Friday, January 30, 2015

Math problems of the week: Common Core-inspired word problems for 5th grade


Extra Credit:

The Common Core Standard on which this problem is based, 5.OA.B.3 (Grade 5, Operations and Algebraic Thinking), is this:
Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule “Add 3” and the starting number 0, and given the rule “Add 6” and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.
Is this problem
(a) an appropriate implementation of  5.OA.B.3?
(b) an appropriate problem for 5th grade?

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.

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?

Tuesday, January 13, 2015

Conversations on the Rifle Range Chapter 21: The Truly Brilliant, the Stupid People, and Egalitarianism

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 21:

Occasionally I would see the mother of the Asian girl, Susan who, at the start of the semester, had cornered me about Susan’s performance and whether she could observe my class. I had made an excuse and she never followed up. On certain days, she helped out Mrs. Perrin, the math department chair whose classroom was near mine. As I passed Mrs. Perrin’s classroom on those days, the mother would look at me, a scowl on her face.

Susan was doing better in my algebra class, but was very distraught over the latest set of standardized tests that I administered. The latest tests were part of the Mathematics Diagnostic Testing Project (MDTP), developed by California State University and University of California. They were straightforward multiple choice exams that had been used for years for placement. (Students already in Algebra 1 took an exam purportedly to determine readiness for geometry.) The MDTP test was in addition to the Silicon Valley Math Initiative (SVMI) test that students had taken earlier as a new facet of the District’s mysterious placement process. In my opinion, the MDTP was a much better measure of math ability than SVMI.

When I administered the MDTP, I was deluged with questions from all my students. “How is it used to place us? If we don’t pass, what class are we placed in? How is that other test we had to take going to be used?” (This last question referred to the SVMI exam). I decided to see Robin, the student counselor who had informed me about two students complaining about my teaching, to seek advice on how to answer the questions.

“There are two tests now; the MDTP and the SVMI test,” she told me. “It used to be only MDTP.”

This I knew.

“And what has been the cut-off for the MDTP?” I asked.

“It varies,” she said. “I’m not sure what it is this year. But for sure, both tests are being used. We're doing away with having so many students qualify for algebra. We simply can’t have this many students taking Algebra 1 in eighth grade. Under Common Core, we want the VERY brightest and most talented in math to be allowed to take that."

The cut-off has been 80% for years; but I wouldn’t find that out until after the school year was over when I submitted a Freedom of Information Act request to the school district to find out what it was, and how many students had qualified for Algebra 1.

Students already taking Algebra 1 were given the MDTP for geometry. What Robin didn’t know when we met was that during this transition year to Common Core the results of the MDTP and SVMI tests didn’t matter for those students, since they had been “grandfathered” in to the system (even though they were still required to take the exams). If they received a B- or better in Algebra 1 and the recommendation of their teacher, they qualified to take geometry the next year.

A few weeks after administering the MDTP, I distributed the summary sheets of the scores to my students. The summary did not indicate what class they would be placed in next year.

At lunch time the next day, Susan came to see me in tears. The results of her test were low. “Does this mean I can’t take geometry next year?”

“I don’t know how they will make that decision,” I said. “I’m told the cut-off level varies.”

“I have to get into the geometry class or my mother will kill me,” she said. “I’m not supposed to be telling anyone this.” She put her head in her hands and started crying. “I have to pass algebra and I have to get into geometry.”

I assured her she was passing the algebra course, but suggested she talk to her counselor about the placement exam. “Will you do that?” I asked.

After Susan left I called Robin and left a voice mail explaining that Susan was extremely distressed about getting into geometry and I was concerned. Towards the end of the day, I received an email from Robin, who had met with Susan. She knew about the problems between Susan and her mother. Robin did some research into the placement process during this Common Core transition year (a phrase she always said with a sigh, and a roll of her eyes heavenward). Robin told Susan that she was grandfathered in; she would get in to the geometry class. “That seemed to calm her down a bit,” Robin said.

Problem solved, I thought. For now.

