Thursday, October 30, 2014

Math problem of the week: more Common Core-inspired math test questions



Extra Credit:

The steps involved in calculating a regression via calculator are shown here. Discuss the extent to which this process fosters conceptual understanding vs. rote learning.

Wednesday, October 29, 2014

Conversations on the Rifle Range 13: Percents, Simple Interest, and Ninja Warriors

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 13:






During my second week of teaching, in my second class of the day, I was having the class do review problems prior to a chapter test the next day, when Robert, a mildly autistic and very bright boy, started crying. The review problems contained many computations and it was easy to make calculation errors. (I didn’t allow calculators) He was continuing to get wrong answers. I heard him muttering “I can’t do this; I’m getting them all wrong. I’m going to fail, I’m going to fail!”

I came over to him. “What’s wrong, Robert?”

“I keep getting the wrong answers. I’m going to fail this test.” He started hitting his forehead.

“You need to take your time,” I said. I searched his paper and found a problem that he had done correctly. “Look, you got that one right. So just keep doing them. Take your time.” He calmed down and continued with the review. Some minutes later he had completed the problems and was busy at work on a drawing of a Ninja warrior that he frequently drew.

When I inherited Mrs. Halloran’s classes (the teacher for whom I was subbing), she gave me a calendar of assignments for last year’s second semester. She said I should transcribe the previous year’s lessons into a blank planner to use this semester, and told me to take care to not give tests or quizzes on a Monday. “Students tend to forget things over the weekend,” she told me. “And I always include a review of the material the day before a test or quiz.”

My first three classes were all pre-algebra. This allowed me to try an approach in my first class, adjust it in the second one, and supposedly perfect it in the third class. In theory this made sense; in practice not always.

The pre-algebra classes were halfway through the chapter on percentages and how to apply them when I came on board. The teacher had taught them to solve percentage problems using the method of proportion. Thus, a problem like “35 is 40% of what number?” was to be solved by restating it as a proportion using the “part/whole” technique. Since 35 is part of a whole, 35 is the numerator. The “whole” is the denominator—“x” in this case. The left hand side is then 35/x. The right hand side is the percentage 40/100. You’re left with 35/x = 40/100. Students solve for x via cross multiplying and dividing.

During my short orientation with Mrs. Halloran, she mentioned this method. I told her that when I was student teaching, teachers had the same book (Holt, “Prealgebra”—as bad as if not worse than Holt’s “Algebra 1”) but had taught students to translate such problems into algebraic equations. The above problem would take the form of 35 = 0.4x, where the “is” translates to the equal sign, 40% to 0.4, and “of what” to “0.4 multiplied by x” or 0.4x. Mrs. Halloran very firmly told me: “We don’t do that; they’ve learned the proportion method. Use that method only; they’ll get confused.”

There are advantages to both methods, of course. But since they’ve already had the “part/whole” technique in sixth grade, I would have preferred having them use the equation form. But since Mrs. Halloran was quite firm about sticking with the proportional method, I honored her request.

For other types of problems for which I was not given an edict, I elaborated. When calculating discounts, students multiply the price by the discount rate, and subtract the result from the price. For example, to find the price of a $200 bicycle discounted 20%, they multiply $200 by 0.2 to get $40, and then subtract to get $160.

I gave the class an alternative—a shortcut—to use if they felt like it. I explained that a discount is like the store offering some money towards your purchase. “Suppose your parents tell you that they’ll kick in 20% towards the purchase of a new bike. What percentage will you pay?”

A few hands went up. Someone volunteered: “80 percent”. Then a little back and forth on how they came up with that—something you’d think would be easy, and in fact perhaps it’s SO easy that they have trouble articulating it, but eventually someone would say “100 minus 20 equals 80” and it’s on from there. Interestingly, only a few students used the short-cut. The rest preferred the long way. Other detours from the main course proved equally interesting. When talking about interest (simple interest—Mrs. Halloran had indicated to skip compound interest, which, in retrospect I wish I had taught), the problems were stated in terms of savings. I brought up the fact that interest can also be something that you pay.

“When you borrow money from a bank, to buy a car, or a house, the bank charges you interest.”

