Take a moment to recall your experience of math in middle school. How were you first introduced to concepts like ratio, exponential growth, or surface area and volume? Chances are, even if you enjoyed your math classes, you probably concentrated mostly on getting the answers “right;” whether you cared personally about it was likely secondary. A good mark on each test was the primary goal. Now imagine you are trying to figure out why honey bees are dying in such large numbers, and you work with an urban beekeeper to shake the varroa mites out of a frame in a nearby hive, noting the number of mites and the number of bees. All of a sudden, figuring out the threshold ratio of mites to bees before exponential parasitic growth leads to bee colony collapse has a more urgent purpose.
Or, imagine you are working with a chocolatier to learn how to make chocolate from fermented cacao beans. You’ve gone through all the steps of the chocolate-making process, including designing and building your own winnower to remove the husks from the cacao beans. Now you have to determine the surface area and volume of your chocolate bar in order to keep packaging to a minimum. You quickly learn that in economies of scale, every small difference in calculation has big financial and environmental implications down the line,.
As part of our approach to math education, Millennium School is committed to embedding important math concepts and mathematical habits of mind within interdisciplinary projects that connect to the real world. And there is plenty of evidence that this works.
The Buck Institute of Education (BIE), whose highest priority is “to help teachers prepare students for successful lives,” is an international leader in promoting project-based learning (PBL) and bringing coherence to PBL practices across grade levels and subject areas. In its research summary of project-based learning in middle school mathematics, BIE finds evidence for many advantages of PBL, including increased problem solving ability, subject matter understanding, and collaboration capacity. BIE also points to research that “students who experience PBL as part of their mathematics coursework develop skills associated with increased expertise such as improved metacognitive and self-regulatory skills.”
Despite the overwhelming positive indicators, not the least of which is improvement in student attitudes toward mathematics, parents understandably express concern that their children might not be getting enough practice with certain skills and concepts, or, even worse, that their child may even be able to avoid the math component of a project altogether, especially when he or she is teamed up with students more comfortable in the mathematical domain.
While Millennium will have ways of ensuring all students engage with the math components of a project, we nevertheless believe that students should experience the joys of pure math, playing with numbers and patterns outside of a project and without an end goal in mind. This is why our approach is a hybrid, combining project-based and pure math. For inspiration in creating this part of our math program, we point to the important work of Ruth Parker and Jo Boaler, as well as a recent TEDx talk by Dan Finkel.
Ruth Parker, of the Mathematics Education Collaborative, promotes holding number talks with students to build “number sense” as a community. Providing meaningful, ongoing practice with computation, number talks require only 5-15 minutes, but the impact of these short dialogues can be profound. When students articulate their own path to a mental computation, not only do they become familiar with their own metacognitive processes, but they also encounter multiple pathways to a solution. For example, a student who explains that she determined 18 x 5=90 by multiplying 10 x 5=50 and adding that to 8 x 5=40 may learn that another student subtracted 2 x 5=10 from 20 x 5=100 to get the same answer. With multiple methods always available, number talks provide built-in differentiation, allowing more advanced students to contemplate alternative approaches while their classmates generate at least one. Plus, the consistent practice of speaking about their mathematical thinking bolsters a student’s ability to write in the language of mathematics.
Another big proponent of daily number talks, Jo Boaler is professor of math education at Stanford University. Her most recent book, Mathematical Mindsets, serves as a fundamental source of inspiration for us. Applying Carol Dweck’s principle of a growth mindset to mathematics, Boaler argues for moving away from performance-based math education, where perfection is the goal, and toward a classroom environment that celebrates mistakes as opportunities for rapid and meaningful learning. (She even suggests saying to your child “I’m sorry to hear that,” when they say they got all their math questions right, regretting they were not given opportunities to learn anything!) Boaler makes the case for shifting the focus of math from computation to pattern recognition and introducing low-floor, high-ceiling math questions students can grapple with at multiple levels. Working in heterogeneous ability groups, which Boaler recommends unreservedly, students share their own mathematical insights—be they visual-spatial, procedural, or conceptual—to push each other toward deeper understanding.
Dan Finkel, founder of Math4Love and clearly aligned with both Parker and Boaler, presents five principles of math education in his talk that resonate with us:
Inherent in these principles is the belief that math should be both challenging and fun, full of time for exploration and moments of discovery. The most compelling example he offers in his talk illustrates #4, “Say yes to your students’ ideas.” He imagines a student believing, for whatever reason, that 2+2=12. After following that path to the logical conclusion that 0 must equal 8, the student “discovers” modular arithmetic (think of a clock face with eight segments) in order to make the statement true. While Finkel doesn’t support leaving students with misinformation, he does encourage letting them follow an idea, even a crazy one, to see where it leads.
These two approaches, embedding important math concepts in real-world projects as well as providing ample opportunities to explore the joys of pure mathematics, form the foundation of Millennium’s philosophy of math education.