Montessori Algebra and Geometry for Adolescents
This article appears in the Winter 2020 issue of MontessoriPublic — Print Edition.
A conversation with creator Michael Waski
Michael Waski has been a conventional public school math teacher and a Montessori high school teacher, and is the writer of two volumes on a Montessori approach to algebra for middle and high school, with a book on geometry in the works. MontessoriPublic sat down with Waski to talk about his work.
AYER: Can you tell us about your background in teaching and in Montessori? How did you get here?
WASKI: I got my B.A. in Elementary Education, and I taught 2nd, 3rd, 5th, and 8th grade math over five years in public schools. I got very frustrated with standards-based education—having to post on the wall what standards were being taught, being held to “students should be able to tell you what standard they are working on,” all of that. For third grade!?? Not for me. And I guess I’ve been part of the AMI/NAMTA Montessori Orientation to Adolescent Studies for about 13 years.
AYER: So what is Waski Math?
WASKI: Do they call it that? I guess I’ve heard that. “Waski Math” started out as book, Teaching Algebra to the Adolescent: A Montessori Approach. Now there’s a forthcoming Teaching Geometry to the Adolescent: A Montessori Approach, and The Math Institute at Great Work Inc., which is a platform for the books, workshops, and supporting materials.
But really, it’s an approach to teaching mathematics for adolescents developed originally for Montessori middle school and high school, but applicable to just about any exploration and discovery oriented secondary school program.
AYER: Tell me the origin story, then. How did this come to be?
WASKI: When I was at Montessori High School, I needed to document my work I had been developing in Montessori secondary environments, which I generally hadn’t done with much detail. The only way I could think to do that was in the style of my Elementary albums. That summer at the Orientation to Adolescence I used my notes to show some work with algebra tiles and participants asked if they could see my documentation. I spent another year cleaning it up, getting it reviewed by AMI Elementary Kay Baker and her husband Terry, a math professor at Yale, and finally got it published.
AYER: And then…?
WASKI: When the Montessori High School closed, I started working for Great Work Inc. (GWI) here in Colorado. We’ve spent the last year trying to define the “Math Institute.” But essentially we’re trying to bring high quality math education, using a Montessori approach, to as many people as we can. Part of that is the books. We are also working on getting materials that accompany those books produced, so there are math materials that are appropriate for the adolescent.
When I was at MHS, I also developed a series of questions (the Daily Reviews) which is a spiraling approach to the curriculum and can be used as an organizing principle and follow-up work to the lessons. Several schools have adopted them, including Compass and Denver Montessori Junior/Senior High (DMHS) and have been using them for the past three years as the core of their curriculum.
I have also been able to offer webinars, on-line classes for math teachers, create instructional videos, and travel to do workshops to help spread the math love!
AYER: So about the book itself—is this a textbook?
WASKI: No, it’s really not! Maria Montessori wrote, “What needs to be known is the little we need to teach. However, it must be taught in an absolutely exact manner.” This is simply a set of lessons based on Montessori elementary material, which has been uniquely adapted and expanded for the adolescent. It is hopefully “the little we need to teach” in our Montessori adolescent communities.
AYER: What is it about adolescents that’s different?
WASKI: Well, for example, the Montessori prepared environment for an adolescent mathematics classroom has to take advantage of students’ social drives. Group work, discussions, hands-on activities, and an atmosphere of collegiality—of “fearless learning”—need to be present.
I use frequent math seminars, and prepare the environment for open exploration and extended work time. Really, it’s the environment and the approach, more than any list of lessons, that makes it work.
AYER: OK, but there is a list of lessons. How do you suggest we use them?
WASKI: The book is organized very much like my Elementary albums. There’s a table of contents, and lessons with prerequisites, materials, and some “teacher talk” and directions. Within topics, lessons are in a basic order of sequential knowledge, but you wouldn’t present them in strict order, first page of the book to the last. Students should have lessons across topics as interest and ability indicate, and many topics cross over one another. I’ve included some sample charts of how one might move through the book, but every student’s path might be unique.
AYER: And you have Geometry for Adolescents coming out, including for middle school. Isn’t that usually for high school?
WASKI: Yes, geometry is usually relegated to tenth grade, which is a shame—it’s such a beautiful and natural subject. When people think geometry is only for older children, they often think of the traditional proofs. Younger students may find these challenging or unappealing, but geometry is so much more than proofs! It’s measurement and experiment; it’s history and puzzles and logic. By “saving geometry for later” we’re missing out on many amazing learning opportunities.
AYER: We do a lot of history, puzzles, and measurement in the elementary—how is this different?
WASKI: The elementary observation of patterns and developing rules continues for the 12-15-year-old as it deepens in sophistication and students apply these rules to the world around them.
Geometry for the adolescent should be alive and dynamic. So much of the work should be project-based and hands-on. Many of the concepts will have been introduced in elementary school, and perhaps early elementary school. Revisiting these concepts in ways that appeal to the adolescent—such as in practical social work, exploring philosophy, or more sophisticated blending of mathematical skills and topics, including algebra—brings the work alive and keeps it fresh. Students don’t have to do formal deductive reasoning to engage with geometric concepts at a deep level. Deductive reasoning is important, but we can build the foundations early on, and revisit later with a new level of sophistication.
AYER: How are you seeing this used in schools? In public schools in particular? Can it be used in non-Montessori schools?
WASKI: It’s always amazing to walk into a school and see the Timeline of Mathematics in the room, or some of the materials that have been developed for adolescents in use. What I love about Montessori is the entire structure of freedom, discipline, and responsibility, the way the day is structured around work, etc., that makes everything else work. To get the maximum out of what I have developed, one needs this whole-school or whole-person approach. Everything makes so much more sense when there are all these components present. It allows so much more for the individualization, the flexibility, the ability to allow students to work deeply and follow their own interests – so many things that ALL teachers want, but unfortunately are locked into a system that doesn’t give them what they really need. It is hard to imagine what students are really capable of, but as Montessorians, we get to see that all the time.
What I hope is that the approach and materials that I can offer appeals to the sensitivities of the teachers who want more, and for them it is useful and helpful. But more so, I hope they see how it still isn’t enough until they have the whole picture, and then this is where it would be great to have people changing over to a whole-school Montessori approach – because they see what is possible and see a path to getting there. And it is so exciting they are willing to take those risks and do something different and unique.
Michael Waski, a 20-year veteran teaching math to adolescents in public and private Montessori schools as well as conventional public programs, is the author of Teaching Algebra and Teaching Geometry to the Adolescent, and is the director of the Math Institute as a part of Great Work Inc.