Sunday, November 30, 2014

Formal proofs vs. intuitive proofs plus jargon

Formal geometry proofs are to some extent the opposite of intuitive. To prove that two figures are similar, for example, you have to throw out any intuitions you have about why they must be similar, and rely exclusively on the axioms and theorems of Euclidian geometry.

Consider, for example, two of the triangles in the problem below.

Intuitively, it’s clear that if you flip triangle BEC over on its bottom vertex, i.e., keeping vertex E stationary, then two of its sides (BE and CE) would lie along two sides of triangle AED (AE and DE).


From the given information, it’s also clear that BEC’s third side, BC, is parallel to the EAD’s third side (AD). That means that in the flipped configuration of AED, if you stretch AED downwards until BD reaches AD, the resulting triangle would completely overlap with triangle AED.

However, a formal proof based on established theorems bypasses these intuitions:

While some people complain that these proofs are overly formal and mindless because they ignore intuition, others, myself included, found these proofs compelling and mentally engaging for the very same reason: i.e., because they force you to abandon all preconceived notions and work things by logic.

Under Reform Math and the Common Core, unfortunately, these formal Euclidian proofs are waning. Replacing them are informal proofs that codify intuition via formal-sounding jargon: “transformations,” “reflect,” “translations” and “dilations.”

And so (as far as I can tell from what I'm seeing in today's Common Core-inspired rubrics) an acceptable proof for the above problem is a formalization of the intuitions I described above. BEC and AED are similar because you can reflect BEC on an axis of symmetry that goes through vertex E and is parallel to BE, and then dilate the triangle (by a scale factor of EA/EB) so that it completely overlaps with AED.

It seems to me that something key is lost when kids no longer get regular practice in deducing what does, and does not, follow from those basic assumptions that we all agree on.

Friday, November 28, 2014

Conversations on the Rifle Range 17: Boundaries of Behavior, Parallelograms, and the Art of Forgiveness

Barry Garelick, who wrote various letters under the name Huck Finn, published here, 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 17:

There are a variety of methods one can use to discipline students: detentions, referrals, sending the student outside of class, contacting the parents. I was confused about most of them and resisted using them. Lunch-time detentions were especially tricky because of a dual lunch schedule at my school. Because of the limited space for lunch there were two lunch periods for the two grades. This meant that during the eighth grade lunch period, I was teaching my fourth period class (pre-algebra).

The first person I ever referred was Peter in my fifth period algebra 1 class. He showed disrespect in a number of ways. He would sometimes say in a sarcastic Eddie-Haskell-like tone: “I think you made a mistake—oh but I know you’re a great teacher,” which would elicit knowing giggles from others. One time when he was particularly disruptive, I sent him outside which in this school meant outdoors. The school was a collection of modules—all classrooms opened to the outdoors. Sandra, another disrupter, waved to him on his way out and called “We love you, Peter.” He has a fan club, I thought—just what I need.

Her seat was next to the wall on the other side of which Peter now stood. She pounded on the wall to get his attention. I heard the pounding, and saw Peter’s head appear in the window as he jumped up to see what was going on. Not knowing the details of the event, I assumed wrongly that Peter had been doing the pounding. I got him back inside and gave him a referral. As I filled out the form, Peter protested and Sandra quickly confessed. “It was me who was pounding on the wall,” she said. I knew Sandra was telling the truth but I decided I had no time for details; the die had been cast. I needed an example. Plus, if the class thought I was acting irrationally or in error, then it was a signal that they better be quiet and not risk my irrational actions.

My referral was thus performed as an irrevocable act. I called the office and said that I was referring Peter and he was on his way. I then turned to Peter and handed him the completed form. “They’re waiting for you.” Butter would not have melted in my mouth.

It worked. The class was like a tomb. I remember thinking “I didn’t mean for them to be this quiet.” But I went on as if nothing had happened and the class noise level eventually increased—but not to its usual level. At the end of class Sandra again confessed and told me she would give herself an after-school detention. “That’s fine,” I said and thought it a good opportunity to give her some of the tutoring that she desperately needed. She would keep putting it off and would not show up; I never pushed the issue.

The next day I dreaded my fifth period algebra class, knowing I would have to confront Peter. I didn’t know how that would go, or even if there would be one. Earlier that day, I sent out a boy named Chad in my third period pre-algebra class. He was noisy, and given to saying “Sweet Jesus!” when amused or surprised or frustrated, and would make what I thought to be frog-like croaking sounds. After hearing such a croaking sound I told him to go outside. He croaked “Why?” and I explained “Those croaking noises are precisely the reason I’m sending you outside.” He left and the class became tomb-like as had my algebra class the day before. After a few minutes, I went outside to talk with Chad.