I know there are parents who pushed to get their children into the algebra class. Perhaps Susan’s mother was one who did. But from what I could see in my algebra classes, with the exception of about 3 or 4 students out of 60, they were doing well, with most getting A’s and B’s. From my perspective, the MDTP was an effective placement tool. But the allure of Algebra 1 in eighth grade did have the potential of creating a student elite—now made even more so by the additional hurdle of the ill-conceived SVMI exam.

I recall in one of my pre-algebra classes after I handed the students their MDTP score summary sheets, a very bright girl named Gail said she hoped she placed into Algebra 1. (She in fact scored higher than 80% on the MDTP, and did well on the SVMI test.) “I don’t want to be with the stupid people,” she said to the girl who sat behind her.

It was probably that attitude that caused some school districts to enlist an “honors classes for all” type of policy, so no one would feel left out. Other school districts such as mine restricted entry as much as possible through their exclusionary tactics (which also kept down the number of students taking geometry in eighth and ninth grades). The eighth grade traditional Algebra 1 class has become an endangered species open only to a newly formed and very small elite. During my assignment at the middle school, about 300 students were enrolled in Algebra 1 in the entire District. This year, the number dropped substantially to 46. Many of the rest would have otherwise qualified, but for the hurdle imposed by SVMI. They were now part of the larger and growing class of the “stupid people” as Gail referred to them. Given how I am seeing Common Core interpreted for the lower grades, her insulting categorization is taking on new meaning. It is a group for whom Algebra 1 will be a watered down Common Core version in ninth grade. All in the name of egalitarianism and the greater common good.

Sunday, January 11, 2015

What's critical about reading?

As the college application process gets under way, certain questions loom:

Should someone with SAT Critical Reading and Math scores, respectively, in the 3rd and 99th percentiles be admitted into an engineering school?

Should someone with similar percentiles on the Common Core-inspired statewide reading and math tests be held back in high school or otherwise barred from getting a diploma until their reading score reaches "proficient?"

These are, essentially, the questions I raised in my earlier posts (see here and here). I've had a couple of thoughts since then.

First, it strikes me that, besides the gigantic spread between the student in question's reading and math scores, there's another spread that is perhaps equally uncommon: a 170 point spread between hid reading and writing scores, or, in percentiles, 3rd vs. 40th.

The writing score suggests substantially stronger literacy than the reading score does. Unfortunately, in the college application process, it often flies under the radar. Many college admissions offices enter only the reading and math scores into their comparison grids.

Of course, for most people, this is not a big deal. Math and reading scores are fairly correlated, and reading and writing scores even more so. For a minority of intellectually gifted outliers, many (most?) hailing from the autism spectrum, even different sorts of reading skills are highly uncorrelated.


Waverly laughed in a lighthearted way. "I mean, really, June." And then she started in a deep television-announcer voice: "Three benefits, three needs, three reasons to buy ... Satisfaction guaranteed . . . "

She said this in such a funny way that everybody thought it was a good joke anti laughed.. And then. to make matters worse, I heard my mother saying to Waverly: "True, one can't teach style. June is not sophisticated like you. She must have been born this way."

I was surprised at myself, how humiliated I felt. I had been outsmarted by Waverly once again, and now betrayed by my own mother.

Five months ago, some time after the dinner, my mother gave me my "life's importance," a jade pendant on a gold chain. The pendant was not a piece of jewelry I would have chosen for myself. It was almost the size of my little finger, a mottled green and white color, intricately carved. To me, the whole effect looked wrong: too large, too green, too garishly ornate. I stuffed the necklace in my lacquer box and forgot about it.
[From a sample reading passage in the College Board Official Blue Book SAT Study Guide]

The CPU interprets and executes instructions stored in RAM. The CPU fetches the next instruction, interprets its operation code, and performs the appropriate operation. There are instructions for arithmetic and logical operations, for copying bytes from one location to another, and for changing the order of execution of instructions. The instructions are executed sequentially, unless a particular instruction tells the CPU to "jump" to another place in the program. Conditional branching instructions tell the CPU to continue with the next instruction or jump to another place depending on the result of the previous operation.
All this happens at amazing speeds. Each instruction takes one or several clock cycles, and a modern CPU runs at the speed of several GHz (gigahertz, that is, billion cycles per second).
To get a better feel for what CPU instructions are and how they are executed, let's consider a couple of examples. This will involve a brief glimpse of Assembly Language, the primitive computer language that underlies the modern languages you have heard of, such as C++, Java, and Python.
[From "Mathematics for the Digital Age and Programming in Python," one of the textbooks for the student's AP computer programming course.]

