Mathematical Education of Teachers, Part I: What is Textbook School Mathematics?

By Hung-Hsi Wu

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

School mathematics education has been national news for at least two decades. The debate over the adoption of the Common Core State Standards for Mathematics (CCSSM) even became a hot-button issue in the midterm elections of 2014. This surge in the public’s interest in math education stems from one indisputable fact:  school mathematics is in crisis.

From the vantage point of academia, two particular aspects of this crisis are of pressing concern: School textbooks are too often mathematically flawed, and in spite of the heroic efforts of many good teachers, the general level of math teaching in school classrooms is below acceptable.

Mathematicians like to attack problems head-on. To us, the solution is simple: Just write better school textbooks and design better teacher preparation programs. I will concentrate on the latter for now and will not return to the textbook problem until the end of Part 2.

An effective math teacher has to know the subject thoroughly (strong content knowledge) and be able to communicate with students (good pedagogy). However, my own involvement with the professional development of teachers in K–8 points strongly to the fact that, as of 2014, students’ nonlearning of mathematics has more to do with teachers’ content knowledge deficit than with deficiencies in pedagogy. If our short-term goal is to get schools out of this crisis as soon as possible, a focus on this content knowledge deficit must be our top priority.

Let me make clear at the outset that the blame for this deficit has to be laid squarely at the feet of the education establishment: the schools of education, statewide and district-wide administrators, and the mathematics community. Its systemic failure to provide teachers with the correct content knowledge they need for teaching mathematics is nothing short of scandalous. When teachers are themselves students in K-12, the mathematics they learn from their textbooks is the same fundamentally and seriously flawed mathematics they will in turn teach to their students. We call this body of knowledge Textbook School Mathematics (TSM). When they get to college as preservice teachers, they are (as of 2014) generally not made aware of the flaws in TSM, and are consequently not provided with any replacement for TSM. It therefore comes to pass that when it is their turn to teach students, they can only fall back on the TSM that they learned as K -12 students. This is how TSM gets recycled in schools, and this is how a body of unlearnable mathematical “knowledge” has come to rule the classrooms of our nation.

TSM has been around for at least four decades. Some may be taken aback by this statement, because didn’t the 1989 mathematics education reform correct all that was wrong with “traditional math”? Isn’t “reform math” at least different from “traditional math”? The short answer is that the two differ mostly in the packaging but not in their underlying mathematical substance: they both suffer from the same mathematical defects. How could this be otherwise when the people who brought us the reform were themselves victims of TSM?

Let us give a brief list of the defects of TSM. Note, however, that these defects are so deep and pervasive that they cannot be fully captured by a short list.

1.  Definitions are absent or mangled. (For example: fractions, percent, rate, graph of an equation, congruence, similarity.)
2. No precision. (TSM blurs the line between reasoning and heuristics: it is never clear whether a statement has been proved, or is being offered as a definition. For example: \( a^{-m} = \frac{1}{a^m}\) for all positive \(a\) and all integers \(m\). Consequently, many teachers do not know the difference between a definition and a theorem.)
3. Reasoning is absent or flawed. (This is the inevitable consequence of having no definition or flawed definitions. For example, why is the graph of \( ax+by=c\) a line?)
4. Incoherence. (TSM makes a disjointed presentation of what should be a tightly organized subject: school mathematics. For example, there is no continuity from middle school geometry (rotations, reflections, and translations) to high school geometry.)
5. The purpose of studying anything is usually well hidden. (For example, why make students learn to write \(\sqrt[5]{7} \) as \( 7^{1/5}\), and why make them learn the laws of exponents for fractional exponents unless it is to simplify expressions on standardized tests?)

One can easily infer from this list that we consider it imperative for teachers to know the mathematics they have to teach in K–12 in a way that is consonant with the normal development of mathematics:  the definitions leave no doubt about what is being discussed; the precise language minimizes misunderstanding; the reasoning and the coherence reduce learning by rote; and finally, knowing the mathematical purpose that a concept or skill serves gives students more motivation to learn it. This list is all about maximizing the learning outcome.

A typical illustration of TSM is the way the concept of percent is presented to students. The meaning TSM gives for percent is out of a hundred. Perhaps “out of a hundred” sounds clear in everyday language, but mathematically its lack of precision is unacceptable and unusable. Is percent perhaps a number? Because TSM gives no clear definition, percent has become one of the most feared topics in middle school. Without a definition for percent, students cannot reason their way to solutions of percent problems. Therefore, when they need to find the percent of a shaded area when 6 of 41 congruent squares are shaded, students are forced to resort to the rote skill of “setting up a proportion”: solve for \(x\) in \(6/41 = x/100\). In fact, rote memorization of procedures is a hallmark of what passes for learning in TSM.

Any hope of improving mathematics learning hinges on our ability to replace TSM in the school curriculum with correct mathematics. For a reason to be explained in Part 2, the starting point has to be helping our teachers shed their knowledge of TSM and learn how to do K–12 mathematics correctly. The mathematics community should take note that this cannot be accomplished simply by teaching future teachers good, advanced mathematics. University mathematics is fundamentally different from school mathematics. In short, the latter is an engineered version of the former. For example, while rational numbers can be dispatched in three lectures in a junior-level abstract algebra course, it would not do to teach ten-year-olds that a fraction is an equivalence class of ordered pairs of integers. Similarly, one cannot tell twelve-year-olds that constant speed means the distance function has a constant derivative. And so on. If we are serious about doing our share to resolve the education crisis, university campuses across the land will have to commit to teaching correct school mathematics to teachers until TSM is no more.

This will be a serious commitment. The absence of such a commitment thus far has frustrated many teachers who are hungry for correct content knowledge but have nowhere to turn for this knowledge. The education establishment has systematically let math teachers down. There should be no illusion, however, about the heavy responsibility that comes with this commitment:  teachers do not shed the habits acquired over thirteen years of immersion in TSM without a protracted struggle and without a lot of help. The help they need translates into sustained hard work on our part. This is hardly glamorous work, but if mathematicians don’t do it, who will?

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