“I want you to control your outbursts. I know you’re a nice boy and can do this work, but I can’t have the class disrupted like this.”

“Well I wish you’d tell Bryan to stop barkin’ like a dog!” he said. While I was unaware of Bryan’s dog imitations, this was the first time I heard Chad in conversation. Except for outbursts, he did not participate in class. I suddenly realized that Chad’s croaking was his natural voice—perhaps the first stage of a changing voice which starts around middle school for most boys. I felt terrible. I told him I’d keep an eye on Bryan.

What with behavior boundaries established (and clarified) we went back inside where I proceeded to demonstrate a more mathematical view of boundaries—namely that the area of a parallelogram is the same as that of a rectangle. I did it by means of magnetic shapes I had made that stuck to the whiteboard. The parallelogram had a right triangle indicated as below. “Now it so happens that this triangle…”

The class didn’t let me finish and many shouted: “You can move it to the other side!”

Which I did, thus transforming the parallelogram into a rectangle without altering its area. “Sweet Jesus!” Chad croaked. I took this as a Q.E.D. for my informal proof.

The day wore on until at last the fifth period that I had been dreading arrived. Peter came into the room and said "Hi, Mr. G!" I greeted him back with a wave and while I began taking attendance he asked "Do you hate me, Mr. G?"

"I don't hate you and I never have,” I said. “I felt you were showing me disrespect so I gave you a referral."

"I wasn't disrespectful yesterday," he said.

"You've been disrespectful for quite some time. I think you know that." He nodded and said nothing else. I realized then that no matter what disciplinary action is taken, it necessitates the equally important act of reconciliation. Our brief conversation was it, and we moved on.

I could feel a chill that day amongst the people who were his friends. Perhaps they felt as I did 50 years ago when, one day during spring semester, I mouthed off to Mr. Dombey, my algebra teacher. He raised his voice and I felt he hated me just as Peter felt I hated him. I remember feeling betrayed, then confused. I don’t recall how it was reconciled; I just remember that the next day it no longer mattered. I knew where the boundary was, and things were both different and the same.


Thursday, November 27, 2014

Turkey Grammar Answer Key

1. Even more ridiculous is the idea of cooking it in a bag.
2. Overstuffing the turkey makes the stuffing come out dense and the turkey difficult to cook properly.

A happy thanksgiving to all!-- And may your turkeys be neither ridiculous nor improperly cooked.

Tuesday, November 25, 2014

Your syntactic toolkit, I: Two tools for Turkey Grammar

Last year I posted a Turkey Math problem; this year it’s time for Turkey Grammar.

Here are your two syntactic tools:

Tool #1: Inversion:

Some sentences contain phrases that can be moved to the front, inverting the subject and verb. For example:

People make pies out of pumpkins only here in America
Only here in America do people make pies out of pumpkins.

Tool # 2: “Tough movement”:

Sentences containing words like “easy” and “difficult” allow a variety of possible word orders. For example:
Cooking turkeys thoroughly without drying them out is notoriously difficult.
It is notoriously difficult to cook turkeys thoroughly without drying them out.
Turkeys are notoriously difficult to cook thoroughly without drying out.

Exercise 1: Invert the second sentence to make it link up better with the first one:
Some people think that soaking a turkey in brine overnight makes it tastier. The idea of cooking it in a bag is even more ridiculous.

Exercise 2: Sharpen this sentence via tough movement.
Overstuffing the turkey makes the stuffing come out dense and makes it quite difficult to properly cook the turkey.

Sunday, November 23, 2014

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

Two side-by-side articles in this past week’s Education Week show a disconnect between what the Common Core authors vs. actual classroom teachers think are the biggest challenges posed by the Common Core. First, there’s an interview with William G. McCallum , the lead author of the Common Core Math Standards. McCallum cites coverage of fewer topics as the biggest change brought by the Common Core, and fractions, ratios, and proportional relationships as the biggest challenges to teachers.

For teachers, on the other hand, what seems to be most novel and challenging is the Math Standards’ emphasis on conceptual understanding and verbal communication. This is particularly true in the case of teachers of language-impaired students and students whose native language isn’t English. The latter are the focus of the other article.