Computer programming and engineering are two fields in which high functioning autistic students often have the greatest potential to thrive. Obviously, even in highly technical fields like these, verbal skills are important. But wouldn't it be nice, for this population, if there were an alternative to the Critical Reading test, call it a Technical Reading test, that specifically assessed comprehension of the kinds of material they are most likely encounter in the kinds of fields to which they are most likely to make the biggest contributions--if only they had the chance to?

Friday, January 9, 2015

Math problems of the week: Common Core-inspired fractions problems


Extra Credit:

Within, the problems on converting fractions to decimals don't get any more challenging than these. Are students missing out on anything if the only fractions that they're ever asked to convert to decimals have denominators of 10 and 100?

Wednesday, January 7, 2015

“I do not want my child prepared for life in the Twenty-First Century.”

In a recent blog post, NPR "math guy" Keith Devlin (also head of the Human-Sciences and Technologies Advanced Research Institute at Stanford University) argues that the Common Core Math Standards align perfectly with 21st century workplace skills. His source on 21st century workplace skills is a 1999 survey of the top desired skills reported by Fortune 500 companies:

The most important skill in the workplace at the start of the Twenty-First Century, according to those leading companies, is teamwork, which in a single generation had leapt up from number 10. The other two skills at the top, Problem Solving and Interpersonal Skills, were not even listed back in 1970.
Here’s the comparison chart:

Of course, one should not assume that every skill a company seeks, no matter its priority, is a skill that schools should teach. Not all skills (creativity? interpersonal skills?) are readily teachable in school settings. Furthermore, the fact that big companies find certain skills important doesn’t mean that they require increased emphasis by schools: not all important skills (interpersonal? motivation? listening?) are in short supply. Other skills lower on the priority lists of companies (writing? literacy? numeracy?), may be in much shorter supply. For this reason, and because they require many years of cumulative instruction, these “lower priority” corporate skills may be top priority for schools.

But that’s not what impresses Devlin. Rather, it’s:
how closely the eight basic Mathematics Principles of the CCSS align to that Fortune 500 list of required Twenty-First Century skills:

1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
So sure is Devlin of this connection between the skills sought by Fortune 500 companies and the goals of the CCSS that he proclaims the following:
The fact is, any parent who opposes adoption of the CCSS is, in effect, saying, “I do not want my child prepared for life in the Twenty-First Century.” They really are. Not out of lack of concern for their children, to be sure. Quite the contrary. Rather, what leads them astray is that they are not truly aware of how the huge shifts that have taken place in society over the last thirty years have impacted educational needs.
I, in turn, was so impressed by Devlin’s proclamation that I posted the following on his blog:
Suppose the CCSS consisted of just one goal:  
1. Be prepared for life in the 21st century.  
If I oppose the CCSS for consisting of that goal, am I, in effect, saying, “I do not want my child prepared for life in the Twenty-First Century”?  
This conclusion excludes other reasons for opposing the CCSS—for example, that the CCSS are much too vague to be useful to those in actual classrooms. Or that, because of this vagueness, the CCSS are easily misinterpreted, allowing the Powers that Be in education to claim that the CCSS support their particular (and often problematic) agendas.
Curiously aligned with these is Devlin’s personal agenda:
To prepare them [children] for that [the 21st century], you need a very different kind of education: one based on understanding rather then [sic] procedural mastery, and on exploration rather than instruction.
Sound familiar? Devlin goes on from here to the virtues of Inquiry Based Learning, which somehow emerge from those 21st century skills and those CCSS math goals--willy-nilly proving my point.

