**Math Teachers Do Not Understand Math**

The mountain of documentation that teachers are not learning the math we expect them to teach, especially at the elementary level, grows higher every day. I touched on this point in my report on calculator research with young children. The National Council of Teachers of Mathematics has gone on record recommending calculator use in the earliest grades with the important caveat, “under the guidance of skilled teachers.” Ah yes, skilled math teachers, where are they to be found? By all accounts, skilled math teachers are not to be found in our elementary schools. Hence, the subtitle of my report, “A Moot Point Without Skilled Teachers.” Targeted efforts to improve the content knowledge of our teachers have proven disappointing.

Dr. Robert MacDuff considered the lack of skilled math teachers and the failure of teachers to master math content in spite of all efforts in his essay entitled, “the Math Problem.” His intriguing title means he believes the problem is with the math content itself. Dr. MacDuff asks, “Could mathematics itself be flawed?” Whatever could he mean? Is he proposing to fix mathematics? In a word, yes, Dr. MacDuff suggests retooling math content from scratch.

Our research suggests that the problem lies in the difficulties imposed on the students by asking them to learn a flawed mathematics content. Their difficulties in turn, lead to a drastic underestimation of their capabilities and thence to shortchanging them in their education. The solution therefore requires a profound re-thinking of the foundations and assumptions of mathematics itself.

Whoa, that's ambitious. Dr. MacDuff would start by considering collections of objects.

The system of ideas that is emerging from this process can be described as a “mathematics of quantity”, which takes as its starting point the consideration of collections of objects.

I have been on the education merry-go-round for quite a while and I remember when one of the carousel horses was New Math, which took for its starting point set theory. What are sets, but collections of objects? Was Dr. MacDuff proposing a return to the detested New Math? Let's see.

This approach separates the learning of mathematics into four major subsections: conceptual understanding (grouping structure), symbol construction (algorithmic manipulations), problem solving and mathematical reasoning. As defined here, conceptual understanding and mathematical reasoning are not to be found in standard approaches to mathematics education. And yet these are the critical components.

**Teach Real-World Math**

Dr. MacDuff is not suggesting the use of real world applications, although he is not precluding them either. “Real-world” means the idea that math is happening all around us. Number sense (and mathematics) starts, not with numerals, but with collections of objects, Piagetan conservation of quantity and relationships between quantities.

Some math teachers do emphasize conceptual understanding and mathematical reasoning in their teaching practice. A student may be lucky to draw only one or two such teachers sometime between kindergarten and high school. However, the traditional American approach makes conceptual understanding and mathematical reasoning the servant of algorithms, not its master. In other words, many teachers try to use mathematical reasoning to help students memorize the algorithms rather than guiding students to learn the math for which the conventional algorithm becomes one of many possible problem-solving strategies.

I deplore the current drive to push first grade curriculum into kindergarten. Teachers give kindergarten children pencils and paper and teach them conventional algorithms. The children spend the first few weeks of kindergarten learning to write numerals. Some of them even appear to master the so-called math they are taught. If they dependably get right answers, they internalize the idea that they are good at math, when in actuality they may not understand math at all.

It is possible to execute an algorithm with no real understanding. How many adults have any idea what is mathematically happening when they perform an algorithm like long division? The math mis-education is further compounded when non-math explanations are routinely substituted for and identified as math explanations. A good example is the typical explanation for the division of fractions. Ask any good math student and you will be led through division of fractions as multiplication by the reciprocal, a non math explanation for why the trick mathematically works. Or how about multiplication of decimals? Mathematically you are not counting decimal places; something mathematical is happening that makes counting decimal places work.

**Start Math Instruction by Avoiding Numerals**

Children can learn lots and lots of math without ever picking up a pencil and paper. In fact, making kids write math before they have mastered the mathematical relationships creates learning obstacles the children may or may not overcome. Meanwhile, many of these same children, the ones who get right answers, have been told by our education establishment that they are good in math, and they believe it. In the plainest terms, schools have been lying to students about their math abilities forever. For thirty years, I have preached this idea about math education like an evangelist.

I teach Algebra. I cannot tell you how many math whizzes show up to Algebra without the slightest clue about the concept of place value. Sure, they can name the place of any given digit, but naming is a far cry from understanding the mechanism of place value and its importance in mathematics. I almost always redo elementary math with students before we launch into the “real” subject matter of my class, algebra.

According to Dr. MacDuff, one part of the brain deals with math, while another part of the brain deals with numerals (like “12”) and yet another part of the brain deals with linguistic representation of numbers (like “twelve”). The traditional approach to math stimulates the wrong parts of the brain. No wonder so few students (and their teachers who were once students) get it.

Dr. MacDuff is not proposing a return to New Math. He is proposing a structural retooling of the math curriculum so that we actually stimulate the part of the brain the deals with math. Some of us have taught that way all along. Dr. MacDuff proposes to make my idiosyncratic methods the system-wide norm. Allelujah!

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