**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.

## 18 comments:

This is awesome! Thank you so much.

"The math educators are needed because they know the students, and what is needed by the students at their various ages."

Do they really know what is needed by the students at their various ages? I don't think so. At best, they are making trade-offs when they mix kids of very wide ability levels in the early grades. I don't turn to teachers to tell me what my child can or can't do.

Everyday Math is designed to allow kids to move from one grade to the next without ensuring mastery. This is a fundamental assumption. They might say that mastery is important, but they don't make sure it happens. You can try to supplement EM, but you can't make it into something that it isn't.

I'm not sure what Wu's goal is here, but I'm not optimistic about it having much effect. It's not just subject knowledge that's needed, but a change in philosophy and a change in expectations. Schools can't view mastery as simply a rote process. They can't unlink understanding from mastery and just talk about conceptual understanding. Math is more than conceptual understanding and guess and check.

When I learned about vector cross products, I didn't learn about them as a rote skill or concept. I learned about them as a (carefully-defined) tool to add to my mathematical toolbox. This has absolutely nothing to do with solving a problem only one way or any other silly view of math. Schools can't look at math as some sort of free-form Zen-like ability to solve problems.

I've come to the conclusion that what drives math curriculum selection in K-6 is full-inclusion and low expectations. Schools know that Singapore Math is better. They won't come out and say that, but they will say that it's not aligned with the state standards. One head of curriculum told me that Singapore Math looks good, but Everyday Math is a better fit for their students. This means low expectations. The assumption is that if kids don't do well in math, it's not their fault. They just point to all of the kids who are doing well.

Schools are supposed to know what kids are capable of at each grade? I don't think so. Just look at the number of kids taking algebra in 8th grade. They think that's normal.

Their job is easier if they lower expectations and place the onus on the student, first as developmentally appropriate, and later as taking responsibility for their own learning.

Sorry, I'd blame your confusion on my writing.

"Do they really know what is needed by the students at their various ages?"

I don't mean "What they need to know" in some philsophical sense. They, the teachers, know the curriculum as it stands, today. Therefore they know the standards. Mathematicians simply don't.

I am not saying, and certainly Wu isn't saying that math teachers know what SHOULD be taught to students.

---I'm not sure what Wu's goal is here

His goal is to fix mathematics education. Pure and simple. He's not naive about what it will take.

He's on standards' boards in California trying to improve them. He spends weeks or months every year teaching seminars and classes to k-12 math teachers so they learn the math that they simply don't know. He writes papers on what needs to be taught in k-12, and what the concepts are that are being missed.

Neither Wu nor anyone else serious in fixing math education is saying that we "just" need anything: this problem is not solved by only varying the value on one axis. It's a straw man to pretend that people are saying that.

Wu agrees with you about the fundamental lack of understanding about mastery. (He has a paper called "Basic skills versus conceptual understanding: A bogus dichotomy in mathematics education". But how would you change their philosophy? First, you have to elucidate WHAT principles in mathematics make their philosophy wrong. That's why he enumerates his five mathematical principles: because until you can name what's missing, what it is that their philosophy fails to address, you can't change it.

You should read Wu's slides and the papers. His slide talk about the corruption of math education in more depth, but I thought we'd hoed that ground pretty well here on KTM so I didn't bother to highlight it. Just reading the titles of his papers will tell you what he's trying to do.

Don't get me wrong. I'm a big fan of Wu. We're all trying to find the right angle or leverage to create the most change. Perhaps he can change the thinking at the NCTM level, I don't know, but I don't think he will have much effect at the local level.

Our schools just select curricula, and certain curicula are off the table. A very important criterion for our schools is the ability of the curriculum to adapt to individual learning needs. This doesn't just mean learning styles, it means levels of ability. Even if the teachers knew much more about math, they still would not want to set higher expectations for mastery. The onus is on the child to show that he/she can do the work.

It's not just a math issue.

Steve,

I believe that Dr. Wu is speaking of how things 'should be' in our schools, not of how they actually are. When he says that

"math educators are needed because they know the students, and what is needed by the students"

he is talking about teachers who have had good training, and have an understanding of math and know how to teach that math to students. Obviously that is not what we have in many of our schools today.

His point is that math educators are needed in developing math pedagogy because they are the ones who are actually in the classroom teaching. They know better what works in a classroom than some university professor. I am willing to bet that EM was not written by a classroom teacher.

Dr. Wu's goal is improved teacher training in this country. Not just in subject matter but in how to teach that subject matter to children.

Our school district adopted Growing With Math a few years ago, another spiral program, and the public school teachers are incredibly frustrated with it. They didn't choose this program, they are just told to implement it. They see that kids are being left behind and for the most part are doing their best to fill in the gaps. But they don't have much training in how to teach math either.

Even with a good program such as Singapore Math, the teacher still needs to know how to teach math. It sure makes it easier to have a well thought out, well organized textbook, but if the teacher doesn't know what they are doing they can still mess it up. They need to know how to teach BOTH understanding and procedural fluency. This is what our education schools are sorely lacking in.

I think it's tragic that education schools are spending so little time on teaching teachers how to teach math in depth while the school districts are going willy nilly spending millions of dollars on the 'latest and greatest' textbooks written by someone who hasn't even set foot in an elementary classroom.

