Fluency/Understanding


Math Teaching Tip #3: An Example of Informal Assessment

A few years ago I tried out a new kind of informal assessment. I had noticed that some of the high school algebra students were confused by expressions like “5a + 3(a + 7).” I wanted to find out what they really understood about the distributive property and the gathering of like terms. So I took a pencil and a clipboard with two pieces of paper on it, and wrote “8a” on the top piece of paper. Next, I lifted up the top paper and drew a vertical line down the middle of the second page—and then let the top page fall back into place. As the students meandered in before class, I went up to one of them and said quietly, “I’m taking a little survey; mind if I ask you a question?” Then I pointed to the “8a” on my clipboard and asked, “What does this mean to you?” The kid said, “I don’t know.” “OK, thank you,” I said politely, and turned away from the student. I jotted down the kid’s name to the left side of the vertical line on the second page; that was where I wrote the names of students who I would gather later in a small group for remedial work. I turned to another pupil, who said, “It means eight times some number.” I thanked him and wrote his name on the right side of my recording page. The next student said, “It means you’ve got eight a’s.” “And what does that mean—that you’ve got eight a’s?” I asked. “I don’t really know,” she said. I wrote her name on the left side. The passing period between classes is short, but it was enough time to identify a small group of students that needed remediation.

Looking back at this experiment, I think there were certain aspects that made it successful. First, my survey focused on a key underlying concept that was essential for success in the students’ current math work. Second, I selected students that I suspected might not know the answers to my survey question (students who usually scored low on homework and tests, or who rarely spoke up in class). Third, I approached the students in a relaxed, informal manner before class, when the ambient noise and movement would provide the mini-interviews with a pressure-removing cover of privacy. Fourth, I wanted them to feel and know that I was sincerely interested in knowing what they thought; and I wanted them to feel at ease, so they would be inclined to honestly tell me what was on their minds. So I purposely used a conversational tone of voice, rather than an authoritative teacher voice—which could have communicated a feeling of “I’m going to ask you something that you really ought to know, so pay attention, concentrate, and answer correctly!” Fifth, to keep the students feeling at ease and to protect the authenticity of future similar encounters, I didn’t tell them if they were right or wrong; I purposely suspended judgment in favor of just getting honest input.

This worked so well with the first class, I decided to try it again with several students from the next class. As before, there were a few students who did not understand the meaning of the “8a” notation. Then it occurred to me to probe a little farther, so I also wrote “5a + 3a” on my clipboard, and asked the students what that meant. Every single student told me that would equal 8a. And when asked again what 8a meant, they still didn’t know. Now that was interesting—they were fluent with the process of gathering like terms, but did not understand what the terms meant!

I tried this same informal assessment technique another time when I wanted to find out who needed help with squares and square roots. When undertaking the Pythagorean Theorem or quadratic equations, it’s essential for students to know what a2 or x2 stands for—and that they know squares at least up to 152. So this time I wrote 52 on the top page of my clipboard, and asked selected students what it meant to them. Some said “twenty-five,” others said “ten,” others waffled between those two answers, one thought it might be “seven,” some were certain, some were not sure, and some had no idea. To probe further, I also wrote 112 and 142 and √100 and √169 and √225 on the clipboard. I found that some students who knew 52 did not know the larger squares; some knew about the squared numbers, but knew nothing about square roots. I also noticed that some students overheard what other surveyed students near them had said, so I added 72 and 82 to my list of questions to prevent them from merely mimicking their classmates. Once again, this survey process produced an accurate selection of students for small-group remediation later in the class period.

