But I Gave them a Diagram!

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.

Alfred North Whitehead and Your Child

Mathematician and philosopher Alfred North Whitehead(1867-1941) understood the role of the intra-parietal  sulcus in higher math even though the existence of this structure didn’t emerge until the end of his life, and the function of this bilateral (left brain/right brain)  structure didn’t emerge until functional MRI studies showed it, half a century after his death.  Whitehead posited that “By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and in effect increases the mental power of the race (Introduction to Mathematics (1911, Alfred A. Knopf) Chapter 5).” Now, with the benefit of this powerful imaging connection, we can actually track thoughts jumping around the human brain.

Recall that the intraparietal sulcus mediated the pathway for facts in working memory to be cleared out of the prefrontal cortex once the concept is grasped. This was a key finding for the study of learning in general, and as we will see, of math learning in particular. The purpose of this region of white matter and synapses is to shunt facts in and out of the higher regions of the brain, reducing the cognitive load on the higher brain and allowing it, as Whitehead said, “to concentrate on more advanced problems,” allowing the higher brain to do what it does best: think.

What happens when the brain as a whole isn’t getting arithmetic and higher math, a condition commonly referred to as “dyscalculia”? While there are many reasons that the brain might not “get” arithmetic, from something in the intraparietal sulcus that doesn’t develop along the lines of a neurotypical child to severe mental retardation, most researchers use the term to mean that something is interrupting the normal process of cycling math facts in and out of the left and right intraparietal sulcus, or through this structure into the anterior gyrus.  Although there are differences in the two sides of the intraparietal sulcus (the left side being stimulated by visio-spacial input and the right by numerosity), this structure serves as a superhighway, a county road, or an uneven bike path for facts to travel in service of the higher, problem-solving brain’s struggle to master more and more advanced math.

I can hear you arguing, “But I thought you were going to talk about my child! What is this intra-parietal technobabble?”  Here’s the point. Butterworth et al. (2011) states,

 Reduced grey matter in dyscalculic learners has been observed in areas involved in basic numerical processing, including the left IPS, the right IPS, and the IPS bilaterally; these learners have not developed the brain areas as much as typical learners.

Is it probable that this structure is a superhighway for the gifted math learner, a county road for most of us, and a rubble-strewn bike path for those unfortunates who would now be diagnosed with dyscalculia? This is an area for intensive research taking place right now. What are the implications for the child learner with dyscalculia?  Patience, dear reader, we will look at that vital topic in some depth next week.

Butterworth (op.cit.) talks about the need to train the growing brain to do roadway improvements on the IPS to make it easier for facts, once synthesized, to make it into the angular gyrus (another grey matter component implicated by fMRI studies in fact recall), and for those facts to be accessed as needed for problem-solving. Mathnook to the rescue! Our mission is to make math facts automatic and to have the right fact appear in an instant when required for more advanced problem solving.


Math Anxiety, Mathnook.com, and Flow

Math Anxiety reduces available working memory and impacts performance (Ashcraft & Kirk, 2001). This is a serious finding. Many people reading this column are teachers, so keep reading, because we are going to propose a possible aid for your math-anxious students.

Working memory has been linked strongly to enhanced arithmetic performance. This doesn’t obtain for fact tables, but with addition/subtraction involving regrouping, participants with math anxiety took three times as long as non-anxious participants to solve the problem correctly. Regrouping is thought to be mediated by working memory. The “central executive” is the part of the working memory that seems to be most affected by math anxiety. Intrusive thoughts that argue for the incompetence of the problem-solver compete effectively for the time and space of the central executive. In this week’s column, I want to look at ways to take the central executive out of the problem, or since that is impossible, to reduce its potential to cause confusion and delay.

Recent research indicates that, while math anxiety doesn’t impact the working memory of subjects whose working memory was low from heredity, trauma, or idiopathic (unknown) factors, subjects with typical or high working memory respond to increased cortisol, the stress hormone, in very different ways. Subjects who display math anxiety lose access to much of their working memory, making them indistinguishable from people with a diagnosed working memory deficit! Clearly, subjects with math anxiety showed a maladaptive response to stress. On the other hand, increased cortisol did exactly what it was designed genetically to do in the other high-working-memory subjects: it increases their already high working memory. That’s an adaptive response to stress.

So far, we can say that math anxiety is a function of available working memory, and that working memory impacts all math higher than pretrained math facts. The logical result of this syllogism is that if you can free up working memory and reduce math facts to a matter of automatic recall, the student can spend whatever working memory is available on the higher-level questions.

Perhaps that is stating the obvious, but how do you do this? At www.mathnook.com, we have hundreds of games that can be played at the level of introduction to the level of mastery. If a student is guided, or finds through her own observation, to the right level and choice of games, she can take routine calculations right out of the working memory. The fact that this kind of practice produced measurable gains, especially when studied after six months (to give the central executive time to process the activity and to feed it back to both the visiospacial and phonological processing loops.

But what about the central executive? Isn’t it still going to throw a wrench into the process?

Short of electrical stimulation, there is no way of actually turning off the executive, but empirical evidence has proven that computer-based training similar to ours improves the interaction between the central executive and the phonological loop. Most of the research on which I base the following hypothesis comes from the study of athletes and musicians, but empirically I can suggest that the reason www.mathnook.com and other game sites with design based on hypnotic motion, competition, and scalable difficulty levels is that mathletes attain the psychological state called “Flow,” first documented by Csikszentmihalyi (1975) in his book Beyond Boredom and Anxiety. According to Csikszentmihalyi, flow is a state of peak enjoyment, energetic focus, and creative concentration experienced by people engaged in play, which has become the basis of a highly creative approach to living. In live descriptions of flow, the author suggests that the central executive is bypassed in a state of flow. The experience is “differentiating,” not “I am struggling with differentiation in Freshman Calculus. I’m doomed.”

While we aren’t aware of research that confirms that our games or any other Computer-Aided Instruction actually induces a state of flow, having observed many children glued to the computer playing these games, we’d bet on it!