School Information System: A statistician's view of constructivist math programs

April 3, 2012

A statistician's view of constructivist math programs

I've had 4 years of undergraduate math courses, two years of graduate math courses, and I have taught graduate level math courses. I had never seen the "lattice" method, the Egyptian method, or any of these other alternative algorithms until last year when I looked at Everyday Math homework. Students don't need them, and those methods will not help a student move onto higher mathematics.

I would have been laughed out of my college classes if I used the "partial sums" method to add. I wouldn't have been able to take differential equations if I hadn't mastered long division. There is a reason why traditional algorithms (the math methods you learned in school to add, subtract, multiply and divide) are needed. Traditional algorithms are needed to understand higher mathematics in college. It is extremely important that they are practiced until they are mastered. In fact, the new Common Core State math standards recommend teaching the standard algorithms.

You might think there is no reason not to offer alternative algorithms, as long as they also teach the traditional methods, but I have three reasons why the teaching of alternative programs is a problem.

Posted by Jim Zellmer at April 3, 2012 4:38 AM
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Comments

(modified 4/4/2012)
I recently played with the lattice method for adding, and thought it interesting, but had to decompose it to determine why it worked. My first impression about the lattice method is that it is a typical result of Gardner's Multiple Intelligences -- Spatial Intelligence, since it seems the requirements these days that teachers must translate concepts into each possible "intelligence". At least they haven't set addition to music yet.

That said, does it have any usefulness in conceptualizing addition and the properties of numbers or addition. I don't know. The appropriateness of teaching this material at elementary school levels is the issue, not if the author has every seen it before. Such methods or at least seemingly strange methods can be useful in other contexts.

For example, there are very "strange" algorithms for addition and other simple operations that are important when implementing these processes in electrical circuits, and in parallel computations.

However, I found laughable that Stouffer argues that learning higher level mathematics requires one to become proficient in the old-fashioned typical long division algorithm. Funny, though I'm not a mathematician, I took 15 - 20 math courses as an undergrad and grad student, and never once that I can remember did the long division algorithm play a part in understanding the material.

I really have no clue what Stouffer is smoking, but her arguments are the stuff of pure irrationality. Long division never played a part in my differential equations course, and in any case, when considering solutions to differential equations using a computer the algorithm used to perform division was never part of any discussion. We did consider issues of error propagation, truncation errors, infinite precision arithmetic, error terms, rapidity of convergence -- but never long division.

In the simple real world though, her argument is silly on its face. If I need to divide a number by 5, Stouffer would want me to use long division. Sorry, I would rather divide by 10 first, and multiple that result by 2; long division doesn't even come into play!

Further, if knowledge of long division is so important for advanced mathematics, I'd opine that not one in ten people know why long division works! Same is true for long multiplication. How knowing the mechanics the long multiplication and division while not understanding the algorithm's foundation can be fundamental to advanced mathematics is beyond me.

Given Stouffer's arguments, I would actually question her fundamental competence.

Posted by: Larry Winkler at April 3, 2012 8:46 AM