When he began working the Common Core State Standards into his instruction three years ago, New York City middle school mathematics teacher Silvestre Arcos noticed that his English-language-learner students were showing less progress on unit assessments than his other students.
"It wasn't necessarily because they didn't have the numeracy skills," recalled Mr. Arcos, who is now a math instructional coach and the 7th grade lead teacher at KIPP Washington Heights Middle School, a charter school in New York. Rather, they were struggling with the linguistic demands of his new curriculum, which was oriented heavily toward word problems and explication of solutions.
To address the issue, Mr. Arcos began incorporating strategies that are typically the province of language arts teachers into his math lessons. Especially when working with his English-learners, he provided detailed instruction in close reading, sentence annotation, and writing fluency.
Nor is Mr. Arcos alone:
Mr. Arcos' recognition that the new math standards may require greater attention to the needs of English-language learners is not uncommon among educators who work with such students. Particularly in the Standards for Mathematical Practice that preface and inform the grade-level objectives, the common core emphasizes the importance of explaining solutions and relationships, constructing arguments, and critiquing the reasoning of others. While such expectations are proving difficult for many students, educators say, they pose unique challenges for those not fully proficient in English.
When I was a 6th grader in a school outside Paris, immersed among native French speakers, math class offered refuge from the linguistic challenges of my other classes. From the beginning, with minimal knowledge of French, I was able to follow what was going on on the chalkboard. And could figure out what to do on homework and tests. Nor do I feel like my math experience would have been any richer had the word problems involved more elaborate French sentence structures and vocabulary, or had I been required to explain my reasoning in French. In fact, the best elementary school math class I had was that 6th grade math class, with its solidly conceptual and engaging French math curriculum. When I returned to the U.S. for 7th grade, I was, in fact, ahead in math relative to my peers. Had my French math class gotten bogged down with “detailed instruction in close reading, sentence annotation, and writing fluency,” I would surely have instead ended up significantly behind.

So is “detailed instruction in close reading, sentence annotation, and writing fluency” in math class really what’s best for our students—whether or not they are non-native English speakers?

Not surprisingly, professors of mathematics education, as opposed to professors of mathematics, applaud this continued dilution of math with English:
In addition, the common core's emphasis on verbal expression and reasoning in math are widely seen as beneficial to English-learners. "The more language you use in the math classes, the more [ELL] students are going to learn, both in math and language," said Judit N. Moschkovich, a professor of mathematics education at the University of California, Santa Cruz.
Especially because it becomes one more excuse to have students work in groups:
At the same time, a Teacher Notes panel provides specific activities teachers can use to help English-learners engage with the language of the lesson. One such exercise says: "Have students work with partners to discuss the graphic organizer and fill in the sentence frames [provided]. Then have them use the word bank [provided] to fill in the summary frame."
As for Common Core Math Standards lead author William McCallum, he seems blind to the problems posed by the Standards’ perceived emphasis on verbal expression. When asked if there’s anything he might change about the Common Core, all he mentions are the geometry progression in the elementary school Standards and the level of focus in the high school Standards:
“I think the geometry progression could be evened out a bit in elementary school. I think in high school there could be more focus. High school was difficult because everybody has their pet topic, and it was difficult to resist those pressures.”
As for challenges of particular Common Core-inspired problems or of conceptual understanding, McCallum blames these on mis-implementations or misinterpretations:
“What's interesting to me is that both the supporters and the critics of the common core, I think, are overemphasizing conceptual understanding—and understandably because everybody's always demanded procedural fluency, and the conceptual understanding is what's new. But that doesn't make the other requirement go away.”
Well, what’s interesting to me is that the lead writer of the Standards (a) thinks that conceptual understanding is something new in math education (b) has written something that he acknowledges is being mis-implemented and misinterpreted, and (c) has failed to do anything to stop this.

Thursday, November 20, 2014

Math problems of the week: Common Core-inspired geometry problem

The Common Core Standard in question:


Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

The source:

The problem:

The justification, solution and scoring:

Student reactions to this kind of problem:

An alternative, more tradition proof:

Extra Credit:

Which proof do you like better?

Use transformations to prove that the two proofs are (or aren't) similar.

Tuesday, November 18, 2014

Conversations on the Rifle Range16: Parallelograms, the Mercy of the Court, and Kit Kat Bars

Barry Garelick, who wrote various letters under the name Huck Finn, published here, 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 16:

A “prep period” –a period in which teachers have no class—is one of education’s most sacred and cherished traditions. Mine was first period and involved making copies and putting finishing touches on lesson plans, as well as pacing nervously in anticipation of the day ahead. The stage fright dissipated when class began but would return if I didn’t pace things right and had slack time left over at the end of class. The resulting restlessness of students spelled disaster and invited clock-watching, students getting out of seats, general disruption, and lining up at the door (though I put an end to that practice quickly).

My fourth period pre-algebra class was the one class I dreaded most, though in the end it turned out to be my favorite. It was the most crowded, and also had six students who were “English Learners”. A Spanish-speaking aide was in class to help them. Mrs. Halloran had explained to me that because they were “low” in ability, they weren’t part of the regular class, and were relegated to the back of the class where they worked on an online course on computer tablets. They were given a special “pass/no grade” credit for the class. With the aide working with six students, the conversations at the back of the room served as a stimulus for other students, many of whom had self-control problems to begin with, to start talking. When the din got to a certain volume I had to raise my voice to quiet them. If that failed, Allysandra, a rebellious Mexican girl would yell “SHUT UP” at the top of her voice. This would generally do the trick.