Monday, January 5, 2015

Conversations on the Rifle Range 20: More Complaints, Factoring, and Grand Master John

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 20:

When I was hired for the long-term sub assignment, the principal told me it would likely last the whole semester. In order not to unduly alarm the parents, he had announced I would be there for just the third quarter. But the day came when I told my classes that Mrs. Halloran would not be coming back and I would be their teacher for the remainder of the semester.

All my classes cheered wildly. But as much as I wanted to believe I was entirely worthy of such adulation, I suspected they were reacting to the news that the super-strict Mrs. Halloran would not be returning.

My doubt was tied in large part to the email I had received from Brian’s mother , which suggested that Brian’s poor performance in algebra this semester was due to me. In even larger part, my doubt was tied to other news I received from one of the school counselors, a young woman named Robin. She had met with me the day before I made my announcement. She started on a complimentary note: “I can't imagine walking in mid-year like you’ve done and trying to figure this all out,” and then got down to business. Two students had complained to her about my algebra 1 class. Who they were she could not disclose. The essence of the complaint was that I didn’t teach like Mrs. Halloran.

“They said she taught things one topic at a time, but you do several,” she said. This made no sense to me at first, until I remembered that in my lesson on word problems, I presented both mixture, and rate and speed problems. Ironically, Mrs. Halloran’s lesson plans called for one more, but I felt that would be too much.

“I asked them if they had talked to you about this,” Robin said. “They said they didn’t want to hurt your feelings.” As touching as this may have been to Robin, I was not impressed. I strongly suspected that 1) Brian was one of the students and 2) they feared retribution rather than hurting my feelings.

Robin closed the meeting as she had opened it. “I can't imagine walking in mid-year and trying to figure this all out.” I felt like saying “Try doing it with only two weeks’ worth of lesson plans.” I thanked her instead and left.

My algebra classes were now on the chapter on factoring. With this chapter, I had started teaching differently. I made up my own problem worksheets for homework. I didn’t like Holt’s presentation of problems, in which they tend to give complex problems at the outset instead of ramping up slowly and building expertise and confidence. My worksheets drew from Dolciani’s algebra textbooks as well as my old textbook from 50 years ago. The reaction from the students was noticeable. When I turned them loose to do their homework, I heard remarks such as “These are easy!” After a few minutes when they got to the harder problems and students started asking for help, the progress continued to be good.

Coming up was factoring of trinomials (for example, factoring x2+ 5x + 6 into the binomials (x+2)(x+3) ). I decided I would first have them practice multiplying binomials again, but this time using the shortcut method known as FOIL (for First terms, Outer terms, Inner terms and Last terms). I had already taught them how to multiply binomials (and polynomials in general) by using the distributive property. That is, (x+3)(x+2) is equivalent to x(x+2) + 3(x+2).

I had not taught them FOIL for two reasons: 1) I wanted them to get used to using the distributive property; 2) In case the “math must be taught with understanding” police came by, I would be on the right side of the law. (Some teachers believe that teaching FOIL harms students, as if it magically wipes away any understanding.) There was actually a third reason for my waiting: I wanted to save the FOIL method in preparation for learning how to factor trinomials. When using the FOIL method, students learn to do the middle term step in their heads, which helps in factoring trinomials.

Students caught on to the FOIL method immediately and some students asked why I hadn’t taught it earlier. “You’ll see,” I said, hoping that would explain things. It did for the most part.

I had students work some problems at the board, but it was a bit slow-going. To speed things up, I picked John, a Chinese boy whose parents started him at Kumon when he was four years old. He was the most adept at algebra in the class. John and Brian were now at the board, and I said my next thought aloud: “This is looking like a competition to me,” I said to see what would happen.

The class responded immediately. “Yeah, Brian against John!” they shouted, and then “I get to play the winner!” The game organized itself rapidly. (Unfortunately no “student-centered classroom police” were there to see the students making their own rules. ) Two people would be up at the board. They would face the class while I wrote the problem twice, so each contestant had the problem in front of them. Someone stood between them so they couldn't see the other's work.

