brain as a whole, they could be on

the way to a better understanding of human intelligence.

The best way to do this, O\'Reilly says, is by developing more

biologically based computer models of the brain to help researchers

understand how the biology of the brain works, and eventually

provide insights into what makes us so smart.

\"Modeling the brain is not like a lot of science where you can go

from one step to the next in a chain of reasoning, because you need

to take into account so many levels of analysis,\" O\'Reilly said.

O\'Reilly likens the process to weather modeling.

\"Most weather models don\'t exactly represent what happens in a

low-pressure system, but they do capture some global features,\" he

said. \"If you capture the essence of it, it tells you a lot about

how the system works. It\'s the same premise when it comes to

modeling of the brain.\"

\-\-\-\--Original Message\-\-\-\--

From: []{#FrankForman2 .anchor}Premise Checker
\[mailto:checker@panix.com\]

Sent: Saturday, February 24, 2007 6:00 AM

To: National Math Panel

Subject: Eureka: Why do black and Latino boys lag behind in math?

Why do black and Latino boys lag behind in math?

www.eurekalert.org/pub_releases/2006-11/uocp-wdb112806.php

Public release date: 28-Nov-2006

Contact: Suzanne Wu

swu@press.uchicago.edu

773-834-0386

University of Chicago Press Journals

Study shows that patterns of inequality in math at the end of high
school

cannot be explained away by early performance

Recent studies and public discussions have focused on female

achievement in math, and an important new study in the November

issue of the American Journal of Education expands the literature

to encompass racial disparity. Using new national data from the

1990s, Catherine Riegle-Crumb (University of Texas, Austin)

explores how Black and Latino males fare in high school math

classes compared to their female counterparts, finding that a

tendency to ignore institutional cues can lead to both positive and

negative outcomes. While Black males are not encouraged by high

grades in freshman math classes, Black females are able to overcome

potentially demoralizing scores.

Compared with white males, African American and Latino males

receive lower returns from taking Algebra I during their freshman

year, reaching lower levels of the math course sequence when they

begin in the same position, Riegle-Crumb writes. This pattern is

not explained by academic performance, and, furthermore,

African-American males receive less benefit from high math grades.

Riegle-Crumb tracked the progression of more than 8,000 students

who enrolled in Algebra 1 as freshmen in high school. Black and

Latino groups have lower enrollment rates in math courses than

Whites and Asian Americans, but attrition was unexpectedly high

even among those who began in comparable positions. Black males

seem to have little response to positive feedback or good grades,

Riegle-Crumb finds, while Black females seem undeterred by low

grades, despite their original disadvantage.

Her findings support the idea that minority students may be less

responsive to institutional feedback whether positive or negative.

Researchers have argued that minority students may reject the

educational system. Black students may feel uncomfortable and

unsupported in academically intense environments dominated by white

students. Furthermore they may experience a phenomenon called

stereotype threat that is, buying into negative academic

stereotypes about their race-ethnicity.

While African American and Latino students of both genders

generally start high school in lower math courses compared with

their white peers, for minority female students, this appears to be

the primary hurdle to reaching comparable levels of math with white

female students by the end of high school, Riegle-Crumb writes.

She continues, The same cannot be said for African American and

Latino males. Like their female peers, they are less likely to

begin high school in Algebra I. Yet their disadvantage does not end

there but is exacerbated by the lower returns from Algebra I they

receive compared with white male peers.

\###

Founded as School Review in 1893, the American Journal of Education

\-\-\-\--Original Message\-\-\-\--

From: []{#FrankForman .anchor}Premise Checker
\[mailto:checker@panix.com\]

Sent: Saturday, February 24, 2007 5:59 AM

To: National Math Panel

Subject: Eureka: Studies yield insight into the numerical brain

Studies yield insight into the numerical brain

www.eurekalert.org/pub_releases/2007-01/cp-syi011207.php

Public release date: 17-Jan-2007

Contact: Erin Doonan

edoonan@cell.com

617-397-2802

Two studies in the January 18, 2007, issue of the journal Neuron,

published by Cell Press, shed significant light on how the brain

processes numerical information\--both abstract quantities and their

concrete representations as symbols. The researches said their

findings will contribute to understanding how the brain processes

quantitative information as well as lead to studies of how

numerical representation in the brain develops in children. Such

studies could aid in rehabilitating people who suffer from

dyscalculia\--an inability to understand, remember, and manipulate

numbers. The researchers also said their findings offer insight

into the mystery of how the brain learns to associate abstract

symbols precisely with quantities.

Both studies reveal in unprecedented detail how structures in the

parietal cortex\--the region of higher cognitive processing just

above the forehead\--activates during perception of both abstract

quantities and numerical symbols.

In one paper, Manuela Piazza and colleagues showed that regions of

the parietal lobe activate in response to numbers, either when they

are presented as patterns of dots or as Arabic numerals.

In their experiments, the researchers asked human volunteers to pay

attention to the quantities conveyed by groups of dots or numeric

digits presented to them. During the process, the subjects\' brains

were scanned using functional magnetic resonance imaging (fMRI), in

which harmless magnetic fields and radio waves are used to measure

blood flow in brain regions, which reflects activity.

The researchers found that the initial presentation of the numeric

stimuli activated the parietal region of the subjects\' brains,

which subsided as they adapted to the stimulus. However, the

activation rebounded when the subjects were presented with an

abrupt change in the quantity, whether it was represented in the

same (dots versus dots) or different (dots versus Arabic numerals)

notation as the original. This rebound indicated that the region

was processing numerical information.

However, to unambiguously establish that the subjects\' brains were

really reacting to numerical quantity, the researcher occasionally

injected a \"deviant stimulus\" into the second presentation of a

quantity as the brain was adapting to it. This deviant stimulus

consisted of a different number that was either close to, or far

from, the number being presented. The researchers found that this

deviant quantity interrupted adaptation more if it was distant from

the adaptation quantity than if it was closer\--conclusive evidence

that the subjects were processing numerical quantities.

The researchers concluded that their findings \"indicate an

important role for parietal cortex in the coding of symbolic and

nonsymbolic quantities.\"

They also concluded that \"crucially, we observed crossnotation

adaptation and recovery, particularly in the right parietal cortex,

supporting the idea that shared neural populations encode

nonsymbolic quantities and symbolic stimuli.\" Piazza and colleagues

also concluded that their findings shed light on how the brain

learns to associate symbols with numbers.

\"Our results show that, at least in the adult brain, numerical

symbols and nonnumerical numerosities converge onto shared neural

representations,\" they wrote. \"Perhaps we attach meaning to symbols

by physically linking populations of neurons sensitive to symbol

shapes to preexisting neural populations holding a nonsymbolic

representation of the corresponding preverbal domain (e.g.,

numerosity).\"

In the other paper in Neuron, Roi Cohen Kadosh and colleagues

conducted experiments demonstrating that the two hemispheres of the

parietal lobe function differently in processing numbers. While the

left lobe harbors abstract numerical representations, the right

shows a dependence on the notation used for a number, they found.

The researchers concluded that \"results challenge the commonly held

belief that numbers are represented solely in an abstract way in

the human brain.\" The authors also concluded that their results

\"advocate the existence of distinct neuronal populations for

numbers, which are notation dependent in the right parietal lobe.\"

In their experiments, the researchers also used the adaptation

phenomenon, that the brain adapts to stimuli by reducing its

initial activity\--and that repeating the same quantity leads to

reduced activation compared to changing the quantity. They asked

subjects whose brains were being scanned using fMRI to view

consecutive numbers presented on a screen that represented either

the same or different quantities. Crucially, the numbers were also

presented either as two words (e.g., two or eight), two digits

(e.g., 2 or 8), or a mixed notation (two and 8).

They hypothesized \"that if the assumption of an abstract

representation of numbers in the \[parietal cortex\] held true, the

adaptation effect would be observed within and across notations. In

contrast, in the case of nonabstract numerical representation, we

expected that the adaptation effect would be modulated by the

notation type. This result would suggest that distinct neuronal

populations for notation exist.\" This meant that if the brain

region was purely representing an abstraction of a number (e.g., 8)

then any notational representation of this number (e.g., 8 or

eight) would cause an adaptation effect. Alternatively, if a brain

region processed a specific nonabstract number (e.g., 8) then

adaptation would only be seen for the same notation (e.g., 8 but

not eight).

Their analysis revealed an effect of notation in the right parietal

lobe, showing that this region appears to harbor neurons that

process nonabstract numerical representations, in addition to

neurons that code for abstract representations of numeric

quantities.

The researchers said that exploring how the processing of numerical

symbols develops could have clinical implications. \"Developmental

studies should focus on tracing the emergence of numerical

representation in the brain, investigating in particular at which

stage such a representational divergence appears. Such findings

could contribute significantly both to the field of numerical

cognition research and rehabilitation of people suffering from

developmental dyscalculia,\" they wrote.

\###

Cohen Kadosh et al.

The researchers include Roi Cohen Kadosh of Ben-Gurion University

of the Negev in Beer-Sheva, Israel and University College London in

London, UK; Kathrin Cohen Kadosh of Ben-Gurion University of the

Negev in Beer-Sheva, Israel and Birkbeck College in London, UK;

Amanda Kaas of Maastricht University in Maastricht, The Netherlands

and Max Planck Institute for Brain Research in Frankfurt am Main,

Germany; Avishai Henik of Ben-Gurion University of the Negev in

Beer-Sheva, Israel; Rainer Goebel of Maastricht University in

Maastricht, The Netherlands.

This work was supported by grants to R.C.K. from the Boehringer

Ingelheim Fonds, the Zlotowski Center for Neuroscience, and the

Kreitman Foundation.

Piazza et al.

The researchers include Manuela Piazza, Philippe Pinel of INSERM,

Service Hospitalier Frédéric Joliot, CEA, DRM, DSV, and IFR49 in

Orsay, France; Denis LeBihan of Service Hospitalier Frédéric

Joliot, CEA, DRM, DSV and IFR49 in Orsay, France; Stanislas Dehaene

of INSERM, Service Hospitalier Frédéric Joliot, CEA, DRM, DSV and

IFR49 in Orsay, France and Collège de France in Paris, France.

This work was supported by INSERM, CEA, a Marie Curie fellowship of

the European Community QLK6-CT-2002-51635 (M.P.), and a McDonnell

Foundation centennial fellowship (S.D.).

Cohen Kadosh et al.: \"Notation-Dependent and -Independent

Representations of Numbers in the Parietal Lobes.\" Publishing in

Neuron 53, 307314, January 18, 2007. DOI

10.1016/j.neuron.2006.12.025. www.neuron.org

Piazza et al.: \"A Magnitude Code Common to Numerosities and Number

Symbols in Human Intraparietal Cortex.\" Publishing in Neuron 53,

293305, January 18, 2007. DOI 10.1016/j.neuron.2006.11.022.

www.neuron.org

Related preview by Ansari et al.: \"Does the Parietal Cortex

Distinguish Between \"10\", \"Ten\", and Ten Dots?\"

\-\-\-\--Original Message\-\-\-\--

From: Premise Checker \[mailto:checker@panix.com\]

Sent: Saturday, December 23, 2006 8:39 PM

To: National Math Panel

Cc: Ginsburg, Alan; De Kanter, Adriana; Jensen, Sarah; Thomson, Kenneth

Subject: Frank H. Heppner, Karen R. Kouttab, William Croasdale: Inquiry:

Does it Favor the Prepared Mind? (with reply and response)

This does not deal explicitly with math education, but the arguments

about \"inquiry\" vs. traditional methods of instruction go over into

math education as well. It\'s good to have a discussion, rather than

just pronouncements of one point of view. So I pass it on to the Panel

for their consideration.

\[I found this grazing the web. First, there was an article linked by

Arts & Letters Daily, http://aldaily.com, a splendid service now run

by the Chronicle of Higher Education that I check regularly. The

linked article was about critical thinking and came from Skeptical

Inquirer. One of its references was to Johnson, Mathew, and Massimo

Pigliucci (2004), \"Is knowledge of science associated with higher

skepticism of pseudoscientific claims?\" The American Biology Teacher

66(8): 536-548. The answer is that the association is all too weak,

when it does exist. Finding that article led me to other articles in

The American Biology Teacher and what you see below. Serendipity

strikes!\]

++++++++++++

Frank H. Heppner, Karen R. Kouttab, William Croasdale: Inquiry: Does

it Favor the Prepared Mind?

The American Biology Teacher

Volume 68, Issue 7 (September 2006), pp. 390-392

\[reply and response included.\]

A. Department of Biological Sciences, University of Rhode Island,

Kingston, RI 02881, B. North Kingston High School, North Kingston,

RI 02852, C. Department of Education, University of Rhode Island,

Kingston, RI 02881, E-mail: birdman@uriedu

\"Inquiry,\" a loosely-defined constellation of constructivist

teaching methods, has in recent years become the best-practice

science teaching method of choice for providing students with the

means to develop higher order thinking skills while simultaneously

acquiring scientific concepts (Haury, 1993). Inquiry is typically

contrasted with \"direct instruction,\" exemplified by the

traditional college lecture. The National Science Foundation offers

strong support for inquiry-based education: from 2000-2003, 279

funded projects in the Division of Undergraduate Education had the

word \"inquiry\" in their title or abstract. Inquiry instruction is

the benchmark for science teaching in the National Science

Education Standards (National Research Council, 1996). Inquiry has

strong advocates (Handelsman, et al., 2004) who argue that the jury

has essentially returned a favorable verdict on inquiry\'s

superiority to \"traditional, passive methods,\" and that inquiry

needs to become standard practice in college science teaching.

However, little attention has been paid to the role of intellectual

development in students\' capacity to learn profitably in an inquiry

environment. The practical difficulties of inquiry instruction,

especially in elementary and secondary school science, have been

pointed out (Costenson & Lawson, 1986), and we believe that many of

the research studies supporting the effectiveness of inquiry,

especially early ones, do not meet contemporary standards of rigor.

Recently, a contrarian view has been presented that suggests that

at least in some circumstances, direct instruction may be more

effective than inquiry learning in science instruction (Klahr &

Nigam, 2004). We suggest that before American science education

accepts inquiry without further question as a universal teaching

method, it be subject to the same kind of examination that should

perhaps have been given to \"New Math,\" programmed instruction, and

the first generation of \"multi-media.\"

A number of research articles published between 1975-2004 featured

qualitative descriptions or discussions of inquiry practices

(Leonard, 1983; Windschitl & Buttemer, 2000). A few articles

provided quantitative evidence generally in favor of inquiry

(McKinnon & Renner, 1971; Schneider & Renner, 1980), but others

reported no appreciable or significant differences in academic

achievement among the groups compared (Cavallo, 1994; Lunsford &

Herzog, 1997). There is the possibility that some of the success

reported for inquiry methods might have been due to expectancy

phenomena, such as the \"Hawthorne Effect\" (Mayo, 1933), where the

attention paid to the subjects in a performance experiment may

produce the favorable effect noted.

If the academic abilities and skills of the students were not

established through any pretest before the comparison study was

begun, then actual measurements remain as cross-sectional

comparisons only and may reflect improvements students would have

acquired as they matured during the course. Some studies were brief

one-time, one-semester investigations involving small groups of

students (Lock, 1992; Graybill, 1975), sometimes in as few as one

or two classrooms (Tamir & Jungwirth, 1975; Germann, 1989). Does

inquiry-oriented teaching provide students with long-term benefits?

We do not know; the initial research did not follow the progress of

these students into other high school or college courses to

determine if any skills acquired during inquiry instruction

persisted or were used in other subjects.

Inquiry instruction is not new to education theorists or

instructors; in one form or another inquiry learning has been a

focus of pedagogical interest and practice since the turn of the

20th century when Dewey (1944) first emphasized learning through

hands-on practice. Dewey wanted students to become more involved in

their own learning and felt they would pursue multiple areas of

study organized around a central theme if the question the students

chose to solve was of interest to them.

