**Statement for the National Mathematics Panel**

**November 6th, 2006**

**Presented by: Dr. Judit Moschkovich**

**Associate Professor, Education Department, University of California,
Santa Cruz**

**Representing TODOS: Mathematics for All**

Thank you for this opportunity to address this distinguished panel. Your
task is extraordinarily important to the mathematics and mathematics
education community, and to the millions of students in our country for
whom your recommendations will have significant consequences. I write as
a representative of TODOS: Mathematics for All, an organization that
advocates for an equitable and high quality mathematics education for
all students, in particular Hispanic/Latino students as well as students
from minority populations such as Native Americans, African Americans,
and others.

My remarks represent TODOS as well as my own professional experience as
a mathematics instructor at the university level and as a researcher in
mathematics education for over 15 years. My career in mathematics
education began when, after receiving a B.S. in Physics, I taught
Algebra courses as a lecturer in the Mathematics Department at San
Francisco State University. I received my Ph.D. in mathematics education
in 1992 and have been conducting research in classrooms since then. I
have been involved in mathematics education at many levels: I have
served as a member of the editorial panel for the Journal for Research
in Mathematics Education and as the Chair of the AERA Special Interest
Group for Research in mathematics Education (2004-2006). I am the author
of many research articles and chapters in edited books. I teach a course
that introduces future secondary mathematics teachers to evidence-based
research in mathematics education and courses for Ph.D. students in
mathematics education. My research for the past 10 years has focused on
the study of the relationship between language and learning mathematics,
especially for Latino English learners.

I will limit my observations and recommendations to address one of the
points listed on the Executive Order with regards to the Report to the
President on Strengthening Mathematics Education. I include a list of
research references at the end. [Point for discussion,
(c)]{.underline}*: The processes by which students of various abilities
and backgrounds learn mathematics*, with implications for instruction,
teacher education, assessment, and materials development.

The main question I will address is "What are the needs of Latino
English learners in mathematics classrooms?" Before I can address that
question, I would like to lay the groundwork by first considering the
following questions:

1\. What is mathematical proficiency?

2\. What is "conceptual understanding" in mathematics? Why is it
important?

3\. Why is communication important for learning mathematics?

**1. What is mathematical proficiency?**

A current description of mathematical proficiency comes from the book
"Adding it up: helping children learn mathematics" published in 2001 by
the National Research Council (edited by Kilpatrick, Swafford, and
Findell). The NRC volume defines the intertwined strands of mathematical
proficiency as: a) Conceptual understanding (comprehension of
mathematical concepts, operations, and relations); b) procedural fluency
(skill in carrying out procedures flexibly, accurately, efficiently, and
appropriately); c) strategic competence (formulating, representing, and
solving mathematical problems (novel problems, not routine exercises);
d) adaptive reasoning (logical thought, reflection, explanation, and
justification); and productive disposition (habitual inclination to see
mathematics as sensible, useful, and worthwhile, coupled with a belief
in diligence and one's own efficacy).

I will address only the first two components, procedural fluency and
conceptual understanding. Fluency in performing mathematical procedures
is what most people imagine we mean when we say "learning mathematics."
Conceptual understanding is more difficult to define and less well
understood by parents, administrators, and beginning teachers. So, what
is conceptual understanding? It involves the connections, reasoning, and
meaning that learners (not teachers) construct. Conceptual understanding
is more than performing a procedure accurately and quickly. It involves
understanding why a particular result is the right answer and what the
meaning of that result is, i.e. what the number, solution, or result
represents.

Another aspect of conceptual understanding involves connecting
representations (such as words, drawings, symbols, diagrams, tables,
graphs, equation, etc.), procedures, and concepts (Hiebert & Carpenter,
1992). For example, if a student understands addition and
multiplication, we would say they have learned to make connections
between these two procedures, and expect that they would be able to
explain how multiplication and addition are related (for example, that
multiplication can be described as repeated addition). If they
understand the procedures for multiplying and dividing negative numbers,
we would say they have learned to make connections between these two
procedures, and expect that they would be able to explain how the
procedures for multiplication and division are similar, different, and
explain why.