John won the first round hands down and continued to beat his challengers. In the meantime, they were getting faster and faster at multiplying binomials using FOIL and computing the middle term in their heads. Pamela, a girl who often smart-mouthed me in class (and who I suspected was one of the students who complained about my teaching) then came up. “Looks like we have an Asian thing going on here,” she said.

I didn’t know what to say but she went on. “I’m Japanese, you know.” I didn’t know. She lost to John.

“Maybe you ought to go against John,” she said.

“I can’t,” I said. “I’ll lose!” John was also unwilling. "I can't compete against the teacher."

The class responded to the dialogue: “Mr. G! Mr. G! Mr. G!”

“Give it your best,” I said to John as we stood on our marks.

“You too,” he said.

We were given the signal to start; he beat me by a millisecond. I shook his hand and proclaimed him “Grand Master John” which became his name for the remainder of the semester.

I then started the class on a worksheet of 43 problems. “Why’d you give us so many?” they asked. “We’ll never finish!” I was worried about that myself, but they finished with time to spare. “I knew I should have given you more,” I said.

They assured me that I had done just fine.

Sunday, January 4, 2015

Favorite comments of '14, concluded: Anonymous, Auntie Ann, and Deirdre Mundy

On Math problems of the week: Common Core-inspired algebra problems:

Anonymous said...
Augmented matrices are covered in the Algebra II texts I've used with my kids (Lial and Foerster). The issue with the problem you showed is that you don't have to know how to do anything with matrices other than set the problem up.
Auntie Ann said...
I don't think I saw matrices until my differential equations class in college.

Deirdre Mundy said...
We had matrices starting in Algebra. Had to crank them out by hand until Calculus, when we were suddenly allowed to use graphing calculators.

They were presented as an easy way to solve systems of equations, but often seemed to take more work than NOT using a matrix!

Katharine Beals said...
I didn't learn matrices until college-level Linear Algebra, and I do not think it would have been useful to have learned about them any sooner than that.

Favorite comments of '14, continued: Anonymous, Hainish, lgm, Jeff Boulier

On Autism Diaries: Strange Stories:

Anonymous said...
For those in your audience who may not be fully NT, can you give the answer?

Is it ... ( )

Zef. Fzvgu vf gelvat gb thvyg Wvyy vagb gnxvat n xvggra? Ohg jul jbhyq Zef. Fzvgu rira guvax gung jbhyq jbex? Rvgure Wvyy vf AG naq xabjf gung Zef. Fzvgu vf gelvat gb znxr ure srry thvygl, be Wvyy vf fbzrjung USN naq qbrfa'g haqrefgnaq jul Zef. Fzvgu jbhyq fnl fhpu n aba frdhvghe.

Or is it ...

Zef. Fzvgu vf frevbhf, naq fur unf gb qebja gur xvggraf gb xrrc gurz sebz tebjvat hc naq zngvat jvgu nyy bs gur fgenl srznyr pngf be ure bja srznyr png gung vf gur zbgure bs gur xvggraf?

Who knows why humans do the things they do? For some of us, every human interaction is an exercise in anthropological/sociological experimentation. The older we get, the more interactions we have, the better we are about guessing other people's motivations, but the secret is, we're still just guessing.
Hainish said...
A high functioning 17-year-old with autism, for example, said that Mrs. Smith's utterance was just a joke.

Can we give this 17-year-old (and others) the benefit of the doubt? While "just joking" may not be the most accurate description of what's going on, it's really not far off the mark either.

Anyhow, Mrs. Smith is a horrifying figure in this story, which manages *at best* to be discomforting. I wonder how many non-NT participants in the study got that, vs. NT controls.
lgm said...
My NT teens would not say the answer involves humor. They would say it involves greed, as Mrs. Smith is trying to profit without regard to Jill's needs and would likely kill the kittens instead of setting them free if she didnt get enough money from a customer. It is highly unlikely she would actually give the kittens away free to a good home if she is making remarks like that to a customer.