We worked through a problem. Then a girl asked: “Why would anyone want to do that? You end up paying more for what you’re buying than if you just bought it without borrowing.”

Ordinarily such questions are used as an occasion to warn of the dangers of credit, getting in over your head, and so forth, which I had done for my first pre-algebra class. But I decided to try a different approach with the second group to see how it worked out.

“Some items like cars and houses cost more money than people may have,” I said. “Like a house, for example. So a loan is one way you can afford to pay for it.” “But you’re paying more for it,” the girl insisted. “Why would you do that?” A rather astute boy named Brian who tended to offer expert opinions at the drop of a hat said “The going price for a house in this area is about $700,000, and not too many people have that kind of money.” I figured someone in his family was in real estate.

“Can’t you just save up?” someone else asked.

“How long would it take you to save up that kind of money?” I asked. “It depends on how much you make,” Robert said without looking up from a drawing of a Ninja warrior that he was working on. “If you invent something that everyone wants, you can make millions of dollars.”

I had similar ideas when I was that age. I let Robert’s statement go unchallenged--and stand as another alternative to consider, like the shortcut for calculating discounted prices.

Monday, October 27, 2014

The homeschooled child and social opportunities in the digital age

Kids these days: their eyes always look downwards at their phones; their words, flowing mostly through their thumbs, reserved for people who aren't there. To parents, this often seems socially sinister. But for us parents of unsocial left-brainers, there's potentially a silver lining. Might reduced practice with real-life social interaction among typical kids level the social playing field a bit? Now it's no longer just our kids: no one's kid is comfortable making eye contact or engaging in spontaneous conversations face-to-face with new people.

On the other hand, for home schooled children like my daughter, things may, in some ways, be more socially challenging than ever.

What's really at issue here, after all, isn't that kids have actually forgotten how to make eye contact and socialize. Rather, it's that they are increasingly selective about who they interact with, both in person and at a distance.

As far as in-person interactions go, kids are increasingly sticking with familiar, routine situations: environments where they encounter the same people over and over again; environments in which social engagement pays off long term because here are peers you'll be dealing with day after day for years. The quintessential example of this, of course, is school, with its daily opportunities--the yard, the cafeteria, the clubs--for extended social interaction. Plus, school is one of the few environments where cell phones often aren't allowed--giving would-be socializers no choice but to socialize with those who are physically present.

Outside of school, incentives and opportunities shift drastically. And presented with a choice between interacting at a distance with a familiar friend, and engaging spontaneously with a new person, kids (or so I've seen) generally choose the latter--especially when any flesh-and-blood peer they might potentially engage with is probably looking down at their own phone.

So here's the problem for unsocial, homeschooled kids. No matter how many extracurricular, group-based activities we set them up with--art classes, musical ensembles, theater class--if they aren't already familiar to their surrounding peers, neither they, nor their peers, are likely to engage much in the dreaded spontaneous, face-to-face, eye-contact linked interactions that are the first steps in making new friends.

Saturday, October 25, 2014

High-stakes testing in Finland

Last week I blogged about a CNN opinion piece by Pasi Sahlberg, former director general in the Finnish Ministry of Education and Culture (and now a visiting professor at Harvard’s Graduate School of Education.)

In that piece, Sahlberg claims that the three things that make the Finnish school system superior to ours are its focus on educational equity, its education spending, and the time it allots for teacher collaboration. Saying nothing about the vast differences in teacher quality and classroom curricula, Sahlberg instead faults American schools for spending too much standardized testing. He notes that Finnish students, in the course of their pre-college years, face only one standardized test. But he doesn’t discuss either the contents of this test, or just how high stakes it is compared to American tests. For that, you have to go over to an article that he only links to here: one he wrote for the Washington Post’s Answer Sheet blog back in March. Here's what he writes there:

The only external standardized test in Finland is the national Matriculation Examination, high-stakes exam that determines college readiness and which all students are required to pass in order to graduate high school exit and enter university. At the time of writing this over 30,000 Finnish high school students are taking this all-important examination that enjoys high esteem as a sign of being a mature, educated person in Finnish society.
A single test whose passage is mandatory for all students for high school graduation and college enrollment; an “all-important examination that enjoys high esteem as a sign of being a mature, educated person in Finnish society”: no current test in America is anywhere near this high stakes—at least for who matter the most, i.e., the students.

