Monthly Archives: October 2006

Review: “If I Only Had One Question: Partner Quizzes in Middle School Mathematics”

The authors of “Partner Quizzes” have followed the advice of the Assessment Principle in the NCTM Principles and Standards by developing and administering assessments which are “not limited to individual, graded, end-of-unit examinations” (Danielson and Luke, 2006). In the middle of a unit, students work in teams of two and produce one quiz for each individual over two days, obeying strict rules of collaboration. According to the authors, “Partner quizzes are useful as formative assessments that help us to monitor and adjust our instruction during the remainder of the unit.”

The authors state explicitly two crucial advantages of these assessments: they demonstrate that working together is important, even (especially) during assessment; and they provide an environment in which “discussion is essential to doing good work.” They claim that because of the structure of the activity, the questions are “deeper and more complex than on an individual assessment.” These characteristics would make the assessments attractive to any teacher.

The process has three strict rules:

  1. Partnerships are completely private and exclusive. Teams are only to work with other team members. The teams can choose their own work style, but there are strict consequences for collaboration outside the team.
  2. Each team is allowed to ask the teacher exactly one question during the quiz.
  3. Teams work on their quizzes on Day 1, and then submit both copies for teacher input. The teacher selects one quiz from each team and provides cryptic feedback, meant to provoke further discussion during Day 2 without guiding the students too directly.

The authors find the “one question” rule valuable because it forces the teams to ration their requests, eliminating the superficial questions such as “What is an outlier?” in favor of the deeper questions such as “Can there be two outliers?”

The article includes two sets of actual quizzes, revealing within one class a range of understanding and sophistication that seems typical of any classroom. The authors analyze the student work as well as the feedback and revision process, making a relatively complex system almost easy to understand.

I’m eager to try this out in my classroom for all the reasons cited above. Certainly this isn’t the most important reason, but I’d like to see my students step up and not ask the same procedural question five times in two minutes. On a slightly deeper level, I’ve been looking for a way to pull the introverts and outsiders into the group so they can participate – and learn.


1. Danielson, Christopher, and Michele Luke. (2006, November). If I Only Had One Question: Partner Quizzes in Middle School Mathematics. Mathematics Teaching in the Middle School, 12(4), 206-213.


Review: “How to Buy a Car 101”

“The style of PBL units is designed for teachers who are willing and able to hand over control of the classroom to the students.” This chilling statement is found in the core of the article “How to Buy a Car 101,” an overview of Problem-Based Learning.

The article describes a project for 7th graders where they are given a few specific criteria for a fictional car purchase and then must research, assemble, present and defend their choices. The final product covers four state math standards, and is graded according to a simple rubric. According to the author, the teachers “no longer act as the experts; they serve as a resource.”

The authors’ students research the requirements and available options (car and financing) on the web, and then present their findings using Powerpoint. This is not feasible in all classes, partly due to the classroom setup (few computers, little experience with Powerpoint) and partly due to limited experience by the students. (It’s been my experience that at the middle-school level, kids are still as a whole relatively unsophisticated regarding technology. In my own class recently, I had to explain what Excel was.)

The project is fascinating, and my impression of it is that if you can pull it off, it’s wonderful for stimulating discussion and high-order thinking. Some drawbacks, as I see them:

  • For 7th graders, the project is relatively complicated, including the commuting distance of the fictional car purchaser, is budget, his target down payment, and the current interest rate.
  • How much time do you want to spend teaching and coaching Powerpoint? And what about showing students how to use the web?
  • In some areas, students have no Internet access at home. Of course, this means they will have to use the library, but does this give an unfair advantage to the kids with broadband Internet in their own bedrooms?
  • Powerpoint is not available on all computers, and actually costs money to purchase. If a student has a computer at home, will he be required to buy a Powerpoint license?

I was happy to see the author list a set of warnings of her own. She says you can never be overprepared; you should get the help of the students for planning future units; and be sure not to do the students’ work for them.