**2. Why is conceptual understanding important?**

One might think, "OK fine, so some researchers think that students need
to be able to draw pictures and explain what they are doing when they
perform a procedure, but what is the big deal about conceptual
understanding? Why can't students just learn their multiplication facts
or learn that the right procedure to divide fractions is to "invert and
multiply" and be done with it? I certainly don't think I understand most
of the arithmetic I learned and yet I have made it through school. Why
does my child's learning need to include conceptual understanding?"

One answer to these questions is that conceptual understanding and
procedural fluency are closely related, even if we, as adults, do not
now remember understanding a particular procedure when we learned that
procedure. Research in cognitive science (Bransford, Brown, & Cocking,
2000) has shown that people remember better, longer, and in more detail
if they understand, actively organize what they are learning, connect
new knowledge to prior knowledge, and elaborate. Children will remember
procedures better, longer, and in more detail if they actively make
sense of procedures, connect procedures to other procedures, and connect
procedures to concepts and representations. Rehearsal may work for
memorizing a grocery list (and even then organizing the list will
improve memorization). Rehearsal, however, is not the most efficient
strategy for remembering how to perform demanding cognitive tasks. The
research evidence is clear. The best way to remember is to understand,
elaborate, and organize what you know (Bransford, Brown, & Cocking,
2000).

**3. Why is communication important for learning mathematics?**

One might think, "OK, I can see why my child needs to develop conceptual
understanding, but what is all the fuss about communication in the
mathematics classroom? I always did math by myself at my desk."
Communication is important because it supports conceptual understanding.
The more opportunities a learner has to make connections among multiple
representations, the more opportunities that learner has to develop
conceptual understanding. But not all kinds of communication will
support conceptual understanding in mathematics. Communication needs to
be focused on important mathematical ideas. Classroom communication that
engages students in evidence based arguments by focusing on
explanations, arguments, and justifications builds conceptual
understanding. Communication should include multiple modes (talking,
listening, writing, drawing, etc.), because making connections among
multiple ways of representing mathematical concepts is central to
developing conceptual understanding.

**4. What are the needs of Latino English Learners in mathematics
classrooms**[^1]**?**

Latino students need to develop both procedural fluency and conceptual
understanding, because these are two central aspects of mathematical
proficiency. They also need to have opportunities to engage in
mathematical communication, because this is a central way to develop
both procedural fluency and conceptual understanding. Now, one might
think "Sure, classroom discussions may support conceptual understanding.
But English learners can't participate in mathematical discussions
because they are just learning English" or "English learners just need
to learn math vocabulary. Then they can participate in mathematical
discussions."

These may seem like common sense claims. However, research shows that
English language learners, even as they are learning English, can
participate in discussions where they grapple with important
mathematical content (for examples of lessons where English Learners
participate in a mathematical discussion see Moschkovich, 1999a and
Khisty, 1995). Instruction for this population should not emphasize
low-level language skills over opportunities to actively communicate
about mathematical ideas. One of the goals of mathematics instruction
for students who are learning English should be to support all students,
regardless of their proficiency in English, in participating in
discussions that focus on important mathematical ideas, rather than on
pronunciation, vocabulary, or low-level linguistic skills. By learning
to recognize how English learners express their mathematical ideas as
they are learning English, teachers can maintain a focus on the
mathematical concepts as well as on language development.

Research also describes how mathematical communication is more than
vocabulary. While vocabulary is necessary, it is not sufficient.
Learning to communicate mathematically is not merely or primarily a
matter of learning vocabulary. During discussions in mathematics
classrooms students are also learning to describe patterns, make
generalizations, and use representations to support their claims. The
question is not whether students who are English learners should learn
vocabulary but rather how instruction can best support students as they
learn both vocabulary and mathematics. Vocabulary drill and practice is
not the most effective instructional practice for learning either
vocabulary or mathematics. Instead, vocabulary and second language
acquisition experts describe vocabulary acquisition in a first or second
language as occurring most successfully in instructional contexts that
are language rich, actively involve students in using language, require
both receptive and expressive understanding, and require students to use
words in multiple ways over extended periods of time (Blachowicz &
Fisher, 2000; Pressley, 2000). To develop written and oral communication
skills students need to participate in negotiating meaning (Savignon,
1991) and in tasks that require output from students (Swain, 2001). In
sum, instruction should provide opportunities for students to actively
use mathematical language to communicate about and negotiate meaning for
mathematical situations.