Dewey felt that \"the engaged learner\" would acquire both knowledge

and skills by becoming more involved in his or her own education.

The hypotheses Dewey proposed intrigued many who wanted to move

away from the very basic practice and memorization methods used in

most schools at that time. Various reform movements of the 20th

century would later alternate between an emphasis on skill versus

content.

After the publication of A Nation at Risk, (United States

Department of Education, 1983) another reform movement began that

again emphasized inquiry-centered instruction in the classroom.

Curriculum once more was modified, this time to incorporate more

\"hands-on\" manipulative experiences for students both in the

laboratory setting and in the general classroom (Moreno, 1999). The

term \"inquiry\" began to be used to describe several instructional

methods that could be described as \"student-centered.\" Inquiry

instruction started to require more involvement or active learning

on the part of students and less focus on the teacher\'s direction

and presentation of material.

As additional instructors began to modify instruction to

incorporate inquiry methods, concerns appeared as both students and

teachers attempted to shift their focus from scientific concepts to

scientific process. Teachers reported that inquiry instruction

required additional time and materials to develop, and took

additional time and effort from students (Moss, 1997).

Intellectual Development & Inquiry Return to TOC

Piaget (1964) developed a scheme of intellectual development in

children wherein higher order thinking develops in a series of

demarcated stages. Piaget\'s scheme essentially stopped at

mid-adolescence. Perry (1999) in the 50s and 60s extended Piaget\'s

concept of a staged intellectual development through the college

years into adulthood. In both schemes, an individual at a

particular stage has a characteristic epistemology, and that

epistemology affects how the student views the nature of knowledge,

its acquisition, and the roles of teacher and student. Both schemes

posit that if knowledge is presented to a student in a way that is

not consonant with his/her current epistemology, a variety of

unproductive reactions can result, from confusion to rebellion.

Many of the adverse reactions to inquiry sometimes reported in

inquiry studies are consonant with students at a stage of

intellectual development where they cannot yet recognize that

knowledge is tentative, and that the role of the teacher is as a

guide, rather than transmitter of received wisdom. A student at an

early developmental stage (which might include many students in

high school or earlier, and possibly many college freshmen) might

well react to inquiry teaching with feelings of insecurity and seek

additional guidance/direction from teachers, which in inquiry

teaching might not be forthcoming. Lawson and Johnson\'s (2002)

finding that neo-Piagetian developmental level predicted success in

both inquiry and expository teaching methods in biology suggests

that consideration of developmental level may be important in

evaluating the results of tests on the effectiveness of inquiry.

If the purpose of science education is only to prepare students to

become future scientists, and it is assumed that these students are

in a state of intellectual development appropriate to the method,

inquiry is probably the ideal teaching method. It is an approach

that likely would have had the most appeal to science teachers when

they were themselves students. If, however, science education\'s

goals include an increase in science literacy among the general

public, and creating a favorable impression of science among that

group, universal adoption of inquiry to teach students who are not

yet ready to receive its benefits may well have a paradoxical

effect, and produce results opposite to those intended.

Before consigning \"conventional\" direct-instruction science

teaching to the dustbin, we ask if the real problem with the

lecture as a teaching technique is a boring lecturer rather than

lecturing itself, or the problem with a \"cook-book\" lab is the

recipes in the book.

References Return to TOC

1\. Cavallo A.M.L. 1994. Do females learn about biological topics

by rote more than males? The American Biology Teacher.

56:348-352.

2\. Costenson K., Lawson A.E. 1986. Why isn\'t inquiry used in more

classrooms? The American Biology Teacher. 48:150-158.

3\. Dewey J. 1944. Democracy and Education: An Introduction to the

Philosophy of Education. New York, NY: The Free Press, Division

of Simon and Schuster Inc.

4\. Germann P.J. 1989. Directed-inquiry approach to learning

science process skills: Treatment effects and

aptitude-treatment interactions. Journal of Research in Science

Teaching. 26:237-250.

5\. Graybill L. 1975. Sex differences in problem solving ability.

Journal of Research in Science Teaching. 12:341-346.

6\. Handelsman J., Ebert-May D., Beichner R., Bruns P., Chang A.,

DeHaan R., Gentile J., Lauffer S., Stewart J., Tilghman S.M.,

Wood W.B. 2004. Scientific teaching. Science. 304:521-522.

7\. Haury D.L. 1993. Teaching Science through Inquiry. Available

online at:

www.eric.ed.gov/ERICWebPortal/Home.portal?\_nfpb=true&rERICExtSearch_SearchValue_0=Haury+Inquiry&rERICExtSearch_SearchType_0=kw&r_pageLabel=RecordDetails&robjectId=0900000b801b4a85&raccno=ED359048.

8\. Klahr D., Nigam M. 2004. The equivalence of learning paths in

early science instruction: Effects of direct instruction and

discovery learning. Psychological Science. 15:661-667.

9\. Lawson A.E., Johnson M.A. 2002. The validity of Kolb learning

styles and Neo-Piagetian developmental levels in college

biology. Studies in Higher Education. 27:79-90.

10\. Leonard W.H. 1983. An experimental study of a BSCS-style

laboratory approach for university general biology. Journal of

Research in Science Teaching. 20:807-813.

11\. Lock R. 1992. Gender and practical skill performance. Journal

of Research in Science Teaching. 29:227-241.

12\. Lunsford B.E., Herzog M.J.R. 1997. Active learning in anatomy

and physiology: Student reactions and outcomes in a

nontraditional Anatomy and Physiology course. The American

Biology Teacher. 59:80-84.

13\. Mayo E. 1933. Human Problems of an Industrial Civilization. New

York, NY: Macmillan.

14\. McKinnon J.W., Renner J.W. 1971. Are colleges concerned with

intellectual development? American Institute of Physics.

39:1047-1052.

15\. Moreno N. P. 1999. K-12 Science education reform: A primer for

scientists. BioScience. 49:569-576.

16\. Moss R. 1997. A discovery laboratory for studying gene

regulation. The American Biology Teacher. 59:522-526.

17\. National Research Council. 1996. National Science Education

Standards. Washington, DC: National Academy Press.

18\. Perry W.G. 1999. Forms of Ethical and Intellectual Development

in the College Years: A Scheme. San Francisco, CA: Jossey-Bass.

19\. Piaget J. 1964. Part 1: Cognitive development in children:

Development and learning. Journal of Research in Science

Teaching. 2:176-186.

20\. Schneider L.S., Renner J.W. 1980. Concrete and formal teaching.

Journal of Research in Science Teaching. 17:503-518.

21\. Tamir P., Jungwirth E. 1975. Students\' growth as a result of

studying BSCS biology for several years. Journal of Research in

Science Teaching. 12:263-279.

22\. Windschitl M., Buttemer H. 2000. What should the inquiry

experience be for the learner? The American Biology Teacher.

62:346-350.

23\. United States Department of Education, National Commission on

Excellence in Education. 1983. A Nation at Riske. Washington,

DC\.

++++++++++++++++=

Rodger W. Bybee: How Inquiry Could Contribute to the Prepared Mind

The American Biology Teacher

Volume 68, Issue 8 (October 2006), pp. 454-457

BSCS, Colorado Springs, CO 80918

The editorial \"Inquiry: Does It Favor the Prepared Mind?\" (Heppner,

Kouttab & Croasdale; September, 2006) argued that inquiry teaching

strategies are ineffective because students are not developmentally

prepared for such educational approaches. The authors suggest that

the research base for inquiry is inadequate and they propose that

direct instruction is the viable alternative. They make the latter

recommendation based on the research of David Klahr and Milena

Nigam (2004). On balance, I found the editorial disconcerting. My

concern is not based on the weakly supported criticism of inquiry

as much as the authors\' failure to construct more than they

criticize. What does favor the prepared mind? Can the authors make

the case that direct instruction favors the prepared mind?

In this brief essay, I will present an alternative by proposing

inquiry-oriented strategies that could contribute to the prepared

mind. I begin with a definition.

Inquiry as presented in science education has several different and

quite distinctive meanings. Beginning with a definition that I

developed using a common dictionary form will help set the

parameters for further discussion.

In.quir.y (In\' kwir\' e) n., pl. ies. 1. An outcome of science

teaching that is characterized by knowledge and understanding of

the processes and methods of science. 2. Outcomes of science

teaching that refer to specific skills and abilities integral to

the processes and methods of science. 3. The instructional

strategies used to achieve students\' knowledge and understanding

of science concepts,principles, and facts and/or the outcomes

described in the aforementioned definitions 1 and 2.

This short statement differentiates between inquiry as teaching

strategies and inquiry as outcomes of science teaching. The

National Science Education Standards (NRC, 1996), for example,

includes inquiry as both content outcomes and teaching strategies.

My discussion here centers on instructional approaches that science

teachers may use to achieve students\' knowledge of science

concepts, an understanding of inquiry, and develop abilities

associated with inquiry. This is how inquiry could contribute to

the prepared mind.

Historically, there always have been individuals and groups

advocating different strategies for teaching science. On one end of

a continuum is direct instruction. Lecture serves as the example of

this teaching method. At the other end of this continuum is full,

unguided inquiry. The extreme position in this view is that

students must discover scientific knowledge themselves without any

guidance from the teacher. In reality, most science teaching is

probably somewhere in the middle of the continuum. Effective

science teaching embodies a variety of strategies and methods. One

difficulty, however, is that terms such as \"direct instruction\" and

\"inquiry learning\" often are argued from either/or positions. This

was the case in the Heppner et al. editorial when the authors

caution against science educators accepting inquiry, without

further question, as a universal teaching method.

Research headed by David Klahr has stimulated review and discussion

of the relative importance of direct instruction and inquiry

learning (Klahr has used the term \"discovery learning\") as

instructional approaches to science teaching (Chen & Klahr, 1999;

Klahr, Chen & Toth, 2001; Klahr & Li, 2005; Klahr & Nigam, 2004).

In a 1999 study, Chen and Klahr investigated an important aspect of

scientific reasoning. They asked the question: What is the

effectiveness of different instructional strategies in children\'s

acquisition of the domain-general strategy, Control of Variables

Strategy (CVS). They had children ages 7-to 10-years-old design and

evaluate experiments after direct instruction about CVS and without

direct instruction, i.e., inquiry learning in the extreme, unguided

form. They reported that with direct instruction children did learn

and could transfer the basic strategy for designing unconfounded

experiments, that is, they could apply CVS (Chen & Klahr, 1999).

Before continuing this discussion of Klahr\'s research I will

introduce a report on the use of the laboratory in high school

science. I later return to Klahr\'s research.

Recently the National Research Council published America\'s Lab

Report: Investigations in High School Science (NRC, 2006). The NRC

proposed the phrase \"integrated instructional units\" to describe

the design of instructional units that carefully combine laboratory

experiences with other types of teaching strategies, including

lectures, reading, and discussion. Research indicates that

integrated instructional units increase students\' mastery of

subject matter compared with other modes of instruction, and, very

importantly, these units aid the development of more sophisticated

aspects of scientific reasoning, increase students\' interest in

science, and somewhat improve students\' understanding of the nature

of science when this goal is explicitly targeted (NCR, 2006, p.

100). All of these are valued goals of science education. Upon

reading this, I immediately made several connections. First,

integrated instructional units had the design features of the SCIS

learning cycle and the BSCS 5E Instructional Model Second,

integrated instructional units were neither exclusively \"direct

instruction\" but may include direct instruction, nor were they

unguided inquiry, but they could include activities and strategies

embodying the essential features of guided inquiry (NRC, 2000).

Third, both the NRC report and David Klahr\'s research claimed

support for their respective strategies as being effective for the

development of some aspects of scientific reasoning, which is a

critical outcome of inquiry-based instruction.

The research methodology used by Klahr and his colleagues actually

paralleled that of an instructional model or an integrated

instructional unit. Although evident in the articles, Klahr and

colleagues concluded that direct instruction was the critical

strategy. The following quotes are from the methodological section

in one of the key articles cited in the direct instruction versus

inquiry learning debate (Chen & Klahr, 1999). The entire

methodology could well be described as an integrated instructional

unit that centers on students learning the key concepts of Control

of Variables Strategy.

The present study consisted of two parts. Part I included

hands-on design of experiments. Children were asked to set up

experimental apparatus so as to test the possible effects of

different variables. The hands-on study was further divided into

four phases. In Phase I, children were presented with materials

in a source domain in which they performed an initial

exploration followed by (for some groups) training. Then they

were assessed in the same domain in Phase 2. In Phases 3 and 4,

children were presented with problems in two additional domains

(Transfer-1 and Transfer-2). Part II was a paper-and-pencil

posttest given two months after Part I. The posttest examined

children\'s ability to transfer the strategy to remote

situations.Chen & Klahr, 1999, p. 4

Dave Klahr and his colleagues present a very well-designed study

that, in my view, most likely used an integrated instructional

approach that closely resembles the BSCS 5E Instructional Model.

With an engagement based on the orientation and hands-on in

introduction to materials, the researchers had the students

continue with an exploration, proceed to an explanation and

demonstration of CVS, and then had the students apply or elaborate

CVS to new situations for which they used the terms assessment and

Transfer-1 and -2.

One could reasonably argue that the research methods employed by

Klahr and his colleagues used instructional sequences that

integrated different strategies but then isolated one strategy,

direct instruction, as the key factor in learning. Others have

generalized these results to claim that direct instruction is the

best way to teach the processes and methods of science and, in the

extreme, all of science. (Adelson, 2004; Cavanagh, 2004; Begley,

2004a, b). Such extreme generalizations based on the methodology

and data of the Klahr studies, in my view, extend beyond the

reasonable limits of the studies. However, the research does

provide insights that may help answer questions about effective

instructional strategies that could be identified as

inquiry-oriented.

How does inquiry-based instruction contribute to the prepared mind?

This, it seems to me, is a reasonable and appropriate question.

Answering the question may advance our understanding of the form

and function of inquiry in science education. Based on recent

reports from the National Research Council (NRC, 1999, 2005, 2006),

I argue that using an integrated instructional sequence that

incorporates varied teaching methods holds the key to a reasonable

and appropriate inquiry-oriented approach that contributes to a

prepared mind.

The design of integrated instructional units requires the careful

selection of activities on the basis of research-based ideas likely

to enhance learning. Laboratory and other experiences are

explicitly linked. As I mentioned earlier, the learning cycle or

the BSCS 5E Instructional Model both meet the design criteria for

integrated instructional units. The strategies used in such units

may include direct instruction, discrepant events, laboratories,

discussions, demonstrations, readings, debates, virtual field

trips, and other activities and methods common to curriculum and

instruction in science.

Table 1 presents linkages among the research of Klahr, the BSCS 5E

Instructional Model, and the essential features of inquiry as

described in the National Research Council report Inquiry and the

National Science Education Standards (NRC, 2000).

Table 1.

Linking Research, Instruction & Inquiry

In conclusion, I have tried to bring some clarity to the term

inquiry as it applies to school science programs and the

preparation of young minds. First, teaching science as inquiry

includes understanding scientific inquiry and developing the

cognitive abilities associated with the processes and methods of

science. Second, inquiry can refer to an integrated and linked

instructional sequence designed with the intention of helping

students learn science concepts, as well as understanding inquiry

and developing cognitive abilities aligned with inquiry. It is past

time to move beyond the old either/or arguments of inquiry vs.

direct instruction. Science teachers have always used multiple

strategies, so we need not make a decision about the one best

strategy for teaching science. There isn\'t one, there are many

strategies that can be applied to achieve different outcomes.

Science teachers should try to sequence them in coherent and

focused ways. This is how inquiry can contribute to the prepared

mind.

References

1\. Adelson R. 2004. Instruction versus exploration in science

learning: Recent psychological research calls \"discovery

learning\" into question. Monitor. On Psychology. 35(6): 34.

2\. Begley S. 2004a, 10 December. The best ways to make

schoolchildren learn? We just don\'t know. Wall Street Journal.

pp B1.