One very interesting note: during the project, the students developed a relationship with local car dealers, who bring new cars to the school for the students to see. They also “kept in contact with the students about any incentives and promotions they might find interesting.” This sounds a lot like indoctrination and commercial promotion to children, and I would want to be careful with this.


1. Flores, C. (2006, October). Using Friday Puzzlers to Discover Arithmetic Sequences. Mathematics Teaching in the Middle School, 12(3), 161-164.

Journal entry: “Breaking the fulcrum

Teacher Journal

10/4/06 My P5/P6 classes today were horrendous to the sub, as I took off some time for some training with a consultant. Besides the sub, there were three other adults in the room that day, and they impressed on me the need to fix things in this class. However, for a brief period, under the direction of the consultant, I saw these difficult kids actually working on math together. Central to her technique is to break the lesson into small pieces and to display the duration of the pieces in an agenda on the board, checking off each piece as it is completed.

10/5/06 Read the riot act today to all classes, including display of “Inappropriate Behavior” page from Binder Reminder. My best-behaved classes shaped up, but my after-lunch period eventually lost it. This was partly my fault since I gave them a small-group activity during their most challenging time. I wanted to have some kind of non-punitive activity for the kids, who are also with me just before lunch. I took the advice of a part-time administrator on campus, and put the repeat offenders in a Chair of Honor, rather than make their days and evict them from the classroom. The chair was pointed directly toward my whiteboard, leaving the victim’s back pointed toward the audience. (I made sure the student was not turned in his or her seat.) This quieted the student in question, but it did not feel like a particularly humane way of dealing with the situation.

10/6/06 Taught a half-baked intro algebra lesson using a balance, plus algebra tiles. The point was to model equations. It was the last period of the week in my most difficult class. Kids were happy to volunteer, and to talk when I told them to discuss, but they didn’t actually do much thinking, and the wild kids kept the whole class from getting anything done. My mistakes: omitting an agenda and not writing clear instructions. The principal and assistant principal popped in in the first ten minutes, saw the class at their best (making me a liar), and split after about 20 seconds.

10/8/06 Returned tests from last week and gave students the option of re-doing five problems to replace wrong answers with correct answers. This turned out to be a major mistake. Even though I gave them exactly five minutes to complete the work, it stretched out to ten minutes most periods as students started asking questions, talking to each other even though I said to work alone, pointing out that they haven’t take the test at all, etc. What a mess. The option for bumping your score was suggested to me by another teacher, but I’m sure I missed a detail. You should only do this outside of class time. Students who are serious about earning more points will be willing to come in before class one day.

10/9/06 Tried to use the balance model in a lesson intended to precede the unit on solving linear equations. The balance fell over (breaking the fulcrum), and the volunteers who were supposed to add things to the balance were uninterested and distracted and only wanted to volunteer so they could stand up. The “x” was a simple folded piece of paper, which I described as an “envelope”; however, the students never saw it as an envelope (which meant the metaphor was defective and not completely useful). Some students understood the concept of “x” but some couldn’t be bothered. I have trouble reaching them other than with a plain lecture about simple procedures (solving one-step equations using addition or subtraction) or with sermons about talking during lectures.

10/10/06 Taught a lesson on solving linear equations using subtraction, this time with see-saws rather than balances. (I resist the see-saw model, vs. balance, since there’s an additional variable of distance from the fulcrum, but I guess the average 13-year-old misses that subtlety.) The students moved into pairs and drew the see-saws on “white boards” (made out of page protectors containing card stock), and displayed their work – including equations and drawings – on their boards. They were totally engaged, for a few minutes. Eventually they lost interest, started chattering and wouldn’t stop, and so I had them put their desks back in rows, and we picked up again with a lecture (and a sermon).

