He was so learned that he could name a horse in nine languages;

so ignorant that he bought a cow to ride on.

*-Ben Franklin*

Can you "over learn" something? Can you bore down on the details and extend and deepen your learning to the point of absurdity?

I believe you can and I also believe we make our students do this.

Take math:

My 8th graders are expected to learn the following algebra topics:

- Meaning of variables in expressions and equations.
- Multiple representations of algebraic expressions using manipulatives, tables, graphs, equations and even words
- Meaning of slope
- Multiple ways to solve systems of equations
- Meaning of quadratic growth
- Factoring quadratic equations
- Quadratic formula
- Inequalities
- Rational expressions (I particularly hate teaching these)

I actually don't have a problem teaching them points 1 through 5, more or less. But even inside of those topics are specifics that can be quite

I don't find inequalities so unreal until they are made so in a variety of problem sets designed to pull out all the complications.

I do find explaining the use of quadratic formula problematic for most 13 year olds and don't even get me started on rational expressions.

**demoralizing and useless**for students. For example, deducing the equations of perpendicular lines, how to find an equation of a line from two points and many other fine points.I don't find inequalities so unreal until they are made so in a variety of problem sets designed to pull out all the complications.

I do find explaining the use of quadratic formula problematic for most 13 year olds and don't even get me started on rational expressions.

In the mean time, my students have either forgotten or never learned some very useful math:

- Statistics: particularly mean, median, mode and box and whisker graphs
- Basic geometry and much about measurement: particularly: easy movement between metric and English standard. (My students are usually quite lost!)
- Long division, which itself is not so important, but plays such a fundamental role, I have decided, in the development of number sense.
- Basic number sense: how to estimate an answer, take short cuts (such as multiplying by 15 with first doing "by 10" then half again.
- Computational skills and mental math!!!! They can't seem to look at a division problem, such as
**84 ÷ 6**, for example, do it without either a calculator or paper pencil. (Just seems wrong to me) - Fractions (I don't expect mastery of every details, but basic idea of equivalence? Basic operational sense? YES)
- Probability: so many decisions we make in real life are based on our intuitive sense of probability, but that intuition, if not trained properly, can be terribly misleading
- Graphs: they abound in our world and they are so poorly understood.

This is just a short list of what my students are "underlearning" in the current

ALGEBRA CENTRIC math education they receive starting in the 8th grade on through high school.

I think Ben Franklin had it right:

**my students are riding on cows while describing horses.**It is quite backwards.
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