RebeccaR

Mathematical engineering, then, is about customizing mathematics for use by students and teachers in K-12.Haven't read his paper yet - but does this mean "use" in the broad sense of "practical" AND "learnable-by-novices"?

Even with a good program such as Singapore Math, the teacher still needs to know how to teach math. It sure makes it easier to have a well thought out, well organized textbook, but if the teacher doesn't know what they are doing they can still mess it up.Yes, absolutely.

With all of my amateur math-teaching around here, I am constantly confronting this question.

I'd say I "learn on the job" -- but Vicki Snider is right: we need a science of teaching.

Of course, we already have a budding science of teaching but no one uses it, which means I'm trying to absorb Wes Becker's book, Doug Carnine's book, etc.

while also trying to learn and teach math(and earn a living).It's an impossible proposition.

well.... not quite impossible

But definitely "unlikely."

Catherine,

I wondered about the use of the word "use" also. My father in law makes a distinction between "using mathematics" and "doing mathematics" and I've seen other mathematicians make that distinction also. At one point I thought it would make an interesting blog entry.

Using math is using the theorems to solve a problem in science, physics, accounting, etc. "DOING" math is about solving problems in the field of mathematics.

Figuring out how fast it takes for two pipes to fill up a swimming pool is (by these definitions) not a math problem but a plumbing problem that uses math. However, figuring out why it is that dividing by a fraction is like multiplying its reciprocal is a math problem, and the act of figuring out the answer to that problem would be "doing math" as opposed to "using" math.

I am not sure what Wu means. I think he must mean that we must help students to use math to solve problems in other fields. The engineer designs the product for the consumer. The consumer does not become an engineer by virtue of consumption of the product, likewise being very good at calculations and word problems doesn't make one a mathematician.

Most people are consumers of math and I think Wu is saying that what we need are those with the knowledge that it takes to design it well for the consumer. At the highest level this would be an applied mathematician proving a theorem for use in physics, but on a much simpler level it would be the teacher designing/engineering math tools that the student can apply to real world problems. Of course, the engineer must know the concepts behind the product in a much deeper way than the consumer ever will.

"I am willing to bet that EM was not written by a classroom teacher."

I don't know about that, but I've met a number of teachers who seem to think it's great. They are the ones who select the curriculum. There might be some teachers who do not like EM, but they are in the minority.

I guess my point is that there is a fundamental difference of opinion and assumptions going on here. I don't think it's just a matter of explaining things in a different way. The people who make the curriculum decisions have different priorities and goals.

Schools might be encouraged to focus more on precision and mastery, but I don't expect them to raise expectations and the population of 8th grade algebra students.

Wu seems to be making a case for the role of mathematicians in the development of K-12 math. (Whereas K-12 educators want to ignore them and define math however they want, especially in K-6.) It's a turf battle, but K-12 educators feel that they have the role of defining (not interpreting) what math is, not just deciding how it should be taught.

Wu seems to be saying that mathematicians need to be in the loop describing how the subject should be translated (engineered?) into easier terms in a rigorous fashion. This might work for grades 7-12 where classes and teachers become more specialized, but I don't see how this approach will be accepted by K-6 schools. This is where most of the damage takes place. You can still screw up Singapore Math with low expectations and no enforcement of mastery. They just don't believe in it.

I'm going to delete the above comment re: Everyday Math.

I think it's OK to say that EM was designed by people with Ph.D.s in mathematics education.

I don't know for a fact that no classroom teachers were involved; I do know that people with doctorates in "mathematics education" were heavily involved.

Well, you could blame me for not making clear the distinction of "use" and "do" in this post, but Wu is clear:

mathematicians DO math. But the rest of us need to USE math. And there's a particular way in which math needs to be turned into something that we can use: this is what mathematics education MUST BEGIN TO DO.

A chemical engineer isn't DOING chemistry. He's using chemistry. He's DOING chemical engineering, but that means customization of chemistry to satisfy human needs.

Math educators need to do the same thing: USE math to DO mathematical engineering. Use math to customize math to satisfy our needs, and that means teaching k-12 math properly enough that math can be used by people who arne't mathematicians.

But, of course, the chemical engineer must KNOW chemistry. And math educators must KNOW math too. That doesn't mean they have to do research mathematics for a living, but they must know and understand it to where they can engineer it for appropriate end users.

And we've reached a point where the corruption of the texts, the attitudes, etc. is so strong that the people in math education simply DO NOT KNOW math. This is why they need to collaborate with mathematicians.

By analogy, it's absurd to suggest that a chemical engineering company or department could exist that didn't know chemistry. That we've gone this far down the road just speaks to the emergency we're in.

--mathematical engineering is about customizing mathematics for use by students and teachers in K-12

---Haven't read his paper yet - but does this mean "use" in the broad sense of "practical" AND "learnable-by-novices"?

Yes, both. It is to be practical, and learnable by novices as is age appropriate WHILE STILL BEING CORRECT.

The point is to simplify without lying, misleading, or otherwise undermining future growth, so that the presentation true for the 4th grader is still true for the 12th, but the 12th can handle more, and can see the connections to the stuff already in his mind. Mastery is then possible.

The point is to simplify without lying, misleading, or otherwise undermining future growth, so that the presentation true for the 4th grader is still true for the 12th, but the 12th can handle more, and can see the connections to the stuff already in his mind. Mastery is then possible.I am SO quoting this at some point. Excellent!

Done. As promised. Thank you again, Allison.

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