Tuesday, July 14, 2015

From the ashes of New Math

[Previously posted at the teaching blog]

One of my big concerns with the education reform debate, particularly as it regards mathematics, is that a great deal of the debate consist of words being thrown around that have a positive emotional connotation, but which are either vague or worse yet mean different things to different participants in the discussion.

As a result, you have a large number of "supporters" of common core who are, in fact, promoting entirely different agendas and probably not realizing it (you might be able to say the same about common core opponents but, by nature, opposition is better able to handle a lack of coherence) . I strongly suspect this is one of the causes behind the many problems we've seen in Eureka math and related programs. The various contributors were working from different and incompatible blueprints.

There's been a great deal of talk about improving mathematics education, raising standards, teaching problem-solving, and being more rigorous. All of this certainly sounds wonderful, but it is also undeniably vague. When you drill down, you learn that different supporters are using the same words in radically different senses .

For David Coleman and most of the non-content specialists, these words mean that all kids graduating high school should be college and career-ready, especially when it comes to the STEM fields which are seen as being essential to future economic growth.

(We should probably stop here and make a distinction between STEM and STEAM – science technology engineering applied mathematics. Coleman and Company are definitely talking about steam)

Professor Wu (and I suspect many of the other mathematicians who have joined into the initiative) is defining rigor much more rigorously. For him, the objective is to teach mathematics in a pure form, an axiomatic system where theorems build upon theorems using rules of formal logic. This is not the kind of math class that most engineers advocate; rather it is the kind of math class that most engineers complain about. (Professor Wu is definitely not a STEAM guy.)

In the following list taken from this essay from Professor Wu, you can get a feel for just how different his philosophy is from David Coleman's. The real tip-off is part 3. The suggestion that every formula or algorithm be logically derived before it can be used has huge implications, particularly as we move into more applied topics. (Who here remembers calculus? Okay, and who here remembers how to prove the fundamental theorem of calculus?)

All of Professor Wu's arguments are familiar to anyone who has studied the history of New Math in the 60s. There is no noticeable daylight between the two approaches.

I don't necessarily mean this as a pejorative. Lots of smart people thought that new math was a good idea in the late 50s and early 60s; I'm sure that quite a few smart people still think so today. I personally think it's a very bad idea but that's a topic for another post. For now though, the more immediate priority is just understand exactly what we're arguing about.
The Fundamental Principles of Mathematics

I believe there are five interrelated, fundamental principles of mathematics.
They are routinely violated in school textbooks and in the math education
literature, so teachers have to be aware of them to teach well.

1.  Every concept is precisely defined, and definitions furnish the basis for logical
deductions. At the moment, the neglect of definitions in school mathematics has reached the point at which many teachers no longer know the difference between a definition and a theorem. The general perception among the hundreds of teachers I have worked with is that a definition is “one more thing to memorize.” Many bread-and-butter concepts of K–12 mathematics are not correctly defined or, if defined, are not put to use as integral parts of reasoning. These include number, rational number (in middle school), decimal (as a fraction in upper elementary school), ordering of fractions, product of fractions, division of fractions, length-area-volume (for different grade levels), slope of a line, half-plane of a line, equation, graph of an equation, inequality between functions, rational exponents of a positive number, polygon, congruence, similarity, parabola, inverse function, and polynomial.

2.  Mathematical statements are precise. At any moment, it is clear what is known and what is not known. There are too many places in school mathematics in which textbooks and other education materials fudge the boundary between what is true and what is not. Often a heuristic argument is conflated with correct logical reasoning. For example, the identity √a√b = √ab for positive numbers a and b is often explained by assigning a few specific values to a and b and then checking for these values with a calculator. Such an approach is a poor substitute for mathematics because it leaves open the possibility that there are other values for a and b for which the identity is not true.

3.  Every assertion can be backed by logical reasoning. Reasoning is the lifeblood of mathematics and the platform that launches problem solving. For example, the rules of place value are logical consequences of the way we choose to count. By choosing to use 10 symbols (i.e., 0 to 9), we are forced to use no more than one position (place) to be able to count to large numbers. Given the too frequent absence of reasoning in school mathematics, how can we ask students to solve problems if teachers have not been prepared to engage students in logical reasoning on a consistent basis?