If humor was the intent of the remark, the author would have indicated so by using more descriptive writing. At the minimum said jokingly' instead of 'said', but more likely with additional wording to set the scene. A fine example of poor writing.

Jeff Boulier said...
For anonymous' rot-13: Pretty sure your first answer is correct. NT doesn't mean you're psychic, it just means you're (usually) more likely to figure out why people do the things they do.

Saturday, January 3, 2015

Favorite comments of '14, cont: GoogleMaster, Auntie Ann, and Jeff Boulier

On December's baffler:
GoogleMaster said...
Since engineering schools already admit non-English-speaking students who cheated on the TOEFL, and may have cheated on the SAT also (see various recent articles on that subject), they should certainly consider a student who already speaks and reads and understands English natively -- as long as it's unambiguous -- who did NOT cheat on his SATs, and who is willing and able to understand and dig into the math and engineering parts.

I would look for a smaller school with a high population of, shall we say, quirky nerds, with a good Disability Services department, e.g.

Also check on -- there's a whole section of the discussion forum devoted to School and College Life, and there are probably some threads with tips for applying to colleges with unbalanced SATs.

Auntie Ann said...
There are trade-offs with small schools, though. If you are looking for a good peer group, sometimes it's easier to find at a large school, with a greater variety of people with different interests, than at a small one. With a small school, sometimes there just aren't enough people around to be friends with.

Jeff Boulier said...
have a friend, and old co-worker, who would partially fit this description. I don't know what the exact difference between this native-born English speaker's Math/Verbal score was, but at least one college requested that he take the TOEFL. He quickly dropped out of the moderately-selective school he was admitted to, but continued to work at the university's computer center. He's wound up doing quite well in life.

Whatever his learning disability was -- or, indeed, if he had one, I never knew. (*) His wife told me once that she was proud that she'd managed to get him reading magazines and credited herself with great improvements in his verbal skills.

Anyway, the point of this story, if there is one, is that with a huge variance between verbal & math scores, colleges may just assume that he's an immigrant. (In a sense, this is problematic, because it could lead to him being admitted to a too-demanding school.)

(*) My friend was definitely not "weird" by IT standards, though admittedly these are kind of loose.

Favorite comments of '14: Barry Garelick and Niels Henrik Abel

On Puzzle Math, II: abstract pattern recognition:

Barry Garelick said...
The "math is about patterns" meme is usually limited to IQ test type problems that depend on inductive (rather than deductive) reasoning. Supposedly this is to develop the "habits of mind" of algebraic thinking outside of the algebra class (like in 6th grade). In real math, the appropriate habits of mind are developed simply by doing the work. The fluency and the skill in recognizing key "patterns" comes from continual exposure to the material and a grounding in appropriate tools of the trade, like subject content, experience, knowledge and skill.So it's one thing to say "math is about patterns" but you still need the subject content and skill to know where to look and what to do with what you find.

By working from routine problems, students can then progress to non-routine problems and beyond. See also this article.

Niels Henrik Abel said...
Unfortunately, a lot of students have trouble with these kinds of exercises. The ones I run across tutoring get stumped with problems that look like #1, which is really rather easy if you know your stuff. The latter ones (e.g., with the radicals) would not even be attempted by such students, I'm sure ~

Favorite comments of '14. cont: Anonymous, Barry Garelick, and SteveH

On More Common Core-inspired issues: the communication skills of non-native English speakers... and of Common Core Authors:

Anonymous said...
What I don't understand is why, instead of having students verbally show that they understand, can't they design problems that can't be solved unless the student has conceptual understanding? The Singapore math books are full of such problems--and interestingly, back when they were first developed, a huge number of the students using them were English language learners.
Barry Garelick said...
Anonymous: Depends what you mean by conceptual understanding. A problem that asks for how many 2/3 oz servings are in 1 3/4 oz of yogurt requires that a student recognize that the problem is solved by division as well as a procedural knowledge of how to do fractional division. If the student cannot explain why the "invert and multiply" rule works but shows he understands what the problem is asking and how to solve it, does that mean he/she lacks conceptual understanding?

