A blog about MathNook, math, math games, and more.

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 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.

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Daily Practice, Daily Play


Daily Practice, Daily Play

Have you ever watched a child who is truly gifted at something go off to the practice room, gym, athletic field, or computer? You can barely hold the child back. He or she wants to dive into that studio. There is no need to scratch your head, thinking of ways to get that child behind his saxophone, on the track, or in the computer lab. The student doesn’t view practice as “work.” If you look at the student carefully, the activity is not exactly “play,” either. It’s more like “flow.”

As you recall, “flow” is the condition of being so absorbed in an activity that the purposes and effort in performing an activity disappears in the sheer performance of the activity. There are top athletes who describe their best efforts in the gerundial form, like “The running swept me forward, and when it was time to kick, the track disappeared, and all I could feel was movement.” Parents of children who show gifts and passion for a musical instrument will overhear their child grouse about the hours invested in practice, but will report that the child plays scales and technical passages beyond all reason.

Every teacher or parent reading this who struggles with a child who performs poorly at math, loathes math practice, and is building up a negative self-image around this vital skill is snorting, “Yeah. Must be nice.” The unusual focus that students experience when passion and talent confluesce guarantees exceptional performance, but how does a child get to this point? One way of making this condition possible is to present the child with tasks in the targeted domain (in our case, math fact fluency up to two-step equations) that are both intrinsically and extrinsically motivating.

A much larger company that plays in the math game space, and therefore has a bigger budget to prove its hypotheses, has established the research on the effectiveness of math games that capture the imagination of the students. In controlled studies, this company has established the efficacy of playing their games as a way to increase fluency with math facts. The results are eye-popping: the classes that used the company’s computer-based learning games to achieve fact mastery soared from the high teens and low twenties in percentile of fact fluency to over eighty per cent in all cases. Our posts have showed why this fluency yields power: the prefrontal cortex is free to learn, not tied up in calculation. The company reports anecdotally that students’ enthusiasm for math reflected this growing power. In my first article writing for this space, I shared the targeted gains and unbridled enthusiasm that my own son exuded when I asked him to take for a spin. While none of this proves the case that Mathnook produces flow, the indications are there, waiting for you to test out with your own child or classes.

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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.


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Math Anxiety,, 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, 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 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!


When Number Sense Fails


In a recent research study, children who showed math deficiency in kindergarten and first grade were given daily practice with two contrasting computer-based interventions using math games , the students showed gains on the subject of the intervention. Why are we not surprised?

In the experiment, the area studied was comparative magnitude of numerical representations. A typical example from one of the interventions was:

1) Here is a bar graph. If the left side is zero and the right side is 100, click the cursor on 65.


The other intervention was typified by examples like this:

2) Click on the bigger number:

a) 75 18 b) 9 21 c) 35 53 d) 29 56

The examples were run in timed and untimed variants.

What do you think happened? Both groups, the one that used graphics and sliders and the other, that asked for a cursor click on a number, benefited the student after a three-week intensive intervention. However, there wasn’t any benefit in any other math areas. Specifically, the young students were tested in magnitude, counting, number/quantity correlation, and simple arithmetic. These skills, taken together, are called “number sense.”

As you explore our site, you will find that at the early grade levels, you can select any of these skills. This means that if a child spends a little time every day on the games on our site, and gets to all of them, she will make advances in the combined puzzle called “number sense” that will carry her through the next grade level. Here are some examples:

The child sees the cards on the screen arrayed like a game of Concentration. The object is to match numerals with the number of objects (1-10) on another card. This timed game changes the array and the object with each play, but the difficulty level remains the same. When the player can win the game before reaching her level of frustration, she can try to beat her best time. This game reinforces the number/quantity correlation of number sense.

When the student can count arrays up to twenty objects or up to ten randomly scattered objects, he has mastered the number sense skill of counting. It is very difficult to simulate counting moving objects on paper, but on a computer, this becomes a simple and straightforward programming task. Try the game “Aquarium Fish,” for example. The game is designed to hold interest by the kid-friendly characters.

The game MathPup Measurement straddles the kindergarten-grade 2 levels, starting with simple size comparisons and ending up with a more sophisticated use of a ruler to make comparisons. Again, this is a timed game, but watch out! One mistake and it’s game over. Kindergarteners who complete the first level repeatedly show mastery of magnitude.

Kids love to blow things up. To develop the skill of adding and subtracting up to ten, you’ll find one of a series of games called Math Lines. Use a cue ball to blow up another ball where the total adds up to ten. As a timed game, this can test even the strongest number sensitivities. The site allows a player to challenge a friend by email. Imagine a war between two kindergarten or first-grade classrooms! Sounds like fun to us.

Children have to develop number sense in order to make math a happy part of their lives. There are, however, four separate dimensions to number sense. Since we still don’t have a magic building block that helps construct competence in all areas of number sense, a child (and parent, teacher, or caregiver) should do something fun that reinforces each of the number sense areas – magnitude, counting, numeric representation, and arithmetic – as often as possible.

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This is a blog about Math Nook, math games, math and other fun and educational subjects.
Math Nook is owned by Jan and Tommy Hall.

Jan is retired from education where she spent 30 years in various positions ranging from classroom teacher to math specialist. She now spends her time working on the website and raising MathPup.

Tommy works full time but spends his free time utilizing his math degree and love of games to create some of the math games found on the website.