3\. Begley S. 2004b, 17 December. To improve education, we need

clinical trials to show what works. Wall Street Journal. pp B1.

4\. Cavanagh S. 2004. NCLB could alter science teaching. Education

Week. 24(11): 1 12-13.

5\. Chen Z., Klahr D. 1999. All other things being equal:

Acquisition and transfer of the control of variables strategy.

Child Development. 70(5): 1098-1120.

6\. Klahr D., Chen Z., Toth E. 2001. Cognitive development and

science education: Ships that pass in the night on beacons of

mutual illumination? In Klahr D., Carver S. Cognition and

Instruction: 25 Years of Progress. Mohwak, NJ: Erlbaum.

7\. Klahr D., Nigam M. 2004. The equivalence of learning paths in

early science instruction: Effects of direct instruction and

discovery learning. Psychological Science. 15(10): 661-667.

8\. Klahr D., Li J. June 2005. Cognitive research and elementary

science instruction: From laboratory, to the classroom, and

back. Journal of Science Education and Technology. Vol. 14(No.

2):

9\. National Research Council 1996. National Science Education

Standards. Washington, DC: The National Academy Press.

10\. National Research Council 2000. Inquiry and the National

Science Education Standards: A Guide for Teaching and Learning.

Washington, DC: The National Academy Press.

11\. National Research Council 2006. America\'s Lab Report:

Investigations in High School Science. Washington, DC: The

National Academy Press.

++++++++++++++++

Frank Heppner, Karen Kouttab, William Croasdale: Response

The American Biology Teacher

Volume 68, Issue 8 (October 2006), pp. 457-457

Our intent in this editorial was to provoke discussion on four

points: 1. \"Inquiry\" is the dominant paradigm for the science

education establishment today, but this one-size-fits-all approach

may not produce beneficial results in all situations. 2. \"Inquiry\"

has been so broadly defined that it is difficult to predict its

usefulness in a given case because different research-based studies

may have been testing different phenomena. 3. Where studies have

shown less-than-stellar results for inquiry, developmental factors

may have been a determining factor; intellectual development has

been understudied in the applicability of inquiry. A study on

effectiveness of teaching methods for college juniors may not have

great utility in predicting its value for 4th graders and

vice-versa. 4. In enthusiasm for inquiry as the most effective

approach to teaching science both to prospective scientists and the

general student population, there has been general disregard for

the outstanding success of traditional methods in producing the

large generation of very capable and imaginative scientists who

grew up in the \'50s through the \'70s, most of whom learned their

science from didactic sages-on-a-stage, some inspiring, most not.

In his very thoroughly prepared response, Bybee does a better job

than we did of documenting the wide variety of ideas that have

appeared under the blanket term \"inquiry.\" However, by including so

many approaches under the heading inquiry, the word becomes almost

synonymous with \"education,\" and any argument over the superiority

of inquiry versus direct instruction or vice versa becomes moot.

For example, he proposes that the following quality be included as

part of the definition of inquiry, \"\... inquiry includes

understanding scientific inquiry and developing the cognitive

abilities associated with the processes and methods of science.\" A

good conventional science lecturer using Socratic teaching methods

can accomplish this, too.

We did not argue, as Bybee suggests, that inquiry strategies are

ineffective because students are not developmentally prepared. To

the contrary, we proposed that in studies where inquiry showed no

clear edge over conventional teaching, these results may have been

due to developmental factors, rather than a problem with inquiry

itself. Further, we did not suggest that direct instruction is the

viable alternative to inquiry. Rather, we posited that good direct

instruction may be more effective than mediocre inquiry activities

in a given population.

We are not inquiry opponents, and we do not believe our piece

suggests this. In our collective 80 years of teaching high school

and college science, we have regularly used (and still use) inquiry

activities. However, we have discovered that inquiry does not

produce positive results for all students all the time, and the

teaching approach has to be tailored to the characteristics of the

intended audience. One of our goals in offering this editorial was

to suggest that in some circumstances, a good, well-prepared (and

perhaps even inspiring) lecturer can be just as effective (or more

so) as a pre-packaged inquiry activity in helping students

understand the process of science, and to force all science teaching
into the inquiry mold may well be counterproductive in specific cases.

\-\-\-\--Original Message\-\-\-\--\
From: Joan Cotter\
Sent: Monday, December 18, 2006 10:14 PM\
To: National Math Panel\
Subject: Comments to NMP

Dear Jennifer:

I am unable to attend the Fifth Meeting of the National Math Panel, but
I\'d to provide some comments.

·       Focus Area: Learning processes

·       Name: Joan A. Cotter, Ph.D.\
·       Title: Curriculum Developer\
·       Organization you represent: Activities for Learning

To get a good start in learning mathematics, children need the
following:

Group in Fives and Tens.

Grouping in fives, as well as tens, makes it possible to recognize
quantities and see them in one\'s mind. This the Romans (VIII for 8) and
composers (5 lines per staff) knew.

Avoid Counting for Adding and Subtracting.

Babies at 5 months can add and subtract up to 3 and they are not
counting. Japanese children learn to add 4 + 3 by first seeing both
quantities mentally; then taking 1 from the 3, combining it with 4 to
make 5 and 2, which they knew. They also learn to add 6 + 7 by seeing 6
as 5 + 1 and 7 as 5 + 2; the two 5s make 10, so the sum is 13. The
Japanese have research that shows the ability to count has little
correlation to math achievement or to progress in the Piagetian
conservation tasks. We must cease to make math attainment dependent upon
one\'s rote memory capabilities.

Name Numbers Explicitly.

All Asian children learn their math by saying \"ten-1\" for eleven,
\"ten-2\" for twelve, \... \"9-ten 9\" for ninety-nine. They understand
place value, the most important concept in arithmetic, early in first
grade. The average U.S. student does not comprehend place value until
the end of fourth grade. By using explicit naming for a few months (my
research shows), U.S. students will understand place value years sooner.

Teach the Thousands in First Grade.

The basic pattern\--math has been called the science of patterns\--that
10 ones equal 1 ten, 10 tens equal 1 hundred, 10 hundreds equal 1
thousand, and so on, cannot be gleaned when the highest number discussed
is 99.  Four- and five-year-olds work with thousands throughout the
world in Montessori classrooms.

Use Correct Vocabulary.

There is no rational reason to use \"number sentence\" instead of
equation. Scientists and engineers do not ponder number sentences. The
word* equation* emphasizes the equality between the two sides of an
equation, a fundamental algebra concept.

*\"*Take away\" limits understand of subtraction because subtraction it
is not always about \"going down.\" For example, when we make change, we
go up.

\"Regrouping\" is not a mathematical term and does not imply equality to
a child. Actually, it is what a military unit does after a defeat. But
every child understands \"trading\"; so, we *trade* 10 tens for 1
hundred.

Who\'s Teaching the Math?

Many teachers assign math homework that often requires a parent\'s help.
For a variety of reasons, over-burdened parents in many families are
unable to provide this help. The achievement gap will not disappear
until the teachers do the math teaching for every child.

Joan A. Cotter, Ph.D.

\-\-\-\--Original Message\-\-\-\--\
[]{#RonAndLindaJohnston .anchor}From: Ron and Linda Johnston\
Sent: Wednesday, November 22, 2006 2:00 PM\
To: National Math Panel\
Subject: The Role of English Instruction in Mathematics\
\
\
Dear Panel Members,\
\
Please accept my paper on what I believe to be the missing link in\
mathematics education. The two page paper is being submitted as an\
attachment in PDF format. Thank you for this.opportunity to bring so\
important a matter to your attention. .\
\
Sincerely, Ron Johnston

\-\-\-\--Original Message\-\-\-\--\
From: John Shacter\
Sent: Sunday, November 05, 2006 1:28 AM\
To: National Math Panel\
Subject: How to Make MAJOR (not just Marginal) Educational Improvements

Dear Members of National Math Panel -\
\
I wonder whether you shouldn\'t make your math-teaching recommendations
in the context of the following, somewhat broader challenge.\
\
After all \-- how can we decide what and how to teach **anything
(including math)**, except in the context of an accepted mission and of
specific performance criteria.\
\
I have sent you several more specific memos on the teaching of math,
money management, savings and stock investments, etc.,\
and I shall be glad to try to answer your questions or expressions of
interest.\
\
Best wishes - John Shacter\
\
=========================

HOW TO MAKE MAJOR (not just MARGINAL) EDUCATIONAL IMPROVEMENTS

Drafted by John Shacter, 11-04-06, for public consideration and,
hopefully, action.\
\-\-\-\-\-\-\-\-\-\-\-\-\-\--\
I shall begin by listing five rather basic assumptions for the reader\'s
consideration and approval.  \
\
To start: \-- Organizations without declared and accepted missions are
condemned to flounder and misallocate their always limited, precious
resources.\
There must be some reason for having ANY educational system.\
In order to develop and implement an acceptable reform package, it is
thus essential that we first agree on the basic mission for
\"education.\"\
\
**1. I PROPOSE THE FOLLOWING OVERALL MISSION FOR \"EDUCATION\":**

TO FACILITATE AND PROVIDE THE MEANS FOR STUDENTS (YOUNGSTERS AND ADULTS)
TO ENJOY QUALITY LIVES AND CAREERS IN TODAY\'S AND TOMORROW\'S
\"SHRINKING\" AND EVERMORE DEMANDING AND COMPETITIVE \"OUTSIDE\" WORLD.

(We shall elaborate on that mission statement, below. I don\'t claim any
originality for this draft-statement. However, there has been
demonstrably a surprising amount of confusion or outright bypassing by
the experts on this essential, basic point.)\
\
\-\-\-\-\-\-\--

2\. FROM NOW ON, WE SHOULD ALL AGREE THAT THE REAL COMPETITION IS NO
LONGER AMONG OUR DOMESTIC SCHOOLS, SYSTEMS OR STATES. IT IS BETWEEN OUR
NATION AND A DIVERSE AND RAPIDLY DEVELOPING GROUP OF LEADING NATIONS,
SUCH AS CHINA, INDIA, AND \-- YES \-- FINLAND.

\-\-\-\-\-\-\--

3\. AS PART OF EDUCATION, \"SCHOOLING\" IS A \"SERVICE\" \-- AND AS ANY
SERVICE \-- WHEN WE REVIEW ITS PERFORMANCE, OR HOW IT COULD BE IMPROVED,
WE SHOULD ALWAYS ASK THE INTENDED CUSTOMERS OF THE SERVICE, ALONG WITH
THE PROVIDERS OF THE SERVICE.

AND IN THE CASE OF \"SCHOOLING,\" SOME OF THE MOST IMPORTANT CUSTOMERS
HAVE GIVEN US THEIR ANSWERS. FOR EXAMPLE, QUALITY EMPLOYERS,
PROFESSIONALS AND LEADERS OF QUALITY UNIVERSITIES HAVE BEEN TELLING US
THAT TOO MANY OF OUR

GRADUATES NEED \"REMEDIAL\" EDUCATION BEFORE THEY CAN BE FURTHER TRAINED
OR EDUCATED.

(We educators don\'t like the term \"remedial.\" We prefer the term
\"developmental\" education.)\
\
\-\-\-\-\-\-\--

4\. BECAUSE OF WHAT THEY READ OR HEAR IN THE MEDIA, MOST OF THE PUBLIC
IS FAIRLY SATISFIED WITH THEIR OWN LOCAL SCHOOLS, ALTHOUGH THEY MAY
EXPRESS SOMEWHAT VAGUELY CRITICAL VIEWS OF THE NATIONWIDE STATUS OR
PERFORMANCE OF \"EDUCATION\" AS A WHOLE.

THEREFORE, ANYONE WHO IS FAMILIAR WITH THESE CONDITIONS SHOULD AGREE
THAT MAJOR IMPROVEMENTS IN OUR SCHOOLING WILL CONTINUE TO BE VIRTUALLY
IMPOSSIBLE TO ACHIEVE, UNLESS A COALITION OF PROFESSIONAL GROUPS, ET AL.
IS FORMED AND WAYS ARE FOUND TO LAUNCH AN EFFECTIVE PUBLIC INFORMATION
AND MOBILIZATION INITIATIVE.  \
\
A BASIC MESSAGE AND SET OF RECOMMENDATIONS WILL THUS ALSO HAVE TO BE
DEVELOPED AND AGREED UPON. MOST OF THE SPECIFICS WILL OF COURSE HAVE TO
BE ADDRESSING VARIOUS APECTS OF ENRICHED AND UPDATED CURRICULUM AND
TEACHING APPROACHES.

\-\-\-\-\-\--

5\. FINALLY, IT IS ESSENTIAL THAT THE MESSAGE AND THE WHOLE REFORM
PROGRAM BE CONSTRUCTIVE AND FUTURE-ORIENTED, NOT CRITICAL AND
PAST-ORIENTED. WE SHOULD NOT BLAME TEACHERS FOR ANYTHING, BUT INDICATE
HOW THEY CAN BE FURTHER DEVELOPED. AFTER ALL, TEACHERS AS WELL AS
STUDENTS HAVE BEEN AMONG THE VICTIMS OF OUR INSTITUTIONAL INADEQUACIES.

