Tuesday, January 27, 2015

Conversations on the Rifle Range 23: The Quadratic Formula Ultimatum, and the Substrate of Understanding

Barry Garelick, who wrote various letters published here under the name Huck Finn, is at work writing what will become "Conversations on the Rifle Range". This will be a documentation of his experiences teaching math as a long-term substitute. OILF proudly presents episode number 23:

It took me about three weeks to learn all the names of my students. Identifiable patterns of behavior took me a little longer. For example, Cindy, who is in one of the two algebra classes, tended to stop me in the midst of explaining a new procedure and say: "Wait, wait, I'm confused, I don't understand."

Over time it got so I could anticipate when this would occur. When showing the factoring of 9x2 –16, for example, I paused after writing down (3x + 4). As expected, I heard: “Wait. I’m so confused. Where did the 3x + 4 come from?” I knew it was Cindy.

“I just don’t understand why it works that way,” she said. I had started the lesson by having students multiply (x-y)(x+y) and other similar problems showing how the middle value drops out. It is not unusual for students to have difficulty extending the pattern of the x2– y2 form to one like 9x2–16. I explained how it worked. Students got it but Cindy persisted. Once she understood something, she got it, but until she did it was painful—particularly when she would get frozen and could not move on until she understood, which was the case here. Students who manage to get it groan when this happens. Someone told Cindy “Because it works out that way; just follow the rule and figure it out later.”

While that exchange tends to bolster my views on how procedures lead to understanding, I can also easily imagine how easy it would be for those in the “students-must-understand-or-they-will-die” camp to make an out-of-context smear campaign against traditional modes of teaching. They would film Cindy saying "Wait, wait, I don't understand!" Freeze frame of Cindy and cue announcer voice-over: "Rote learning is no way to learn algebra. Paid for by Friends of the Common Core.”

Cindy eventually got it as she did most things; she aced all her tests and often came to my classroom before first period to get help. And all-in-all, the algebra classes did fine on the factoring unit, and on both the quiz and the test, the class average was 90. They seemed happy with themselves and with me, and we moved on to quadratic equations. There I was able to have them use what they learned in factoring to solve quadratic equations, all the while continuing with worksheets that I made, drawing from older algebra textbooks. We went through how to “complete the square” and then used that technique to solve quadratic equations that could not be factored.

The day came for presenting the quadratic formula, which, in case you have forgotten, is
x=(-b±√(b2-4ac))/2a. It was fifty years ago when I saw Mr. Dombey present it in my algebra 1 class. I watched dumbstruck as I realized that the solution of a generic equation with tools we had been using yielded a much more powerful tool. For people who think that math education should be about “patterns” rather than “meaningless algebraic symbols that bore students”, I point to the derivation of the quadratic formula as an example of taking pattern-finding ability to the next level. Some problems can be solved from a few small examples, but solving every quadratic by completing the square is too time-consuming. That's where the magic of formalism comes into play. The intuition lets go and math does the work of creating a formula by solving ax2 + bx + c = 0 via completing the square. In my opinion it is a large part of what mathematics is about. I wanted to give at least a few students that same epiphany.

My classes were patient with the explanation and with the predictable interjections from Cindy of "I don't get it" and "I'm so confused". Were I to do it over again, I would start with showing the formula and how it’s used to solve any quadratic equation. But I prefaced my presentation by saying that those who could present the derivation on the next quiz would get 10 percentage points added to their score. This tended to focus concentration.

My presentation went fairly well, though I knew that it was the sort of thing that only a few truly followed, and others would put in the effort afterward, if only to learn it well enough to get the extra credit. Others wouldn’t bother. This is my version of differentiated instruction.

Pamela (one of the students who I suspected complained about me to the counselor) tried to negotiate for more. “Can’t you give the extra credit on the chapter test? You’ll be giving a quiz next and it won’t count as much.”

I gave what I felt was a measured response.

“Well, compared to the ultimatum I gave my daughter regarding the quadratic formula when she was taking algebra, I think this is all quite fair,” I said.

“What was that?” Pamela asked.

I described how in order to entice my daughter into learning the derivation, I told her that once she starts dating, her future dates would have to show me they can derive the quadratic formula. “Now this gives you a choice. You can either date a boy who knows how to derive it, or if he doesn't, you can learn how to derive it so you can show him how, and then he can demonstrate it to my satisfaction when he picks you up."

