Making Math Meaningful

Use Cognitive Instruction in Math Modeling
by Sue Henninger

(Reprinted with permission from the Summer 2010 issue of “Ithaca Child”, The Paper for Parents)

“Math is a subject that convinces students that they can’t do it and that something is wrong with them. Students that experience failure in math often begin to see themselves as failures in other areas as well … CIMM (Cognitive Instruction in Math Modeling) effectively breaks that log jam,” says Dr. Christopher Horton (aka the Math Doctor), who uses the CIMM method to tutor students from his home in Massachusetts.

Sophia shows her work using the CIMM method.

What makes CIMM a better teaching tool than other math programs? In layman’s terms, Horton replies, “It works because it makes sense!” He continues, “No matter how abstract the math concepts become, as you advance with CIMM, you’ll always have a solid foundation to refer back to. Once students really believe that quantity doesn’t change no matter how you regroup and that they can always represent numbers as relationships between quantities then they can clearly see for themselves what makes sense and what doesn’t.”

Horton began his journey with CIMM when he was teaching failing ninth-grade algebra students in Springfield, Mass. After three years of searching for an effective way to help the teens master math, he was frustrated because, “No one could tell me how to teach these kids so they could learn.”

Researching his dilemma, he discovered that Rob MacDuff at Arizona State University was developing a method for teaching math more effectively and was in fact in the process of successfully introducing CIMM in the public schools in Phoenix area and on the Navajo reservation.

MacDuff’s work on CIMM began when David Hestenes, a renowned physicist at Arizona State University, asked him to try to come up with a sensible way to integrate math and science for students. MacDuff was researching this when, he says, “Suddenly it hit me! The divide between math and science is really a divide between numbers and relations between things so the way that math is traditionally taught makes it impossible to ever integrate the two subjects. My only solution was to come up with a brand new, relations-based way to teach math!”

“You can’t just engineer the curriculum for kids; they have to build their own understanding of math.” MacDuff explains. For example, “If you built your own house, you would know where each nail and each board was and how they all fit together. If a student builds a math foundation then they will have the same type of understanding.” He asserts that CIMM gives kids the tools they need to build a strong math foundation, and these tools can help students understand concepts which can then be transferred to other academic subjects as well.

Still dubious? Try this exercise. Draw two circles next to each other. Put two dots in each circle. Ask yourself if the two circles are the same. When my family tried this, we each put different sized dots in each circle so our answer was “No”. From an artist’s perspective that’s the correct answer, MacDuff agrees. However from a mathematical point of view, the two circles are exactly the same because each is one group containing two dots. Emphasizing that neither answer makes the student (or adult!) a failure or success, MacDuff explains that, depending on how you look at the circles, the idea of what it means for something to be the same can be significantly different. He believes that CIMM opens doors to new ways of looking at traditional problems by using an exciting and positive approach rather than the “this is right, that is wrong” system favored by other mathematical approaches.

MacDuff is able to offer a variety of exercises which illustrate how CIMM works. Though he clearly loves math, he readily acknowledges that eighty-five percent of the students he deals with are “math phobic” with significant level of anxiety related to anything mathematical. Laughingly he adds, “If I’m out at a social event and I happen to say that I teach math, the responses are predictable. People either tell me how much they hate math or how math was their worst subject at school or they talk about how they just can’t understand it.”

“What if eighty-five percent of students left school unable to read or write?” continues MacDuff. “Yet we allow this to happen all the time with math, which you can’t live without either.”

From the beginning Horton has felt that helping students understand math is just a matter of learning how to teach the subject in a way that “taps into the incredible power of the brain to reorganize itself.” He asserts that the CIMM method of teaching develops and strengthens interconnections between specialized parts of the brain and cross links it, allowing kids to understand what they’re learning on a much deeper level. CIMM is taught using principles of guided enquiry (modeled after a physics program) where students are given both the tools to work with and the tools to express their thoughts. There are lots of small group tasks and presentations and CIMM’s flexibility and expressive aspects keep students engaged which according to Horton, results in an unexpected benefit in the classroom. “We’ve found that there is less need for discipline or constant redirecting by the teacher because the students are hooked and don’t want the math session to end,” Horton explains.

Based on his experiences with school children, MacDuff further hypothesizes that if CIMM were introduced in second or third grade, by the time those students were in sixth grade, they would be perfectly capable of learning calculus and perhaps middle and high school remedial math could be eliminated entirely, resulting in major savings for school districts.