\-\-\-\-\-\-\--\
\
Please allow me to elaborate on these points.\
\
\
It has become obvious that \-- for the sake of our quality lives and
careers, if not survival \-- we shall have to prepare most of our
public-school (and college) students more fully for quality jobs,
careers, and any meaningful and enjoyable participation in the ever
wider open and competitive \"flat outside world.\"\
\
So far, we have been losing ground to old and new foreign competition.
For example, whereas the Chinese and Indians turn out hundreds of
thousands of engineers per year, the U.S. is preparing only about
70,000, and a substantial portion of them are either foreign-born or the
children of foreign born parents. (However, we are by far the world\'s
undisputed number-one producer of LAWYERS per capita!)\
\
We must also meet this challenge to our school systems, communities, and
homes if we wish to prevent serious slippages in our economic, political
and security positions, as well as in our share of the world\'s quality
jobs. And we need to provide effective \"catch-up\" and \"enrichment\"
programs for interested adults, as well.\
\
However, this time \-- before we \"invent\" another round of educational
planning and improving \-- let\'s all commit to the following
proposition:\
\
IT SHOULD BE THE CORE MISSION OF OUR PUBLIC SCHOOLS TO PROVIDE THE
FULLEST POSSIBLE PREPARATION OF OUR GRADUATES FOR THE \"OUTSIDE
WORLD.\"\
\
Other interests can be added. However, they should not be allowed to
compete with this core mission.\
\
With 13 years of mandated schooling, not counting any pre-K programs, we
should be able to include the following three target areas as key parts
of the \"outside world\":\
\
A. QUALITY EMPLOYMENT AND CAREER\
B. QUALITY HIGHER EDUCATION OR ADVANCED CAREER TRAINING, AND\
C. QUALITY PARTICIPATION IN FAMILY, COMMUNITY AND SOCIETY.\
\
Once we can assume agreement on the above, let us reconfirm and resolve
that the public-school curricula of all states should include the
following basic and \"spice-up\" categories of knowledge and skills. I
have not attempted to tie each of the following items to just one of the
above three mission targets. Rather, I feel that many of the following
items would meet the requirements of more than one of the above mission
targets:\
\
1. EFFECTIVE COMMUNICATIONS \--\
INCLUDING EFFECTIVE LISTENING AND READING, WITH UNDERSTANDING AND
CRITICAL EVALUATION, FOLLOWED BY CLEAR AND EFFECTIVE SPEAKING AND
WRITING.\
\
Without effective communication, there can be hardly any expression or
payoff for any level of education. We currently list some \-- not all
\-- of these topics under titles like reading, vocabulary, spelling,
language arts, and writing. All too frequently, any or all of these
topics are inadequately presented, appreciated, and practiced. A very
simple spot interviewing process \-- say of customers in a mall \--
could quickly establish the existing gaps among the youngsters and
adults of any community.\
\
(I have in fact developed a list of rather simple questions which could
be applied in this kind of sampling process on this topic and any the
following ones. For example, one or a couple of the earliest questions
should test for effective LISTENING. Obviously, students should be
evaluated for levels of understanding and reasoning, and for clarity and
effectiveness of expression \-- not for the particular views they
express. Some teacher development will undoubtedly be required.)\
\
2. EFFECTIVE QUANTITATIVE OPERATIONS AND REASONING (MATH), including
\"numbers sense\", and including sound choices involving the development
and selection of preferred alternatives for a future with uncertainties
or \"probabilities,\" as well as with facts and data. (See also the next
topic.)\
\
In today\'s rapidly advancing and highly competitive world, this second
category of teaching and learning is almost as essential as the first
area.\
\
3. MONEY MANAGEMENT, including understanding budgeting, determining
profit or loss, assets and liabilities, savings versus stock
investments, etc.\
\
Properly presented, this and the following categories can be also
regarded as highly interesting and challenging \"spice-up\" areas. They
can be combined or interspersed with any current curriculum.\
\
4. INNOVATION, including scientific, technical/engineering, business,
societal/government, and artistic innovations, and successful project or
business startup requirements.\
\
5. PERSONAL SUCCESS REQUIREMENTS, including reliability, punctuality,
consideration for laws and morals, consideration for members of the
family, work-teams, neighborhood, society at large \-- and a willingness
to insist on the adoption of these personal qualities on the part of all
responsible individuals and groups.\
(This item to be applied at once, including a spirit of collaboration
and proper behavior in our schools, buses, etc.)\
\
6. AWARENESS OF THE BASIC ELEMENTS OF OUR COMMUNITY, STATE, NATION AND
WORLD, including key constitutional, political, societal, economic and
cultural factors, and current events, issues, and choices.\
\
7. AWARENESS OF MAJOR, PAST WORLD-WIDE AND U.S. DEVELOPMENTS from
ancient to current times.\
\
8. REPEATED REVISITATIONS OF CHALLENGING CAREER CONSIDERATIONS AND
PLANNING, starting in the elementary grades, with clusters of careers,
interesting and well informed outside volunteers, self-evaluation, etc.\
\
This outline of the proposed next round of educational planning and
improving will be continued in the form of additional memoranda.\
It is not intended to formulate an entirely new curriculum to take the
place of the current state or local curricula, BUT IS INTENDED TO
\"SPICE UP\" THE RATHER BORING AND SEEMINGLY PIECEMEAL OR OVERLY COMPLEX
APPROACHES THAT WE ARE TAKING IN SO MANY OF OUR TEXTBOOKS AND
CLASSROOMS, TODAY!\
AND IT IS ALSO INTENDED TO RELATE THE CURRICULUM AND TEACHING TO THE
REAL, OUTSIDE WORLD.\
\
More complete success will also depend upon the existence of ADULT
LEARNING PROGRAMS, of early PRE-K PROGRAMS, and of enriching AFTERNOON
AND SATURDAY PROGRAMS (like boys and girls clubs), particularly for
children who are in need of community subsidies and support.\
\
Obviously, current and future teachers will have to be introduced to
some new topics and approaches which would greatly broaden their
preparation in today\'s teachers colleges. Until these colleges enrich
their own staff and curriculum, this further development could be
arranged in combination with experienced, perhaps retired, local
volunteer-professionals or military retirees in communications, in the
sciences, in engineering, in enterprise and innovation planning and
management, etc. At least hundreds of age-appropriate, introductory
videos are also available. They should be prescreened and accessible to
every teacher at every public school. (If youngsters know anything, they
know how to watch television, and teachers can push the \"pause\" button
for discussion purposes. Most teachers or supervisors would want to
preview the videos before introducing them to the class. A considerable
fraction of the materials may be almost as \"new\" to the teachers as to
the students.)\
\
The professional volunteers could also assist the teaching process by
participating in teachers workshops as well as in regular classrooms.
Current certification requirements need to be broadened for this
purpose.\
\
The whole program is intended to be implemented in a positive and
constructive \-- not critical \-- school and community environment.
Let\'s make our teaching and learning as exciting, challenging,
profitable and enjoyable, as possible. One of the aims should be to make
our teachers and students look forward to their next day of \"work.\"\
\
©2006 John Shacter; semi-retired engineer, management-and-technology
consultant, and still very active volunteer-teacher and educational
consultant.\
Additional background information can be found in the Who\'s Who volumes
of Science and Engineering, and of Finance and Business.\
(By the way, John received his early primary and secondary education in
Vienna, Austria.)

\-\-\-\--Original Message\-\-\-\--\
From: Judit N. Moschkovich\
Sent: Monday, October 30, 2006 3:55 AM\
To: National Math Panel\
Subject: statement for National Math Panel

Dear Ms. Graban,

I am attaching a statement I wrote to represent TODOS at the November

6th session. Please let me know that you have received this document

and were able to open it.

Thank you for time and consideration,

Dr. Judit N. Moschkovich

Associate Professor

Education Department

University of California

[Moschkovich Math
Panel.doc](../../../../../../../../userdirs/Kevin.McIntosh/TEAMSITE/wdchqdtmsp01/cache/default/main/ED/en/ABOUTED/WORKAREA/work/about/bdscomm/list/mathpanel/documents/moschkovich-math-panel.doc)

\-\-\-\--Original Message\-\-\-\--\
From: TJ Treloar\
Sent: Friday, October 27, 2006 7:53 PM\
To: National Math Panel\
Subject: Nat'l Math Panel

As a speech-language pathologist in the Ventura Unified School District
in Southern California, I work with special ed children every day.  I am
aware of how much energy and funding is spent on these children. 
However, as a mother of two children identified as gifted, I have a
vested interest in the energy and funding spent on their education. 

 

My daughters love math and science, and I am pleased with how excited
they are to learn these subjects!  They need to have teachers who are
well able to keep them interested.  We need my little girls to grow up
to be scientists and leaders of our world. 

 

Please fund programs for gifted education.  As much as we need to raise
the bar for our lower functioning students, it is equally important to
inspire our children who love to learn and are capable of excelling
academically!

 

Thank you,

TJ Treloar

\-\-\-\--Original Message\-\-\-\--\
[]{#ChrisCarthel .anchor}**From:** Chris Carthel\
**Sent:** Sunday, October 22, 2006 10:36 PM\
**To:** National Math Panel\
**Subject:** Comments for the National Math Panel

Dear Sir or Madam:\
\
I am a parent of two math students who are both in high school now. In
addition, I recently completed the mathematics necessary for a Bachelor
of Science degree in Chemistry. I experienced some difficulties in my
earlier math classes, including classes prior to (and including)
algebra, partly because of what I believe were shortcomings in the way
the courses were organized (i.e., curriculum) and taught (i.e.,
pedagogy).\
\
**Problem-Solving: Point A to Point B**\
\
One of the most frequent complaints I hear from my children and college
peers is \"I don\'t see how you got from point A to point B in that
solution.\" I have also had this compliant myself, and I see this as a
recurring problem in both teaching and textbook design.\
\
The better math textbooks (and teachers) I have seen include annotated
solutions that provide a clear explanation of how a problem progressed
to a solution through each intermediate step (e.g., what rule was
applied, what manipulation was performed, etc.). Good annotations of
solution steps provide not only an explanation of how to progress from
step to step, but can even serve as a sort of built-in remediation in
some cases. For example, a step in a calculus problem may require the
utilization of a trigonometric identity. Identification of this in an
annotation not only explains the transition but can also provide a
remedial effect. Annotations are universal in that they can serve every
student's needs. The advanced students can simply ignore them, while the
struggling students can use them to build up their skills. In fact, a
web-based textbook could even be configured so that every student could
decide for themselves whether to turn on or turn off the annotations.\
\
**Use of Technology to Enhance Math Education**\
\
I have seen some truly impressive web-based technologies, such as Java
applets, living graphs, etc., that could enhance the learning of
mathematics. But I have been surprised by the slow infusion of these
technologies into classrooms and textbooks. The value of these new
technologies is that they permit the student to make real-time, two- or
three-dimensional observations of the behavior of equations at different
values and limits. It makes the learning experience more real and
understandable. Although I am aware of some copyright concerns regarding
the use of electronic (e.g., PDF format) textbooks, students badly need
the ability to search and retrieve information as quickly as possible in
an electronic format.\
\
We are, in my opinion, long overdue for an electronic textbook approach
that resembles a web page. I am not necessarily advocating the complete
abandonment of physical textbooks, but perhaps an approach where the
textbook is bundled with an electronic version available via perhaps a
web account that contains the ability for word searches; quick linking
from tables of contents, glossaries, and indexes; interactive JAVA
applet-based figures and graphs where appropriate, etc. (e.g.,
<http://mathworld.wolfram.com/>).\
\
**The Need for National Math Standards**\
\
It is my sincere hope that by defining what is meant by "competence in
algebra" and "readiness for higher levels of mathematics" as described
in Executive Order 13398, Sec. 4.(a), the Panel will be in a position to
provide meaningful guidance for developing national math standards.

The American Chemical Society (ACS) publishes national standards
(including testing standards) for chemistry. I have been surprised to
discover that there is not, at least to my knowledge, an analogous set
of national standards for mathematics.\
\
**Teacher/Student Diligence**\
\
I have seen several brilliantly knowledgeable math teachers who
displayed only mediocre skill at *conveying* their knowledge in an
absorbable way. I have come to realize that student achievement in
mathematics is not simply a function of the teacher's knowledge of
mathematics, although teacher knowledge is certainly important. In my
opinion, the real magic of student achievement occurs as a result of a
teachers' *skill* at *conveying* their knowledge in an interesting and
organized way that can be easily absorbed by engaged students.\
\
Note that I limited my statement to engaged students. There is a
obviously a certain degree of diligence required of math students
themselves. This is what I think of as the \"you can lead a horse to
water, but you can\'t make him drink\" syndrome. The best teacher,
textbook, and curriculum in the world are all worthless to a student who
is not making the conscious choice of engaging themselves in the
learning process by showing up, paying attention, and absorbing,
applying, and practicing as much as they can. Students cannot be
overlooked as participants in the process.\
\
I mention students because I have seen some evidence in our American
culture, in particular K-12 math classes, of what I refer to as
"glorification of mediocrity" or "antagonism of success." In other
words, a peer pressure environment sometimes exists that utilizes
harassment and embarrassment to prevent some promising students from
achieving their full potential. I have seen potentially excellent
students make a conscious choice to perform badly in order to "fit in"
with their less engaged peers. It seems to fit with the old maxim
"misery loves company" This is perhaps better described as "laziness
loves company."\
\
Every child deserves to be freed from the bondage of what President Bush
has described as the "soft bigotry of low expectations," regardless of
whether that bigotry arises from a teaching institution, a specific
teacher, or a fellow student.\
\
Thank you for the opportunity to comment. I look forward with great
anticipation to the panel's conclusions and recommendations.\
\
Chris Carthel

\-\-\-\--Original Message\-\-\-\--\
From: John Shacter\
Sent: Tuesday, October 17, 2006 11:07 AM\
To: National Math Panel\
Subject: Re: Ideas for the National Math Panel

Hi Jennifer -\
\
Thanks a million for your info. on the above.\
Could you distribute my memo (below) to the participants and to Sec.
Spellings?\
\
Thanks, John\
\
John Shacter\
Semi-retired engineer, management-and-technology consultant, and still
very active volunteer-teacher & educ. consultant\
===========================\
\
Hi Folks -\
\
Let me try once more:\
\
(This is not limited to just \"math\") \--  but I cannot agree that we
can discuss what and how to teach without first agreeing on the \"what\"
and then on the \"how.\"\
\
Overall, I believe we tend to omit or play down some of the most
interesting and useful topics and then seem to be succeeding in making
too much of the rest of \"studying\" too boring and laborious.\
\
\-\-\-\-\-\-\-\-\-\-\-\-\-\--\
\
Generally, students also get too little opportunity to help in the
understanding and explanation process, or to express themselves more
generally \-- orally or in writing.   \
\
As teachers, we should be also able to illustrate to the students WHY we
are teaching them the things that we are teaching them.\
\
We should do a lot of teaching by asking a series of leading and teasing
questions (\"Socratic Method\").\
\
Use of common sense, the KISS principle (Keep It Simple, Stupid), a
light touch or sense of humor, and introduction of some competition can
also help to engage the students in the teaching and learning process.\
\
\-\-\-\-\-\-\-\-\-\-\-\-\-\--\
\
Let me illustrate the approach with the teaching of some basic math:\
\
Effective teachers require their 3rd-grade students to memorize the
multiplication table. For example, they want their students be able to
tell them \-- eyeball to eyeball \-- what 7x8 is. But the best teachers
teach more. They might want the students to tell them also what 56
**[DIVIDED BY]{.underline}** 7 or 8 is. And then, when they receive
correct answers, they might also ask the students to tell them what
5**[7]{.underline}** divided by 7 or 8 is \-- EXACTLY, AND STILL
EYEBALLING THEM. (Today, most **[COLLEGE GRADS]{.underline}** and
professors might not know the answer to that one! Chances are, they
might ask for permission to use their calculators!!)\
\
Or the best teachers might ask their students whether integers or whole
numbers could be treated as decimals, as well. (The answer is of course
that they CAN. All you have to do is to place a decimal point at the
right end of the number. After that you can use the decimal rule of
moving the decimal point, say, a couple of places to the right, when you
wish to multiply the number by 100, or to the left, if you want to
divide the number by 100. It takes a movement of six places if you
multiply or divide by one million. And if you don\'t have enough digits,
just add a sufficient number of zeros on either end of the number\....)\
\
Or the best teachers might introduce their students \-- also in third or
fourth grade \-- to the use of question marks or letters along with
numbers in equations \-- without scaring them by the use of the word
\"algebra.\"\
\
\-\-\-\-\-\-\-\-\-\-\-\-\--\
\
Many additional examples could be given in such areas as \"societal and
character education\", \"money management\", \"starting your own
business\", \"U.S. form of government\", \"current events and issues\",
etc. etc. which should be (but are not always) taught and discussed
during the 13 long years of primary- or secondary-school curricula.\
\
In my view, today\'s U.S. teachers aren\'t taught what and how to teach
in more interesting, challenging and effective ways because their
professors in the colleges of education are often themselves lacking in
their selections and understanding of the topics \-- and they are likely
to practice too much \"pseudo-psychology\", and tend to take themselves
much too serious.\
\
In any case, it is my guess that few of them would dare to face an
average primary or secondary classroom and try to demonstrate their own
proposed, conceptual approaches to \"effective teaching\" in qualitative
or quantitative (student-gains) terms.\
\
Fortunately, there are effective answers to these challenges, and there
are some teachers who have taught themselves to apply them.\
\
Cordially, John\
\@2004 John Shacter\
Semi-retired engineer, management-and-technology consultant, and still
very active volunteer-educator.

\-\-\-\--Original Message\-\-\-\--\
**From:** John Marshall \[mailto:jm0603@gmail.com\]\
**Sent:** Monday, October 16, 2006 6:09 AM\
**To:** National Math Panel\
**Subject:** RE: Some views of US math

Ida

 

I had hoped to make further comments for the National Math Panel to
consider but life seems so hectic at present that time doesn't allow.
However, I would like to offer the attached articles from KAPPAN written
on the MATHEMATICS WARS theme, which I hope will make a point.

 

If time permits in the not too distant future I will try a write
further.