Then there was Trevor, a disruptive boy, not well-liked by the other teachers. He got into fights and was even suspended for a week for one of them. He had a gift for arguing convincingly even when wrong. I first noticed this when he put some make-up work in the wrong place. When I told him where the correct bin was, he said “Chill.” My reaction was so swift it caught both me and the class by surprise: “Don’t you EVER tell me to chill!” I said. The class became a tomb.

“No, I didn’t mean it like that,” he said. “I meant ‘chill’ in the same way you say ‘cool’ or ‘OK’. That’s how I meant it.” He was convincing so I backed off and the class din resumed after about a minute. When I related this tale the next day to a teacher, she told me “Oh, that’s Trevor! He always has some excuse about how he didn’t mean this or that. He’s good at that.”

I mentioned to her that he was on the school’s mock trial team. This team competed with other schools in a mock trial, judged by a real court judge. “Given his gift for arguing and being on the mock trial team, I would guess he’ll end up being a lawyer,” I said.

“I hate to think who his clients will be,” she said.

I put that thought aside and tried to reach out to him. He would sometimes read a book quietly if he either 1) finished his homework early or 2) was avoiding doing the homework. Rather than try to find out which it was, I would ask him about the book he was reading and he would tell me. I sensed a more cooperative side to him, but wasn’t sure whether it was because I was showing an interest in him, or because of the impending championship mock trial coming up that was helping him focus his energies. Since I had members of the mock trial team in all of my three pre-algebra classes, I decided to tap into these students’ argumentative gifts during the unit on geometry.

I was finishing up a chapter on parallelograms. I decided to put a figure up on the screen in each of my pre-algebra classes and asked if anyone could tell me if two particular line segments in the figure were congruent and why. Some parallel lines were marked as such. The two line segments of interest were marked as being perpendicular to one of the two parallel lines and were, in fact, opposite sides of a parallelogram and therefore congruent.

I offered a Kit Kat bar to any student who could answer the question. Students immediately rose to the challenge. In all classes, some student would inevitably say “Can’t you just measure the two segments?” to which I would reply “Inadmissible evidence! The court will not allow rulers or any type of measurement devices in this trial. Demonstrations must be made using definitions and theorems only.”

As expected, the members of the mock trial team rose to the challenge. Some tried to get around my restriction of no measurement devices by saying “They look equal” but I easily put that to rest. “Not adequate. Visual comparisons are not allowed.”

In fourth period, upon hearing the “Inadmissible evidence” warning, Trevor rose to the challenge. He stood up and said “I got this! I got this!” and then proceeded to make spirited, breathless demonstrations that didn’t quite make the case.

I gave him some hints. “Do you think these two line segments are parallel?” I asked.

“Yes, definitely,” he said.


“Because they’re both at right angles, at right angles!” Trevor said as if pleading to a jury that his client did not deserve the death sentence. “What about the right angles?” I asked.

“It proves it,” he said.

“Proves what?” I asked.

“Proves that the lines are congruent.”


“Because they’re right angles, they’re right angles!”

I paused as if giving the matter great thought and the class quieted.

“Do you mean to say if two lines are perpendicular to the same line they are parallel?”


“So why are they congruent?”

Someone shouted “Because it’s a parallelogram!”

In the best spirit of courtroom drama, Trevor protested: “Unfair! I was going to say that!”

“There will be silence in the courtroom,” I ordered to no avail. “Counsel will be seated, please,” I said and continued: “The court will show mercy and recognize that counsel’s observations and arguments have merit and has provided indications that he knows that opposite sides of a parallelogram are…what?”

“Congruent!” Trevor shouted.

“One Kit Kat bar is awarded.” The class applauded, though the person who identified the figure as a parallelogram wanted one also.

“Come up and claim your Kit Kats,” I said and presented the awards. To Trevor’s credit and my satisfaction, after he took his Kit Kat, he shook my hand.

Sunday, November 16, 2014

Autism Diaries: reversing heart rates

J spent at least the least the first 15 years of his life relentlessly raising the heart rates of everyone around him. In years 1-5, he'd constantly throw things around and break them; turn appliances off (the lights in the evening; the refrigerator and/or freezer) or on (the heat in summer; burners); run into other people's yards or ahead of us into the street (face turned towards us, grinning); push and grab people and poke them in the eye; and vanish in pursuit of ceilings fans. In years 5-10 he'd disrupt his classes and alienate his teachers; charge through crowded hallways and thoroughfares, force random people to sign two; bother the heck out of his siblings; and vanish in pursuit of ceilings fans. In years 10-15 he continued to disrupt his classes and alienate his teachers and charge through crowds, as well as engaging in increasingly sophisticated mischief and vanishing in pursuit of ceiling fans.