Not only is it high stakes; it’s also academically rigorous. It requires, not just critical thinking, but actual content knowledge in, for example, history and math. Here’s the sample history question that Sahlberg cites:
Karl Marx and Friedrich Engels predicted that a socialist revolution would first happen in countries like Great Britain. What made Marx and Engels claim that and why did a socialist revolution happen in Russia?
And here, courtesy Sahlberg and Google Translate, are some sample math questions from the mathematics part of test. There are 15 problems in all, of which students must do 10. I’ve chosen the ones that had the clearest translations and were easiest to format.

1 c. Simplify the expression (a2-b2)/(a - b) + (a2-b2)/(a + b) with a not equal to b or –b.

5. A circle is tangent to the line 3x-4y = 0 at the point (8, 6), and it touches the positive x-axis. Define the circle center and radius.

6. Let a1…an be real numbers. What value of the variable x make the sum (x + a1)2 + (x + a2)2 + ….+ (x + an)n as small as possible?

9. The plane 9 + x + 2y + 3z = 6 intersects the positive coordinate axis at the points A, B and C.
a) Determine the volume of a tetrahedron whose vertices are at the origin O and the points A, B and C.
b) Determine the area of the triangle ABC.

13. Let us consider positive integers n and k for which n + (n +1) + (n + 2) + + (n + k) = 1007
a) Prove that these numbers n and k satisfy the equation (k +1)(2n + k) = 2014.


No wonder Finish students who come to America end up having to repeat the school year upon their return.

Thursday, October 23, 2014

Math problem of the week: Common Core-inspired 3rd grade test questions

From the New York State Testing Program Mathematics Common Core Sample Questions Grade 3:


Extra Credit:

1. Is a piece of string mathematically comparable to a number line?

2. Should alternative answers for A and C be, respectively, 3/4 and 1/4?

Tuesday, October 21, 2014

Conversations on the Rifle Range 12: Teaching to the Authentic Assessment

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 12:



Back in September, when I was doing my sub-assignment for the high school, I attended a math department meeting the day before school began. Sally from the District office presided, and among the many things she told us at that session was that this year the students in the District would not have to take what is known as the STAR test, by order of the superintendent of the District. “And as you know, the Superintendent is like the Pope. What he says goes.”

While this last was uttered partly in jest, the reaction in the room was celebratory. The STAR exam has been an annual ritual in California and in May of each year about two weeks are devoted to a review and prep for this test, which is keyed to California’s pre-Common Core math and English standards. Such activities inspire accusations that schools are “teaching to the test”. But now in the midst of a transition to implementing Common Core math standards, California was looking at the Smarter Balanced Assessment Consortium (SBAC) exam that would be given officially starting the following school year. (Actually, they don’t call it an exam; they call it an “assessment”. You’ll forgive me if I call it an exam.) For now, however, the state would be field testing the exam. What this meant was anyone’s guess: perhaps this first go-round on SBAC would be to provide a baseline to see how students scored prior to full implementation of Common Core. Or perhaps it was to fine tune the questions. Or both. Or neither.

In any event, when I started my new assignment at the middle school, I had to have my classes take a practice SBAC exam. The day before I was to take all my classes into the computer lab for the practice exam, I attended an after-school faculty meeting.

I had started my assignment at the school earlier that week, so the principal introduced me to the group. I was welcomed by applause, and urgings by fellow teachers to help myself to the tangerines that were brought in for the occasion. I took two tangerines, and as if he were taking that as his cue, the principal started the discussion.

“As you know, we are in transition to the Common Core, and one aspect of this is the SBAC test that will be given this spring. We want students to have a chance to practice with some sample questions. This does two things. It will get them used to the computer interface, because the test is given entirely on computers. And secondly it will get them used to the questions which are not the typical multiple choice question like on the STAR tests. The SBAC is more of an ‘authentic’ test.”

