Collaboration is a core value at Illustrative Mathematics. Creating a high quality instructional system — with curriculum and professional learning — is complex work. The demands of mathematical coherence and pedagogical appropriateness often pull in different directions; you can have a curriculum that is mathematically correct but not engaging for students, and you can have a curriculum that students enjoy but where they are not learning grade-level mathematics. We think IM K–12 Math has achieved the perfect balance between coherence and engagement, and we got there by having mathematics experts and educators working together, reviewing and critiquing each other’s work, and coming to a consensus around tough questions.
A good example of the balance between mathematical and pedagogical priorities is the tuna casserole activity in Lesson 2.6 of Grade 6 in IM 6–8 Math. Recipe contexts are good for learning about equivalent ratios because the ratios between various quantities in the recipe have a real world meaning (the flavor of the recipe) and because recipes are often scaled or cooked in containers of different sizes. The tuna casserole example provides a rich arithmetic context, particularly with fractional quantities, thus affording important skill building as students work with the ratios in the recipe. Furthermore, the extension activity, Are You Ready For More?, depends crucially on the fact that the vessel is rectangular, and gives students an opportunity to reinforce and use prior knowledge about area and volume. This is an example of the sort of collaborative thought that went into all the IM lessons.
Another sort of blending of expertise occurs when we try to put research about pedagogy into practice. Research recommends a problem-based approach to instruction where students have a chance to work on problems for themselves and the teacher synthesizes learning afterwards. But the practical experience of the teachers involved in writing our curriculum reminded us that you have to make the problem-based instructional model explicit and learnable. This led us to develop a carefully curated set of instructional routines, which help teachers and students manage problem-based instruction without getting bogged down in logistics, and which teachers can learn over time as they become more familiar with the curriculum.
“Because students are sharing their thinking, students using less efficient strategies will see other students using more efficient ones and learn from them. It also works the other way around. Students using more efficient strategies deepen their understanding as they explain those strategies.”
The principle of diverse teams collaborating extends to what goes in a classroom using IM. Many of the activities are designed so that students can use a range of strategies to solve them. Because students are sharing their thinking, students using less efficient strategies will see other students using more efficient ones and learn from them. It also works the other way around. Students using more efficient strategies deepen their understanding as the explain those strategies.
The Mathematical Language Routines in IM K–12 use collaboration to help all learners, including English learners, produce mathematical language to enable rich discussion of mathematical ideas. For example, in the Information Gap students work in pairs where each student has different parts of the mathematical problem and they ask each other questions to collaboratively solve the problem. The structure of the routine is designed so that students must formulate specific mathematical questions in order to get the information they need.
The collaborative learning embedded in the IM instructional model is particularly important in supporting culturally responsive pedagogy. Collaboration comes naturally to many cultures that are often marginalized in the classroom. Giving students an opportunity to share what they bring to the classroom builds their sense of belonging and self-efficacy.
Another way in which we live out our value of collaboration is in working through IM Certified distribution partners such as Imagine Learning. Again, each partner brings something different to the collaboration. IM brings its expertise in curriculum and professional learning, whereas Imagine Learning brings a digital platform that makes teachers’ lives easier and supports student engagement with additional features such as Student Spotlight Videos.
The next phase of IM’s journey involves collaborating with schools and districts around implementation support. We plan to build an implementation support ecosystem around our curriculum and professional learning that provides schools with a coherent suite of products and services that all work together to help teachers bring about our vision of a world where all learners know, use, and enjoy mathematics. Stay tuned for more exciting news about these plans over the next few months!
Bill McCallum, co-founder of Illustrative Mathematics, is a University Distinguished Professor Emeritus of Mathematics at the University of Arizona. He has worked both in mathematics research, in the areas of number theory and arithmetical algebraic geometry, and in mathematics education, writing textbooks and advising researchers and policy makers. He is a founding member of the Harvard Calculus Consortium and lead author of its college algebra and multivariable calculus texts. In 2009–2010 he was one of the lead writers for the Common Core State Standards in Mathematics. He holds a Ph. D. in Mathematics from Harvard University and a B.Sc. from the University of New South Wales.