Guided Discovery


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!”

Since this post is part of a series, it may be helpful to read over A Model for Ongoing (Formative) Assessment – To Sit Beside and Observe #3 and Beginning a Lesson Using Guided Discovery to put it in context. You may also want to look over the podcast posted earlier on the Partner or Folding Pages to see a model of the content of this post.

Earlier this month, we posted an outline approach to integrating assessment into a math lesson. Teacher and student ongoing assessment run throughout that initial step. Equally important, doing something about what is observed, with every student, is also integrated into that step.

This post takes up the second step in the guided approach – working with a folding or partner page of problems. This engages students in quickly reconstructing and re-experiencing the work they did on the graphic manipulative. Both the graphic and partner pages are deceptively simple looking, while being highly flexible and engaging tools for building understanding and fluency together – in a way that can be directly observed.

On the left is an image of a negative number partner page with the problems at one end and the answer key facing the opposite direction at the other end. (Our March 2 post has an image of a Soccer Teams partner page.) The same teaching approach laid out in the post on Guided Discovery should be used here – 1. use short phrases, 2. begin with an action verb, 3. model your directions as needed, and 4. look around to make sure everyone is on track. There is no point in having students construct and build visual memories of a wrong procedure or amount!

These steps for delivering a lesson with the partner pages follow on the approach with the graphic pages. While not prescriptive, they do help keep students actively engaged in the work.

Start by saying:
1a. “Fold the partner page in half and set it down on top of the graphic page.” The partner page takes the place of the blank paper described in the earlier post.

1b. “Read the first problem on the partner page and move it to show that many…” teams, weeks, squares or roots, dollar bills, tickets, shirts, gallons, fractions of a cake, cheese slices, negative temperatures, “quadratic hearts”, or whatever graphic starts the lesson.

Then
2. Ask “How many players (teams, fractions, weeks…) do you see?”

3. Respond to correct answers with a simple “good” or “correct.” Respond to every correct as well as incorrect answer. By doing this, there is no question as to whether the answer is correct or not. For an incorrect action or response, never give the answer. Instead, guide students to correct themselves with: “Try again” or  “Show me on the chart how you got that answer.” As they reconstruct and rethink their solution, many will see for themselves what they did wrong and how to do it right. A few may require further guiding questions.

Do the next problem or two as a class as needed. Then ask the class to work the rest of the problems like that individually. Ask them to read the problem; move their paper; and say the answer in a quiet voice. So that students won’t copy each other, have some work the problems going across the page; some going down (from top to bottom); and so on. Circulate around the room checking on their work responding to correct and incorrect answers.

4. Ask for a volunteer to do the first row of problems for the class. Remind them that it is not cheating to use the graphic page. Ask for others to do the next rows. Have the rest of the class follow along with the answer key to make sure these students are answering each problem correctly.

5. Ask for two volunteers to model working in partners, one doing the problems, the other checking the accuracy with the answer key. The partner checking the answers follows the model in step #3, never giving answers and saying “good” for every correct answer. And the partner simply says “try again” if the answer is incorrect. They have the graphic page to refresh their understanding of the procedure and the basic facts. The rest of the class follows to make sure these two are doing it correctly.

6. Now have everyone work in partners following the model in step #5.

Summary: The partner pages build understanding and fluency together – in a way that can be directly observed. They involve cooperative learning with individual accountability, students receive immediate and ongoing feedback (but no answers), so they engage teacher, peer, and student in student self-correction. Going in different directions on the partner pages enables non-rote repetition. Like music students, math students benefit from going over a familiar exercise more than once. Getting better at what they are starting to get good at builds confidence along with competence.

At this point, students have achieved the first two levels of success: 1. they understand what they are doing, and can accurately solve problems slowly, with the aid of the graphic page; 2. using a partner page, they connect manipulation with computation and gradually wean themselves from the graphic page, accurately solving problems with greater fluency. The next level, developing independent speed, accuracy, and understanding, using the progress chart, will be addressed in an upcoming post.

In an earlier post, we gave an outline for integrating assessment into a math lesson. And we said that in subsequent posts we would address several aspects in more detail. Here is a closer look at the first step – starting the lesson with a graphic representation of the concept and procedure handed out to each student.

In the post that followed, we put up a video (The Soccer Teams Graphic) that demonstrated exploring the graphic in numerical order and re-exploring it out of numerical order. Through tutoring and classroom application of the graphic pages, we have observed more effective and less effective ways of guiding a lesson using a graphic manipulative. While there are lots of manipulatives and number arrays out there, how they are implemented and integrated makes all the difference. As always, it comes down to effective teaching. Here’s an approach that works when guiding students with a graphic page and a blank piece of paper:

1. Use short phrasessay only what is needed for students to construct, count, and observe on the graphic page. AVOID explaining. “Now you see class, what you just did was…” Guide and allow for their discoveries. Short phrases move the lesson along, keep students actively engaged with the graphic, and help language learners and students who have a hard time focusing.