But wait, there’s more! MacDuff quotes a student who told him proudly, “I’m privileged to have this program.” When he asked her to elaborate, she continued, “I’m lucky because we’re the ones piloting it,” adding, “I understand math that my friends don’t, even though I’m the remedial one!” Both MacDuff and Horton acknowledge that having a special program, designed “just for them” may have an invaluable secondary psychological benefit for academically struggling students who are not usually offered the opportunity to take thought-provoking classes or accelerated programs.

On a video made about the CIMM program, a teenage girl and her mother talk about their experience with math in the public school system. The mom describes math homework as “absolutely horrible at times,” saying that every day her daughter would get progressively more frustrated until she would retreat to her room, saying that she hated math, hated her life, and wanted to drop out of school. Her daughter admits that she hid flashcards and books her mom bought her an even resorted to having her grandmother do her math homework for her. She says, “I felt really bad about myself, like I wasn’t going to make it in the world.” Since CIMM, her perception of her future has changed and she believes not only that she can graduate from high school, but that she has more options for higher education if she decides to pursue it.

Interesting, the CIMM method seems to work equally well for all students, no matter where they’re located on the academic ladder. Vermont resident Juliana Parker home-schools her eleven year old daughter, Sophia. Parker recalls that as a young child Sophia loved math but that as she got older her enthusiasm for the subject began to dwindle and she started to view math lessons as “drudgery”. A dedicated home schooling parent, Parker has researched lots of math programs and tried to put together an interesting lesson plan for Sophia, and her daughters growing dissatisfaction with math began to feel like it was related more to the curriculum than to the subject itself. Parker found herself remembering, “I started to fade out of math when I hit algebra. I began to lose the coherence of understanding what I was doing and to feel like there was a gap in my knowledge that I was always trying to compensate for. Eventually I began to lose my confidence in my math ability to do more advanced math.” Like Horton she is convinced that for many students math is a “backbreaker of confidence” and that instead of eroding confidence, CIMM actually reinforces the belief that there is an answer to whatever questions you might have. “Rob (MacDuff) put the program together with no holes,” she says. “CIMM” doesn’t ask you to take principles on faith. It actually shows you why it works.” Parker and Sophia both agree that another benefit of the program is that their brains feel more awake and active when they’re doing CIMM and Parker even goes so far as to say that the program seems to “wake up my networks of neurons!”

It’s not only children who can benefit from CIMM, says Horton, who taught a CIMM course to elementary school teachers at International College in Springfield a few years ago. He reports that fifteen out of twenty teachers failed the math exam that he gave them at the beginning of the course but that, after he introduced CIMM and taught topics of arithmetic that referred to CIMM methods, by the time the course ended, nineteen out of twenty teachers passed the exam. Horton adds that many of the teachers even thanked him, saying they had never understood math so well before.

Robin Rosenthal, who taught everything from pre-algebra to advanced calculus in Arizona’s Paradise Valley School District, also believes in the power of CIMM. On the CIMM video she says that, not only does she see her students understanding math concepts and being able to explain their answers to others, but she’s observed that their increased confidence in math has led to more self-confidence in other areas as well. Another plus for Rosenthal has been gaining a new understanding of what her students “get and don’t get” about math.

MacDuff and Horton realize that it can be difficult to implement new programs or teaching styles in school systems where they may be met with some resistance. Rather than starting with the school administration and having them impose the CIMM program on the teachers, the two offered CIMM training to the teachers themselves with the goal of empowering them to judge, further develop, and recommend the course to the administrators. Introducing CIMM into schools in this grassroots way has been much more effective says MacDuff, because the people actually responsible for teaching it to students “really buy into it.”

Clearly it’s not possible for most Finger Lakes residents to travel to Massachusetts or Arizona to try CIMM for themselves. But Parker, who cant’t say enough about Horton as a tutor, has a solution. She and Sophia practice distance learning with Horton two times a week for an hour and a half sessions. Parker calls their classes a “low maintenance system,” explaining that you can do the lessons either on iChat or Skype or use your libraries webcam. The only other materials they need are whiteboards with erasable markers so they can all see each other’s work. Parker has been very satisfied with this arrangement and has recommended it to a number of homeschooling parents in her network.

Sue Henninger is a full-time freelance writer and a regular contributor to familiy magazines. She lives in Trumansburg, New York with her husband, three teenage boys, and several pets. Contact her at or

Is your curiosity sparked after reading this article about CIMM?

Would you like to find out more?

Feel free to contact Rob MacDuff


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