When helping out another teacher with his class, I asked if I could do a similar informal survey with his students. I had recently helped one of his students with some other math work, and discovered that she knew nothing at all about squares and roots. I was hoping to help her and some other students with that on this day, and wanted to gently and accurately identify students who would benefit from a small group remedial lesson. The teacher didn’t like the survey idea, and instead wrote a dozen simple expressions containing squares and roots on the board, explained that he wanted to find out who needed help on this sort of thing, passed out paper, and asked students to write their answers. This approach seemed to be simple and direct, but it had less than the desired effect. Students were immediately troubled: “Why were they being tested on this?! No one said anything about there being a test today! Everyone knew that stuff! Who on earth would need help with that?! Did he think they were stupid?! What a waste of time! Etc.” Finally the class quieted down and wrote their answers. Then they exchanged and corrected papers. Guess who got all the answers right but one? Yes, the girl who didn’t know any of it. She was very proud, very protective of her ignorance—and an absolutely consummate cheater. No way was anyone going to find out that she didn’t know that stuff; she made sure that one way or another she got those answers. I wished that we could have just taken the informal survey approach; even though it was less comprehensive than a written quiz, it was more gentle, more accurate, and less time-consuming.

Judging from the number of low-achieving math students in almost every school, it’s clear that even an excellent teacher can give a splendid lecture to an interested, involved class—and the students’ subsequent class work, homework, and tests can nevertheless be riddled with conceptual and factual mistakes. “I taught it, but they didn’t learn it” is an oft-repeated teacher lament.

Trying to teach students lessons that they are not prepared to learn is an exercise in futility. To be successful, lessons must address the minds of the students exactly where they are—not where they are supposed to be. For that to happen, teachers must be aware of what their students really know. In an effort to put their finger on the pulse of their pupils’ minds, teachers usually utter two obligatory words at the end of every lecture: “Any questions?” But all too often, students do not ask questions when given the opportunity to do so. So the teachers are left to wonder: Who doesn’t know what?

If they knew the answer to that question, they would have a realistic chance to do something about it. But even better than the typical end-of-the lesson query—the before-the-lesson mini-survey is an informal assessment tool that teachers can use to get a more accurate picture of the current state of their students’ mathematical thinking. This awareness can help them to adapt the content, delivery style, and pace of the lesson in ways that fit more comfortably to the developing minds of their students—right where they are, not where they should be. When the teachers know the mathematical contents of their students’ minds half as well as they know the content of the math lesson they’re about to teach, then it will be much more possible for them to proudly proclaim, “I taught it, and they learned it!”

Here is our podcast on the progress chart as a way to both measure and purposefully improve fluency and understanding. See our post, A Model for Using Guided Discovery in a Math Lesson, for when to use it in an effective lesson.

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more about “The Progress Chart“, posted with vodpod

As we wrote in previous posts, fluency is important. Basic math facts do not go away but are used throughout the upper grades in more abstract and involved contexts. If the students’ minds are burdened with having to reestablish basic facts and concepts, they’ll be like music students not fluent with their basic scales, burdened and frustrated later when they come across those scales in a piece of music.

The progress chart, found in the middle of every partner page, builds on the good practice started with the graphic and partner pages. It uses those pages to keep practice grounded in experience, so there are no tricks or drills separated from a physical context that is understood. It takes what is good about timing exercises and leaves behind the problems. It both measures and improves performance. It develops further the understanding, speed, and accuracy needed for work in the upper grades.

The overall time and increments in the progress chart differ between lessons based on our experience in the classroom and tutoring. You may find it helpful to read the earlier posts in this series, A Model for Using Guided Discovery in a Math Lesson, if you haven’t already, as these steps are sequential.

Here’s a model for working with the progress chart.

1. After students are oriented and comfortable with a graphic and accompanying partner page, ask for a volunteer to do a timing experiment with you. While they say the answers on the partner page, time them to see how many seconds it takes them to get a perfect score (with the help of the graphic page if needed). You can say that it’s their first time doing this so it doesn’t matter if they’re fast or slow; this helps relieve a sense of anxiety. To avoid the tension sometimes associated with racing, do not begin timing with “ready, set, go!”; simply begin the timing when they say the first answer. Their partner should continue checking to make sure the answers are correct as explained in the post on the partner pages. When they finish, tell them their time and have them circle all the times on the progress chart that they beat.

There is no target time. However long it takes to get a perfect score is the objective at this point. Because of their previous work with the graphic and partner pages, many students will move comfortably into this timed approach. The problems are better understood and they’re becoming more familiar to them. To keep this from being a rote exercise, change the direction in which they solve the problems.

