Prefrontal Cortex and Math Facts do not Mix
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, are being forced to pretend they are 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.
1. set divisor outside the box and dividend inside it
2. Estimate how many times the divisor will fit into the dividend, or into the appropriate place value of the dividend
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
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.
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 http://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.
NOTE TO WORLD: What are the STEM fields if not the epitome of seeking?
The centerpieces of the event honoring Arthur Waskow and Gloria Steinem, put to good use in front of vacant Philadelphia property.
Chapter 1. School for Parenting
I tasted the sweat rolling down my cheeks. The big green car my mama and this man I knew as Vader drove cooked me like a loaf of cornbread when I stayed inside, so I got out and leaned against a telephone pole and opened the door for my little brother, who was sleeping.. Dressed only in shorts and flip-flops, I figured out that my shirt did more for me as a parasol than as a garment, and beside, in Ghost Town, Oakland, USA, who cared what I looked like? Most people here were like my ma, just thinking about where they were going to get high and who they owed money to.
My ma’s clear, dark chocolate skin and natural, untamed Afro are what I remember most clearly. That, and the fact that when she stood up straight, which wasn’t often, I could see her head above the Dodge Dart that Vader got at a police auction in Fremont. As far as I knew, anyone whose head rose up above the roof of a car was big. All grown-ups were big. Especially my ma. How wrong I was, and not just physically, either.
Ma had told me to sit tight, that she’d be right out. I knew there was something wrong when my head started to swim. I didn’t know how long I was roasting in the Dart before I opened the door and hung up my shirt on the telephone pole. I was six years old. I couldn’t tell time. I stood underneath my shirt, hoping that it would serve as a sail and catch any wayward breeze. I kept time to an imaginary song by wearing a crease in my flip-flops with my toes.
A rebellious blade of grass squeezed through a crack in the concrete sidewalk. There wasn’t much nature in Ghost Town. Aside from the blade of grass in front of me, and its equally wayward cousins, there were a few bramble bushes and a rare rosebush. Down Crawford Street, at the very end of town, one angry palm tree stood guard, stretching its jagged fronds threateningly at anyone who ventured that far. That palm tree dominated the cityscape of squat cement-brick warehouses and dented, rusting aluminum-sided one- and two-story houses. Around any commercial building stood a chain-link fence; at the top of the chain-link fences, chances were there would be barbed wire.
Out of boredom, I flipped off my right flip-flop and started sliding that blade of grass between my toes. I could tell just how much pressure I could put on the little green sliver before I would tear it off. My mind wandered, far out of Ghost Town, out of the nasty poverty of Foster-Hoover, all the way up the steps of my grandma’s 4-plex apartment building on North 42nd. Street, where one of our neighbors had a new Ford Thunderbird, and nobody who had their own place was in poverty, or at least not too deep in. There, Ma would go out into the laundry, and come back all shaking, but when we stayed with Grandma, we ate, slept, wore good clothes, and went to McDonald’s. And we didn’t have to share our space with Vader, who had moved in with Ma.
The earth was turning, but Ma wasn’t coming out. Shadows took little baby steps forward, finally giving the blade of grass a welcome midday shade. I crossed the sidewalk and hugged the building, wondering whether I should go inside. I was a six-year-old boy. I had to take care of my four-year-old brother. I felt so alone. I stood in that shade, and I wondered about my mom. She had done this to me before, but usually my older sister, who was nine, was there to keep me company. I didn’t yet know what these trips into Ghost Town were all about. I was, like I said, a six-year-old boy.
I couldn’t tell time, even if I had a watch, but I could guess that, based on the stretching of the shadow to the end of the sidewalk, off the curb, and onto the hubcap of the Dart, that it was two or three hours that Ma was in the house. When she came out, I was surprised, because I had stopped watching the door. I was dreaming of McDonald’s. Filet-O-Fish. Almost too hot to eat. Tartar sauce, ketchup. French fries.
“Sorry, baby,” she stammered.