The classroom became strangely silent and Pamela looked at me in disbelief. “You actually told her that?” she asked.


“What did she say?”

“She made a fist, held it in front of my face and said ‘I will hurt you!’ ” The class was generally appreciative of this and someone said “Good for her!”

Lonnie, a bright boy, asked me how old my daughter is. I saw in his notebook, he had every detail of the derivation copied down. “Too old for you, Lonnie,” I told him. “Go for the extra credit.”

Some students asked what if they simply memorize the derivation? I suppose I could have told them they had to supply reasons for each step, but I decided not to. Anyone willing to put in the time to reproduce the derivation was going to pick up something. Even things one learns by rote represent the substrate, the raw material, of understanding. Not the popular view, I realize. In my case, Mr. Dombey didn’t require us to derive it. But his presentation of it fascinated me enough that I tried to reproduce the derivation on my own. It also played into my decision to major in math.

Sunday, January 25, 2015

Autism diaries: remote control via iPhone

J’s greatest moment of unadulterated happiness—the one time he ever screamed for joy—was when he got an iPhone for his 17th birthday. Suddenly he had at his fingertips, anytime, anywhere, a means to:

1. film ceiling fans

2. text us about future outings that might involve fan viewing (and filming) opportunities

3. impersonate us while texting others about fan visits.

But, as we all know, an iPhone is a double-edged sword. For all it has empowered J vis a vis fans, it has empowered us vis a vis J. More directly, it has empowered our own iPhones, which, linked to his, have become extremely powerful remote tracking and control devices. With a few strokes of the thumbs or pinky we can convey queries, requests, threats, and incentives, such that:

1. no longer do we have to constantly chase after him, and worry about finding him, whenever he runs off

2. no longer do we need to repeatedly climb the stairs to visit him in his bedroom or to keep vigil outside of it until he finishes a task

3. no longer do we need to slide handwritten notes underneath locked bathroom doors when he decides to turn off his cochlear implant, rendering himself deaf to all spoken commands.

Instead, we now control J with often minimal effort, sometimes at great distances, reigning him in when he runs off, and causing homework to be completed, rooms to be cleaned up, showers to be taken, teeth to be brushed, clothes to be put on—even when we ourselves are far away from home. For proof of compliance, we can request that photographs be attached to text messages. And for refusal to answer text messages (or other kinds of “phone abuse”), we can threaten to stop paying his phone bills.

I sometimes wonder to what extent J appreciates the downside of having an iPhone. Is the convenience vis a vis fans really worth the restrictions on freedom and privacy?

Of course this is a question that applies, in one form or another, to us all.

Friday, January 23, 2015

Math problems of the week: Common Core-inspired tessalations problems for grades 3-5

From Kidspiration:

Extra Credit:

What are 3rd to 5th graders in more traditional programs, and in other countries, missing out in terms of 21st century skills for college and career by not doing tessellations problems?

Wednesday, January 21, 2015

Conversations on the Rifle Range 22: Tesselations, Border Crossings and Guess and Check

Barry Garelick, who wrote various letters published here under the name Huck Finn, is at work writing what will become "Conversations on the Rifle Range". This will be a documentation of his experiences teaching math as a long-term substitute. OILF proudly presents episode number 22:

Tessellations are repeating interlocking designs, made famous by the Dutch artist M.C. Escher. They are also included in many math textbooks –becoming more prevalent over the last two decades. While they are interesting in their own right, I’m not a fan of teaching about tessellations when there are more relevant and useful topics that will prepare students for algebra. But because my school district had dropped California’s standardized STAR test in order to field test the Common Core aligned SBAC exam, the two weeks normally devoted to prepping for STAR were gone. My pre-algebra classes faced a two week gap that I had to fill. I came up with various lessons, assignments and activities. On the first day, I had them construct tessellations.

The tessellation activity involved drawing a design on one side of a square card, cutting it out and taping it to the opposite side of the card to form a template, which was then traced repeatedly onto a piece of paper to make the interlocking design.

I had the students work in groups of their own choosing. Their choices were sometimes surprising. Trevor, the boy who was on the mock trial team, chose to sit with Jacob, a Chilean boy with whom he argued frequently—so much so that Trevor himself had earlier requested that he be seated far from Jacob. Little Esteban, a Mexican boy, who was quite bright but frequently did not do his work, joined them. (He had the habit of singing Mexican songs during tests and quizzes to combat his test anxieties.) When I checked in on them, they were doing the tessellation activity but were engaged in a lively debate. “Listen to this, Mr. G,” Trevor said. “Jacob says the Russian and Chinese military can beat the U.S. in a war.”