In conclusion, I would like to thank the National Mathematics Panel for
this opportunity to address these concerns and represent TODOS.

Dr. Judit Moschkovich

Associate Professor

Education Department

University of California, Santa Cruz

1156 High Street

Santa Cruz, CA 95064

[jmoschko@ucsc.edu]{.underline}

**References**

Blachowicz, C. and Fisher, P. (2000). Vocabulary instruction. In M.
Kamil, P. Mosenthal, P. D. Pearson R. Barr, (Eds.), [Handbook of Reading
Research]{.underline}, Volume III. Mahwah, NJ: Lawrence Erlbaum
Associates, 503-523.

Bransford, J., Brown, A., Cocking, R. (2000). [How people learn: Brain,
mind, experience, and school]{.underline}. Washington, DC: National
Academy Press.

Hiebert, J. and Carpenter, T. (1992). Learning and teaching with
understanding. In D. Grouws (Ed.), [Handbook of Research in Mathematics
Teaching and Learning.]{.underline} New York: Macmillan, pp. 65-97.

Khisty, L. L. (1995). Making inequality: Issues of language and meanings
in mathematics teaching with Hispanic students. In W. G. Secada, E.
Fennema, & L. B. Adajian (Eds.), [New directions for equity in
mathematics education]{.underline} (pp. 279-297). New York: Cambridge
University Press.

Khisty, L. L., McLeod, D., & Bertilson, K. (1990). Speaking
mathematically in bilingual classrooms: An exploratory study of teacher
discourse. [Proceedings of the Fourteenth International Conference for
the Psychology of Mathematics Educator, 3,]{.underline} 105-112. Mexico
City: CONACYT.

Kilpatrick, Swafford, & Findell (2001). [Adding it up: helping children
learn mathematics]{.underline}. National Research Council. Washington,
DC: National Academy Press.

Moschkovich, J.N. (in press) Bilingual Mathematics Learners: How views
of language, bilingual learners, and mathematical communication impact
instruction. To appear in N. Nasir and P. Cobb (Eds.), [Diversity,
Equity, and Access to Mathematical Ideas]{.underline}. New York:
Teachers College Press.

Moschkovich, J. N. (in press). Using two languages while learning
mathematics. To appear in [Educational Studies in
Mathematics]{.underline}.

Moschkovich, J. N. (2002). A Situated and Sociocultural Perspective on
Bilingual Mathematics Learners. [Mathematical Thinking and
Learning]{.underline}, 4(2&3), 189-212.

Moschkovich, J. N. (1999a). Supporting the participation of English
language learners in mathematical discussions. [For the Learning of
Mathematics,]{.underline} 19(1), 11-19.

Moschkovich, J.N. (1999b) Understanding the needs of Latino students in
reform-oriented mathematics classrooms. In L. Ortiz-Franco, N.
Hernandez, and Y. De La Cruz (Eds.), [Changing the Faces of Mathematics
(Vol. 4): Perspectives on Latinos]{.underline}. Reston, VA: NCTM, 5-12.

Pressley, M. (2000). What should comprehension instruction be the
instruction of? In M. Kamil, P. Mosenthal, P. D. Pearson R. Barr,
(Eds.), [Handbook of Reading Research]{.underline}, Volume III. Mahwah,
NJ: Lawrence Erlbaum Associates, 545-561.

Savignon, S. (1991). Communicative language teaching: State of the art.
[TESOL Quarterly]{.underline}, Vol. 25, No. 2, 261-277.

Swain, M. (2001). Integrating language and content teaching through
collaborative tasks. [Canadian Modern Language Review]{.underline},
Volume 58, No. 1, 44-63.

[^1]: I address this question in more detail in a chapter titled
    "Understanding the needs of Latino students in reform-oriented
    mathematics classrooms" (Moschkovich, 1999b).