Fast-forward a few years. The mother of a dear friend who works with J has been in the ICU all week. And so she, too, has been in the ICU all week, at her mother's side. But she misses J tremendously--as she often does when separated from him for more than a few days. She misses, in particular, the fresh air and levity he provides when times are tough: his innocent questions ("How is your mom's heart?") and hopes ("When she gets better, do you think we can go to the restaurant with fans?"). So, two days ago, she asked if I could drop him off near the hospital so that she and her mother (another fan of J's) could "get some J time."

Afterwards she wrote me a text message commemorating what has to be one of the biggest milestones we've seen in his 18 years:

I am literally in tears over how sweet J is. Thanks for sharing him with us. He was a calming presence and brought my mom's heart rate down to the lowest it's been since she got here.
Who could ever have predicted that, 18 years on, J could not only provide comfort when times are tough, but enter an ICU and bring someone's heart rate down?

Friday, November 14, 2014

Math problems of the week: Common Core-inspired geometry test questions

Here is the breakdown of the Common Core standard that has inspired this problem, CCSS.Math.Content.HSG.GMD:

Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments.

Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.

Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

Extra Credit:

(Follow-up to last week's Extra Credit question)

1. Will a student who has never heard the phrase "Cavalieri's principle" know how to proceed on this problem?

2. Should a student who has never heard the phrase "Cavalieri's principle" end up with fewer points on this problem than one who has?

3. Should a student who explains without reference to "Cavalieri's principle" why the two volumes are equal get full credit for this problem?

4. Is it acceptable to argue that the volumes are equal because they contain the same number of equal-volume disks?

5. To what extent does knowledge of "object permanence," typically attained in infancy, suffice for grasping why the two stacks built from the same number of equal-sized building blocks have equal volume?

6. To what degree does this problem test knowledge of labels rather than mastery of concepts?

Tuesday, November 11, 2014

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

Barry Garelick, who wrote various letters under the name Huck Finn, published here, 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 15:

I had come to the point in the chapter on systems of linear equations in my algebra 1 class where the book presented mixture, rate and current, and number problems. To prep them for the onslaught, I included a word problem into one of the warm-up problems I had them do as I checked in their homework.

The problem was: “The length of a rectangle is 3 units more than the width. The perimeter is 58 units. Find the length and width."

Students asked me “How do you do this problem?” as I came around to check their homework. I offered one hint: “You can solve it using the substitution method”.

"What does this problem have to do with the substitution method?" a boy named Lonnie asked.

I answered his question when I went over the warm-up questions. “If you solve the problem to find length and width you will have two equations in two unknowns which you can solve by substitution."

Many students shouted at once.

“I can’t answer you when you all shout at once. Raise your hands.”

I called on Lonnie. "How do you get the two equations?" he asked.

I drew a rectangle on the board. “How do I express length (L) in terms of width (W) if length is 3 more than width?”

One person ventured an answer, a very smart and quiet girl named Anna. “L = 3 + W” she said.

“Now give me an equation for the perimeter.” After some struggling to remember the formula for it, they came up with 2L + 2W = 38. From there it was obvious how substitution played a role. With this framework now in place, I segued to rate and current problems (which I discussed in Chapter 7), some of which leant themselves to solving by substitution, and others solving by elimination.

After we worked through a few problems I said: “These problems seem hard now, but the problems in the book are really all the same. If you learn how to solve one, you’ve solved them all.”

Which is one of the many complaints that math reformers have against traditional math, though I hasten to say here that such an approach is traditional math done poorly. And Holt Algebra 1 is guilty of providing problems that are exactly alike in structure and vary only in the given values in the problem. I would also like to see problems that up the ante in difficulty, so that subsequent problems force students to extend their thinking from the worked examples.

“I expect that you will find these very easy after a while,” I continued. “So easy, you’ll find them boring. And when that happens, I’ll give you some that are a bit more interesting.”

They didn’t like this idea, and Naomi, a seventh grader, asked "When will we stop doing word problems?"

"We won't,” I said.

“So the answer is ‘never’?”

“Correct,” I said.

And with that I launched into the other types of word problems. Over the next two days we covered the various word problems. For mixture problems, they learned to put the information into a particular format, whether working with nickels and dimes summing to a certain amount, coffee beans selling for various prices per pound, or mixtures of acids of varying strengths. For example:

“A grocery store sells a mixture of peanuts and raisins for $1.75 per pound. Peanuts cost $1.25 per pound and raisins cost $2.75 per pound. Find the amount of peanuts and raisins that go into a 6 pound mixture.”

