Janet Banks, multiage author and educator, posted on the national Multiage Listserv this excellent and very thorough overview of how to teach mathematics in a multiage classroom. In this short article she tells how to incorporate math strands from the NCTM standards, real math situations, and your district's current textbooks into a multiage setting. In addition, she has listed a number of resources for further reading on the subject. With her permission I have included her post here.
Mathematics in a Multi-age Classroom
Teachers of multi-age classrooms have expressed concerns when it comes to teaching math, one of the most difficult areas to plan for, when you have a wide range of achievement levels. Some have posted questions (on the multiage listserv) about math instruction for multi-agers, and I have been receiving many private requests for help, so it may be helpful to put some of my beliefs and suggestions on the (internet).
I believe that the best way to teach math in any classroom is to follow the strands, as outlined in the NCTM standards. If you teach your math by strands, working with one at a time, you can diagnose what your students know and are able to do in relation to essential learnings for that strand, then determine the correct placement for each student on just that strand. It is much easier, for instance, if all of your students are working on measurement at the same time, even when at different levels, since so many concepts overlap and can be taught to everyone. You can also have all of the manipulatives, charts, bulletin boards, etc. related to measurement while you are working with it. Problem solving menus can be created, with activities on many different levels, so students can choose activities which fit their own achievement levels. Cooperative learning and peer tutoring possibilities are also great. Less able students will also learn a great deal just by observing what the more able students are doing on menu activities.
If you have math programs with adopted textbooks, you can consider the following: Textbook activities can be used, looking at several different levels and listing activities from each textbook according to the difficulty of the material. Assignments can be given according to difficulty, with students completing activities from several different books, without concern over which grade level the book is intended for. Books on higher levels always have review lessons, and books on lower levels usually have enrichment lessons. Students should be taught that it doesn't matter what grade level the book is, for that reason. They will just be working on activities that are appropriate for them.
Many of the concepts can be taught to the whole group, or to fewer groups, when everyone in the group is working on the same strand (instead of on whatever topic is covered on the next page of a textbook). Having all of the materials available that are related to the strand gives children a chance to explore with them and see what others are doing with them. Start by designing a diagnostic test for each strand, following essential learnings standards and objectives, and make note of the areas each student needs help with. Problem solving activities should be included in these tests, in order to see if the students can use their mathematical knowledge, as well as being able to work the abstract problems. Students should then be given instruction and assignments according to their needs.
Make a collection of as many activities as possible to teach the concepts. Find activities that are active, hands-on, and interesting. Pull out enrichment lessons and reteaching lessons from textbook series. These lessons tend to be more interesting and usually contain more active participation and problem solving than the regular text book pages. Let students work together and discuss what they are doing, and share what they are finding out. This helps them to learn concepts faster, as they learn so much from the thinking of others. Concepts must be understood before abstract problems are assigned, in order for students to really understand what they are doing and retain the methods for future use.
Create problem solving experiences that are related to real-life, so students can see a purpose for completing them. Design problems that have more than one answer, or more than one way to find the answer. Problems might involve such things as taking family trips, ordering food at fast food restaurants, earning allowances, attending sports events, finding the best deal on film to buy for a family trip, determining how much money they can make at a summer job, determining which pizza restaurant gives you the most pizza for the least amount of money, etc. When possible, let students design their own problems.
Use such items as snap cubes and bean sticks for teaching adding and subtracting, use geoboards for teaching geometry terms, use spinners for teaching probability, draw pictures to show symmetry, divide groups of objects to show division, etc. Design problem solving activities that contain work on various levels of difficulty. Everyone can have a part to do to help solve the problem, but the students with more knowledge can do the most difficult work, then explain to others how they have found the answers. When you are ready to teach concepts, you will find that students will catch on faster since they've seen and heard something about it before.
Numbers and operations, including computation skills, are important as a strand of their own, but also need to be included in activities from the other strands so that students get enough practice with the operations in order to master them. Stress mathematical reasoning and communication. Students must be taught the reasons behind mathematics and how and why it works the way it does. They need to talk about what they are doing and write about their individual and group work. This also helps them to clarify their thinking and understand concepts.
Seat children heterogeneously while they are working on practice-type assignments and encourage them to ask for and give help to others as needed. Using this method, you will find that a great deal is also learned just by being exposed to the higher level work that others are doing in the group. Peer tutoring is valuable both for the one asking and the one giving the help. It really helps students to clarify their own thinking when they explain it to others. When you are ready to teach skills lessons, pull out groups of students with the same needs.
Find math games that students can play with partners or in small groups. A great deal of learning takes place through playing games.
Here are some books that I often recommend when working with other teachers:
Math By All Means, by Marilyn Burns. (a series) Each book concentrates on a specific strand, with separate books for curriculum of different grade levels.
Writing in Math Class, (2-8) by Marilyn Burns. Explains how to make writing a part of math instruction.
Math as a Way of Knowing, (1-6) by Susan Ohanian. Gives ways to organize your curriculum and classroom to support heterogeneous grouping, collaborative learning, and interdisciplinary studies, ways to connect math across the curriculum, the use of manipulatives to solve problems, etc.
Teaching Thinking and Problem Solving in Math, (2-6) by Char Forsten. Covers how to use cooperative learning, graphing, logic, working backwards, drawing a picture and other proven strategies to build problem solving skills, etc.
Getting More from Math Manipulatives, (K-2) by Birgitta Corneille. Gives dozens of great, classroom tested activities to go with your favorite math manipulatives: pattern blocks, snap-cubes, colored links, geoboards, counters, floor mats, clocks, money, and more. Includes assessment ideas.
Essential Learnings of Mathematics, (3-6) by Janet Caudill Banks. Outlines student learning objectives, vocabulary, specific concepts, and procedures for each of the strands of mathematics, for grades K-8, along with suggestions for concrete, real-life problems and activities to develop them. Also includes group management ideas, strategies for teaching through problem solving, mathematical reasoning and communicating, math games and suggestions for performance assessment, etc.
I am the author of the last book listed above, Essential Learnings of Mathematics. It is based on teaching math with a concept approach, concentrating on the strands, one at a time. This book can be ordered by you or your local book store through the publisher: CATS Publications, 8633 233rd Pl. S.W., Edmonds, WA 98026-8646 (425) 776-0344
I wish all of you success with your math programs. Let me know if I can help.