Math education IS mathematical engineering

Math education is mathematical engineering, so says Professor Hsu-Shieh Wu, professor of mathematics at UC Berkeley, topologist, gifted teacher, member National Mathematics Advisory Panel, etc. He is clear: this is not an analogy, but a definition.

Wu gave the plenary talk at the National Council of Teachers of Mathematics (NTCM) Annual Meeting in 2007. The comments here are taken from the slides of that talk (available here) and from recent papers he has written on the same subject, that of the relationship between mathematicians and math educators (see here). The errors or confusions should be considered due to me, rather than Wu. Consider the comments in italics to be straight from his talk or papers, and the rest to be from me.

Wu uses this definition of engineering: engineering is the customization of abstract scientific principles to satisfy human needs.


Mechanical engineering, then is about turning the laws of classical physics into pulleys, cranes, refrigerators, etc. It's how you build your cel phone or IPod so you can drop it without it breaking. Likewise, chemical engineerings is about turning chemistry into the plexiglas for your aquarium, the non toxic antibacterial cleanser you use on kid's toys, the gas you put in your car.

Mathematical engineering, then, is about customizing mathematics for use by students and teachers in K-12. That is the human need of mathematical engineering. Without the customization of math to students and teachers in K-12, math cannot be used by adults. It is as if you were unable to make any use of chemistry or physics. Math education, and its pedagogy, matters deeply. You cannot simply throw adult math at school age children.

Engineering gets better over time, with refinements, but the underlying principles don't change. Engineering must find its way between two principles: the inviolable scientific principle, and user friendliness of the end product.

Likewise, mathematical engineering must improve with better pedagogy, better adaptation to the students, etc. but it can not change violate the principles of mathematics: 1) precision, 2) definition, 3) coherence, and 4) reasoning, 5) purposefulness.

Precision means making clear, unambiguous mathematical statements. There is no unspoken context in math: you are expected to make clear what is known/given, and what is unknown.

Definition: concepts in math have specific definitions. A specific concept as a specific definition--one, and all others can be shown equivalent. But the structure of math demands a definition for all concepts.

Coherence: math builds on prior knowledge. Nothing comes out of the blue, but it unfolds from what is known already.

Reasoning: mathematics cannot proceed without reasoning. Reasoning must be illuminated.

Purposefulness: math is goal oriented. It solves specific problems.

Mathematical engineering, that is, math education, to date as lost these principles. While there is some pedagogy available for teachers, over and over again we see that the teaching of basic ideas (geometry, fractions, etc.) of math violate most or all of the above 5 principles. This is unworkable engineering: it is the equivalent of chemical engineering labs that don't know enough chemistry to create stable chemicals.

In other papers as well as this talk, Professor Wu elaborates on the failures of current pedagogy to address those 5 principles. He does this by showing the failure in the presentation of fractions, geometry, and in what he calls the Fundamental Assumption of School Mathematics. I will address the fractions presentation in another post.

For geometry, he states that the mathematics of Euclid and Hilbert goes from axioms to theorems to proofs in an un-user friendly way. However, the major school presentations either teach it this way, without addressing students' learning capacity, or they present it without definitions, theorems, and proofs, as if it is experimental geometry, and so it lacks precision, coherence, reasoning, etc.

For the Fundamental Assumption of School Math: mathematically, it is true that all arithmetic operations on fractions (i.e. rational numbers) can be extrapolated to work on the reals. Why this is true is nontrivial mathematics. School math is only about rationals, and then presentation of irrationals is done without any such claim or explanation that it can be done. The assumption is left unstated: a vioation of the precision, definition, reasoning, etc.

So mathematical engineering requires both mathematicians and math educators, just as chemical engineering requires engineers and chemists. The math educators are needed because they know the students, and what is needed by the students at their various ages. Math educators know what the school math curriculum is for a given level, if not how to present it. They, too, are familiar with what the maturity level of their students is. Mathematicians are needed because correct mathematics must be taught at each of these levels, even though what is known at each level is different. That means a variety of correct explanations must be made available. That requires deep subject knowledge--deep enough to understand what's true and correct from a variety of view points. And in truth, these issues come together as we attempt to find the best approach for each student: you need both pedagogy and mathematics in order to reach students and still teach them math that adhered to the above 5 principles.