 

Good Luck

 

John Marshall

Inspector of School, UK

Lecturer University of South Florida. USA

[Marshall.Math Wars Taking
Sides.pdf](../../../../../../../../userdirs/Kevin.McIntosh/TEAMSITE/wdchqdtmsp01/cache/default/main/ED/en/ABOUTED/WORKAREA/work/about/bdscomm/list/mathpanel/documents/marshall.math-wars-taking-sides.pdf)

[Marshall.Math Wars 2 It\'s the Teaching
stupid.pdf](../../../../../../../../userdirs/Kevin.McIntosh/TEAMSITE/wdchqdtmsp01/cache/default/main/ED/en/ABOUTED/WORKAREA/work/about/bdscomm/list/mathpanel/documents/marshall.math-wars-2-its-teaching-stupid.pdf)

\-\-\-\--Original Message\-\-\-\--\
From: []{#DavidGrebow .anchor}David Grebow\
Sent: Friday, October 13, 2006 4:39 PM\
To: National Math Panel\
Subject: Math Panel Comment

Dear Jennifer:

 

The issue of mathematics readiness needs to being at the Pre-K level.
Mathematics needs to be regarded as a language. And the current research
points to the fact that languages are best learned early, practiced as a
part of one\'s daily life, i.e. made relevant and heard at home as well
as at school. I would throw technology into the mix since many parents
do not know and would not learn basic or advanced math. So the \'in loco
parentis\' in this case would be the mathematics website where math
would be \'spoken\' and, like learning a language, be incorporated into
age and grade appropriate songs, games and other enjoyable activities.

 

I am an educator and have worked at the intersection of education and
technology for many years. Since I know this will work, and that the
technology and pedagogy are in place, all that\'s missing is the will to
make it happen. Hopefully the meetings of the National Math Panel and
their final recommendations will look at the issue from this perspective
and programs will be started to see if this is a good path to travel.

 

Sincerely,

 

David Grebow

CEO, KnowledgeStar. Inc. 

\-\-\-\--Original Message\-\-\-\--\
**From:** Norman Berger \[mailto:nwberger@comcast.net\]\
**Sent:** Friday, October 13, 2006 2:48 PM\
**To:** Graban, Jennifer\
**Subject:** Public Comments on Math Education

Ms Graban,

 

Thank you for providing the opportunity to comment on math education.

 

1\) Early elementary teaching.

 

Children need to be taught that math requires a lot of concentration,
unlike reading. (In reading, running a test of comprehension by randomly
omitting letters on a page allows the reader to still get the gist of
the material until 50% of the letters are removed thus scanning is
possible. This is not true for math. Every single number or operator
needs to be comprehended.)  My suggestions are:

 

A\) In a long list of numbers in a matrix ask the children to circle all
of the 3\'s. Then count how many the found. Then find the missing 3\'s.
On and on.

 

B\) Line up numbers. For instance, if a list of numbers are to be added
together with many columns, demand that each column be in a straight,
vertical line. Use cross-hatch paper if necessary. No slanted lists. The
reason for this is that children get confused when going from column to
column and this eliminates that. It will reduce their frustration
greatly.

 

C\) When adding the above, make them write the \'carry-over\' number at
the top of each column, but lightly.

 

2\) Algebra

 

A\) The abstract concept of algebra if not dealt with in the very
begining haunts students. Some type of visuals representing unknowns
should be used. (I don\'t have a good idea how this should be done,
however.) Lots of time should be spent on discussing algebra before many
problems are given for homework. Let them get used to the language and
to talk about it. The abstractness of algebra should be dealt with.

 

B\) The concept of what can be done with \'equals\' is not properly
imbedded in students. Equals can be added, subtracted, multiplied by and
divided by equals. This concept is clear in teachers\' minds but they
must make absolutely sure that it is understood. Algebra cannot continue
until this is done.

 

C\) Make sure that all work is shown, period. There should be a series
of specific steps shown on paper that leads to an answer. No shortcuts
in the brain because when problems finally get complicated confusion
occurs. For instance the work should look like this:

 

Problem:                                                  4a + 8 =  2a -
2,  Solve for a.

                                                              [-2a     
 = -2a]{.underline}   ,  

                                               Add this:    2a
+ 8 =       -2

                                                            
  [     - 8  =      -8]{.underline}    

                                               Add this:     2a      
=     -10

                            

divide each side by 2,      a       =      -5 

 

You get the idea, step-by-step-by-step.   This allows for clear
thinking.

 

3\) Geometry.

 

Not really a suggestion, but my high school teacher did away with the
textbook. We had to preserve our classroom notes and in effect write our
own text book. It worked, we concentrated.

 

Best of luck on getting kids to be good at math. Nothing raises
self-esteem better than success. We have to make these children
successful

 

Norman Berger

\-\-\-\--Original Message\-\-\-\--\
From: Lisa Brady Gill\
Sent: Thursday, October 12, 2006 6:59 PM\
To: National Math Panel\
Subject: NatMathPanel_TIcomments.pdf

Hi Jennifer:

TI and Melendy Lovett were pleased to receive the invitation to provide
written comments to the National Math Panel.  We are very supportive of
the work of the National Math Panel and appreciate the took the
opportunity to support their work. 

I\'ve enclosed our written comments for your review.  In addition, we
are sending 22 hard copies by Federal Express this evening to you at the
U.S. Department of Education for distribution to the National Math Panel
members.  Can you let me know if they will receive them prior to the
November meeting?

In addition, you were kind enough in your letter to Melendy to suggest
she might have the opportunity to give oral remarks at the November
meeting and we\'d like to formally request that she be able to do so.  I
mentioned this to Tyrrell at the meeting in Boston and let her know
we\'d be following up with this request. 

TI is honored that Richard Schaar has been invited to share
effectiveness research related to graphing calculators at the November
meeting, as well.  And look forward to working towards our shared
missions of improved mathematics education for all students in the
future. 

Thank you for your consideration of these written comments and of our
request for Melendy to give oral remarks for the Math Panel members to
consider as they prepare their report.  And please don\'t hesitate to
contact me should you have further questions or need more information.

Best Regards,\
Lisa Brady Gill

Lisa Brady Gill\
Executive Director, Office of Education Policy and Practice\
Texas Instruments, Incorporated

[NatMathPanel_TIcomments.pdf](../../../../../../../../userdirs/Kevin.McIntosh/TEAMSITE/wdchqdtmsp01/cache/default/main/ED/en/ABOUTED/WORKAREA/work/about/bdscomm/list/mathpanel/documents/natmathpanel_t1comments.pdf)

\-\-\-\--Original Message\-\-\-\--

From: []{#JackFretwell .anchor}Jack Fretwell

Sent: Friday, September 08, 2006 2:51 PM

To: National Math Panel

Subject: RE: June 29th Public Comment

Hi Jennifer,

I can\'t make the Boston meeting and I know it\'s late, but  will you be
able to do anything with the attached comment?

 

Jack Fretwell

Starboard Training Systems

[Not
Teaching.doc](../../../../../../../../userdirs/Kevin.McIntosh/TEAMSITE/wdchqdtmsp01/cache/default/main/ED/en/ABOUTED/WORKAREA/work/about/bdscomm/list/mathpanel/documents/not-teaching.doc)

\-\-\-\--Original Message\-\-\-\--\
[]{#JohnMarshall .anchor}**From:** John Marshall\
**Sent:** Monday, September 04, 2006 4:44 PM\
**To:** National Math Panel\
**Subject:** Some views of US math

 

I would like to make a comment about Mathematics teaching in the USA but
before I do I should introduce myself. My name is John Marshall and my
main residence is in England where I spent the majority of my working
life in the field of mathematics. For my sins I was a teacher, college
lecturer and inspector of schools. More recently I was a 'visiting
lecturer' at the University of South Florida. In recent years I
contributed to the Minnesota K-12 Mathematics Framework as an expert
reviewer, keynote speaker and author. The American magazine Phi Delta
KAPPAN has published four articles of mine, the most recent being in the
January 2006 edition - *Math Wars 2: It's the Teaching, Stupid*. The
NCTM journal Teaching Children Mathematics also published *Educating
Hannah: It's a What?,* in the Geometry special of February 1999, done
with Sir Wilfred Cockcroft.  My wife, of 45 years, and I have a home in
Oldsmar, Florida. Our son spent a year at an American high school before
his University course in the UK.

  In terms of the colloquial Math Wars I would be seen as being on the
reform side but I have to say at the outset that I do not recognize much
of what I see as 'reform based mathematics' in the USA as being
compatible with my experience. As I said in my KAPPAN article of
November 2003; "I find myself in complete agreement about the need for
reform. Let there be no doubt about that. We cannot go on killing so
many young minds like we have for they will grow up and replicate the
problems we now have and nothing will have changed. And yes, the aims
and aspirations of current NCTM Standards are to be commended. I just
don't see how many of the exemplars offered match up with those aims. To
me they just do not seem to reflect the attitude to the
teaching/learning process that the aims talk about and as such are
surely not 'what children need'!"

Personally reform was about offering children a better deal, period. A
better deal because what they were getting was not doing the job for the
world was changing and mathematical education was not keeping up. In the
UK a response was made to the Sputnik issue by looking at what young
children were doing. The Nuffield Foundation Mathematics Teaching
project of the early 1960s was created to produce a contemporary course
in mathematics for children between the ages of 5-13 that would produce
in them a "critical, logical, and creative turn of mind." It was
probably the beginning of MATHEMATICS for young children as before they
had only been offered arithmetic. It is interesting to note that the
parent's book from the project, said, amongst other things, "Whether we
like it or not, our children will be concerned in the future with more
abstract mathematics than their predecessors. The world of computers and
computer programs, of automatic production line processes, or of
operational research by managements, is a far cry from the world of the
nineteenth century clerk, mill-hand or small industrialist. Our most
important task must be to teach children to think mathematically for
themselves. From a gradual awareness of the patterns of ideas lying
behind their practical experiences, there must be built up a willingness
to accept the underlying mathematical ways of thinking which are proving
so vital in the development of modern technological society." That was
written in the mid 1960s and I wonder how it would sit today with those
anti-reform groups who appear to see the aim of school math as getting
their children into University. I am reminded of that Brian and Greg
Walker cartoon which shows a child asking father "Why do we have to do
Algebra?" and getting the reply "So you can help your children with
their homework"! And on the subject of entry into university I would
have to say that all my students in Florida had passed the test but few,
very few, very very few, knew mathematics. They were truly excellent at
passing tests though!

Let me go back in time and make a brief comment on what the UK has done.
In the late 1970s the (Labor) Prime Minister of the day, the Rt. Hon.
James Callaghan. MP, was concerned about the teaching of Mathematics in
schools and set up a panel to look at the problem. The terms of
reference for this committee were : ***To consider the teaching of
mathematics in primary and secondary schools in England and Wales, with
particular regard to the mathematics required in further and higher
education, employment and adult life generally, and to make***

***recommendations."*** The committee, chaired by Sir Wilfred (Bill)
Cockcroft, reported in January 1982, when Mrs. Thatcher (Conservative)
was the Prime Minister. Cockcroft looked at the 'past', looked at the
'present' and looked into the 'future' and then said in paragraph 800 of
the report: "We therefore believe major changes are essential."

(I was happy to receive a signed copy of the report from Sir Wilfred,
who wrote: "To John: With thanks for giving me the chance to see and
listen. Bill" Sir Wilfred could see 'reform' in action because it had
been happening in my development school for well over 20 years.)

The major change that was required was to teach for understanding rather
than by rote, which had been the style for years and years, and this was
addressed in paragraph 238 of 'The Cockcroft Report' -- "We have had
several submissions which have urged that more emphasis should be placed
on 'rote learning' The Oxford English Dictionary defines 'by rote' as
'in a mechanical manner, by routine; especially by the mere exercise of
memory without proper understanding of, or reflection upon, the matter
in question; ... ... we do not believe that it should ever be necessary
in the teaching of mathematics to commit things to memory without at the
same time seeking to develop a proper understanding of the mathematics
to which they relate. As our discussion of memory shows, such an
approach is unlikely to meet with long term success."

Mathematics needs long term success as children move through each stage
of the education system and into adult hood. How often do we hear that
the next stage in the learning process feels let down by the previous
stage. (And kindergarten has been known to blame the parents!!!) An
Inspectors job is strange but privileged. I must have visited over 500
classrooms and in everyone I was told the mathematics was perfect.
(Rather like the situation described in the Stigler and Hiebert book
*The Teaching Gap*, page 123/124)  The overall teaching style I saw was
'rote'. It was how teacher was taught by teachers who had been taught by
rote and it was the way 'she' taught. Principals were proud of the
outcome and could often produce test scores to prove their point. But
all too often the mathematics atrophied by the next stage. This was
perfectly illustrated in one city where the educational provision was in
4 stages: 5-8 yrs, 8-12 years, 12-16 years and 16-18 years. ALL the
subsequent stages found the entry levels were 'not what was expected'!
From within this system I even received complaints that bright
university students struggled to take their mathematics with them: -
"Indeed it is a common, and sometimes somewhat disconcerting, experience
to those embarking on degree courses in mathematics to find that their
understanding of topics which they have tackled with apparent success at
school is questioned and shown to be insufficient." (Mathematics Counts.
page 68 HMSO. London.)

But teaching for understanding is not easy for one has to know
mathematics and how children learn.

In their (US) paper *Reaching for Common Ground in K--12 Mathematics
Education* (Focus January  2006) Deborah Loewenberg Ball, Joan
Ferrini-Mundy, Jeremy Kilpatrick, R. James Milgram, Wilfried Schmid, and
Richard Schaar look for agreement in different sides on the Math Wars.
Although I am not sure who is on which side of the argument, I do
recognise that Joan Ferrini-Mundy has her name on the NCTM's *Principles
and Standards for School Mathematics 2000***^ ^** book and that Richard
Schaar is from Texas Instruments, who make calculators. In their joint
paper the authors assert, quite rightly, that "Mathematics requires
careful reasoning about precisely defined objects and concepts.", and
that students need to "understand the operations". "Precisely defined
objects and concepts" and "understand the operations", who could
disagree? But what does that mean and how does it manifest itself in the
classroom? As far as the key operation of multiplication is concerned
where do NCTM and Calculators stand? Sadly not together although they
claim to be! Let me explain: NCTM says on page 151 of *Principles and
Standards for School Mathematics* "It is important that students
understand what each number in a multiplication and division expression
represents. For example, in multiplication, unlike addition, the factors
in the problem can[^1]^1\]^ refer to different units. If students are
solving the problem 29x4 to find out how many legs are on 29 cats, 29 is
the number of cats (or number of groups) and 4 is the number of legs on
each cat (or the number of items in each group) and 116 is the total
number of legs on all the cats. Modeling multiplication problems with
pictures, diagrams, or

concrete materials, students learn to be clear about what each number in
the problem represents."  I should add that no pictures, diagrams, or
concrete materials are used to make the point. In fact the classroom
picture used (page 142) is very much 'as it used to be' where the
results of mathematics are displayed and not the mathematics itself,
which would include the results! However, this is NOT the definition
that is found in American Dictionaries -- "Multiplication: the process
of finding the number or quantity (product) obtained by repeated
additions of a specified number or quantity (multiplicand) a specified
number of times (multiplier); symbolized in various ways (ex. 3x4=12 or
3.4=12, which means 3+3+3+3=12, to add the number three together four
times)." (Webster's 3rd Edition College Dictionary) - nor in Japanese
text books.

In the statistics section of *Principles and Standards for School
Mathematics* *2000* it says (page 49) "Recognizing that some numbers
represent the values of the data and others represent the frequency with
which those values occur is a big step." It is a huge step because if
one follows the NCTM definition of multiplication and take it forward
into a calculator one finds that 'it doesn't work' as the calculator
follows the definition of multiplication given in American Dictionaries.
Let me quote from my KAPPAN paper of February 2001 *(Dear Verity, Why
are all the dictionaries wrong?)* where I gave an example in which it
was necessary to find the mean height of 10 children, *"6 of whom were
120 cms. tall and 4 were 110 cms. Setting the calculator to the
statistics mode, we keyed the data in as (6x120) + (4x110). You know, '6
lots of 120' \[as per Standards 2000\], etc. Pressing x-bar, to get the
mean, we got 5.04 cms. Now nobody is that size. That's silly. So we then
keyed it in as (120x6) + (110x4), that is 120, 6 times etc., \[*as per
the dictionaries!\] *and got 116, which made sense."* How can this
possibly mean that the Common Ground authors agree?