Vader stumbled through the doorway next. Ma fumbled around in her purse for a moment, and drew out Vader’s key ring. Vader, quieter than he was when he went in that bungalow, didn’t even yell at Ma when the keys slipped off his hand and hit the pavement. . The big man stumbled forward and missed the keys. He grunted, turned around, and lifted up the keys. He stumbled toward the door of the Dart. I noticed that even though I was standing on a convection oven and he was inside a house that was dark and cool, he was sweating. Ma went around the front of the car to the passenger side
“It’s unlocked,” I told her.
She opened her door, and I did as well. That man cursed again, and I heard the jangle of the keys hitting the car floor. He reached down, grunted a few times, opened the door again, and stuck his feet out. So did Ma. She was looking on the floor boards and under the seat when she froze. I got back out of the car, and tapped her on her thigh.
“Can I help you, Gwen?” We called her by her first name. Maybe I would have smelled trouble with that if I were a little older.
“I got it, baby.”
She pulled the keys out from under the seat. I noticed that she was sweating, too, and that her eyes were huge, unnaturally wide open. “Ma, are you hot?” I asked. I knew that she was in a cool basement, while I was in the bright sun.
“No, I’m cold, but I’ll be OK. You never mind.”
We got back in the Dart, and she started the car. We were parked on the wrong side of the street, and Vader started out driving the wrong way. Thankfully, he corrected his mistake. After a few blocks, he found a 7-11 and pulled in front.
We walked in. Vader grabbed a bag of Halls menthol lozenges. Ma picked out a Dr Pepper and a moon pie. I had to ask. “What about me?” She rooted around in her purse, finding two crumpled dollar bills. She opened her change purse and frowned.
“We’ll have to share, baby.”
There were a lot of things that didn’t add up, but I would have time to figure this out soon enough. Too soon. Now I just wanted the cool, sweet Dr Pepper on my lips.
Chapter 11: With Boots On
The rest of that Sunday passed with a predictable off-rhythm. Visiting my dad hurt; I loved everything about him, and I hated leaving him. There was one consolation: Mom took us shopping for toys, and I got to buy a new R/C rock climber. For the uninitiated, that’s a remote control truck with a high carriage, oversized wheels, and independent suspension. None of the stupid crap Mom buys, thinking it’s good enough but always being wrong. I just asked her one time, when we were talking about it.
“Mom, would you buy a car made of the cheapest parts from a country that doesn’t make cars? Like the Yugo?”
“How do you know about Yugos?”
“Mom,” I said with barely concealed disbelief. ”Cars II?”
“Right, dumb parent.”
“Well Chinese R/C trucks are Yugos. Take whatever money you would have spent on it, and I’ll match it, and we’ll get something I can take out to FairmountPark.”
She actually bought this idea! Now I got a nice Delta A/C R/C that could go over rocks and climb uneven terrain. I even promised Ru that she could play with it if she asked me first. She got Beywheels, which are like the metal tops that are launched very fast from a serrated rip cord but have rubber bumpers instead of metal alloy fusion wheels.
I’m happy that Ru is playing with my toys, even though I don’t let on. We can play together, I love Ru, even if she gets me in trouble on purpose sometimes. But she dislikes golf, I dislike the falling down and getting bruised part of ice skating, and there is no way I cam make my body do anything in the same gym with a kid a third of my weight who can do a spinning dismount from uneven parallel bars at age 5. It’s almost like we’re four families, an Ezra family and a Ruchama family at each parent’s house. Mom doesn’t allow me to use the computer unsupervised because she thinks I’ll talk to Aba, so if Ru couldn’t play above age level every now and then, it could be really boring around my mom’s house.
It was a chilly November day, so we put on our light gloves and took the RC truck back to the half-pipe our former tenants had built as much for themselves as for me. Brandon couldn’t have been less like Branden the bully if his name were Hikaru. Brandon was a gentle man, who made himself useful to people all around him, including my mom, my sister, and me.
Ru and I walked back to the back yard, and I drove the new supertruck out over the rough, rain-soaked ground. It moved slowly, but successfully> I guided it onto the half-pipe, where it performed just as I had imagined. Ruchama started whining about my promise, bugging me for her turn on the controls until I had managed to complete the half-pipe, to turn the truck around, and come flying back. As the truck moved faster and faster, it slid up the curved wooden ramp. The engine is overcoming gravity! Cool!