10/16/06 Homework rate for P1/2 was a dismal 50%. Reminds me of high-school. I’m partially responsible for the low turnout because I didn’t practice enough examples during Friday’s lesson and because I assigned a few problems requiring skills slightly beyond where we are. I say “partially” because the students still are required to do their assigned work. In any case, I will now need to come up with some kind of “tough love” policy on homework. I’ve heard of middle-school teachers who have been successful with notes home or not allowing the students take tests unless they turn in the homework. This seems extreme, but it makes sense, since I claim and believe that few of these students (Foundations of Algebra, with a second “Support” period each day) can have any success on tests without regular practice.

10/16/06 Despite ongoing frustration with behavior in P5/6 – I sent two outside, and assigned notes to be signed to four others – a few of the seeming knuckleheads who will never learn anything except how to get Juvenile Hall were actually trying. One of them, a guy who shouts at me if I correct his behavior the wrong way, actually said, “Could you put up a few more so we could practice them?” (He was asking for single-step equations with division.) I couldn’t get him to see the simplicity and universal truth of the “undo the operation to isolate the variable” approach, but he was happy to memorize the steps I was teaching. Unfortunately, we needed to get on to the day’s lesson (multiplying by the reciprocal if the coefficient is a fraction), and he got a little short-changed. I’m going to try to catch up with him tomorrow.

10/16/06 In one period today, at least ¼ of the class didn’t have their homework because they forgot it at home, in their textbooks. “What can you do next time, class, to remember your homework? That’s right: when you finish it, get up and walk all the way across the room and put it in your backpack!” One guy never has a backpack and so is constantly in fear of having his things “jacked” from his binder. I’ve asked him why he has no backpack; his answers seem to be circling around some sort of family-deficiency thing that I’d rather not force him to reveal.

I’m now displaying just the even-numbered answers, as the students grade their own homework, because the odd-numbered answers are in the back of their books. I’ve explained how to do the work – do each problem, check the answer in the book, make corrections – but they are either too lazy to look or have some lingering antipathy toward “cheating” and so won’t check.

Review: “Using Friday Puzzlers to Discover Arithmetic Sequences”

The author teaches eighth-grade mathematics and writes about using related puzzles to lead students to discover the formula for the sum of an arithmetic sequence. The puzzles are presented each Friday, and students look forward to the experience each week.

The guidelines are presented before the first puzzler:

  1. Each puzzler can be completed within a class period.
  2. Your current mathematics knowledge is all you need to solve the puzzle.
  3. You will never be given a puzzle that is unsolvable.
  4. Often by first thinking about it, a technique will emerge that allow you to complete the puzzle in less time than if you plod ahead without thought.

Yolles uses the following five core Puzzlers:

  1. Gauss: A Child Prodigy. This is the story of Gauss’s 3rd-grade teacher assigning the class “find the sum of the numbers from 1 to 100” and having Gauss almost instantly produce the answer.
  2. Handshakes All Around. Ten friends attend a party where each person shakes everyone else’s hand, exactly once. How many handshakes occur?
  3. Lighting the Chanukah Menorah. Over the course of eight days, how many candles are lit in all on a menorah?
  4. The Twelve Days of Christmas. How many gifts were sent on Day 12?
  5. Clock Face Puzzle. Can you locate one straight line to split a clock face in two so that the sums of the numbers on the two parts are equal?

Students work on each problem for an entire class period, earning stickers if they solve the problem. The author says that few succeed at first, but the proportion of students solving each problem grows through the year. Puzzlers are not provided each Friday; the students take a few weeks between sessions.

The author provides samples of student work; her students show surprising creativity in their approaches to these problems of counting. She also describes Socratic conversations which attempt to bring together the results of multiple weeks’ work.

By the end of the Friday Puzzler sequence, she says, “all students have earned an ‘I got the Friday Puzzler!’ sticker,” and the class has discovered the formula S = (n/2)(a + l), where n is the number of terms in the sequence, a is the first term, and l is the last term.

I found the article fascinating and inspiring, and I look forward to using it in my own classroom.


1. Yolles, A. (2003, November). Using Friday Puzzlers to Discover Arithmetic Sequences. Mathematics Teaching in the Middle School, 9(3), 180-185.