A student demonstrates that 1/2 = 6/12, 3/6, 2/4 and 5/10
“EXPLAIN HOW YOU WOULD TEACH YOUR LEARNERS PLACE VALUE” and “Explain in detail how you teach ABET level 1 learners fractions” are two recent requests from readers of this blog. (ABET stands for “Adult Basic Education and Training” and is a term used in South Africa as well as other places.)
When I first saw the comments, I began by being a little miffed—had they read my manual? Had they read my many posts? Had they looked at my articles? Did they want me to start again from scratch? Could they not put two and two together? It was this last question that brought me back to my calmer self.
Maybe they couldn’t put two and two together because I hadn’t set out clearly enough the connections between my strategies for teaching the math content and the context in which I like to teach: a class that is as safe for risk-taking as I can make it, where student resistance is honoured and accommodated, and where students are part of the teaching team; when all these are in place, there is a willingness to work, a desire to learn, and a lightheartedness that makes teaching a pleasure.
So besides a couple of files that set out in detail the strategies I use to teach fractions and place value, in this post I’ll talk a little more about how the other things work together to make the strategies successful.
I have two main strategies for teaching both place value and fractions: first, asking students to demonstrate concepts using manipulatives, and, second, group work at the board that practices and reviews concepts and algorithms. The precise details, along with student hand-outs, are in these files: Strategies for Teaching Fractions and Strategies for Teaching Place Value.
What follows are some thoughts on the why and the how of the strategies set out in the files, a little extra context to fit the bald details into.
1. Using Manipulatives
I use manipulatives because they work. There are difficulties—the messiness, the expense, the resistance from students—but they always work to make the math concepts and operations clearer, and students learn and are confident in their understanding.
- Manipulatives slow down the action, slow down the explanation, so students have more time to absorb what is happening before instruction moves on to the next step.
- The student controls the pace of the work.
- Manipulatives provide visual cues to help students remember (shape, colour, size, etc.)
- Teachers can watch students work with manipulatives and get valuable information about the state of their understanding, especially when students can’t say where they are having problems.
- Students get the right answer, so that the instructor can work on extending understanding, or on helping the student articulate the concepts, rather than having to deal with an error.
In most classrooms where manipulatives are availaable, students use them to get an answer to a question posed by the workbook or the teacher. When they get the answer, it is checked against an answer key, and the student moves on to the next question. The focus there is getting the right answer.
My strategy is quite different. It doesn’t matter how students get the answer—using the manipulatives, using pencil and paper, by mental math, or by asking their neighbour. Their assignment is to use the manipulatives to demonstrate or prove that their answer is right. They set up their demonstration, I look at it, initial their handout to record its completion, and we have a little conversation (often only 30 seconds or so) to solidify or extend their understanding. The focus here is on understanding and talking about the concepts.
Students resist using the manipulatives. Some think they are childish; some have had their fingers rapped too many times, and are not willing to put themselves forward again; some think that it is cheating to use manipulatives; some think it is not manly. I try to defuse the negative emotions by saying out loud what they may be feeling. “Some people are embarrassed because they think these things are just for babies,” I say, or “If you feel stupid using these math tools, let me tell you that the smartest mathematicians in the world make models to figure out math problems. We use these tools to make models of fractions.” Speaking the emotions into the room often releases them enough to melt the resistance.
Students from previous classes are always willing to come in to talk to new students about the manipulatives—how they too started out feeling stupid or embarrassed to use them, and how they came to change their minds. It seems that new students are more willing to believe former students than to believe me. Go figure!
To provide further motivation to use the manipulatives, I offer them a choice: they can do the usual workbook (90 pages of questions, check the answers, make the corrections, etc.) OR they can do the 12- page handout I give them (included in the files linked above), demonstrating all their answers with manipulatives. Both groups, students who use the workbook and students who follow the handout using the manipulatives, write the same tests.
In fact, I have never had a student choose the workbook over the hand-out/manipulatives combination. They look at the 12 pages of handout, and the 90 pages of workbook, and they opt for the shorter way.
2. Practice at the Board
I ask all the students to go to the board for about 20 minutes for a practice session every time the class meets. I use this time to practice things they already know, to extend and solidify work they are doing with the manipulatives, and to articulate concepts and explain how and why algorithms work. As the days go by, the questions remain the same, but the work gets harder. (For those non-math-teachers who are still with me, “algorithm” is a fancy word for the method we use to do things like add, subtract, multiply and divide.)
Why do I have students work at the board instead of at their desks? Because I can see all of their work at once, and can quickly go to help students who are having difficulty. Because students get instant feedback on their work. Because students can see each other’s work and talk to each other, or copy if they need to. Because it provides all the benefits of social learning.
Practice at the board has to be relatively pain free, or students will refuse to do it. Students working at the board are working in public, so I work to make it safer for them to take that risk:
- I make sure that every student has easy access to an eraser so that when they make a mistake they can quickly erase it. This means an eraser per student, or at most two students share an eraser between them.
- I ask them to make themselves comfortable at the board—get a nice long piece of chalk, or a coloured marker—stand beside someone who is good at math, or beside someone they can talk to—stand near a set of manipulatives and use them as needed—stand near the door so they can take a walk in the hall if they need a break—whatever they need.
- The work I ask them to do is not new. It is practice and review of work we have already covered.
- I remind them frequently that this is practice, not a test. It is fine to look around at what others are doing. It is okay to ask for help. It is a good thing to help your neighbour.
- If a student is having trouble, or is slow at writing the answers, I never ask them to explain the procedure. If they knew the procedure, they would be able to get the answer quickly. So I say the procedure, step by step, to guide them through the process.
There is usually a wide range of skill and ability in any ABE, ABET, or GED class, so it is challenging to get people to work at the board together on the same questions. You need to go at the pace of the slowest student, so what can you do to challenge those who can do the work quickly and easily?
- I ask them to explain what they did, how they know their answer is correct, and to elaborate on the math involved in the question. For students who can do the work, the next step is to be able to articulate what they are doing. This explanation also helps those who are struggling to do the work.
- I remind them of the class rule “Refuse to be bored,” and invite them to make the questions more challenging—to do more in the time allowed, for example, or to write bigger numbers, or fractions with bigger denominators.
- I ask an advanced student to take over from me for a few minutes and read the questions, another to help check that the work on the board is correct.
- Sometimes I offer two questions, one harder and one easier, and ask students to pick which one they will do, and offer the option of doing both and commenting on the similarities and differences between the two.
A Balanced Approach
These two strategies taken together provide a balance of individual and group work. Work with manipulatives gives time and attention to understanding, and the board work provides review and practice.
Twenty minutes working with the manipulatives brings greater understanding than twenty minutes watching the teacher do math and explain it. Twenty minutes with the rest of the class at the board provides a greater quantity of meaningful practice and review than twenty minutes at a workbook doing problems on a single operation.
Implementing the Strategies
The strategies I’m talking about here may need a little creativity or extra work in the beginning, but the payoff for you is less marking and a more active dynamic in your classroom. If your program doesn’t have manipulatives for all students, you and your students can make some (directions in Changing the Way We Teach Math, page 26). If you don’t have enough board space, or any board space, can you come up with some alternatives? Maybe doing the work on large sheets of paper, students sitting together and writing big enough for all to see? I think the board work for the place value exercises could be done in a computer lab, with students using size 50 font so their screens can be seen from a distance.
Getting students on the teaching team, so they will be willing to try out and evaluate these strategies, will be a necessary first step. You don’t want their resistance to rain on your parade.
Working with Student Resistance to Math Tools from Literacies
Changing Practice, Expanding Minds from Focus on Basics (page 20)