2. Now ask the class to look at their partners and decide who is partner A and who is partner B. When you say “begin,” all the A’s will say the answers and the B’s will check the answers and keep track of the time with you. When they’re done, they can look up and you’ll give them their time.

3. The times that are beat are circled and the partners switch roles.

4. The partners continue taking turns, comfortably and joyfully improving their times by increasing their familiarity with the material. After the initial work on the graphic page, reinforced and expanded on with the partner page, the progress chart creates even greater competence and confidence with the concepts, procedures, and facts.

This is very different from enforced time tests, where students do as many problems as they can in a predetermined time—which often means that they fail to answer some problems, and answer others incorrectly. Our emphasis is on correctness, thoroughness, and understanding every bit as much as time. The students correct themselves as they go; all of the problems are answered correctly. Therefore, no wrong answers are written and reinforced, and no answers need to be given to them. They don’t miss needed practice on certain problems because of not finishing the exercise in a predetermined time. The voluntary character of working with the progress chart creates a happy feeling. And our experience continues to be that students’ natural inclination is to keep going to improve their time, and in so doing, their understanding and fluency.

Following on the previous two posts, here is the Partner or Folding Pages podcast. With the partner pages, students rediscover, reinforce, and extend the work they did on the graphic pages. They connect manipulation with computation and continue building memory by taking more spins around the city.

Four additional features of the partner pages that support students who are not doing well in math are:
1. They involve cooperative learning with individual accountability
2. Students receive immediate and ongoing feedback
3. They engage teacher, peers, and students in student self-correction
4. They use no-stress timing focused on improving rather than merely measuring performance (See the earlier postings on fluency for a discussion of this.)

Again, because of the lesson design, we can observe and assess how students are doing each step of the way – and make the adjustments our observations indicate.

Vodpod videos no longer available.

more about “Partner or Folding Pages“, posted with vodpod

Standard timing practice has more to do with measurement than improvement. Measurement is useful, so is improvement. They can be accomplished together.

The standard approach to timing along with the fact that it is typically done without any graphic or manipulative support to refresh knowledge of the facts creates more problems then it solves.

Students write some incorrect answers, which can reinforce memories of wrong answers. Students leave answers blank, which does little for their interest in mathematics, reinforces a negative self-concept, and causes them to miss needed practice on certain problems. Getting the paper back sometime later when the students are no longer engaged in the exercise means they probably will not go over missed problems to see where they went wrong. When it is corrected by someone else and not self-corrected, they are another step removed from the whole process.

Again, measurement and improvement can be accomplished together. Consider the posting “Measuring AND Improving Fluency.”

What exactly is fluency? With languages, fluency is the ability to clearly express thoughts without having to grope for words. With mathematics, fluency refers to a student simply knowing a fact or procedure, without having to stop and think about it. Many first-graders can instantly tell you that 2 + 2 = 4. But too many older students hesitate with say, 9 + 5.

If the students’ minds are burdened with having to go back and reestablish 9 + 5, when they should be free to think about more complicated concepts (x + 9)(x + 5), their progress will be hampered. Their feelings about math in general will take on an aura of stress and clutter, and the feelings of curiosity, the joy of discovery and confidence will fade away. Tasks such as homework and tests will take longer than they should.

The first graders and older students may understand the concept. While understanding provides a basis for fluency, it does not automatically produce it. What enables learners to acquire fluency? Practice. But practice of the right sort is required. See our post, A Model for Using Guided Discovery in a Math Lesson.

There are two levels of accomplishment with understanding and fluency. The first level: students slowly solve problems with understanding and accuracy. The second level: students quickly solve problems with understanding and accuracy.

Many think that the job is done when students have reached the first level. After all, they got the problems right, and understood what they were doing—does it really make a difference if they were fast or slow? It does. Students who instantly know that 9 + 5 = 14 have a clear advantage in more complicated contexts: 49 + 65 = ?   91 + 52 = ?   98 + 57 = ?  976+458 = ?  9x + 5x = 280   (x+9)(x+5)= ?

As students grow older, 9 + 5 and all the other basic facts do not go away. They are used repeatedly throughout the upper grades.