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!”