In THIS article, “The Faulty Logic of The Math Wars” in the New York Times, Alice Crary, a philosophy professor, and W. Stephen Wilson, a math professor argue that the ideals that math reformists claim to be working towards when removing standard algorithms (such as long division) from elementary math textbooks are actually being moved away from if the thought processes behind the algorithms are analyzed differently.
Today the emphasis of most math instruction is on — to use the new lingo — numerical reasoning. This is in contrast with a more traditional focus on understanding and mastery of the most efficient mathematical algorithms. …
The reformist’s case rests on an understanding of the capacities valued by mathematicians as merely mechanical skills that require no true thought. The idea is that when we apply standard algorithms we are exploiting ‘inner mechanisms’ that enable us to simply churn out correct results. We are thus at bottom doing nothing more than serving as sorts of “human calculators.” …
This mechanical image of calculation is the target of a number of philosophical critiques. … Wittgenstein suggests that analogies between mathematical computations and mechanical processes only seem appealing if we overlook the fact that real machines have parts that bend and melt and are invariably subject to breakdown. … That is, it seems clear that, in order to be justified in believing that I have mastered an algorithm, I require a type of mental activity that isn’t simply causally generated. Far from being genuinely mechanical, such calculations involve a distinctive kind of thought.
That the use of standard algorithms isn’t merely mechanical is not by itself a reason to teach them. It is important to teach them because, as we already noted, they are also the most elegant and powerful methods for specific operations. This means that they are our best representations of connections among mathematical concepts. Math instruction that does not teach both that these algorithms work and why they do is denying students insight into the very discipline it is supposed to be about.
While reformists claim the need to change traditional math is based on the need to promote independent thought, the authors draw parallels to other subjects which are not being argued (usually) to start requiring creative thought in the early grades.
To begin with, it is true that algorithm-based math is not creative reasoning. Yet the same is true of many disciplines that have good claims to be taught in our schools. Children need to master bodies of fact, and not merely reason independently, in, for instance, biology and history. Does it follow that in offering these subjects schools are stunting their students’ growth and preventing them from thinking for themselves?
Further, the reasoning the reformists so desire needs to be built upon a solid foundation of “bodies of fact” such as parts of cell, the dates of wars fought in America, and the basic algorithms of math. If you do not know the basics, it will be difficult for someone to think deeply about more difficult topics within a subject. No one is arguing that the why should not be taught; let’s just not do a disservice to our students by not showing them the best way to do something. In Gideon we always aim to teach the why AND the best. In higher addition for example, visual movement exercises among place value (such as changing 12 ones into 1 ten and 2 ones) are used before carrying or regrouping, but the basic algorithm of putting a 1 above the tens column to aid in speedy addition is practiced until mastered.
Even if we sympathize with progressivists in wanting schools to foster independence of mind, we shouldn’t assume that it is obvious how best to do this. Original thought ranges over many different domains, and it imposes divergent demands as it does so. Just as there is good reason to believe that in biology and history such thought requires significant factual knowledge, there is good reason to believe that in mathematics it requires understanding of and facility with the standard algorithms.
Read the entire article HERE.
This article received a lot of comments such as these below:
John, an outraged mathematician from NY remarked:
How would the reform template work if applied to learning a musical instrument? What would the result be if beginning guitar students were not taught the basic chords? A lot of wasted time and a cacophony. Creativity can and should come later, once the basics are mastered. The same is true in mathematics.
(Another) John, a college professor from VA said:
I am … a multilevel modeler — aka, fancy statistics. … it would have been impossible for me achieve what I have without their instruction in what I am now being told was a method that limited my creativity.
The ability to perform basic arithmetic operations without thinking too much about them frees me … to think more about more complex mathematical operations/concepts.
Occasionally, I have students who have difficulty with simple arithmetic operations. These students also have difficulty understanding how to set up the models that they need to analyze their data and perhaps more importantly, they have difficulty interpreting the results.
Basic arithmetic operations are the simple sentences of mathematics and by extension, of disciplines such as statistics that rely on arithmetic operations. If you do not know how to construct a simple sentence, paragraphs would seem to be out of reach.
Read the rest of the comments HERE by scrolling the bottom of the page.
For more thoughts on this article and subject go HERE to Joanne Jacob’s blog where she links up with articles from Barry Garelick who discusses that reformists claim to be teaching the superior ‘understanding’ instead of a traditionalist’s mere ‘skills’:
. . . The reform approach to “understanding” is teaching small children never to trust the math, unless you can visualize why it works. If you can’t “visualize” it, you can’t explain it. And if you can’t explain it, then you don’t “understand” it.