I thought about a demonstration Mr. Gillum made about centrifugal force. He took a tennis ball, tied it in a string nest so it wouldn’t fly out, and stood in front of the class and spun the ball around him. I think he said that the string was like gravity and the ball would fly out straight at a right angle to the string if the nest came undone. I reversed the direction of the truck again, and then the fourth time, Ruchama yelled, “EZRA!” I got it.
“Your turn,” I replied, and handed the control over to Ru.
I resisted the urge to help her with the controls. Instead, I focused on the truck, my new acquisition, my pride and joy. The designers did a good job with the details. They remembered the cowl on the cab; instead of a glued-on plastic afterthought that would break with the first roll-over, this one was hinged, and could be raised or lowered. It was painted a shiny jet-black, and when Ruchama got the truck turned around to face us, the sun glinted off it.
The grille work was the truck’s weakness. Very nicely painted, but plastic. I had to decide that the truck would still make me happy if the grille, the hubcaps, and the bumpers got damaged or broken off. I decided that I could live with the damage that would come with the rollovers and spinouts that I knew were in its future. A big selling point for me was that the doors opened, so we could put little toys in the cab as drivers and passengers. Where truckers paint silhouettes on the doors and the hood, the manufacturers painted lightning bolts with curves.
The bay was undersized, reminding me of pickup trucks that still clattered around the roads in Guatemala City when we went to bring Ruchama home. But I really focused my attention on the tires, and the way they bounced independently on the places on the ramp where the wood has started to warp. Ruchama didn’t have my touch on the controls. So when she struggled to execute the turns at the end of the track, I could see the truck doing most of the work. Four wheels on the wood as all times. No flip-overs, even in the hands of a five-year-old child, This was going to be fun. I can’t wait for Aba’s weekend, when we might go all the way out to French Creek to let the truck take on a tough trail, or at least up to FortWashington to go up and down some pretty steep hills.
Ouch. I remembered, I was only going to see Aba for two hours with all the other men who wore a “B” on one butt cheek, and a “D” on the other. Branded by the court. Bad Dad. They didn’t look that bad to me earlier that day. One dad changed his little boy’s diapers in less time than it took Aba to beat me in Connect Four. Another read to his boy and girl so expressively that all the kids on his side of the room grabbed their own dads and formed an impromptu story circle. Five or six dads went happily. They took turns telling stories. These were not the people Mom told us to protect ourselves against. But their kids don’t get to play with their R/C trucks, or go hiking, or fishing, or play football from any continent.
I talked about the truck during dinner. I took my homework up to my room and did a model for distributing emergency supplies for a hurricane disaster area. Mr. Gillum had pulled some information on disaster preparedness from a FEMA website the past week. Mom had bought a map of New Jersey which I glued onto poster board. On it, I marked the path of Hurricane Sandy, and plotted the most likely places to put emergency relief supplies where they would be safe, but would be close enough to get to the possible disaster areas in a hurry. This was so interesting that I stayed up until 11 to finish it, just using my desk light, with a blanket under the door to block the light just in case my mom walked by.
Remembering my plan at last, I changed into my monster truck flannel pajamas, but I put my hiking boots on without socks. If I fell off my boat, I figured that my boots would dry during my dream. The socks would probably make me uncomfortable, and might make me wake up early.
How does this scene remind you of supervised visits with people in your life?
Chapter 10: Supervised Custody
I hadn’t seen Aba in weeks. That was what my mom wanted. Here’s a typical conversation about it. We are at the dinner table. Dinner is just served, and I am not pleased. Mom sets out the apricot chicken with steamed broccoli. Great. The “atmospherics” – a campaign word in an election year – are just right for a talk.
Mom: “Yes, Ezra?”
Me: “Don’t you miss Aba, even a little bit?”
Mom: “I thought I told you not to talk about him.”
Me: “I had asked you to let me stay with him more. Is that why you won’t let him see me?”
Mom, raising her voice: “The judge says it’s not safe for you to see him. That’s all, Ezra!”
Me, raising my voice back:: “I want you to tell him that you’re the mom, and you want to take it out of his court! It’s just what Aba warned me about!”
Mom: THAT’S ALL, EZRA!
Me, in full anger mode: NO! IT! ISN’T! GIVE ME BACK MY ABA! I’M GOING OVER THERE RIGHT NOW!!
