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.

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 www.mathnook.com 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.

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.

But Will it Work for My Kid?

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.

Apples and … Basketballs?

Apples and…Basketballs? I recently had the opportunity to talk with Alfie Kohn, one of my favorite theorists of education. In fact, I asked him if he might have a look at some of the games on our site. It was with a sense of dread that I opened up his email later that afternoon.

Mr. Kohn (If he holds a Ph. D., he is self-effacing enough never to mention it, even on his blog, www.alfiekohn.com) is a leader in a counterrevolution in American letters, whether kindergarten mathematical reasoning or applied political philosophy. He’s a fan of situated learning, which is to say, giving children tasks that are inherently worth doing, and working the specific fact learning in at the margins. I appreciate his point of view, and I really, really wish that American education would have taken that course when it played with going that direction in the post-McCarthy Era through the Sixties. Situated learning is associated with thinkers like John Dewey, and it brought us victory in the Space Race with the Soviets – that same empire that is piecing itself back together over the bodies of Ukrainians, and before that, Georgians (that is, the kind whose last name ends with “-vili,” not the kind that keeps winning the National League East). Clearly, setting children loose with meaningful tasks is a great way to help them think about math, and at www.mathnook.com, we mostly turn fact recall and algebraic problem-solving into a matter of rapid-fire muscle memory.

So would Alfie (as he prefers) rip me a new one, given that what we have done on Mathnook.com has nothing in common with what he proposes as a way to educate students to be able to apply mathematical reasoning to real problems great and small? No. He said, “While in principle I think that this kind of game is counterproductive, it is so only if the game is used to replace situated learning, not to supplement it (emphasis mine).” In other words, the corporate education system is screaming for kids to learn math by rote so that they can fill in those bubbles quickly and accurately, which Alfie rejects (see the abstract from “The Schools Our Children Deserve” on his website at http://www.alfiekohn.org/teaching/math.htm), but at www.mathnook.com, we don’t prepare kids to do that! We simply take advantage of the fact that anything that sucks you in and grabs your attention is going to make an impression.

Is it “worthwhile mathematics” to know by fast-twitch muscle response that 7×6=42, or equally hateful, that 7×8=56? Well, no. However, if the student who could devote an hour to Candy Crush Saga spends it instead on the mastery of math trivia to the point of not having to spend a scintilla of effort on how many sides in a hexagon, the product of 7×6 (or 7×8, for that matter), or what x makes 3x-1=20 true, might that student be free of “skill, drill, and kill” forever and be free to explore math in context of life? I think so. The second-youngest President of the United States attributed his ability to succeed at what mattered to his ability to routinize everything that doesn’t. You would never catch this man dead designing his daily workouts, figuring out what to eat for breakfast, mixing and matching his wardrobe, or any of a host of tasks in which you and I sink precious energy.

So what about comparing apples to basketballs? Last week, we discussed the fact that a highly complex math simulator, DimensionM (not a Mathnook product, alas), has been shown in at least one peer-reviewed study to increase skill, aptitude, and interest in high-school level algebra among middle-school students, family background and economics factored out. DimensionM is a highly sophisticated simulation that has more to do with James Cameron’s Avatar than with the simple designs that allow me to get so many games up so fast. Yet, I get similar results. Why? Let’s say that DimensionM is an apple, crisp, cold, and healthy. If you want a healthy math mind, you want to evangelize DimensionM and similar products as opposed to wasteful social media and mindless entertainment – the potato chips and Milky Way bars of consumer electronics. However, developing a healthy math aptitude requires a good diet and a healthy dose of exercise. Consider Mathnook.com the basketball. Let’s play!

P. S. For those of you who want to teach oddball facts like 7×6 and 7×8, let me give you an idea. Two, actually.

1) Divide and conquer. Most people have 7×3=21 and 7×4=28 (fewer this one) committed to memory, and may even be able to produce an array of seven rows and three or four columns. Using the associative property, 7×6 = (7×3)x2, or (7×3)+(7×3) = 21+21 = 42. Similarly, 7×8=(7×4)x2, (7×4)+(7×4), 28+28, or 56.

2) Nearest square: Most people can give you 6^2=36, 7^2=49, and 8^2 (chessboard) = 64. Adding one more six to 36 depends on knowing what multiplication means, but I am all for learning the meaning before practicing the facts. Similarly, adding a seven to 49 or subtracting an eight from 64, while a little more arithmetically cumbersome, amounts to the same thing.

STEM Teachers in a Box

How many science, tech, engineering and math (STEM) teachers do you think enter the US workforce every year? A few hundred thousand? Surely, at least fifty thousand, right? Try again. Since 2004, both the present Democratic administration and its Republican predecessor have begged Congress to enact into law a plan to graduate a measly 10,000 new highly-qualified (more on that term in a future posting), jolted me out of my chair when I read it; the US actually falls short of even that modest goal. The persistence of the 10,000-a-year goal for STEM educators demonstrates how hard it will be to develop even this human resource.

On the other hand, games like Worlds of Warcraft reach millions of new users every year (in the case of WoW, four million in 2012, according to www.statista.com ), and Food Force, the food security simulator from the UN got played a million times in its first six months after launch. Who does the student listen to more, the poor algebra teacher (who might not have as much as a math minor in college), or the flashing, blaring, addictive video game (see last week’s post on “Addicted to Math?”). Many people are lining up on the side of the video game here, noting the smashing success of Khan Academy and the “flipped classroom” model – homework in class, lectures at home from Khan.

What about effectiveness? Surely, the personal touch does better than the Max Headroom approach. Well, maybe not. A far more complex game than we offer (yet!) at www.mathnook.com, DimensionM, recently received a peer-reviewed grade by a major UK journal. The gamers showed a lasting boost in algebra skills and yes, in interest. Interest in math! Even math teachers report that their worst day of the year is that dreaded “What do you like or dislike most about math?” day. That’s the day when students get to kill their math teachers, lumping them in the category of oral surgeons at the dental clinic. Just the very possibility that electronics might dissolve the emotional barrier against STEM learning has us tickled and giggling (for a counteropinion, albeit earlier than the Dimension M paper, click through to this paper which thinks math games are a mixed lot).

Another research paper raises a more pithy question: “How can learning design maintain a sense of the wonder and joy of learning, minimize math anxiety, and improve performance on standardized tests?” No, really. They wrote that last clause, not us. The point is that we have to find ways to teach that fit the brain’s natural way of learning, which means that “development of left-brain skills that depend on sequential action and thought (reading, writing and arithmetic) must be complemented by development of the holistic, creative processes by means of right-brained activities such as visual support, story-telling, and role playing.” This includes dealing with the emotional component of learning, too. There is a negative feedback loop between seeking behavior and fear, anger, and panic. Seeking behavior is reinforced by play and attention, creating a positive feedback loop with more seeking as the result.

READERS RESPOND: What are the STEM fields if not the epitome of seeking? If you find this question pithy enough, visit us and let us know.