**Presentation to the National Mathematics Panel**

Chapel Hill, NC, June 29, 2006

By **Hyman Bass**

Professor of Mathematics and Mathematics Education, University of
Michigan

Mr. Chairman, I thank you and the Panel for this opportunity to speak
here. I am a research mathematician, specializing in various branches of
algebra. In addition I have been collaborating, over the past several
years, with educational researchers in Michigan, focusing largely on the
nature and measurement of the mathematical knowledge needed for
teaching.

I want to offer the Panel some brief perspectives on the following three
themes:

I.  The nature of algebra in the school curriculum

II. How the Panel will proportion its attention to issues of equity and
    of mathematical enrichment.

III. The norms of discourse and reasoning of the Panel

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I.  **The nature of algebra in the school curriculum**

Algebra is rightly seen to be a central concern in mathematics
education, for example because:

a.  Algebra ­-- its content, methods, and symbolic language -- is
    foundational for all of mathematics and science.

b.  Algebra is a gateway subject, through which many, far too many,
    students [fail]{.underline} to pass, thus foreclosing further access
    to technical fields, with all the attendant economic and other
    consequences.

c.  The students who are thus disempowered are disproportionately
    identified by race, ethnicity, and socio-economic status.

This is not a new problem and concern, and there are several competing
hypotheses, implicit in new curriculum materials, standards documents,
and assessments about how best to intervene on the problem. Much of the
debate about these alternatives has -- inappropriately in my view --
been framed as an argument about what algebra "really is." I would like
to propose a reframing that, I suggest, will help clarify the discourse.
I see two prevailing views of what algebra is.

**Algebra as the theory of (polynomial) equations ("Symbolic algebra")**

This "traditional" view emphasizes systematic use of symbolic notation
for variables and functions, and the rules for their use and
manipulation, sometimes referred to as "generalized arithmetic." A
function is generally presented by a formula, y = f(x), expressing a
relationship between two numerical variables. This view also emphasizes
the coordinate plane, and graphs of functions, equations, and
inequalities. Detailed treatment is generally restricted to linear
functions and lines, two linear equations in two unknowns, and quadratic
equations, plus perhaps some discussion of exponential functions of
integer (or perhaps rational) exponent. There is also some treatment of
the formal arithmetic (+, -, x, ÷ ) of rational expressions.

**Patterns, functions, and algebra ("Algebra as modeling")**

This is a relatively new curricular environment in which some educators
have had algebra take up residence with new neighbors. Their
motivations, which have a certain plausibility, flow from some popular
(and compelling) slogans about mathematics and its role in the
contemporary world. "Mathematics is the science of patterns." "The
notion of function is one of the central concepts of mathematics." "The
world is now saturated with data, and a primary role of mathematics is
to find patterns in (or functions to model) these data." For example,
students in such a curricular regime typically encounter a function
(even a linear function) not as given by a formula, but rather as an
inferred "pattern" in a sequence of numerical or geometric data, in
other words, as the product of a process of mathematical modeling. While
some deep truths lie behind the slogans above, the curricular
incarnations given to them by some educators have (appropriately in my
view) been the subject of vigorous debate.

It is not my purpose here to argue the pros and cons of these two views
of algebra, but, instead, to make clear that they represent *two
different mathematical subject areas*, each of which can present
substantial and legitimate mathematics, but *not the same* mathematics.
To feud over which one has the right to be called 'algebra' seems to me
a pointless legalistic debate. Its implied effect is that validation of
one perspective is at the cost of displacing the other, a result that is
neither necessary nor desirable.

I would say that *preparation for university level mathematics
definitely requires fluency with symbolic algebra*. At the same time,
mathematical modeling is an increasingly important context of
mathematical applications, and students should be exposed, *in
developmentally appropriate ways*, to its methods. This is a complex
curricular challenge, and likely requires deeper thought and research
than it has so far been given.