The information is placed into a table and information arranged in such a way that the associated equations become obvious.

1.75*6= 10.50

What I wanted to do was to introduce harder problems of my own eventually, once they caught on and found them easy. I wanted to progress to variations such as: “A grocer blends teas worth $0.66 and $0.48 a pound. If he interchanges the amounts, he saves $6 in a blend of 100 pounds. Find the ratio of the weight of the two teas in the original blend.” This is varied enough that they cannot simply plug the values in to a handy table.

But I never got to do this. Looking back, this would have involved giving them a bunch of word problems interspersed between some of the lessons. As it was, the class had started a little behind where other classes were, so I was playing catch-up. Not to mention writing lesson plans only about a week ahead of where I was. And as it was, not as many students achieved the facility with these problems that I wanted.

And then there was the eye opening revelation that came when doing number problems. These are the type such as: “The sum of the digits of a two-digit number is 3. When reversed, it forms a number that is 9 more than the original number. Find the number.” When going through these in class the day before the quiz, I asked a girl who got the answer correct (the original number is 12) how she did it.

“I guessed,” she said.

“What do you mean you guessed?”

“I just can’t do it the way you showed us. But I can get the answer really quick if I try certain numbers out till I get the right answer.”

An argumentative girl named Sandra who did very little work and had a knack for hijacking conversations (e.g., “What’s your favorite color, Mr. G?”) came to the other girl’s defense. Her teacher last year (Mrs. Perren, the math department chair) embraced some of the math reform philosophy and Sandra apparently absorbed what her teacher told her. "Guess and check is a perfectly legitimate mathematical procedure that requires deductive reasoning to narrow down the choices," she told me.

My response: "You need to show the equations for the problems on tomorrow's quiz or it will be marked wrong. Guess and check will not be allowed."

“Even if we have the right answer?” she asked.

“Correct,” I 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.

In the meantime, with my position firmly stated, I went over once more how to solve the number problems using algebra.

Monday, November 10, 2014

Autism Diaries: An unexpected wildcard on the SAT math test

J has just taken the SATs—hopefully for the last time. To anyone in the know, his scores—with the several hundred point gap between verbal and math sections--cry out “autism” (or, possibly, “gifted math student from a non-English speaking country who only had a few years of English instruction”). Given that J’s autism-related communication difficulties have him reading at a 7th grade level, there’s only so much his verbal scores might budge upwards. But an 800 in math is theoretically possible, if only J would avoid the sorts of careless mistakes—there are always 2-3 of these--that he invariably makes on practice tests.

The wildcard here are the 10 “student produced response questions.” Here, instead of selecting among 5 multiple choice options, the student enters a numerical answer by filling in bubbles in a number grid. Everything else being equal, these SPR questions are much easier to get wrong than the multiple choice questions: they allow, after all, a much broader range of possible answers. But they’re also easier to get wrong for stupid reasons in particular. If you misread a multiple choice question, your misreading often becomes obvious to you when you look at the 5 choices and see that none of them fits. If you misread a student produced response question, there’s only the number grid to clue you in, and it will do so only if your response is, literally, off the charts.

Mathematically speaking, the SPR questions strike me as generally easier than many of the multiple choice questions. Despite this, they’re the only sort of question J gets wrong on practice tests. He misreads the question, and then has no clue that he’s misread it.

You can sort of see why SPR questions are theoretically appealing. Multiple choice questions always get a bad rap, and SPRs, by comparison, look refreshingly open-ended. But this open-endedness goes only so far—and remains a far cry from the open-endedness one finds in college entrance exams in many other countries (for example, Finland). The end point of America’s SPR questions is still a single, right-or-wrong response, with no opportunity to show your work (not to be confused with “explaining your answer”) and get partial credit. In a way, therefore, SPR questions combine the worst of both worlds: they don’t rule out stupid mistakes, as multiple choice questions do, and they don’t allow partial credit, as truly open-ended responses do.

So I can only hope that J managed somehow to show an unprecedented level of vigilance vis a vis the 10 SPR questions that confronted him this past weekend.

Saturday, November 8, 2014

My daughter can now date Barry's daughter!

Barry Garelick once told me that anyone who wishes to date his daughter must first successfully derive the Quadratic Formula. A few days ago my daughter proved up to the task.

In fact, she was able to derive it on her own the first time around, with minimal assistance: without having first had it demonstrated to her. In Constructivist parlance, she "discovered" it! But only after lots of cumulative, guided practice placed it squarely within her Zone of Proximal Development. (Cumulative, guided practice of the sort that, incidentally, is entirely missing from Reform Math Algebra, which, if it asks the students to learn the formula at all, has them do so not via conceptual understanding, but via rote memorization).

