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.
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 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!
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.
Educational computer games have become valuable tools that can help teachers reach 21st-century students. Unfortunately, many educators still don’t know how to implement modern technology into their lesson plans. If you are a teacher who’s been looking for ways to better connect with students, learn how you can incorporate computer games into your classroom.
Experts Offer Tips
Recently, a group of educational publishers issued a report offering tips on how teachers can use computer games to educate their students. Members of the Software & Information Industry Association (SIIA), these experts broke their advice into three key phases to help educators effectively deploy a program involving educational computer games and simulations.
- Sell the Idea: According to the report, educators shouldn’t feel threatened by or uncomfortable with computer games. Instead, they should take the lead in helping others feel more accepting of these modern teaching tools. It’s important for educators to help parents understand the pedagogical benefits of computer games, which have been proven to serve as very useful tools to help supplement traditional educational materials. One of the best ways to do this is by altering the way parents see these activities. Instead of viewing educational gaming as recreational, teachers should equate them to lab time. Ultimately, if teachers want to effectively incorporate computer games into their classrooms, they need to gain parental support by diffusing common misconceptions. This means providing regular reports explaining the scope, purpose and results of the program.
- Preparing: Most educators are unfamiliar with computer games; so, it’s important for them to take time to thoroughly familiarize themselves with the games before they deploy a program. According to the report, it helps when teachers work with one another to create a safe place where they can ask so-called “dumb” questions. Once they develop a good understanding of the concepts related to each game, teachers can assess which ones best fit their current lesson plans. They’ll also be able to effectively determine the best pace in which to introduce games, while also being able to offer effective guidance to students who may need help along the way.
- Implementation: To make computer games work in the classroom, teachers need to focus on blending them with other teaching strategies. Activities should also be organized so they fit well within the day-to-day classroom schedule. Students should know exactly when they will be playing computer games, whether it’s at the end of every class, when they’ll have a chance to use the day’s lesson plan in a gaming format; or, whether the gaming is limited to just one specific day every week. Likewise, teachers need to know how to get the games they need. Sadly, most schools can’t afford the ongoing expense of purchasing gaming software for every single classroom computer. On the other hand, websites such as Mathnook provide online access to a virtually unlimited number of games that are well-suited for students across virtually all grade levels.
One of the concepts that any math student must master if they hope to succeed in algebra or other advanced math courses is the coordinate grid. If you are looking to help your child learn about this concept, you should simply invest in some graphing paper and a pencil.
First, have your child fold a sheet of graphing paper in half lengthwise. Then, fold it again along the width of the paper. When you open the sheet up, you should have two folds forming what looks roughly like the four different directions of a directional compass with the lines all intersecting roughly at the center of the paper. Explain to your child that you are going on a journey and this is the origin point. Its value is zero (o). Then show them that everything heading from the right or left of this point is traveling along the folded line called the x-axis. If the point is to the right (in the easterly direction of the compass), then the value of x is positive. If the point is to the left (in the westerly direction of the compass), then the value of x is negative. So if you move five squares to the right the value of x is 5 but if you move eight squares to the left the value of x is -8.
Then explain to your child that the vertical folded line is called the y-axis. Any number above the origin (in the northerly direction) is positive while anything below the origin (in the southerly direction) is negative. Therefore, a point that is six squares above the origin has a y value of 6. However, a point that is 4 squares below the origin has a y value of -4.
Now, you should point out to your child that points on a coordinate grid are expressed by writing two numbers in parentheses separated by a comma like this: (5, -8). This means that the x value is 5 and the y value is -8. In order to plot this on the same piece of graph paper, the child would move five spaces to the right of the origin point and then eight spaces down.
The idea here is to have your child practice this by plotting a variety of points until it almost becomes second nature to him or her. After that, it is easy to find activities online such as fun math games to help reinforce these skills. These games give your child the extra bit of practice to make them more comfortable with the concept so that it becomes ingrained.
The coordinate grid may sound like “scary” math but it doesn’t have to be. It is actually a quite easy concept that visual and kinesthetic learners can easily pick up with some simple materials and great resources.