4.  Mathematics is coherent; it is a tapestry in which all the concepts and skills are logically interwoven to form a single piece. The professional development of math teachers usually emphasizes either procedures (in days of yore) or intuition (in modern times), but not the coherent structure of mathematics. This may be the one aspect of mathematics that most teachers (and, dare I say, also math education professors) find most elusive. For instance, the lack of awareness of the coherence of the number systems in K–12 (whole numbers, integers, fractions, rational numbers, real numbers, and complex numbers) may account for teaching fractions as “different from” whole numbers such that the learning of fractions becomes almost divorced from the learning of whole numbers. Likewise, the resistance that some math educators (and therefore teachers) have to explicitly teaching children the standard algorithms may arise from not knowing the coherent structure that underlies these algorithms: the essence of all four standard algorithms is the reduction of any whole number computation to the computation of single-digit numbers.

5.  Mathematics is goal oriented, and every concept or skill has a purpose. Teachers who recognize the purposefulness of mathematics gain an extra tool to make their lessons more compelling. For example, when students see the technique of completing the square merely as a trick to get the quadratic formula, rather than as the central idea underlying the study of quadratic functions, their understanding of the technique is superficial. Mathematics is a collection of interconnecting chains in which each concept or skill appears as a link in a chain, so that each concept or skill serves the purpose of supporting another one down the line. Students should get to see for themselves that the mathematics curriculum moves forward with a purpose.
At the risk of putting too fine of a point on it, this approach tends to produce extremely formal and dense prose such the following (from a company Professor Wu was involved with):
Dilation: A transformation of the plane with center O and scale factor r(r > 0). If
D(O) = O and if P ≠ O, then the point D(P), to be denoted by Q, is the point on the ray OP so that |OQ| = r|OP|. If the scale factor r ≠ 1, then a dilation in the coordinate plane is a transformation that shrinks or magnifies a figure by multiplying each coordinate of the figure by the scale factor.

Congruence: A finite composition of basic rigid motions—reflections, rotations,
translations—of the plane. Two figures in a plane are congruent if there is a congruence that maps one figure onto the other figure.

Similar: Two figures in the plane are similar if a similarity transformation exists, taking one figure to the other.

Similarity Transformation: A similarity transformation, or similarity, is a composition of a finite number of basic rigid motions or dilations. The scale factor of a similarity transformation is the product of the scale factors of the dilations in the composition; if there are no dilations in the composition, the scale factor is defined to be 1.

Similarity: A similarity is an example of a transformation.

1 comment:

  1. Let's follow the lead of the mathematicians here and first be clear about our definitions. Common Core is a set of standards: it is a list of behaviors that students are supposed to achieve at each grade level. And it is the intention that those who attain those standards will, at the end of high school, be prepared for college or for certain non-college-degree-requiring careers.

    That is quite a separate matter from issues of textbooks and teacher-training. These are key for successful implementation of the Common Core standards, and I do not deny the importance of these things. But unless you want to argue (and perhaps you do) that the Common Core standards are inherently impossible to implement, you cannot rationally attack the standards by criticizing specific textbooks, or even the present lack of any adequate textbooks.

    I think it would be helpful to your readers if you would make it clear whether you disagree in any substantial way with the Common Core Math Standards. I personally have read them and they strike me as quite appropriate. Do you agree or not? If not, what are your concerns?

    Then if you want to blog about the inadequacies of Eureka math or other textbooks, do so--but don't cast it as a problem with the Common Core standards. My own daughter is learning math under the new Common Core standards--and, in plain English, her textbook sucks! So I'm with you on this.

    But let's be clear what we're talking about: the standards themselves, the implementation of the standards in the classroom, or the assessments of achievement of those standards, or the utilization of those assessments to evaluate students, teachers, and schools. These are all separate issues and nobody is truly served by conflating them.