My daughter's cumulative, guided practice included solving dozens of quadratic equations of varying complexity (including so-called "literal equations" in which the coefficients themselves are variables): first by factoring, and then by completing the square. I credit in particular her honing the technique of multiplying the equation by four times the co-efficient of the squared term before completing the square. Not only does this simplify the process by eliminating the need for fractions; it also makes the Quadratic Formula derivation a tad more elegant.

It's the difference between this:

And this:

Or, in her own hand:

Of course, either way works--whether for handling quadratics, or for dating Barry's daughter.

Having to resort to rote memorization, on the other hand, substantially limits your future prospects: both mathematical and romantic.

Thursday, November 6, 2014

Math problems of the week: Common Core-inspired test questions


A problem purported to address the following Common Core standard for 8th grade geometry:
Understand congruence and similarity using physical models, transparencies, or geometry software.  
CCSS.Math.Content.8.G.A.1 Verify experimentally the properties of rotations, reflections, and translations.

Extra Credit:

Discuss the possibility that 8th graders already know what happens to figures when you drag them, stretch them, rotate them, or flip them.

Relate this possibility, along with the demands of this problem, to the issue of labels vs. concepts.

To what degree does this problem, rather than assessing attainment of CCSS.Math.Content.8.G.A.1, merely test vocabulary?

Tuesday, November 4, 2014

Conversations on the Rifle Range 14: Late Start Mondays, Debby Downers, Nervous Nancies and a Tiger Mom

Barry Garelick, who wrote various letters under the name Huck Finn, published here, 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 14:

Like many school districts, mine instituted “late-start Mondays”, in which school started an hour later and periods were shortened from one hour to 47 minutes.  We reported at 7:30 (as always) and had to attend a meeting or other activities as announced.
On one of my first late-start Mondays, the math teachers were told to meet for a discussion of the upcoming parents’ “math night” scheduled for later that week. The District would present and discuss the various pathways in math under Common Core and answer questions. There was growing concern among parents regarding the increased limit on the number of eighth grade students who can take Algebra 1, and questions about how students could progress to calculus in twelfth grade. 
We met in the core area of the module my classroom was in—a hallway/workspace common to all the classrooms in the module. Sally, the District person I had met at the high school earlier, led the meeting. Her talk was similar to what she told us when I last heard her at the high school in the fall: Algebra 1 for eighth graders would be limited to the “truly gifted”. 
“I imagine we’ll have the usual Debbie Downers and Nervous Nancies in the audience on ‘math night’. We want to make two things clear: that there’s no shame in taking Grade 8 math; under Common Core it’s equivalent to the traditional Algebra 1.  And secondly, placement in eighth grade Algebra 1 will be more difficult. Fewer students will qualify—Common Core is very challenging.” Which all sounds plausible but leaves open the question of why an elite corps of students is then allowed to take the traditional Algebra 1 course in eighth  grade.
In fact, in addition to the one placement test that had been used for years, a new test would be administered. This one—never given anywhere before—was written by a group called the Silicon Valley Mathematics Initiative (SVMI). And while the title of the group conjures images of people of high calling in math, science and engineering who reside in Silicon Valley, it is actually a group of math reform types funded through the reform-minded Noyce Foundation, and who believe in 1) themselves, and 2) authentic assessments—not necessarily in that order. 

