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Every teacher who has taught a new concept in math has been told, “First, make a diagram.” In fact, as a problem-solving strategy, this is often listed as the indispensable first step in getting to an answer. Would you believe that the research proves that this is all wrong? The study in question, published in 2009, shows that the presence of a diagram is no help to test-takers, either children or adults. But wait, there’s more.

If a test problem gives the taker a completely irrelevant diagram, it slows the test-taker far less than if a diagram with essential information without which the problem could not be solved! That means that the better the information, the worse the student does! Every math educator is cringing to read this. The study has been used to create a great deal of mischief in math education, supporting the trolls who think that math education is the process of piling tips, tools, and techniques on the back of the student to see if she breaks before she memorizes her rules for differentiation. I’m not naming names, but a curriculum with a lot of credibility in the charter school world uses this paper to support its anti-constructivist philosophy.

What is really at play here? Let me quote from the paper in question:
The present results, however, demonstrate that illustrations can slow down processing, but not necessarily affect the learning process. In fact, when integration of information is needed (as in the ‘‘essential’’ illustration) then the child may not reach a correct solution because understanding the association between features of arithmetic word problems and solution schemas is becoming difficult.

The study highlights the difference between learning and performing mathematics. When you are performing on a test, you need to create the maximum space and flexibility in your working memory, and then to maintain that space by clearing out the working memory as you move from step to step. When you are learning the mathematics, you must go the exact opposite direction – you must engage with the material in the greatest depth possible. That diagram should not only be designed so that it is essential to the problem, but it should be embodied in three dimensions and even four dimensions if the passage of time is part of the problem.

More often than not, the kind of deep dive into the meaning of a math problem is something that happens in the formative stages of learning. Students work out the representations in groups, show how they could build a physical model of the problem and its solution, and present their models to each other in what is known as a formative assessment. The point of what we do here at www.mathnook.com is to clear out the working memory so the higher brain is free to bring in more advanced concepts.

Think about this approach like the difference between the barre exercises in dance and a solo turn at the Bolshoi Ballet. Every member of the corps de ballet is able to perform at the barre, but the best of the best, the soloists, have the technique ingrained in their kinesthetic memory. Out leaps the Black Swan, and all the technique that young aspiring soloists work on hour after hour become absolutely transparent. The technique is not forgotten; rather, it has been learned to the point where the performer subitizes these mechanics the same way that a five-year-old subitizes a group of six marbles or counting bears. By working at this level, the math student develops the ability to create with math the way that the ballerina creates with movement.