2. Start with an action verbcount (“Count how many…”), move (“Move your blank paper to show…”), cover (“Cover all but the first team”), uncover, and so on. This aids in fulfilling the first step. Students know right away what to do.

3. Model your directionswhen you ask them to uncover the next row of soccer players or members of a rowing team or weeks with the blank paper, do it with them at first – in a way that they can see your movements. A picture is worth a thousand words. This is helpful for everyone and especially for those learning the language – whether English or math! – and those who have a problem paying attention to aural directions. Your observations will tell you how long to continue modeling. It has proved more effective to model one step at a time rather than to give students an overview to recall and execute.

4. Observe (assess)look around to make sure everyone is engaged and following your direction on the graphic correctly. Since it involves physical activity, it is pretty easy to see who is where they need to be for the direction you gave and who is slow or looking around to see what to do. Assessing determines the speed and direction of the lesson – does a prior skill need to be addressed – who needs a little more support – who is ready for more of a challenge – and so on. Creatively using a graphic page enables you to derive multiple lessons and multiple challenge levels from the same page. Each student matters.

5. Correct student responses nowsince each student matters, something needs to be done when you see inaccurate or continually slow responses. But never give an answer. Guide individual students to correct themselves: “Show me on the chart how you got that answer.”  As they reconstruct and rethink their solution, they see for themselves what they did wrong, and therefore how to do it right. Some may require further guiding questions. It is not efficient to give them the answer and move on. You will have to retrace this same concept or procedure eventually. Do it now before it becomes an unlearned prior skill needed for another procedure the student is trying to learn.

6. Begin the transition to print - say “show this many teams” as you write a “5” on the board. Don’t say the word “5.” Or say, “show this on your page” as you write on the board, “7 x 11.” By doing this, students see numbers and symbols as action prompts and begin making the connection between manipulation and computation. It prepares them for the problem pages that follow.

7. Ask questions - transition to mini-word problems. “3 teams are coming to the practice tomorrow, how many water bottles will we need?” After the students have some experience with a concept and procedure, then begin to make the process conscious by asking them questions about what they did.

Our experience has been that new vocabulary and notation should be introduced gradually in conjunction with the work on the graphic pages and reiterated during practice with the partner pages that follow. It is better to solidify (talk about) concepts with the correct nomenclature after students have actually experienced using them with the graphic pages. They own something concrete then to hang those words on. We will look more closely at working with the partner pages in a later post.

Summary: Guide work on a graphic manipulative using short phrases. Model how to do it. Make sure they’re doing it correctly; help them if they aren’t. Guide the work using numbers and math notation on the board. Give further written and aural challenges at appropriate levels. Ask them questions about what they did.

How to learn the way around a new city? In a more traditional approach, the teacher, in the classroom, explains some basic turns and landmarks and even models how to get to a specific location, then hands the student the keys to find someplace specific for homework. In this approach, the teacher is pulling on past experience with the city, and the explanations and modeling on the board are a short-hand for that experience. The student needing extra help (intervention) has no or at best a mixed experience to make accurate sense of the short-hand.

In a second approach, the teacher and student get in the car together but the teacher drives, pointing out the turns and landmarks he or she knows and again asks the student to find a specific location for homework. Better, but it is a second-hand experience for the student.

A third approach is to have the student drive with the teacher in the passenger seat not telling where or when to turn. The teacher first orients the student to a map of the city to be experienced. With good questions, the student notices landmarks on the map and is able then to begin making connections between the map and what is experienced while driving, “Oh this turn coming up is right here on the map.” “Good,” says the teacher, and that’s it, no lecture on the history and use of gazetteers. The teacher keeps the student in the driver’s seat.

If the student makes a wrong turn, the teacher simply asks the student to show on the map what they did and where they wanted to go. The student retraces the steps and notices that a left turn was needed not a right turn. “Good,” says the teacher and the student turns left. The city is slowly internalized as it is experienced, guided by the teacher and the map.

A good graphic representation of the math, not explained but oriented to and guided through, can be the map of the conceptual and procedural processes students have to navigate through.

Students are their own best teachers, including those who need extra help. This is true even though they will not take a very efficient approach to learning on their own and they will probably miss some essentials. This does say that teachers and parents have an important function to perform. But that function is far too often seen as sharing an insight or explaining something.

We can explain to our students that if an altitude is drawn to the hypotenuse of a right triangle, then either leg of the given right triangle is the mean proportional between the hypotenuse of the given right triangle and the segment of the hypotenuse adjacent to that leg, or of course the projection of that leg on the hypotenuse.  The amazing thing is not that we get some blank stares for our efforts, it is that we get as many students as we do looking like they got something. We then hope the homework will solidify it for them and make it clear to a reasonable number of the others who gave us the blank stare.

And we can explain to them the FOIL approach to working with quadratic equations and some of them will solve many of the problems correctly. But most of them will not know what they’re doing or why it works. They are not fishing; they are eating the fish we gave them.

In between the extremes of unaided student discovery and teacher explaining and modeling is guided student discovery. See the post on “Guided Student Discovery” for a little more on this topic.

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