Ruchama, crying: DADDY! I WANT DADDY!
Mom: EZRA: UP TO YOUR ROOM! NOW!!
Me, throwing chair against the wall: “DAMN YOU!!”
I storm out the front door and slam it for all I’m worth.
Mom, calling out the door: “You go and I call the police! Then you can visit your father where he belongs, in jail!!
Ruchama runs to the front door. “EZRA! COME BACK!!” I turn around, and when I get to the door, I see Ru on the floor, sobbing. I try to comfort her.
Mom’s last line in these conversations was not a variable; it was a constant. “EZRA!! TO YOUR ROOM, NOW!!!!”
I don’t know who will break first. After the third or fourth time of hitting the repeat sign, Mom put me on dinner – room protocol, took away my phone, and eliminated my email account. And no TV, except that I would always get Ru and come down to watch Nick at Nite when it was 2 or 3 am and I couldn’t sleep.
It was after the third of these idiotic farces wearing a Halloween costume called conversation that we were scheduled to see my dad for the first time since Mom took him away. Mom said we were going to the court to see him, so we didn’t get to go to Hebrew school. Ru asked, “Is Daddy in jail?” She knew that’s what happened to people who lost in court, because Aba said so. I don’t think that was very smart of Aba, but how was he supposed to know that his words were coming back at him on an express train in a dark tunnel?
Mom replied, “No, Bunny Rabbit, but there will be all kinds of bad men there, but they have people in uniform to protect you.”
“Protect me from Daddy?” Ru asked.
“Just protect you, that’s all.”
Ru got as silent as a corpse. The color drained out of her face too. You don’t want to see a Central American whose color has drained out of her face. She looked sickly green-yellow to me. I nodded my head really hard to get her attention, and then put my finger over my lips. She didn’t say a word until we were around the corner from Supervised Custody.
“Is this the jail?” Ru asked.
I reached out and put my hand over hers. She fell silent again. Neither of us said “boo” until we saw Aba on the steps of the courthouse. Ru didn’t know what to do. Aba split that dilemma by bounding up the stairs and sweeping us both into his arms. I guess the court would say this was “inappropriate.” I say it was Aba. Ru said nothing, but threw her arms around his neck. In a stage whisper, I gushed, “Aba!”
Mom’s eyes drilled holes in the back of my neck. Aba didn’t notice; he was too busy hugging us.
Supervised custody happens in a place that might as well be a jail, for all the warmth it has. We trudged in to a big, huge, cavernous room with tree-high ceilings, dingy beige paint, and grimy linoleum tile. There must have been a hundred wooden chairs, all connected, all anchored to a baseboard that itself was riveted into the floor every ten feet. What did they think was going to happen, Aba would steal a chair? Or maybe that some parent or child would erupt and smash one of the chairs into splinters of pain and fury? That must be it, if they thought at all. I don’t think anybody in that court thinks. They just get bullied around by lawyers.
We sat in those high-security chairs, waiting for Aba. I didn’t tell you that he had matching bouquets for us from his garden, and sweet matza brie. So he tossed the flowers and the food down on a chair with his right arm, and gave us more hugs with his left. We talked about almost nothing. At some point, we even had to play Connect Four. You don’t want to play any strategy games with my dad. You will lose. Ruchama hates to lose, but even she didn’t care this time. We talked about how awful this was, and how much we missed each other. I told Aba that I was still golfing, using borrowed clubs because Mom wouldn’t buy me a set for her house. Ru was wait-listed for gymnastics for the winter. Aba didn’t ask. He didn’t have to. If one gym wait-listed her, and he had anything to do with it, she’d go to another gym.
I lied about borrowing the clubs.
You never would have known that this was a room full of bad parents. All I saw was tip-top parenting. I think that every parent there, from the crack user to the prostitute to the wife-beater, was auditioning for a Seal of Approval to take to their own judge to get their children back. The only difference with Aba is that he was always this way.
On the way home, all Ru wanted to talk about was Aba, and why did we have to leave, and could we see him tomorrow, and could we miss regular school to go see him like we had missed Hebrew school, and… “Stop it!” Mom snapped.
It was time for me to pay a visit to the angry red holes.