**II. How the Panel will proportion its attention to issues of equity
and of mathematical enrichment**.

By "equity issues" I refer to the persistent achievement gaps in
mathematics (and science) along race and class lines. This problem, with
all of its economic and social consequences, is perhaps the most urgent,
yet still intractable, problem facing American education. My concern is
that this issue may be too weakly represented in the National
Mathematics Panel -- in the rhetoric of its charge, in its discourse so
far, and in the profile of its membership.

In contrast with this, the Panel's membership includes strong
representation and advocacy of special mathematics enrichment programs
for "gifted" students. One risk of such programs, and related practices
of tracking, is our notoriously unreliable, often inequitable, ways of
identifying "gifted" or more accomplished students, and the often
invidious effects on those students otherwise characterized. Let me
emphasize that I am not opposed to such programs. Indeed, as President
of the American Mathematical Society, I protested the ending of the
Young Scholars Program at the NSF, and I helped set up an AMS endowment
fund to support such programs.

My point here is that, while this is an important, and high leverage,
kind of investment to make, it is really at the margins of the central
problems of mathematics education in America, which have to do with
raising levels of achievement for the full diversity of American
students. It would be a sad and distorted use of the National
Mathematics Panel to treat these two issues as of commensurate priority
now, or to attend to concerns for enrichment in ways that have adverse
effects on efforts to improve equity. I urge the Panel to do all it can
to identify the best knowledge and expertise to support learning and
achievement by all students, with particular attention to issues of
language, culture, and poverty as they impact opportunity to learn and
learning outcomes.

**III. The norms of discourse and reasoning of the Panel**

I would like to sensitize the conversation to what I shall provisionally
call the

\[content/resources =\> pedagogy\] - fallacy,

which often obfuscates otherwise important debate. The format of this
argument is, "One should [not]{.underline} teach/use X because doing so
means enacting Y with students in classrooms," (and Y is presumed to be
abhorrent). Here are some examples:

+---------------------------------+------------------------------------+
| **Teaching Practice**           | **Presumed Consequence**           |
+---------------------------------+------------------------------------+
| Teach long division (or some    | Long pages of examples for drill   |
| other standard algorithm)       | work,                              |
|                                 |                                    |
|                                 | or                                 |
+---------------------------------+------------------------------------+
|                                 | The presumption that the only      |
|                                 | reason to study an algorithm is to |
|                                 | be able to execute it.             |
+---------------------------------+------------------------------------+
| Teach how to solve quadratic    | Factoring long lists of quadratic  |
| equations                       | polynomials with integer           |
|                                 | coefficients, the quadratic        |
|                                 | formula being out of reach.        |
+---------------------------------+------------------------------------+
| Situate mathematics problems in | The contexts will be artificial,   |
| "real world" contexts           | mathematically insignificant, and  |
|                                 | obscure the mathematics to be      |
|                                 | learned.                           |
+---------------------------------+------------------------------------+
| Eventually present mathematical | This will be austere, and          |
| ideas in their abstract and     | alienating to students, because    |
| general form.                   | they cannot relate it to their own |
|                                 | experience.                        |
+---------------------------------+------------------------------------+
| Use calculators in grade *n*.   | This will lead to disabling        |
| Or graphing calculators in      | reliance on the calculator, and    |
| grade *N*.                      | loss of technical skill and        |
|                                 | judgment.                          |
+---------------------------------+------------------------------------+
| Use manipulative materials in   | These are little more than a       |
| mathematics instruction.        | playful diversion of children from |
|                                 | their serious mathematics          |
|                                 | learning.                          |
+---------------------------------+------------------------------------+

People on all sides of the education debates are guilty of this kind of
fallacy. They often draw general, even causal, conclusions from
anecdotal examples.

Finally, I would urge that, in keeping with your announced norms for
evidence and intellectual integrity, that blanket criticisms of Federal
agencies, like NSF EHR, or of professional organizations, be ruled out
of order. Criticisms should be reserved for specific documents or
actions, and be accompanied by evidence-based support for the
evaluations being made.

Thank you.

Hyman Bass