“It’s true,” Jacob said.

“It is NOT true,” Trevor said and gave his evidence of troop numbers, weapons and military know-how. “No one can penetrate our borders,” he said.

Little Esteban listened quietly and then chimed in. “OK, tell me this, then! If the U.S. has such great protection at the borders, how do all the Mexicans get in?” As excellent as this question was, Trevor and Jacob ignored it, Little Esteban folded his arms in frustration, and I moved on to other equally interesting conversations too numerous to mention here.

All in all, the tessellation exercise worked out better than I thought it would. Doing such activities also had the advantage of making it look to any border guards of education who happened to be passing through my classroom that I believed in and subscribed to “group work”, “collaboration”, “student-centered learning” and other fads that pass as relevant to education and/or “21st Century skills”.

The last group-based activity I did before returning to the textbook was a “Problem of the Month” activity. I had asked Mrs. Halloran for some ideas, and she sent me some “Problem of the Month” (POM) exercises. These were problems developed by the Silicon Valley Math Initiative (SVMI)—the same folks who constructed the test that was now being given as an extra barrier to taking algebra 1 in seventh or eighth grade. Touted as being “aligned with the Common Core standards”, each POM is a set of five related problems at different levels of difficulty. I decided to use “The Wheel Shop” POM, which involved determining the number of bikes, tandems and tricycles in a shop. The first two levels (A and B) were fairly easy and involved straightforward arithmetic. The explanatory material for teachers stated that the third level (C) “may stretch sixth and seventh grade students” and that “students use algebraic thinking to solve problems involving solving for unknowns, equations, and simultaneous constraints.” Roughly translated, this meant that my students didn’t have the math needed to solve the Level C and above problems efficiently and would have to resort to guess and check among other methods.

To solve the Level C problem efficiently required knowing how to solve systems of linear equations. The problem stated that “There are a total of 135 seats, 118 front handlebars (that steer the bike), and 269 wheels. How many bicycles, tandem bicycles and tricycles are there in the Wheel Shop?”

Indicative of the prevalence of guess and check thinking and instruction in school, most of the students knew immediately to use this technique. Papers filled up with diagrams and tally marks of trials and errors. A very bright boy named Bill called me over. He chose to work alone rather than in a group. “How do I solve this?” he asked

“Most people are using guess and check,” I said.

“I hate guess and check. Can’t you teach me how to solve it using algebra?”

Bill was extremely likable; he had many friends, was on the mock trial team and could argue persuasively about almost anything. He had helped me out of a jam one time when students had finished their work early and I had several minutes dead time before the dismissal bell rang. I wasn’t too good about what to do in such situations. Bill, sensing that I was getting nervous about the rising noise level suddenly stood up and announced “Let’s sing ‘If You’re Happy and You Know It’! ” The class, warming to the spontaneity of what seemed to them a rebellious act, sang along with him; not just once, but two times until the bell rang. “I owe you one,” I told him as the class filed out.

I decided to pay him back by teaching him the algebra necessary to solve the problem. “You know how to solve equations but you haven’t had a lot of what I’m going to show you,” I warned him. “So follow close.”

We established that B, T and R equal the number of bicycles, tandems and tricycles. Knowing there is one set of front handlebars on each type of bike, one seat on bicycles and tricycles and two on a tandem, and two wheels on bicycles and tandems and three on a tricycle, I coached him through setting up the following equations:

B + 2T + R = 135
B + T + R = 118
2B + 2T +3R= 269

I led Bill through the elimination method to solve for T by subtracting the second equation from the first, eliminating the variables B and R. Then, substituting the value of T into the second and third equations, I led him step-by-step, in solving for B and R. In the end, he solved it: 68 bicycles, 17 tandems and 33 tricycles.

I knew I was at risk of criticism for “telling” rather than “facilitating” and not letting Bill discover the solution by himself. But there were neither recriminations nor accolades of praise. I passed through all borders unobserved, which suited me fine.

Monday, January 19, 2015

Autism diaries: yet another Internet page on autism and repetitive questions

“If I tell the waiter I’m feeling hot, what do you think he will do?”

