Mathematical education of teachers Part II: What are we doing about Textbook School Mathematics?

By Hung-Hsi Wu

This two-part series is a summary of the longer paper, Textbook School Mathematics and the preparation of mathematics teachers.

TSM (Textbook School Mathematics) has dominated school mathematics curriculum and assessment for the past four decades, yet, in mathematics education, TSM is still the elephant in the room that everybody tries to ignore.

We will look at three examples of this phenomenon.

Example 1. The volume on The Mathematical Education of Teachers II ( MET2) rejects the temptation of teaching teachers only advanced mathematics and leaving them to find their own way in school mathematics. That is good. But its recommendation for the preparation of high school teachers, for example, is the completion of the equivalent of a math major and three courses with a primary focus on high school mathematics from an advanced standpoint. The suggested organizing principles for the three-semester course sequence are (MET2, p. 62):

Emphasize the inherent coherence of the mathematics of high school.

Develop a particular mathematical terrain in depth.

Develop mathematics that is useful in teachers’ professional lives.

There is no mention of teachers’ thirteen years of mis-education in TSM, much less what to do about it.

For example, we know how definitions are ignored or mangled in TSM:  definitions are not important.  A definition, teachers are told, is nothing more than “one more thing to memorize”. How then can they learn to start emphasizing the importance of definitions?

Take the concept of congruence: in K-8 TSM, students are taught that any two figures are congruent if they have the ‘same size and same shape,” but in high school “curvy” figures are forgotten and only polygons are considered: two polygons are congruent if corresponding sides and corresponding angles are equal. Now come the CCSSM which want congruence to have one definition in terms of reflections, translations, and rotations all through middle school and high school.  Similar remarks hold for all other concepts, such as similarity, expression, graph of an equation, graph of an inequality, etc.

When we ask for such a sea change, would a genteel discussion in general and an in-depth investigation of a particular terrain be enough to bring it about?

I suggest that these three courses be used to give a systematic exposition of the high school mathematics curriculum at a level as close to the school classroom as possible, but in a way that is mathematically correct. Such an exposition will show teachers how definitions can be used productively in the school classroom as well as how school mathematics differs from TSM in terms of coherence, reasoning, precision, and purposefulness.  If we want a sea change in teachers’ conception of mathematics, let us show them the way, from the ground up.

Few math departments have the resources to offer such courses, but one of them is at UC Berkeley: see the description of “Math 151–153″ in the Appendix of this article. Until we can provide teachers with a knowledge of correct school mathematics, the more esoteric recommendations in MET2—such as research experience for high school teachers—can wait.

Example 2. The CCSSM have made significant inroads in steering many topics away from TSM, but the CCSSM have also prefaced the content standards with eight  Mathematical Practice Standards (MPS) for students. A confluence of unusual circumstances has created the misconception that equates the CCSSM with the MPS. The idea that, in order to implement the CCSSM, all it takes is to study the MPS has taken root. Let us take a reality check.

If teachers know correct mathematics, the substance of the MPS would be a natural side effect of this knowledge. Mentioning the MPS somewhere in the content stanards is definitely a good thing. Unfortunately, putting MPS front and center in the transition from TSM to the CCSSM puts the cart before the horse. Let us consider, for example, the second and third MPS that state:

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

How would these work out for the task of writing down the equation of the line passing through two given points, \( (1,2)\) and \((3,4)\)?

TSM only teaches how to do this by rote, because the slope of a line is incorrectly defined in TSM as the difference quotient of the coordinates of two points that are a priori given on the line. But the CCSSM want slope to be defined correctly so that any two points on the line can be used to compute its slope. Then it is self-evident that both of the following lead to an equation of the line: For any point \((x,y)\) on this line,

\[ \frac{y-2}{x-1}\ = \ \frac{4-2}{3-1} \quad \mbox{and} \quad \frac{y-4}{x-3}\ = \ \frac{4-2}{3-1}\]

Now the MPS exhort students to explain how the equations come about and to critique each other’s reasoning. Given that we have only provided teachers with a knowledge base of TSM and that students continue to get TSM from their textbooks, studying the MPS will help neither the students nor the teachers in this task. Instead of encouraging this fixation on the MPS, how about first helping teachers to replace their defective knowledge (TSM) with correct mathematics?

Example 3. A recent volume Principles to Actions (NCTM 2014) has the goal of describing “the conditions, structures, and policies that must exist for all students to learn.” We will refer to this volume as  P-to-A.

P-to-A makes no mention of TSM or the need to help teachers overcome the damage done to their thinking by TSM.  Nevertheless, it asks teachers to use “purposeful questions” to “help students make important mathematical connections, and support students in posing their own questions”, (P-to-A, pp. 35, 36). Given teachers’ immersion in TSM, what can they say when students ask for the purpose of learning the laws of (rational) exponents?  In TSM, it could only be to ace standardized tests. P-to-A also says “effective teaching of mathematics builds fluency with procedures on a foundation of conceptual understanding” (P-to-A, p. 42). Since TSM gets it all wrong even in something as mundane as solving a linear equation (see p. 22–25 of this article), teachers who know only TSM will be able to do nothing but transmit TSM’s pseudo-reasoning about the procedure of solving an equation to their students. There goes conceptual understanding out the window. And so on.

P-to-A enthusiastically recommends actions to realize these and other worthy learning goals seemingly without realizing that, given their damaged knowledge base, our teachers are not ready for these actions. On the issue of how to help teachers, all that P-to-A has to say is that they should be provided with all the necessary resources and professional development they need. Nothing about TSM. Since current professional development mainly recycles TSM, how can this possibly help?

TSM is the elephant in the room that everybody tries to ignore. This cannot go on.

Let us bring closure to this discussion. TSM comes from school textbooks, so why not just concentrate on getting rid of TSM by writing better textbooks? Two reasons: (1) The vicious circle syndrome: Staff writers in major publishers are themselves products of TSM. (2)  The bottom-line mentality: In order to maximize the sales of their books, publishers do not publish anything teachers (products of TSM) don’t feel comfortable reading.

At the moment, the only hope of getting better school textbooks is for teachers to reject TSM-infested textbooks. Then, and only then, will publishers listen.  Helping teachers to eradicate TSM is therefore not only imperative for improving their content knowledge, but it may also be the only way to get better school textbooks written.

The author is very grateful to Larry Francis for his many suggested improvements.