I have heard it said that because the operation of multiplication is
commutative then 'it doesn't matter'. I note that neither the
dictionaries nor the NCTM guidelines say that for they are quite
specific. The Schodor Foundation website, a group that supports the NCTM
*Principles and Standards for School Mathematics* *2000*, actually
builds this confusion into what looks like a lesson on introducing
multiplication where the definition is given á la the dictionaries,
including its own, and then it is ignored as follows:-

Student: So now that I understand addition and subtraction, are there
any more operations?

Mentor: Yes there are. The next operation is called multiplication. We
write multiplication problems in the form axb, or a times b. For example
we will consider 5x3. What this means is that 5 is being added to itself
3 times (5+5+5), but a better way to think about it is that it means 5
groups of 3 units each. Therefore to solve 5x3 you would count the
number of units in each group.

Student: I feel a little confused.

  Mentor: Alright, we will use an example you can visualize. Picture 5
separate plates, each with 3 quarters on them. How many quarters are
there altogether?

Student: Well, if there are 3 quarters for each of the 5 plates, then
there are 15 quarters altogether. So 5x3=15.

  Mentor: Exactly. Now, what if we had 3 plates, each with 5 quarters?

  Student: Then there would be 5 quarters for each of the 3 plates,
meaning 15 quarters altogether. So 3x5=15. Does that mean that
multiplication is commutative like addition? \[ Etc.\]

(www.shodor.org/interactivate/discussions/intmult.html)

  It looks to me as if our 'Mentor', like many others, is confusing
'meaning' with 'operation'! Whilst the **[operation of
multiplication]{.underline}** is commutative, the **[meaning of
multiplication]{.underline}** is surely not. I wonder if those who take
this 'doesn't matter' stance  would say taking 3 pills a day for 21
days, 3x21, is the same as taking 21 pills a day for 3 days, 21x3 -- NOT
to be tried at home I should add.

  This is not my idea of reform. This is not my idea of teaching. This
is not even my idea of mathematics. Is it really America's?

  The definition of multiplication means that one cannot multiply
'things' by 'things' yet traditional mathematics teaching has engraved
"area \[of a rectangle\] equals length times width" in the minds of
generations of students world wide. Is it any wonder that American
researchers find that students are confused - 'many elementary and
middle-grades children have difficulty with

understanding perimeter and area. Often, these children are using
formulas such as P=2l+2w or A=lxw without understanding how these
formula relate to the attribute being measured or the unit of
measurement being used Kenney and Kouba (1997) and Lindquist and Kouba
(1989) -- or so it says in the NCTM Standards! I am not convinced that
student teachers who read their Math Methods text which says "Now we are
multiplying two lengths to get an area" and a text book that says "rows
x columns" is the way to go, are going to put any minds at rest. As for
myself, I daren't look at what advice was on offer when it came to
volume! In teaching about a=lxw we need children to understand that 'l'
and 'w' refer to different things. (See Math Wars 2 : It's the teaching,
stupid. Marshall. KAPPAN January 2006)

  I did though consider how the meaning of multiplication was applied to
fractions and was advised to consult a research based book edited by 'an
expert on rational numbers'. The chapter on multiplication is quite
clear that in 6x4, the 6 is the multiplicand and the 4 the multiplier,
fitting in with the American Dictionaries. However as the next chapter
discusses 2/3 x 4/5 I am told that the repeated addition nature of
multiplication cannot be taken forward when looking at the
multiplication of fractions. Surprisingly, no indication is given as to
just what multiplication experience children do take with them when
attempting such problems. It all looked very much 'yours is not to
reason why, just invert and multiply' (almost literally!) as a series of
bullet type instructions are given! Against this confusion how do we
expect children to 'read' statements such as 2/3 x 4/5 with feeling?
What images are created in the mind when children see such statements?
Where could it come from? What is the story? Is it really too much to
say that we have 'two thirds, four fifths of a time' -- a carry over
from previous teaching? The point being, like in our area problem, the
language of multiplication is taken along with them. After all,
understanding is about tackling new problems -- "There is general
agreement that understanding in mathematics implies an ability to
recognize and make use of a mathematical concept in a variety of
settings, including some which are not immediately familiar."
(Cockcroft, Sir Wilfred. Mathematics Counts. HMSO. London. (1982) page
68)

  Many years ago I was asked for advice about introducing young children
to Geometry. The key here was 'young children', young children who were
at the concrete operational stage of their development. This recognition
of how children develop, so vital in the teaching of mathematics,
demanded that we start with 3-dimensional shapes. This seems to be at
odds with the reform of NCTM for a cursory glance at the illustrations
in *Principles and Standards for School Mathematics* *2000,* and in the
methods book I was given*,* indicates that in the K-2 Geometry section
2-dimensional shapes are the way to go. But is it? In the section of
*Principles and Standards for School Mathematics* *2000* entitled
Reasoning and Proof 3-dimensional shapes are given for children to
handle but there is an overwhelming desire to call them by their
2-dimensional names, at least I think that is so. After all, what is a
'thick circle'? A pattern is shown in figure 4.27 (page 123) that a
child has made and is proud to see the shapes as 2-dimensional. Where is
this going? My children see cylinders and rectangular prisms (cuboids),
etc., all over the place. The door and the cereal packet are the same
shape -- perhaps even being at odds with Pierre van Hiele (e.g."
Children might say, 'It is a rectangle because it looks like a box' !!!)

  Looking at the Geometry issue of *Teaching Children Mathematics*
(January 1999) there is an article (Shape Up) where the authors are
critical of some misconceptions some children acquire. The advice
offered suggests readers should "1) Emphasize the properties and
characteristics of a concept, 2) Provide many examples and non examples
....., 3) Play close attention to language use, and 4) challenge
understanding and broaden generalities." The reader is then invited to
give students "a collection of geometric solids and thin attribute block
pieces ..." Evidently these thin attribute block pieces can be handled!
It is my view that journals should not publish such papers. I don't
think they do so to deliberately confuse teachers/children but rather
because the editorial staff genuinely believes that 'a circle has a
thickness'. (The editor once told me where I could buy some!)
Incidentally, the same magazine carries a paper that claims there is no
such thing as a plane shape with thickness (Education Hannah. It's a
what? : Cockcroft and Marshall)

  In my travels I have spoken with many suppliers of educational
materials and this has been quite illuminating. It is clear that they
have a different, and in some way understandable perspective, to me. I
see good math products and bad math products. They see profitable

products and non profitable products. It seems that bad products sell.
The bottom line is that if 'the market' wants 'thick circles' etc.,
 then they will supply 'thick circles' etc., by the tonne for it pays
the mortgage. (My neighbor even has a book that encourages these
misconceptions claiming, amongst other things, that a carrot is a
triangle, and her young (preschool) grandson loves it!!!! When he gets
to school he will get an 'A' I am told. When he gets to my class at
college he will get 'F'.) Only recently I had a conversation with a
supplier who told me quite bluntly that the 'in thing' is now
'probability' but his buyers are not interested in concepts but rather
knowing that the probability of getting a '3' is 1/6 for the test. (I
assume this is a 1-6 dice!!) As I said earlier, it pays the mortgage.

  I could go on. Quite naturally friends in both the USA and UK ask me
what education is like in the other country. I would have to say that I
haven't seen a good math lesson in the US neither has my wife finding
ourselves rather like the professor  in the Hiebert and Stigler book I
mentioned earlier -- "In US lessons, there are the students and there is
the teacher. I have trouble finding the mathematics; ..." Having visited
something like 50 students teaching 6 lessons (not all math) I asked her
how many lessons she would be pleased if our son had been in when he was
young. The answer was none. However, a recurring theme in my
conversations is the quality of those student teachers we met. There is
a small group of truly outstanding people wanting to teach and I say
this without wanting to add any 'grade inflation' to the word. Not all
we met are in this category, far from it. If things are going to get
better it is these people who will do it. They must be given their head.
Why America feels it so necessary to have such central control when all
it does is make sure everyone gets mediocrity is beyond me. When the
county (or State) says this is what must be done then they had better be
right. Setting schools free may, I say may, just allow some quality to
come through which can then be replicated.  Getting this workforce
excited about teaching mathematics, and retaining them, is going to need
a vast rethink across the board.

  Teaching mathematics is a huge challenge and an enormous task. It has
been a lifetime work for many who would claim to only have scratched the
surface. Getting it right for our children is vital as Thomas Friedman
says in his book; The World is Flat, and Senator John Glenn, in his
report, Before it's Too Late. The bottom line as I see it is clearly
expressed by Keith Devlin in his book The Math Instinct. (Thunder's
Mouth Press. New York. 2005. ISBN 1-56025-672-9 Page 241) where he
writes: "The problem is that humans operate on meanings. In fact, the
human brain evolved as a meaning-seeking device. We see, and seek,
meaning anywhere and everywhere. A computer can be programmed to
obediently follow rules for manipulating symbols, with no understanding
of what those symbols mean, until we tell t to stop. But people do not
function in that way." I look forward to your conclusions.

 

John Marshall

Monday, September 04, 2006

[Math Wars Taking
Sides.pdf](../../../../../../../../userdirs/Kevin.McIntosh/TEAMSITE/wdchqdtmsp01/cache/default/main/ED/en/ABOUTED/WORKAREA/work/about/bdscomm/list/mathpanel/documents/math-wars-taking-sides.pdf)

[Math Wars 2 It\'s the Teaching
,stupid.pdf](../../../../../../../../userdirs/Kevin.McIntosh/TEAMSITE/wdchqdtmsp01/cache/default/main/ED/en/ABOUTED/WORKAREA/work/about/bdscomm/list/mathpanel/documents/math-wars-2-it's-the-teaching-stupid.pdf)

\-\-\-\--Original Message\-\-\-\--

From: []{#DaveMarain3 .anchor}Dave Marain

Sent: Friday, September 01, 2006 8:41 PM

To: National Math Panel

Subject: Statement to the National Math Panel

This email is being sent by the deadline date of 9-1-06. Pls confirm
receipt of this email.\
\
The teachers of the mathematics department of Ramapo H.S. wish to
express their strong sentiments embodied in the attached statement. We
know the Panel will read this and the hundreds (thousands?) of other
similar statements and hopefully will accept these as a mandate for
change in mathematics education in this country. It's not about reform
or preserving tradition. It's about what we feel is best for our most
precious national treasure, our children. Do not abandon another
generation. Listen to our teachers -- no one knows better what is needed
in our classrooms.\
\
Sincerely,\
Dave Marain\
Supervisor of Mathematics & Business

[Nat_Math_Panel_Statm_9-1.doc](../../../../../../../../userdirs/Kevin.McIntosh/TEAMSITE/wdchqdtmsp01/cache/default/main/ED/en/ABOUTED/WORKAREA/work/about/bdscomm/list/mathpanel/documents/nat-math-panel-statm-9-1.doc)

\-\-\-\--Original Message\-\-\-\--

From: []{#BertBeck .anchor}Bert Beck

Sent: Friday, September 01, 2006 1:47 PM

To: National Math Panel

Subject: REVISED paper submitted for National Mathematics Advisory Panel

Good morning Jennifer,

 

I apologize. A small change has been made in the text of the document
submitted for the National Mathematics Advisory Panel. Please delete the
previous submission and download the attached documents instead.

 

Please call or e-mail with any questions or concerns.

 

Thank you.

 

Bert Beck

Office of the Superintendent

Orange County Department of Education

[01_NatlMathPanelCoverLetter.pdf](../../../../../../../../userdirs/Kevin.McIntosh/TEAMSITE/wdchqdtmsp01/cache/default/main/ED/en/ABOUTED/WORKAREA/work/about/bdscomm/list/mathpanel/documents/01-natlmathpanelcoverletter.pdf)

[02_OCDE_NatlMathematicsPanelCover.pdf](../../../../../../../../userdirs/Kevin.McIntosh/TEAMSITE/wdchqdtmsp01/cache/default/main/ED/en/ABOUTED/WORKAREA/work/about/bdscomm/list/mathpanel/documents/02-ocde-natlmathematicspanelcover.pdf)

[03_R9106_OCDE_NatlMathematicsPanel0906.pdf](../../../../../../../../userdirs/Kevin.McIntosh/TEAMSITE/wdchqdtmsp01/cache/default/main/ED/en/ABOUTED/WORKAREA/work/about/bdscomm/list/mathpanel/documents/03-r9106-ocde-natlmathematicspanel0906.pdf)

\-\-\-\--Original Message\-\-\-\--

From: []{#ArthurHu .anchor}Arthur Hu

Sent: Thursday, August 03, 2006 10:51 PM

To: National Math Panel

Subject: Parent for real math

I am a parent from Kirkland/Bothell WA, and I am appalled at the NSF
funded textbooks my teachers were forced to use by the district. The
homework took much more time, and taught much less. They took an entire
booklet to cover median and average, yet did not teach the simple
formula average = sum of items divided by number of items. The whole
movement seems devoted to making students work harder to learn less. If
the US is to maintain technical leadership, it cannot abandon the
teaching of mathematics as it has been taught for generations since the
invention algebra. The new methods refute, rather than teach classic
time tested methods of computation and mathematics.

 

Arthur Hu

Parent of 3 boys grades 5 through 9

\-\-\-\--Original Message\-\-\-\--

From: []{#DaveMarain4 .anchor}Dave Marain

Sent: Thursday, August 03, 2006 3:39 PM

To: National Math Panel

Subject: Re: Questions to the Panel

HI Ida!\
Thank you for your quick response. I'm still in shock that you're taking
the time to reply! You're my only reason to still have some faith in
this panel.\
\
I did read the executive order and there were enough references to
secondary math (up to calculus) that I still see a real need for a
current secondary teacher on the panel.\
\
There is also a section devoted to culling opinions from others not on
the panel, including parents, other experts, etc. I will certainly try
to voice my opinion in person and attend the next meeting but that
remains unlikely.\
\
You need to understand the source of my frustration and cynicism
regarding bureaucratic processes. It's been almost 20 years since NCTM
came out with their first set of Curriculum recommendations. Math ed
professors have written new books, districts have made several new
textbook adoptions and students' arithmetic and algebra skills continue
to steadily erode. There's more than anecdotal evidence here. The
educators I work with every day, savvy students and parents echo these
sentiments ad nauseam. Still, new curriculum committees, researchers and
textbook companies continue to ignore the obvious that Liping Ma has
been calling for -- a 'profound understanding of fundamental
mathematics' is essential. Perhaps her presence will turn things around,
but who knows. And there's more\... Higher-order problem-solving in
which students from other nations successfully engage require one to
apply their knowledge, just like on our SATs and math contests. The
problem is that our students DO NOT HAVE THE KNOWLEDGE TO APPLY! We have
a generation of clever problem-solvers who are tech-savvy, who know how
to work-around many issues, but YOU CANNOT WORK-AROUND THE ESSENTIAL
SKILLS OF ARITHMETIC AND ALGEBRA. You can't fake the Laws of Exponents!
You can't consistently achieve accurate solutions to problems by
pressing the 'Solve' key on a graphing calculator. Parents know this,
educators know this yet educational 'experts' IGNORE THIS!\
\
Yes, there are some members of the panel who seem to share some of my
concerns. However, it takes bold courageous individuals to make a sea
change happen and that is what I believe is needed here. From my
contacts with many other educators and parents, my views are not so
extreme and are not isolated.\
\
Understand that I will continue to endeavor to be heard by this
committee. However, I really don't believe my lone voice will amount to
much. I do believe a blog that gets the attention of education
journalists can reach thousands. I know you will wish me all the best
with this! I know Tyrrell will send me another boilerplate response. I
know that real change will only occur if one can be heard. Thank you for
caring\...\
\
\
Sincerely,\
Dave Marain\
Supervisor of Mathematics & Business

\-\-\--Original Message\-\-\-\--\
From: []{#JohnShacter2 .anchor}John Shacter
\[mailto:jsplg@comcast.net\]\
Sent: Friday, July 28, 2006 10:48 AM\
To: Graban, Jennifer\
Subject: Teaching Math with the \"KISS\" Approach (rev.)

