An Active and Engaging Approach to Learning

Educators have often lamented that the majority of students do not understand mathematical concepts, or see why mathematical procedures work, or know when to use a given mathematical technique. According to the National Research Council's Everybody Counts, "Much of the failure in school mathematics is due to a tradition of teaching that is inappropriate to the way students learn." ¹

In the traditional teaching model that has dominated our schools for many years, a teacher demonstrates an algorithm or technique, assigns a set of problems for students to do on their own, and then tests the students a week later on their accumulation their skills.

Students in such a situation often do not understand what they are doing because they are simply following instructions. They typically see no need for the mathematics, other than to pass the test. The result is a system in which students "view mathematics as a rigid system of externally dictated rules, governed by standards of accuracy, speed, and memory."²

This process of "personal intellectual engagement" lies at the heart of IMP's view of learning and is shared by educators around the world. NCTM's Curriculum and Evaluation Standards elaborates on what this means in terms of what should happen in the classroom:

An approach to learning that maximizes student involvement in thinking through important mathematical issues leads to a different role for the teacher. It deemphasizes the teacher's role as creator of concepts and disseminator of algorithms, and sees the teacher more as a facilitator of learning. The teacher provides learning opportunities, asks thought-provoking questions, and allows students to develop their own mathematical frameworks. The teacher uses his or her expertise to provide the "glue" needed to help students tie ideas together and to clarify any misconceptions that may arise. The teacher no longer dictates which steps students will take to solve a problem. An approach which promotes lifelong learning combines the mathematics at hand with thinking and reasoning skills, and encourages risk-taking and perseverance.

IMP Promotes Understanding

The Interactive Mathematics Program lets students be active, engaged learners, using what they already know, making conjectures and learning from errors. The IMP curriculum presents students with rich mathematical contexts and gives them well-designed opportunities to discover and develop mathematical concepts as well as to prove important results. This approach offers students meaning for abstract concepts, gives them ownership of mathematical ideas, and heightens this interest in mathematics. Research shows that IMP students are apt to take more mathematics in high school than their peers in traditional programs.

Josie: "Josie was a tenth grader, with a shallow math background but good verbal test scores. She came into IMP to understand math--the Whys and Hows. Josie came back to talk to us after her first semester in college. She talked to my seniors and told them about her first traditional math class in three years. She said, I always ask questions. The others don't. I thought it was because they knew it already, but then after class they would ask me questions. I realized that they are just scared to ask. They don't know what is going on. Oh yes, I got an A.'"

Byron Hildebrand, Highlands H.S., San Antonio, Texas

How to Promote Mathematical Thinking

We teachers were most likely taught mathematics in a system where mastery of skills was the focus. We were the ones who succeeded in that system. IMP is not based on a mastery approach to learning. While the IMP curriculum seeks to attain the same goal of long-term mathematical understanding as mastery approaches, IMP promotes that understanding through a series of spiraled mathematics experiences which result over time in mathematical proficiency. It may be exceptionally hard for us to let go of the idea that students take more responsibility for their own learning. You will have to remind yourself, often, that you are trying to build independent thinkers and reasoners. As you see your students struggling with an awkward approach to a problem, you may have to work hard to keep from giving them the most elegant way. Easier said than done!

For the IMP teacher who finds it hard to let go of his or her traditional role in the classroom, my number one piece of advice is: Be patient! Have faith in the students and have faith in the curriculum. Your students are creative and capable! You will be surprised and delighted by the variety of ways students attack problems and investigations, yet come to successful conclusions, when given the opportunity to think independently and together. You will have to bite your tongue as you watch a student play with a topic for which you know the conventional super-formula. Don't steal that student's "Ah-ha!" experience.

Developing knowledge experientially with an activity-oriented curriculum takes more time than "delivering" knowledge through lecture. Think about why that is true. A student developing his or her own mathematical ideas to solve challenging problems has to do many things--see a need for the math, try a problem with tools already at hand, look for a pattern, investigate a conjecture, and convince himself or herself of the findings. Of course that takes more time--it is a much longer process than listening to a lecture and practicing an algorithm. The process is what gives the student ownership of the end result.

Although you don't want to steal the "Ah-ha's," that doesn't mean that you sit by idly and let your expertise go to waste. You, as the facilitator for the learning process, have to be a skilled questioner. For example, as groups begin work on a problem, you will need to ensure that they understand the directions so that all students can access the problem. Perhaps you will have one student read the directions aloud and then have groups paraphrase them. Once kids get started, you will want to walk around the room asking questions of all kinds--probing questions to promote thought, testing questions to see where the students are, and focusing questions to have students explain or justify their ideas. The teacher's guide for each unit includes samples of such questions for you to use in the context of specific problems and activities.

Your Own Growth

As you watch your students become active and engaged learners, you will most likely be learning yourself. Be open and share your own learning experiences with your students. If you were trained in a different school of thought, you will grow from seeing the various approaches students offer. You will also benefit from attending teacher inservices that give you opportunities to learn as your IMP students are learning--through exploring, questioning, guessing, estimating, arguing, and proving. As you begin each IMP unit and activity, take time to talk with at least one of your fellow IMP teachers. Together, identify the goals of the unit or activity and think through how you are going to assess student progress toward such goals and grade student performance.

¹National Research Council, Everybody Counts: A Report to the Nation on the Future of Mathematics Education (Washington, DC, National Academy Press, 1989), p. 6.
²Everybody Counts, p. 44.
³Everybody Counts, p. 6.
⁴Curriculum and Evaluation Standards for School Mathematics (Reston, VA: National Council of Teachers of Mathematics, 1989), p. 5.