“For this new assessment, students will be required to use their prior knowledge to solve new types of problems—types they’ve never seen before,” Sally explained. I spoke up at this point and asked “If it’s never been given before, how is it going to be considered in making the decision for placement?”
“That’s something the District is going to have to determine once we see the results.” Though sounding like it answered my question, it didn’t.
Discussion continued about pathways to calculus by twelfth grade. Students who take algebra 1 in ninth grade and who wanted to take calculus in twelfth grade could double up their math courses during tenth grade. There! Every question that a Debby Downer or Nervous Nancy could ask was covered.
With that, we dispersed to our various classrooms and Sally bustled off to another destination. My first period was a “prep,” meaning I had no class. I spent some time getting materials in order and was about to go to the copy room when there was a knock at the back door of the classroom. A small Asian woman asked if I was Mr. Garelick. She introduced herself and said her daughter Susan was in my algebra 1 class.
 “I am subbing for Mrs. Perren” –the math chair—she said. “I sub for her quite often. And I also volunteer at the school.” She went on about how she had done work for my teacher (Mrs. Halloran) and wondered if she could drop in to see what was going on in her daughter's algebra class. While I had not worked long enough as a teacher to form the us/them defenses that would cause me to classify this woman, though I was leaning toward Tiger Mom. I definitely did not want her sitting in on my classes. I told her that I was still getting organized in my classes, so it would be better to wait on that. “I understand,” she said.
Her daughter was a very shy, nervous girl—a seventh grader who had placed into Algebra 1. I recalled returning Susan’s Chapter 5 test which she was to have a parent sign. She scored 63%, and I not yet received the test back. This appeared to be her first bad grade.
The mother hadn’t seen the test.  Obviously Susan was keeping it a secret. I couldn’t be party to that so I had to tell the mother her score.  “Oh this is terrible!” she said. “She did not tell us!”  She looked at me as if examining an insect.
"When was this test given?"
 “Mrs. Halloran gave the test before I started. I didn't teach that chapter,” I said
 “Oh.  This is embarrassing.”
I thought to myself: “I should say so, blaming me off the bat!”, but her embarrassment lay elsewhere. Her husband was a math professor at the nearby university, she told me. “So embarrassing,” she said again.
 “I’ll make sure Susan gives you the test,” I said. She left with a scowl on her face.
I told Susan what happened when she came into class later that day. We were alone; she was generally the first one there.
 “Oh no!” she said. “Was she mad? Did it look like she was mad?" I tried to assure her, and said that with corrections she could bring her score up to a 70% (my teacher didn’t allow corrected tests to exceed that score). 
"You will get this material; I’ll work with you."  She covered her face with her hands. Students came into the room and she took her seat. As she regained her composure, I realized that the traditional Algebra 1 course I was teaching was becoming an artifact to be reserved only for the “truly gifted” as measured by questionable means. Students who didn’t qualify would be relegated to supposedly deeper treatment of algebra “concepts”.  A student like Susan whose difficulties may lie outside of ability would be cited as evidence of why eighth grade Algebra 1 was not for everyone. Parents who questioned this dichotomy would be labeled as “one of those” by personnel whose pronouncements were to remain permanent and unassailable—but always full of good cheer.

Sunday, November 2, 2014

Critically applying Common Core Standards and innovative best-practices to real-life situations

“The Common Core is pedagogically neutral”; “The Standards are guidelines, not a curriculum”; “Teachers can use whatever tools they want to help students meet the standards”. Or so we hear, repeatedly, from Common Core authors and advocates.

And yet, whenever we look up close at an assignment or activity or classroom showcased as having been inspired by the Common Core Standards, we see the same old Constructivist imprints: student-centered; discovery-driven; group-based; real-life-relevant; and “critical-thinking”-fostering. (“Critical thinking”, for the initiated, means applying concepts to new, “real-life” situations, reflecting on your thought processes, and communicating those thought processes to others).

One of the most recent examples of this—many thanks to Barry Garelick for the heads up—is seen in a recent article in a local Utah newspaper called the Spectrum. As the Spectrum reports:

Now in its third year of implementation in Utah public schools, the mathematics Common Core, a set of standards for students and teachers, is completely immersed in the classroom and is continuing to be an effective way to teach students, some teachers and administrators said.
“Some teachers and administrators” include Kris Cunningham, a School District math coordinator [here and throughout, all bold-faces are mine]:
"It's rewarding for the students to get to the correct answer and be able to explain what that answer means.” A staple of the Common Core standards is having teachers utilize the students' critical thinking skills in terms of how they'd apply mathematics principles to scenarios they'd find in real life.
As for the teachers, however free they are to teach the way they want, the Powers that Be spend many hours training them in the right sort of critical thinking: i.e., the sort required to apply Common Core principles to scenarios they’d find in real life. 7th grade teacher Janelle Warby, for example, reports that:
Her school administrators have sent her to trainings along with other teachers to learn how to best embrace new teaching methods.
As a result, Warby has learned that:
"There (are) a lot more critical thinking skills required now. The students do a lot of group work, and I've seen that be a really good and positive thing. The students are learning and retaining more."
Warby admits that:
Group work isn't ideal for all students, especially those who tend to be shy or prefer to work on their own.
But, she reassures us, there's a social benefit: it often helps bring those students out of their shell.

Also free to teach how they want are college professors. Our trainings, for now, are only optional:
Dear Colleagues,  
The Center for Excellence in Teaching will be hosting a workshop on Problem-Based Learning at _____. Lunch will be served.  
Problem-based learning is a student-centered experience which promotes critical thinking and problem-solving skills by asking students to apply theories and discipline-specific knowledge to situations relevant to the student's area of study. This pedagogical best practice is currently being used, with much success, in various colleges across the university.
At the college level, this sort of instruction gets no “Common Core” imprimatur (the Common Core applying only to K12 education). Rather, it gets labeled as “best-practices,” “research-based,” and, above all, “innovative.” And, free as we may be to teach how we want, we face increasing pressure  to adhere to the notions that certain people have of what these three terms mean when applied, oh so critically, to real-life scenarios.