In September of 1996 Debbie Hackett posted on a multiage listserve a description of the math program that she created along with her teaching partner. It is substantially different than what my partner and I have set up, but it is another good way to "do math" in a multiage setting. With her permission I have reposted it here.
I teach in a multi-age class for 6-10 year olds. As you would guess, the range in math ability is a far greater span than 4 years. There is no textbook that is going to address the needs of all these children. There is no one program (Math Their way, Marilyn Burns, etc.) that has all the answers or even comes close to providing all that these children need. I think the best way to go is to use a variety of materials. A textbook could certainly be one of them, but for us, it is no more important than any other book we use.
In hopes of convincing you that learning math can be done without the guidance of a textbook, let me explain how we approach the teaching of math in our classroom.
I co-teach with a wonderful person named Marie. ( Marie is very organized and efficient. I say this only to prove that life without a text book does not have to be haphazard or chaotic. Marie would never put up with that.) The time of day that we do math is called, Investigations. Investigations is an attempt to coordinate the teaching of math, science and technology (making things). It arose out of a desire to 1. make the purpose for learning skills apparent 2. spend more time DOING science and 3. incorporate the use of technology. We break the children into two groups for Investigations. We call the groups "youngers" and "olders". I work with the 6-8 year olds and Marie works with kids who are 8+ - 10+. Using a science topic as the context, we design, build, explore and investigate a myriad of things. Built in to most investigations is a project. Through the project children learn and apply math skills. For the purposes of this discussion let's just talk about the math piece.
Let me say at the outset that neither Marie or myself are math wizards. However, we can both use math to solve everyday problems. If you can do that, you can teach without a textbook. You get comfortable with it by doing it.
First, get your hand on your school's math curriculum guide. Copy all of the pages that relate to the grade levels you teach. Go a grade higher and a grade lower. Go through these pages with the infamous red pen and cross off things that are repeated. There will be plenty. Then cross off those things that are senseless. There will be some, like: students in grade three will add and subtract like and unlike fractions. Sensible for older children but not for 8 year olds IMHO. You will be left with a list of skills deemed important by you and the school for the children you teach. Be careful how you use this list. It is only meant as a guide.
Next, decide which strands will receive emphasis when. For us this depends on the science topic. For example, at present the youngers are learning about the migratory patterns of certain animals. Therefore our math strand of focus is pattern. However, this does not mean that we only work on the concepts of pattern during our migration study. Number is always a focus. Problem solving is a constant. I design a project for the children that will present them with real reasons for learning the skills. During this study we are working with the University of Kansas tagging Monarch butterflies. Analysis of our tagging data and the data of other "scientists" helps the entomologists to identify the migratory patterns of the Monarch. Children can appreciate the need to know about pattern when they need to put it to use. We will also look at symmetry, be involved in measurement activities as we watch our caterpillars grow, learn to collect, graph and analyze data, use numbers, talk about the probability of our tagged butterflies making it to Mexico and much more. Our next study which will be motion, will be heavy on the measurement.
I use clinics, one on one conferences and whole group lessons to teach concepts (Math Their Way and Marilyn Burns is very helpful here) and skills if the students show a need. These are usually short, with the children returning as soon as possible to their project to use their new knowledge to complete their work. For example, last year we did a systems study. We were looking at pond life and in order to be able to observe more often we decided to build ponds for the classroom that could sustain life for at least a week. I wrote up the criteria for each pond and the children went to work building them. Some of the criteria were: the pond must contain at least 1 1/2 liters of water at all times. Your pond must contain: 1 amphibian not more than 3 inches long, 10 times as many insects as amphibians, at least one half of the insects mut be in the larval stage, at least one fifth of the insects must be beetles. Maintain a temperature of 60-70 dgrees farenheit. The criteria I developed was based on knowing what my students needed to learn. I'm sure you can see what I had to teach before many of them could complete this project. (This particular project was done with 8-10 year olds.)
My points are that teaching without a textbook can be done, is loads more fun, and allows the teacher to focus on the needs of her children. I firmly believe that math learned in the context of real work is easier to understand and stays with the kids longer. Kids can spot a cutesy activity a mile away and respond accordingly. Give the children real work to do and they will keep coming back for more. And I guarantee that no one in your class will ever ask you, "Why to we have to learn this?"