I agree with Katharine that McCallum and others should be making more public statements about what CC math standards require and what they do not. After I wrote my first article offering alternative interpretations of CC math standards (in Heartlander), McCallum showed some interest and even blogged about it. He even acknowledged that the standard algorithm for multidigit addition and subtraction can be taught in grades earlier than 4th--the grade in which that algorithm is mentioned in the CC standards: "By the way, the standard algorithms for addition and subtraction are “strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.” The standards do not forbid them being introduced in Grade 2—nor do they require that."
SteveH said...
Conceptual understanding is just that - conceptual. Too many educators mistake it for proper mathematical understanding. Conceptual is at the level of motivating why you are learning the material in the first place. It's at the pie chart level, and that simple kind of understanding of fractions provides no support for understanding rational fractions. Many teachers say that the toughest math class in high school is Algebra II because that's when those conceptual understandings fall apart. Understanding should be built on identities, not pies or bars.

Some students need more motivation than others, but it is neither necessary or sufficient for real understanding. That is built from doing and understanding nightly individual problem sets.

As for testing, how can standardized tests ever test conceptual understanding or problem solving in the sense of applying mathematical ability to new problems - ones not covered in all of the problem variations encountered in homework?

What are teachers, potted plants? Are they incapable of making those judgments even with seeing the kids every day? What is the purpose of yearly standardized tests? Are they used because we can't trust teachers or are they used just as a safety net? If they are used as a safety net, then it would be much simpler to just test the basics - results that can give specific things to correct. As it is now with NCLB, our schools get vague scores on "problem solving" and no other details on why the results aren't as good this year. Shouldn't they already know what the problems are?

Why on earth would anyone expect a problem solving yearly standardized test be part of a critical thinking feedback teaching loop?

Friday, January 2, 2015

Favorite comments of '14, cont: Ze'ev Wurman, Barry Garelick, Auntie Ann, eddie sacrobosco, and concerned

On Conversations on the Rifle Range 15: Word Problems, No Guess and Check, and a Sound Bite for an Interview:

Ze'ev Wurman said...
"This experience would serve me well, I thought. If I ever got to interview for a teaching job and I was asked to describe how I would work within the Common Core standards, I could say “Getting the right answer isn’t enough; students have to show their reasoning” or some such language."

Nice! :-)

But why to insist on equations? Aren't pictures, "visual fractions," and "area models" good enough? We need no stinkin' equations in Common Core! Another :-)

Barry Garelick said...
Thanks for the smiley, Ze'ev. And not to torpedo my own work, but it occurs to me that the interviewer could say "And what do you think of 'guess and check' as a reasoning strategy?" Oops. Guess I better seek employment in a car wash!

Auntie Ann said...
Answer: I believe it is important for students to work real-world problems, and the real world doesn't come with an answer key to check against. An engineer often has no way to check their calculations. If an aircraft engineer is wrong, the plane crashes. People in the real world who use math have to have a reliable method for attaining the answer.

They also will be fired if they spend all day guessing and checking instead of reaching a quick and reliable answer.

Guess-and-check is simply not a real-world method.

Anonymous said...
Guess and check is only an acceptable way to reach an answer if the possible options are VERY limited. Adults use it all the time (in our heads) when there are only 2 or 3 possible answers. So you can introduce it to children on that basis, using a few examples that show that this is an OK method ONLY if it gets you to the answer very quickly.

eddie sacrobosco said...
@Auntie Ann

When I used to teach word problems in Beginning Algebra I would tell students, "You may be able to solve these word problems without equations, but I'm not so much interested in the answers as I am in the writing of the equations. If I were only interested in answers the questions would be A LOT harder. And if you're interested in just answers, go take an engineering class because they're mostly interested in answers - but your answers better be right!"

And the equations are a great way to convince someone else that your answers are right!

concerned said...
You might provide students with an example that's prepared in advance, for addressing this question, with solution (5+5/7, 2-2/7). One in which guessing the solution is highly unlikely.