I can’t tell you many times J, with absolutely no idea how creepy he sounds, has told waiters that he’s “feeling hot” in order to get restaurant staff to turn fans on fast. Or how many times he’s asked me, “If I tell the waiter I’m feeling hot, what do you think he will do?”

Every so often I get sick of this question. So sick, at times, that I depart from the usual “autism mom” ideals. The last time this happened I found myself saying:

“Do you know why you like to ask the same question over and over again?”

No response.

“It’s because you’re autistic,” I explain, opening up my laptop. “Let’s see what happens when we google autism and repetitive questions.”

I type in “autism repe” and the rest is automatically filled in for me. I show him the 108,000 results. At the very top is this from the Indiana Resource Center for Autism:

Family members and professionals are often puzzled about what to do when an individual begins to ask repetitive questions. Like most things that involve individuals across the autism spectrum, the answer is not simple and clear cut. Instead, it is dependent on the circumstances surrounding the usage of the repetitive questions.
After reading this out loud to him, I have him look at the list that follows:
Possible Functions or Reasons for Repetitive Questioning:  
Inability or difficulty adequately communicating ideas via oral speech.
Difficulty knowing how to initiate or maintain a conversation.
Lack of other strategies for gaining attention in a positive way.
Need for information.
Need for reassurance.
Need to escape a situation that is boring or unpleasant.
Need to avoid transitioning to a new situation.
Desire to be social.
Need to be in control of the situation and/or attempt to keep the social interaction within his/her level of understanding.
Fascination with predictable answers.
Desire to demonstrate knowledge or competency by content of questions.
A motor planning problem which makes novel utterances more difficult to produce in affective situation.
“Which of these reasons do you have?” I ask him.

A gimme: “Fascination with predictable answers,” he says right away.

“But why are you fascinated with predictable answers?”

Silence. This question, not a gimme, is perhaps one for me to ask repetitively.

Some of the 108,000 pages are discussion boards. These are dominated by comments like: “Autism and why all the repetitive questions? I am going crazy!” and “My son is driving me crazy with all his repetitive questions!” J chuckles.

Whoops... Is this giving him a huge new incentive to keep asking me about “I’m feeling hot”? And to start asking me, repetitively, whether his questions are driving me crazy?

As it turns out, not at all. Not once has he uttered this hypothetical new question. Could it be that our Internet research made him ever so slightly uncomfortable? If so, I’m guessing that’s all for the good. And so, perhaps, is repetitive Internet research.

After all, there are another 107,995 pages, or so, to go.

Saturday, January 17, 2015

Yet more reasons for hands-on group activities: "Students study harder if professors hold them accountable!"

I just finished passively re-reading an article published by the New York Times at the end of last year. Entitled Colleges Reinvent Classes to Keep More Students in Science, it reminds us just how passive an activity it is to listen to extended prose—and, by extension, to read extended prose. As I began to passively read this article, having already spent about 80 minutes passively reading a host of others, my unengaged brain began to drift off.

Hundreds of students fill the seats, but the lecture hall stays quiet enough for everyone to hear each cough and crumpling piece of paper. The instructor speaks from a podium for nearly the entire 80 minutes. Most students take notes. Some scan the Internet. A few doze.
Me, too [snore]. But then a shocking dichotomy jolted me out of my stupor:
In a nearby hall, an instructor, Catherine Uvarov, peppers students with questions and presses them to explain and expand on their answers. Every few minutes, she has them solve problems in small groups. Running up and down the aisles, she sticks a microphone in front of a startled face, looking for an answer. Students dare not nod off or show up without doing the reading.
How could two classes be taught in such a contrasting fashion? What kind of out-of-the-box thinking, what gall, did it take to teach the second class in such a revolutionary way? My eyes widened when I learned that these are two sections of the same class:
Both are introductory chemistry classes at the University of California campus here in Davis, but they present a sharp contrast.
Nor could I believe that so many of the changes that have proved so unequivocally fruitful in K12 schools were actually beginning to gain ground in colleges:
Many of the ideas — like new uses of technology, requiring students to work in groups and having them do exercises in class rather than just listen to the teacher — have caught on, to varying degrees, in grade schools and high schools.
Riveted, I read on:
In their classes, Dr. Singer and Dr. Uvarov walk up to students, pace the aisles, and eavesdrop on working groups. They avoid simple yes-or-no questions and every query has a follow-up, or two or three.  
Before each biology discussion session, students are supposed to go online to do some reading and answer questions. The teaching assistants then know who has done the reading, who has understood it and whether the group is weak in some spots, so they can tailor lessons accordingly. Students complain about being unable to escape scrutiny, but they acknowledge that they learn more. “I don’t like getting called on like that,” said Jasmine Do, a first-year student who was one of those singled out by Dr. Uvarov. “But it makes you participate and pay attention because there’s always something new going on, and it makes the time go by really fast.”  
Faculty members have smartphone apps that let them call on students at random, rather than just on those who volunteer. When the instructors post multiple-choice questions on big screens, students answer with remote controls, providing instant feedback on how much information is sinking in and allowing faculty members to track each student’s attendance and participation, even in a class of 500.
I couldn’t believe it. Who could have predicted that students would be more likely to do the reading if you called on them and held them accountable for it in class?

