MathNook

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

Alfred North Whitehead and Your Child

September1

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, Mathnook.com, and Flow

August24

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!

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When Number Sense Fails

August11

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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But Will it Work for My Kid?

July29

The last few posts have focused on the general advantage of turning math facts into a kind of video game, like we do at www.mathnook.com, and how doing so creates room for higher-order thinking skills in operational memory. I compared a brain with snap access to the needed math fact to a computer that is loaded with RAM, in which swapping data with the hard drive takes an infinitesimal amount of resources from the central processing unit (CPU). I created the cautionary tale of the reverse: a brain that has to spend all its time figuring out 18/3=6 is like a dinosaur of a computer, like a monster running Windows XP with 2G of RAM trying to run Adobe Creative Suite and crashing like a drinking glass hurled in frustration. Now, I decided to look at the research for kids with learning differences, particularly for those who are considered to have math handicaps to the point of earning the tag, “dyscalculia.”

First off. Let’s try a definition. The National Center for Learning Disabilities says that dyscalculia is a wide range of lifelong learning disabilities involving math. There is no single type of math disability. Dyscalculia can vary from person to person. And, it can affect people differently at different stages of life.

As opposed to dyslexia, which refers to a very specific impairment in image processing, dyscalculia has at least two roots: visual-spatial difficulties, in which the brain misinterprets what the eyes see, and auditory language processing difficulties, in which the brain doesn’t interpret what it hears in the absence of physical or language handicaps. As a parent, or as a teacher who teaches students with dyscalculia, this means that the problem is not set in stone. As the brain evolves, and all brains do, even mine, the neural pathways will change. Some will be reinforced, some will be backed up by roughly parallel ones, and some will be pruned if they fall into disuse. A person who “doesn’t get math” at age ten may develop a different toolset to apply to math at age sixteen, and become a STEM professional at twenty-four.

Still, our job is to provide something that works for your child or student(s), preferably last week. The reason why an attention-gripping, addictive video-game-style experience works for students with dyscalculia is that such children need access to their random-access memory even more than neurotypical kids. Back in 1990, when URLs looked something like this:

[email protected],”

Research papers had to be photocopied and carted around in your backpack, they knew that attention-disordered and learning-disabled (the term “dyscalculic” hadn’t been invented yet) children suffered from the ability to access and retain math facts. Among paired findings of a study out of the University of Missouri (http://web.missouri.edu/~gearyd/Aphasiology.pdf), one conclusion is that the development of the prefrontal cortex, which governs what we call “executive functions,” is no different for dyscalculic kids with no further deficits than for the neurotypical kids (the other has to do with a pathway through the left occipital-parietal-temporal region, reinforced by several subcortical structures, but if you want to go this far into the weeds, you can click on the link above). Skip forward to 2005 (http://cercor.oxfordjournals.org/content/15/11/1779.full), when it was shown that younger children and children with dyscalculia rely more on the prefrontal cortex to solve arithmetic problems than older children and children who function at a higher level in arithmetic. The latter groups don’t need the involvement of executive function to the same level. They use that left occipital-parietal-temporal region, from the weeds of the University of Missouri paper.

What does that mean for us? Remember that virtually all the games at www.mathnook.com make grade level math facts reflexive, thus getting them out of the province of the executive function needed for higher-order thinking, and into that occipital-parietal-temporal sweet spot. Drilling and killing could do that for the few students who would submit to such discipline willingly, but the usual victim of skill-drill-and-kill is the student’s curiosity and affinity for math. What is true for neurotypical students is manifestly more true for students with learning difficulties from dyscalculia to mild mental retardation. On the upside, turning math fact acquisition into a game, even if the gamer is playing two or more years below grade level, supports just the kind of automaticity that leaves precious prefrontal cortical “head-space” available for integrative, higher-order thinking, learning, and synthesis.

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Prefrontal Cortex Should not be Bothered

July23

PROCESS:

Prefrontal cortex recognizes need for data

Math fact data “delivered” to prefrontal cortex

Prefrontal cortex combines fact recall with rules and patterns, resulting in skilled problem solving

Pictures at www,mathnook,com/blog

Why Fact Games Work

Readers of this column know that we at www.mathnook.com believe that there is value to rote memorization, but that “skilling, drilling, and killing” students with facts and procedures simply kills kids’ motivation to learn, or even to try. It seems like an impossible dichotomy, but in fact, there is a simple analogy to something we all encounter on a daily basis. Do you have an old computer, a warhorse relic from ten or more years ago? Other than warning you to check continuously for viruses on your XP machine (please don’t pass your viruses to us!), I want you to remember what it is like to try to stream a movie, write music, or develop your own graphics with a machine that was built long before Netflix and the Adobe Creative Suite™.

You are sitting there fuming as your machine keeps saying things like, “Adobe Creative Suite (not responding).” You are tempted to yell at it, “You’re a machine! Stop ‘not responding,’ respond!” Finally, you slam down Ctrl-Alt-Delete and find out that your CPU usage is stuck at 100%. Permit me this geek moment, but I can explain just why this is happening.

Let’s say that your computer has something like 2 gigabytes worth of random access memory (RAM). Adobe Creative Suite™ requires almost all of that for the program to work. The closer you get to full utilization of your computer’s RAM, the more your CPU (central processing unit) takes over the work, which slows your computer down like the carapace on a giant turtle.

Is your child, or your classroom if you are a teacher, trying to compute with too little RAM? If so, the part of the brain that we think is responsible for storage and recall of math knowledge is underutilized, while the parts which should be retrieving the data from centers like the parieto-occipital sulcus, is busy pretending to be RAM. You can buy more RAM quite cheaply for your computer. Why not buy some more RAM for your problem-solving centers to query, by getting the math facts out of the way? Relegating facts to the random-access memory part of the brain frees the prefrontal cortex to organize itself around problem-solving, not fact recall.

For an example, let’s look at the long division algorithm. On the left of the table below, you will find the step, and on the right, the brain processing step that should go into applying stored data. We are going to assume a neurotypical student with at least an adequate storage for math facts and rules.

Step Brain Process
1. set divisor outside the box and dividend inside it Rule recall
2. Estimate how many times the divisor will fit into the dividend, or into the appropriate place value of the dividend Higher-order processing
3. Multiply the divisor by your estimate in step 2 Fact recall; maybe recall of multiplication subroutine
4. Subtract result in Step 3 from dividend Fact recall, application of place value (higher order knowledge) and regrouping rules
5. Repeat (iterate) steps 2-4 until there is no remainder or the desired level of accuracy is reached Higher-order processing, fact recall, and assimilation of rules, facts, and applications.
6. Report the results Mathematical language

Even as I look at this, I’m astonished that some students who manage to become proficient at math remain unable to zap you with “42” when you ask them, “Seven times six?” If you have to work out the staggering number of math facts in every long division problem, by the time you reach algebra, your prefrontal cortex is going to be like that dinosaur computer running Windows XP that we met at the beginning of this entry.

The games at www.mathnook.com don’t claim to train your prefrontal cortex for higher-level functions. A regular visitor to this site will, however, reduce the cognitive load on the part of the brain that needs to send out data requests and integrate the responses into an answer.

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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.