Math Strategies in Context

Math Strategies in Context

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.

Two Strategies

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.

Student resistance

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)


Back to Basics

Back to Basics

A year ago today I began writing this blog, with the goal of sharing some of the things I’ve learned about teaching adult literacy and numeracy. On this anniversary, I’m re-playing my first post–still relevant, I think.

Slowly, over the years, because I was willing to learn, my students gave me a fresh take on the three R’s. I learned that to teach well, I needed to think about respect, resistance and reality. Continue reading

Power Share

Power Share

all in it togetherHere’s another story about sharing power with adult literacy students, to go with the one I posted last week called “Who’s in Charge Here?” Continue reading

Survival Strategies Come First

Survival Strategies Come First

Jenny HorsmanThe assignment was to make a graphic representation of the plot development in a novel we were reading together in class. To this end, I had assembled some supplies on a table in front of the room: various kinds of large sheets of paper, felt pens, pencil crayons, glue sticks, stickers and labels of the kind scrapbookers use, some collage materials, etc.

We talked about various possibilities, such as diagrams, time lines, and flow charts, Continue reading

If They Come, They Care

If They Come, They Care

www.katenonesuch.comI expected it to be an interesting activity. I was sure people would take part, and hoped they would enjoy it. But they didn’t seem to care.

I gathered some objects on a table in the classroom–modelling clay, bread dough, a crumpled plastic bag, rubber bands, pebbles, a plastic mug and a ceramic mug, Continue reading

Mandated Students

Mandated Students

judge 2They come because they have to.

Any of these people may have mandated students to your adult education class: the judge or their parole officer; their lawyer, hoping to make a good impression at a sentencing hearing; their social worker, financial aid worker, workers’ compensation officer, or other professional with the power to deny their request for benefits; parents who say if they want to live at home they have to go to school.

Unlike other students in your class, they are not self motivated; their motivation comes from someone outside the class, someone you have little influence on. Continue reading

Not a Fairy Tale

Not a Fairy Tale

“Ruby slippers” by RadioFan at English Wikipedia. Licensed under CC BY-SA 3.0 via Wikimedia Commons –

In a recent post I told the story of Naomi, who said “I pass” for more than three months in our basic literacy class, refusing all invitations to take part in group reading, writing and math sessions, or to do any private work in those areas; who instead spent her time making and colouring banners.

She was able to refuse to take part because of the classroom rule “Just say pass,” which is one of my mainstays in teaching adult literacy. She sat on the outskirts of the class, watching, until she could find a way to participate that was comfortable for her. She tested us for three months until she decided she could trust the situation, until she decided it was safe for her. Continue reading

What’s in It for Me?

What’s in It for Me?

Two people commented on my last post, about how working with Bernice got me started on marking for confidence.

First, Evelyn said that she thought many teachers would have seen Bernice as “resistant or difficult or careless or smartass.” I think most of those judgements are accurate.

She was resistant–she did not want me to “go over” her work with her; she wanted to keep herself out of a situation she had doubtless been in many times before, where a teacher pointed out where she had gone wrong and expected her to fix her errors. Continue reading

Respect, Resistance, and Reality

Respect, Resistance, and Reality

Slowly, over the years, because I was willing to learn, my students taught me a new set of three R’s—Respect, Resistance and Reality.

First, Respect. Respect for them, and for every decision they took, every choice they made. Respect for myself.

Second, Resistance. They come back to school, that place of previous failures and humiliation, because they want what they think I offer—the key to a better life. I offer them different ways of learning: group work; choice; meaningful work; I invite them to join the teaching team and make decisions about how and what they will learn. But they resist my best efforts to do things differently, because it is not what they expect, and it scares them. They refuse to risk going back again into that position of failure and humiliation. I need to acknowledge their resistance publicly, to honour it, and to work with it. I need to recognize my own resistance, too, because it gets in the way.

Third, Reality. My work is most successful when I listen to what is really going on. Speak the unspoken thoughts and feelings. Do reading and writing that is real, in the real world. Find an audience for the voice; find information in response to questions; say yes to every chance to move the literacy work into the community, and bring the community into the class.