Hi Jennifer - Thanks v. m. for your kind message.\
\
Below is a slightly extended version of my letter. The additions are in
the fourth to last, large paragraph.\
\
I cannot do a lot of traveling for a five-minute presentation. However,
I will send you additional, rather brief e-mails, and I would like to
submit to more extensive questions by and discussion with the panelists.
I like to believe, that they will not be wasting their time with me, and
that it is unlikely that any other advisor will have the same set of
practical math-user ideas for them.\
\
Thanks again for your interest. \-- You have been very kind \--\
\
John

\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\--\
\
Please pass this on to the Math Advisory Panel. I\'ll  be glad to
elaborate, including visits, if requested.\
\
By the way, the  acronym KISS refers to: \"Keep It Simple, Stupid.\"\
\
I am a semi-retired  engineer, management-and-technology consultant, and
educator. You can find  more background information in the Who\'s Who
volumes on Science and  Engineering, and on Finance and Business. I have
thus been a user of math, for  many decades, rather than a
mathematician. I am a bit concerned, that the Math  Advisory Panel will
come up with a more complex approach than needed.  \
\
For example, the panel which doesn\'t seem to include a single
 engineer, technologist, or business \"types,\" could actually place
barriers  against the KISS approach and make students hate math even
more than they  already do today, if it were to set overly ambitious
objectives of trying  to teach the complexities or rote-vocabulary of
math in an attempt to convert  the maximum number of public-school
students into future mathematicians.  \
\
I am also a frequently invited local and regional substitute teacher
 and prefer the teaching of math in any public-school grade, but
especially  grades 3-12. As a user of math, I firmly believe in the
\"KISS\" approach of  teaching any subject, especially math. The
students seem to like my style and  ask for me to be re-invited. Some of
the teachers have asked me to teach or  tutor their own children.\
\
(By the way, this is not very relevant, but I  received my early
education in Vienna, Austria. There are some differences in  the way
math is being taught in different countries or regions, and our own
 students seem to fall behind, the more years they spend in our
classrooms.  However, our parents and students seem to score highest in
 \"self-esteem.\")\
\
Generally, students (and some teachers) claim that they  \"hate\" math.
However, all of them just \"love\" money, and I am shameless enough  to
take full advantage of that combination. So when I see the students\'
eyes  fog over, I just switch for awhile and teach them about \"budget\"
or \"profit\"  

concepts, differences between savings and investments, or why \"money
 management\" doesn\'t just consist of knowing how to sign a credit
 card!\
\
Students need to understand how to reason qualitatively and
 quantitatively (with math), and how to express any aspect of this
reasoning  clearly and effectively, orally and in writing.\
\
Moreover, for  illustrative purposes \-- most students in third grade
need to know instantly  what 7 times 8 is. Memorizing the
multiplication-division table for math is  just as important and basic
to more advanced math-learning, as memorizing the  alphabet is for
learning English vocabulary. Thus, they need to know,
 eyeball-to-eyeball, (no pen, paper or calculator) what 56 divided by 7
or 8  is, and what 5[7]{.underline} divided by 7 or 8 is, with and
without the use  of \"remainders\".\
\
In third or (latest) fourth grade, they also need to  know how to handle
decimals, fractions and percents, and how to proceed to  translate
quickly and smoothly from one system to either one of the other two.
 They also need to know, again eyeball-to-eyeball, what 3 divided by 1/3
is  (and the answer is [NOT]{.underline} 1!), or how to multiply or
divide integers and  decimals by 10, 100, 1000, etc.\
\
(Unfortunately, the way some of our  colleges educate our teachers, some
of them may not be able to supply all of  the above simple, elementary
arithmetical answers. Moreover, many of today\'s  enormous-size text
books may tend to cause more delayed hernias rather than  simple
understandings.)\
\
Finally, still illustrative of my teaching in  the third or fourth
grade, I introduce my students teasingly to simple  algebra, and when I
tell them that they had just been introduced, I finally  use the word
\"algebra\" which adults seem to be so proud of. I start with the  use
of a question mark \-- like: 2 + ? = 5. What is the question mark? (Most
of  them know immediately.)\
\
Then I claim that I hate question marks \-- will they allow me to ask
instead: 2 + A = 5. Now what is \"A\"? (Most of them  think that is a
silly question, since the answer is still the same.) And then  I may ask
them about 2 + A + B = 5, pointing out that this is like a detective
game. If you only have one \"clue\" or equation, but two \"unknowns\", A
and B, then there are an infinite number of valid combinations of A and
B, until and unless you get a second \"clue\". If I have a bit more
time, I also show them how the infinite number of valid combinations can
be represented by a line in the x/y plot or graph, and that there is an
even \"larger\" number of infinite combinations which are not valid
combinations\... They seem to delight in all of  this teasing and go
home bragging to their parents that this crazy substitute  teacher
taught them \"algebra\" which they weren\'t supposed to be ready for,
until well into middle school \-- like 7th or 8th grade.\
\
I am going to send you a simple, short  memo telling the reader all
he/she needs to know to operate with \"fractions\".  The way we teach
them, many students seem to think that \"fractions\" are rather
 difficult or complex.\
\
You have many other advisors. So I may offer a  bit of pleasant
competition. Let\'s select, say, 45 fourth or fifth graders.  Give your
favorite expert 15 students by random selection, and give me 30.  Then
let\'s see who can add greater gains to his group in a defined area of
 math \-- say solving problems with integers, decimals, fractions and
percents,  and being able to translate any one of them to the others. We
can  then  analyze the effectiveness of competing approaches in
conveying math and  hopefully making students like it, as well.\
=============\
\
Would  any of this type of practical experience be of any value to the
math advisory  panel? If so, I shall be glad to elaborate or assist the
panel, including  visits, as well as I can. I have many other ideas on
how to make major, not  marginal improvements in curricula and teaching.

However, I also believe that  many of our gaps in quantitative reasoning
and other subjects can be traced to  an inadequate understanding of
basic math.\
\
Cordially, John\
John  Shacter

\-\-\-\--Original Message\-\-\-\--

From: []{#JohnShacter .anchor}John Shacter

Sent: Thursday, July 13, 2006 11:12 PM

To: National Math Panel

Subject: Teaching Math with the "KISS" Approach

Please pass this on to the Math Advisory Panel. I\'ll be glad to
elaborate, including visits, if requested.\
\
By the way, the acronym KISS refers to: \"Keep It Simple, Stupid.\"\
\
I am a semi-retired engineer, management-and-technology consultant, and
educator. You can find more background information in the Who\'s Who
volumes on Science and Engineering, and on Finance and Business. I have
thus been a user of math, for many decades, rather than a mathematician.
I am a bit concerned, that the Math Advisory Panel will come up with a
more complex approach than needed.\
\
For example, the panel which doesn\'t seem to include a single engineer,
technologist, or business \"types,\" could actually place barriers
against the KISS approach and **make students hate math even more than
they already do today**, if it were to set overly ambitious objectives
of trying to teach the complexities or rote-vocabulary of math in an
attempt to convert the maximum number of public-school students into
future mathematicians.\
\
I am also a frequently invited local and regional substitute teacher and
prefer the teaching of math in any public-school grade, but especially
grades 3-12. As a user of math, I firmly believe in the \"KISS\"
approach of teaching any subject, especially math. The students seem to
like my style and ask for me to be re-invited. Some of the teachers have
asked me to teach or tutor their own children.\
\
(By the way, this is not very relevant, but I received my early
education in Vienna, Austria. There are some differences in the way math
is being taught in different countries or regions, and our own students
seem to fall behind, the more years they spend in our classrooms.
However, our parents and students seem to score highest in
\"self-esteem.\")\
\
Generally, students (and some teachers) claim that they \"hate\" math.
However, all of them just \"love\" money, and I am shameless enough to
take full advantage of that combination. So when I see the students\'
eyes fog over, I just switch for awhile and teach them about \"budget\"
or \"profit\" concepts, differences between savings and investments, or
why \"money management\" doesn\'t just consist of knowing how to sign a
credit card!\
\
Students need to understand how to reason qualitatively and
quantitatively (with math), and how to express any aspect of this
reasoning clearly and effectively, orally and in writing.\
\
Moreover, for illustrative purposes \-- most students in third grade
need to know instantly what 7 times 8 is. Memorizing the
multiplication-division table for math is just as important and basic to
more advanced math-learning, as memorizing the alphabet is for learning
English vocabulary. Thus, they need to know, eyeball-to-eyeball, (no
pen, paper or calculator) what 56 divided by 7 or 8 is, and what
5**[7]{.underline}** divided by 7 or 8 is, with and without the use of
\"remainders\".\
\
In third or (latest) fourth grade, they also need to know how to handle
decimals, fractions and percents, and how to proceed to translate
quickly and smoothly from one system to either one of the other two.
They also need to know, again eyeball-to-eyeball, what 3 divided by 1/3
is (and the answer is [NOT]{.underline} 1!), or how to multiply or
divide integers and decimals by 10, 100, 1000, etc.\
\
(Unfortunately, the way some of our colleges educate our teachers, some
of them may not be

able to supply all of the above simple, elementary arithmetical answers.
Moreover, many of today\'s enormous-size text books may tend to cause
more delayed hernias rather than simple understandings.)\
\
Finally, still illustrative of my teaching in the third or fourth grade,
I introduce my students teasingly to simple algebra, and when I tell
them that they had just been introduced, I finally use the word
\"algebra\" which adults seem to be so proud of. I start with the use of
a question mark \-- like: 2 + ? = 5. What is the question mark? (Most of
them know immediately.)\
\
Then I claim that I hate question marks \-- will they allow me to ask
instead: 2 + A = 5. Now what is \"A\"? (Most of them think that is a
silly question, since the answer is still the same.) And then I may ask
them about 2 + A + B = 5, etc. etc. They seem to delight in all of this
teasing and go home bragging to their parents that this crazy substitute
teacher taught them \"algebra\" which they weren\'t supposed to be ready
for any of that until middle school.\
\
I am going to send you a simple, short memo telling the reader all
he/she needs to know to operate with \"fractions\". The way we teach
them, many students seem to think that \"fractions\" are rather
difficult or complex.\
\
You have many other advisors. So I may offer a bit of pleasant
competition. Let\'s select, say, 45 fourth or fifth graders. Give your
favorite expert 15 students by random selection, and give me 30. Then
let\'s see who can add greater gains to his group in a defined area of
math \-- say solving problems with integers, decimals, fractions and
percents, and being able to translate any one of them to the others. We
can  then analyze the effectiveness of competing approaches in conveying
math and hopefully making students like it, as well.\
=============\
\
**Would any of this type of practical experience be of any value to the
math advisory panel? If so, I shall be glad to elaborate or assist the
panel, including visits, as well as I can. I have many other ideas on
how to make major, not marginal improvements in curricula and teaching.
However, I also believe that many of our gaps in quantitative reasoning
and other subjects can be traced to an inadequate understanding of basic
math.\
**\
Cordially, John\
John Shacter

\-\-\-\--Original Message\-\-\-\--

From: []{#DougJohnson .anchor}Doug Johnson

Sent: Monday, June 12, 2006 5:14 PM

To: National Math Panel

Subject: math ed

National Mathematics Panel Members:

 

I am pleased to see the government form the National Math Panel to
support and improve the delivery of math instruction in our nation.  I
have taught math, science, and English for twenty-six years in the state
of Washington.  My assignments have included levels from second grade to
college.  Most of my work has been at the high school level.  I have
been the chairman of the math department at Ellensburg High School with
an assignment of algebra through AP Calculus for eight years now.

 

I was a little disappointed to see the K-12 teachers underrepresented on
the panel.  There are thousands of us out here getting it done day by
day, many with outstanding programs, working hard, challenging kids, and
changing their lives and futures.  Hopefully, a balance more toward the
teachers "in the trenches" will be promoted.

 

My main concern, however, has more to do with curriculum than with panel
personnel.  For approximately two decades, the National Council of
Teachers of Mathematics has "led the charge" on the math reform effort. 
Their efforts have led to some positive change in the math education
world, specifically emphasis on problem solving and concept
development.  These gains, however, have come in my opinion at a
debilitating cost to students.  Excessive attention to these skills has
resulted in math programs (often labeled as "integrated") that present
material in a seemingly random order that does not make sequential sense
to the students.  Students are not immersed in any particular topic
sufficiently, and mastery is not achieved.  But that's not the main
shortcoming of most of these programs.  This major shortcoming is the
lack of drill and practice.  Most students need to work repeatedly on
problems on one concept before they "get it."  Without mastery on each
topic, math becomes a confusing hodge-podge of material, and the kids
drop out of mathematics, psychologically or literally.  In Ellensburg,
we get students from other schools with integrated programs.  They need
considerable remediation, lots of drill and practice, and extra
attention to get up to speed. When I teach at the local university, it
is common to get students from integrated programs with major skills
shortcomings.  It is now a conversation among some college math and
science departments about the lack of basic high school math skills in
freshmen.

 

Here in Washington, we are subject to the Washington Assessment of
Student Learning, or "WASL" test.  It is given in the sophomore year,
and passage is a graduation requirement.  The math test is based mostly
on NCTM type standards, and has little emphasis on algebraic skills. 
Here in Ellensburg, we view this as a mistake.  We refuse to adopt
integrated texts. Instead, we use a more traditional text and supplement
it with WASL-type work twice a week throughout the year.  A few weeks
before the test, we place special emphasis on WASL work, to the
exclusion of all else.  The system is working, and this spring we scored
approximately twenty percent above the state average on the test.  We
think this supports our contention that traditional programs, taught
well, lead to success.  And we don't have to compromise the futures of
those Ellensburg kids who want to go to study math, science, engineering
and other math-related fields. These students need basic algebraic and
other math skills. We believe integrated math programs often cripple
potential mathematicians and scientists.  In our experience,
well-structured traditional programs, supplemented by "reform math"
practice, lead to success on both.

 

Thank you,

Doug Johnson

\-\-\-\--Original Message\-\-\-\--

From: []{#KirkAvent .anchor}Kirk Avent

Sent: Sunday, June 11, 2006 6:54 PM

To: National Math Panel

Subject: High school math

Dear Committee:

 

My son recently graduated from Phillips Exeter Academy. His math courses
were extraordinary. No textbooks, just problems. Principles of
mathematics emerged as students worked problems, rather than the usual
teacher-based approach where the principles are presented by the teacher
then reinforced by problems worked by the student.

 

"Oh, sure", you say. "Fine for an elite school like Exeter but
impossible in an inner city school".  Wrong! The math department has a
program for urban schools called the Exeter Mathematics Institute.
Please check it out at:
<http://math.exeter.edu/dept/development/EMI.html>.

 

 

I suggest you contact the math department chairman at Phillips Exeter
Academy: Joyce Kemp, <jkemp@exeter.edu>.

 

Kirk Avent

\-\-\-\--Original Message\-\-\-\--

From: []{#DianeHirakawa .anchor}Diane Hirakawa

Sent: Friday, June 09, 2006 7:22 PM

To: National Math Panel

Subject: Suggestions for Elem. Math

Good Morning,

My passion is math in the elementary schools.

 

I recently retired with 23 years of teaching experience in southern
California.  Presently, I work with children and their parents in my
home on a smaller scale.  For these weekly tutoring session, I use many
of the same ideas which provided highly successful math students in my
grades 1-6 classrooms.

 

In  June, 2006, issue of \"The Achiever\" magazine I read that you have
established a national advisory panel on math.  I am hoping that you
will consider my highly successful, 23 years of experience and the
insight I am about to share with you.