Even more compelling is the underlying research. It turns out just one study, but multiple studies support this approach:
Multiple studies have shown that students fare better with a more active approach to learning, using some of the tools being adopted here at Davis, while in traditional classes, students often learn less than their teachers think.
Well… actually one of the studies is about tutorials in recitation sessions; not about making lecture classes interactive. But the other study (one I blogged about earlier) demonstrated (as an earlier NYTimes article explains) how giving students more in-class activities, as well as online activities “assigned to be completed before class along with textbook reading” and “intended to force students to think about the material”—with the instructor able to see which students had completed these activities—resulted in [drumroll…] higher scores on posttests. Furthermore:
Surveys of students who had taken the class showed that those who had the more active approach were far more likely to have done the reading, and they spent more hours on the work, [and] participated more in class...
The eye-popping takeaway of one the study’s authors:
“In a traditional lecture course, [students are] not held accountable for being prepared for class, and they really don’t need to be, because an instructor is going to tell them everything he or she wants them to know. Would you read a report for a meeting if you knew your boss was going to spend 15 minutes summarizing it for you? I know I wouldn’t.”
Equally compelling is the takeaway of the Times:
Given the strength of the research findings, it seems that universities would be desperately trying to get into the act. They are not. The norm in college classes — especially big introductory science and math classes, which have high failure rates — remains a lecture by a faculty member, often duplicating what is in the assigned reading.
Noah Finkelstein, a physics professor and the director of Colorado’s overhaul efforts, agrees, adding that:
“Faculty don’t like being told what to do, and there are people who push back and say they can figure it out on their own and they know what works for them. There’s plenty of data that says they’re mistaken.”
To this, the Times adds:
Of course, telling experienced teachers that they need to learn how to teach does not always go over well, especially when they have tenure.
Yet another thing that astounded me was how unusual my own college experience was. Even in classes in which the (often tenured) professors summarized things and “told us everything we wanted to know,” and in which
Hundreds of students fill the seats, but the lecture hall stays quiet enough for everyone to hear each cough and crumpling piece of paper. The instructor speaks from a podium for nearly the entire 80 minutes. Most students take notes… A few doze. [No Internet back then]
… even in these environments, we still learned stuff. Weekly discussion sections and frequent papers kept us on top of the material. Pop quizzes would have done the same thing. Duplication of the reading material by the lecture was reinforcing (“multi-modal learning,” anyone?). Some professors even asked questions and led back and forth discussions from the podium! Student centered group activities would have detracted substantially from all this expert-driven instruction. Of course, today, we know that students are the real experts; they, alone, can construct their own learning.

But here I must interject a giant disclaimer. All the reflections I just shared were thoughts I had during the stupefying process of passively taking in the Times' extended prose (as opposed to actively engaging in hands-on group activities). I therefore have no confidence that these solitary thoughts of mine involve any higher-level thinking whatsoever, let alone real-life relevance and real-world application.

Thursday, January 15, 2015

Math problems of the week: 4th Common Core-inspired 4th grade word problems vs. Singapore Math

I. From MathWorkSheetsLand:

2. From the Singapore Math placement test for the first half of 4th grade:

3. Extra Credit:

How do the 4th grade "Common Core"-inspired word problems compare with the Singapore Math problems in terms of mathematical vs. non-mathematical demands?

Which problem set involves more of the kind of algebraic thinking that Common Core authors want to see starting in elementary school?

Does the lack of illustrations in 4th grade Singapore Math problems deprive students of opportunities to apply math to real world contexts?