He went on about how Common Core will change the way we teach, but the words all blurred together in my mind amidst phrases like “critical thinking”, “higher order thinking”, and “deeper understanding”. I do recall a conversation between two teachers at my table. One mentioned she saw some of the questions and said "Yes, there are still multiple choice questions on the test. I was very disappointed to see that."

Well, OK, I like open response questions too, but I get rather tired of the “it’s inauthentic if it’s multiple choice” mentality. I took the CSET math exam required in California to be certified to teach math in secondary schools. The multiple choice questions were not exactly easy; I would hesitate to call the exam “inauthentic”. What I find inauthentic is judging seventh and eighth graders’ math ability based on how well they are able to apply prior knowledge to new problems that are substantially different than what they have seen before or have worked with.

On the day of the practice exam the assistant principal took charge of my first group—the first of my three pre-algebra classes and probably the most cooperative of all of my students. He spoke in a loud, commanding voice and gave instructions on how to log on, what page to go to, what things to click on, and had everyone do things at the same time. I only know that I could not duplicate this feat for any of my classes; students would rush ahead, ask me to explain again what I had just said, and inevitably asked “Will the test affect our grades?” I explained that it was for practice and did not affect their grades, nor would the actual test given later in the semester, but the question kept coming up. When it came time to take my fifth period algebra students to the computer lab, I had written on the white board: “No, this test will not affect your grades.”

A boy named Peter exuberantly agreed. “Yes, Mr. G, that’s a great idea, because…” I couldn’t hear the rest amidst the noise of the class, which then followed me outside the classroom to the computer lab. The instructions had to be repeated several times, as I had done throughout the day.

Because this was a practice test, I felt no compunction about giving students help in answering the various questions. For the most part, questions were reasonable, though the students found some difficult. One question I recall on the seventh grade test was “Enter the value of p so that 5/6 - 1/3n is equivalent to p(5-2n). Seventh graders have only learned about how to distribute multiplication, but not to factor. While the answer of 1/6 seems to jump out at adults, this problem presented difficulty to most of my seventh graders, probably because they hadn’t seen anything like it. The variable “n” also was a distraction. I gave them hints like “Can p be a fraction? What fraction would you multiply 5 by to get 5/6?”

On the eighth grade test, one open-response item was quite complex, involving pulse rates versus weights of various animals, which students had to analyze in terms of slope and a “trend line”. One of the questions was “Interpret the slope of the line from Item 1 in the context of the situation.”

At the end of sixth period, I dismissed the students, and went back to my classroom. I realized that when Common Core kicked in students would be “taught to the test” for these all of these particular types of questions. I have no problem with teaching to a test if the test covers material that should be mastered. I do have a problem when part of this is learning how to write explanations that will pass muster according to scoring rubrics.

As I got ready to leave for the day, one of my sixth period students popped her head in the door. “Mr. G, will the test today affect our grade in the class?” I said it wouldn’t, but not for the last time that week.

Sunday, October 19, 2014

Teaching what can be discovered instead of what can't be

Barry Garelick's recent piece Education News, Undoing the ‘Rote Understanding’ Approach to Common Core Math Standards, got me thinking about one of the things that Reform Math has backwards.

Barry talks about the emphasis by today's math educators on ad hoc methods like "making tens" at the expense of traditional algorithms like borrowing and carrying. And so we see more and more worksheets like this one:



And fewer and fewer like this one:


And, in promotional, Common Core-inspired videos like this one, we see just how painfully slow the "making tens" method can be--as well as how it, by itself, does not give students a general method for solving more complex addition problems.

As I wrote in a comment on Barry's piece, I don't remember ever learning officially how to make tens. I remember it instead as something I discovered on my own--in the course of computing all those long columns of sums that students used to be assigned (sometimes upwards of six addends!) and eagerly looking for shortcuts.

The standard algorithms, on the other hand, I most certainly did *not* discover on my own, and am quite glad to have had teachers that were willing and able to teach it to me.

It's ironic how "discovery-based" Reform Math spends more time showing students how to do stuff they might discover on their own than it spends showing them how to do stuff they almost certainly won't learn on their learn own.