 

Please feel free to contact me if you would like more in-depth
descriptions of my brief summaries listed in this email.

 

1)  TIME:  Teaches must allocate additional minutes  for math
instruction and exploration in the elementary school classrooms,
throughout all grade levels.

 

2)  SMALL GROUPS:  Teachers must review and check math concepts in small
groups rotated after their whole-group instruction.  The children not
with the teacher can be exploring already taught concepts with
manipulative tools either at their seats or with a volunteer, until it
is time for their math group with the teacher.

 

3\) MANIPULATIVE TOOLS:  Classrooms must have the tools for the children
to manipulative to make sense of the algorithms.  Paper and pencil
algorithms should not be taught at the beginning of the new learning,
but after the manipulative tools has been used.  Many of the expensive
tools can be made with 60 lb. weight paper.

 

4)  ERROR CORRECTIONS:  Every single missed test question must be
corrected one on one with the teacher or with a trained, competent
volunteer.  This is essential for each individual student.

 

5)  CONNECTIONS:  The teacher and the parents must make connections to
the children\'s lives outside of the classroom using the math
standards.  Each student must see how the math can be used in his/her
life.  (Example: For the concept of \"square area\", a standard in all
grade levels, some students might consider the size of a soccer field,
others the size of a tennis court, swimming pool, or playground
hopscotch square.)  It must make sense to them.

 

If each teacher in America would follow these 5 guidelines, as I have
for 23 years, we would be doing a great favor to the children learning
math in our school systems.  They would leave the elementary schools
prepared for junior high algebra and higher level math in the high
schools. 

 

By the way, my younger son is a high school math teacher and my older
son is a research scientist.  Specific teachers and myself helped them
through the maze of math.

 

Thank you for your time

\-\-\-\--Original Message\-\-\-\--

From: []{#PamelaGood .anchor}Pamela Good

Sent: Tuesday, June 06, 2006 10:30 AM

To: National Math Panel

Subject: Math Skills commentary

Hello,

 

My name is Pamela Good.  I am a pharmacist, and I am very interested in
the work of the National Math Panel.  My 17-year-old daughter will be a
high school senior this fall, and she will be taking AP Calculus.  So
far, she has been an A student in math, except for a B in one semester
of Geometry.  Our journey to get her there has not been easy.  She is an
honor student at a Wichita, Kansas public high school with an enrollment
of about 1800 students.

 

One thing I would like to encourage the panel to do is to contact the
Sylvan Learning Center.  They have broken down every math skill into
it\'s tiniest part.  When they initially assess a student\'s math
skills, they know which tiny skills are in place, and which ones are
missing and need to be taught.  They get results by teaching these very
small skills, repeating them, making sure there is retention of these
skills, and then building on them.  Without our tutoring experience
there in earlier years, I\'m sure my daughter would not be where she is
today in her math education.  They have alot of experience in tutoring
for deficient math skills, and I feel that they have alot of valuable
insight on why students are not learning math in school.

 

Another thing that I feel is really important is to look at how much
support outside the classroom it takes to get students to learn math. 
My daughter has gone in before school and stayed after school frequently
to get additional help from her teachers.  She has also had a
highly-educated math parent at home who was willing to help her when she
had trouble learning new concepts, which was quite often.  One hour a
day in a math class is not sufficient to teach new skills to most
students.  It takes alot of demonstration and repetition to get the
skill mastered.  At Sylvan, they teach based on \"mastery\" of a
concept, and then they repeat the skill and retest the skill at
intervals until there is \"retention\", which is key.  It\'s too bad we
can\'t translate that into our school classrooms.

 

One other problem that we have had is that the math teachers of higher
level classes tend to teach at a higher level, assuming that only the
brightest of the bright will be in those classes anyway.  My daughter is
bright, but she needs alot of repetition to master new skills.  Teachers
need to teach math, especially in high school, at an appropriate level
so that these students can learn.  They also need to slow down when they
are teaching.  You can\'t teach at breakneck speed and assume that they
will \"get it the first time.\"  They need to allow time for the new
information to soak in.

 

I hope these comments are helpful.

Thanks very much for listening,

Sincerely,

Pamela Good

\-\-\-\--Original Message\-\-\-\--\
**From:** []{#NormanBerger .anchor}Norman Berger
\[mailto:nwberger@comcast.net\]\
**Sent:** Tuesday, May 30, 2006 6:45 PM\
**To:** National Math Panel\
**Subject:** Some basics

To teach kids math here are some super-basics regarding math education:

 

1) Young children need to have paper that has vertical and horizontal
lines to do arithmetic because if they write a list of numbers on a
slant they cannot be expected to add them up.

 

2) Math isn\'t like reading. Studies have shown that if you take a full
page of a document and systematically take out one letter at a time
(testing lots of people at various stages of this exercise) that you
have to remove almost 50% of the letters before it is incomprehensible.
This is unlike math where one number taken out or misread or skipped
over makes it impossible to do the problem Thus, childrent need to be
taught to concentrate on every number and instruction and not \'read\'
the problem.

 

Start with these fundamentals and you\'ll see math scores improve and
children smile.

 

Norman Berger, Menlo Park,CA

\-\-\-\--Original Message\-\-\-\--\
**From:** []{#LarryMorris .anchor}lammorris@att.net
\[mailto:lammorris@att.net\]\
**Sent:** Friday, May 26, 2006 9:17 PM\
**To:** National Math Panel\
**Subject:**

Gentlemen,

As a parent with a degree in mathematics, I was appalled to learn that
my child was struggling with algebra because the teacher was trying to
make it a \"feel-good\" class.  Mathematics takes work, and especially
concentration on learning the definition of terms, by rote if
necessary.  I learned that I could not communicate with my child because
she had not been taught the meaning of technical terms in mathematics. 
The teacher told me, \"Oh, we don\'t bother memorizing definitions, we
just do it\". 

It is time to stop this nonsense and get down to work.  I hope you
agree.

Larry Morris

<lammorris@att.net>

\-\-\-\--Original Message\-\-\-\--

From: []{#JohnRaeth .anchor}John S. Raeth

Sent: Thursday, May 25, 2006 4:31 AM

To: National Math Panel

Subject: Algebra Reform

Mr. Faulkner,

    This note is in response to the article in AAAS\'s U.S. Math
Education, \"Well-Balanced Panel to Tackle Algebra Reform\" in the 19
May 2006 edition of *Science* on page 982.

 

    It is refreshing to hear that an intelligent debate is ongoing in
reference to math education.

 

    As a high school math educator, it is significant that I agree with
the points of view of what might be referred to as both sides of the
discussion.  It is significant because both points of view, algebra
reform and \'more rigorous instruction on basic skills\', are very
important.

 

    It is an unfortunate fact that students entering high school do not
have the arithmetic skills they need to perform satisfactorily in my
algebra classes.  In fact, I have monitored this closely as have many of
my colleagues.  This is one area that I have counseled my students on
during the course of the year.  It is fairly easy to note after
observing verbal responses and written results that the students are
picking up the algebra concepts but are struggling with advancement
because they cannot perform at the basic level in arithmetic.

 

    This is one result of the lack of rigor in elementary and middle
school arithmetic education techniques.  By the time these students
arrive in the high school setting, they have \'learned\' two bad
habits:  they have not learned to do the basic arithmetic and they have
not learned the disciplines they need with higher concepts.  This is
true whether or not they will be moving on to science, engineering, or
other subject area where mathematics will play an integral role.

 

    Algebra reform is also very important.  It is critical that algebra
be taught in a way that will be more relative to modern lives and
circumstances.  In other words, more realistic.  As I am sure you are
aware, one of the first and loudest questions heard is:  \'How or why
are we ever going to use this.\'  Therefore, it is a challenge to
encourage and motivate learners, especially from the beginning of the
curriculum.  That needs to start early. 

 

    We cannot wait until High School.

 

    So, instead of being adversaries, rigorous arithmetic and rigorous
algebra education need to be partners instead of critics.

 

    Another area of concern is the distinct lack of practical and
applicable professional and commercial secondary level math education
materials.  For example, I just attended a conference that was supposed
to be designed for secondary level mathematics educators.  It was led by
a well-known and experienced math educator, a doctor with many years of
experience in education \-- elementary education.  One of his first
comments was that his experience was in elementary and special education
and that we would need to adapt what we experience to our secondary
education classes.  This is typical of what we face in High School.

 

    Finally, I encourage your panel to foster this forum in a public
manner \-- encourage input and suggestions from all that experience out
there.  It will be successful and meaningful.

 

    Thank you for your significant and much needed contributions.

 John S. Raeth

Harlem High School, Georgia

\-\-\-\--Original Message\-\-\-\--

From: []{#MartaGray1 .anchor}Marta Gray

Sent: Sunday, May 21, 2006 7:49 AM

To: National Math Panel

Subject: RE: Curriculum Directors

Since you are listening\...\...\...\...\...

The Fordham Foundation report, \"State of the State Math Standards
2005\", was written by mathematicians that evaluated the state standards
of 49 states. Most states standards were graded a C, D, or F. Many
states rely on teachers and math educators to write these standards. If
every state was required to subject their standards to a panel of
mathematicians for review, with the stipulation that standards must
receive a grade of B or better, we could improve the standards greatly.
This would filter down to classrooms, as curriculums and teacher
training would be consistent with high standards. Having a low target to
begin with guarantees poor student performance.

Another issue that has a tremendous impact on the classroom is the
constant and steady stream of interruptions ranging from announcements,
to phone calls, to impromptu assemblies. Some of these interruptions
cannot be helped such as fire drills, evacuation drills, and the like,
but the cumulative effect is staggering. I am not exaggerating when I
tell you that it is not uncommon to have 6 interruptions in an hour and
I have heard this from many other teachers as well. This has a negative
effect on student concentration and takes the teacher away from students
that require his/her attention. I have read that top achieving nations
have gone as far as requiring \"quiet zones\" in the surrounding areas
of schools. Learning takes immense concentration. I don\'t know if the
panel can address this issue but it\'s toll on productivity is huge.

This may sound extremely negative but it is the \"truth from the
trenches\". Adolesence is an emotioal time and many students go through
a period of rebellion against authority and the establishment. Kids are
quite sophisticated these days and understand that middle school grades
do not \"count\" toward high school graduation. Parents with skills
adequately counter this issue and keep their children on track through
this tumultuous time. Unfortunately, many parents are ineffective in
dealing with this situation. Their children make little or no effort
during these years. It is tragic when these kids grow up enough to
decide that they want to do well in school but are so far behind at that
point, the odds are almost insurmountable that they will overcome lost
skills in their quest for a high school dimploma. This is a problem many
teachers face with students in \"at-risk\" schools. We try to motivate
kids but this job would be much easier if grades counted toward
graduation starting in 6th grade instead of 9th. It is amazing that many
of the kids most resistant to trying in school during adolesence, tend
to be \"stars\" if we are able to get them back on track before too much
ground is lost.

Thank you for your consideration of these ideas. I realize that solving
these issues may be beyond the reach of this panel but I thought you
should be aware of issues that many teachers see on a daily basis.

Respectfully,

Marta Gray

\-\-\-\--Original Message\-\-\-\--

From: []{#MartaGray2 .anchor}Marta Gray

Sent: Thursday, May 18, 2006 5:01 PM

To: National Math Panel

Subject: Curriculum Directors

Dear Esteemed Panel Members,

As you consider the best interests of students in the U.S., please
remember that many speak English as their second language. These
students can succeed in math as long as their parents are able to
support them at home. It has been my experience that textbooks with lots
of examples/visuals help parents to remember the concept being taught,
which enables them to assist their son/daughter in their native
language. Many of the math programs on the market are so wordy that the
math gets lost and students that might otherwise perform well, suffer.

Thank you for your time.

Marta Gray

McLoughlin Middle School

Vancouver, Washington

\-\-\-\--Original Message\-\-\-\--

From: []{#JoanCotter .anchor}Joan Cotter

Sent: Wednesday, May 17, 2006 8:40 PM

To: National Math Panel

Subject: First grade mathematics research

National Math Panel:

I have done research with children in the first grade employing some
techniques?used in Asia. The results were outstanding; the attached
article published by the NCTM (National Council of Teachers of
Mathematics) details the results. It shows a great disparity between the
way children are traditionally taught in the U.S. and what they could be
doing.

Some examples follow:

1\. The children could determine quantities up to 100 visually without
counting, by grouping in 5s as well as 10s.

2\. Children can learn addition and subtraction visually
[without]{.underline} counting. Five-month-old babies can add and
subtract up to three. Japanese children are discouraged from counting
when they start school. This approach is also much easier for children
with learning problems. (In the U.S. we traditionally spend a year
teaching children how to count and spend two years telling them not to.)

3\. It has been known for decades that flash cards and timed tests are a
poor way to teach the facts. The children in the experimental group used
visual strategies combined with games for practice. The AL abacus
enabled them to see and practice the strategies.

4\. Place value is the single most important topic in arithmetic.
Average American students don\'t understand it until the end of fourth
grade. The experimental children used the ?math way? of number naming;
they understood place value in first grade as do Asian students.

5\. The children learned to add four-digit numbers with carrying on the
AL abacus. A few days later they had figured out how to do the standard
paper and pencil algorithm.

6\. The children could mentally add two-digit numbers (with carrying).

7\. By using part/part/whole circle sets, the children easily solved
problems, including those with missing addends.

My background includes a degree in electrical engineering, a Montessori
teacher for children ages 3-6, tutor for children with learning
problems, mathematics teacher for grades 6-8, a?doctorate in mathematics
education and early childhood, and the author of the K-Gr 4 *RightStart
Mathematics* program. Currently I am writing middle school mathematics
curricula.?

I am very interested in improving mathematics in the U.S. What we do in
the early years has a profound influence in later years. Please read the
attachment.

Joan Cotter, Ph.D.

[Using Language and
Visualization.pdf](../../../../../../../../userdirs/Kevin.McIntosh/TEAMSITE/wdchqdtmsp01/cache/default/main/ED/en/ABOUTED/WORKAREA/work/about/bdscomm/list/mathpanel/documents/using-language-and-visualization.pdf)

\-\-\-\--Original Message\-\-\-\--\
**From:**[]{#LisaJohnson .anchor} LisaJson@aol.com
\[mailto:LisaJson@aol.com\]\
**Sent:** Monday, May 15, 2006 2:54 PM\
**To:** National Math Panel\
**Subject:** Math Panel Inquiry

To Whom It May Concern at the National Math Panel,

 

I would like to contribute in any capacity I can to the National
Mathematics Advisory Panel.  I just noticed this morning that Dr. Larry
Faulkner, the former president of the University of Texas, where my
middle son attends, is leading this panel.

 

I have over 12 years of hands-on experience tutoring elementary school,
secondary school, and college students in mathematics.

 

I have developed my own teaching program where I believe there needs to
be four fundamental steps in teaching mathematics: 1) Learning
Definitions, 2) Learning Rules (such as equations, theorems, etc.),  3)
Working through Examples, and 4) Applying the math to Real World
problems.

 

I believe the teaching of math is a delicate balance of the memorization
and understanding of concepts; the first 2 fundamentals above require
more memorization, and the last 2 require more understanding.  Once this
balance is understood and practiced for each particular student, the
math can be learned more easily and truly enjoyed!  I\'ve found that by
recognizing the balance, students can learn math at a much faster pace
than they are used to.

 

I have my own tutoring business after having worked in the telecom
industry for over 25 years as a systems engineer.  I received my B.A. in
Mathematical Science from Rice University in 1978, and my M.S. from
Stanford University in Operations Research in 1979.  I also have 3 kids
who have gone through the award-winning Plano, Texas school system.

 

Please let me know if there\'s any way I can contribute in any capacity
to this exciting new program.

 

Thank you,\
Lisa Johnson\
5849 Bedrock Drive\
Plano, Texas  75093\
www.planotutor.com

[^1]: ^\[^