This blog post is about how the math department at my school in Suzhou, China implemented changes to the way we taught Math 10 and 11 to incorporate data and research from cognitive science to help our students learn better. I include a summary of what we learned, and some ideas for improvement.

 

It began last summer, at math camp. Yes, I attended math camp as a fully-fledged adult! Yes, there were other adults present. And yes, it was awesome! (Officially named the “Summer Math Conference for Teachers” but let’s not get into the nitty gritty). One of my favourite sessions was the one led by Sheri Hill, Arian Rawle, and Lindsay Kueh on the grade 10 course redesign they have implemented in at their school in Ontario. The course redesign is based on research and best pedagogical practices outlined in the book Make It Stick, The Science of Successful Learning by Peter C. Brown, Henry L. Roediger III, and Mark A. McDaniel.

Book Synopsis: Why is it that students seem to understand what is being taught in class but end up failing when it comes to test day? How does one progress from fluency to mastery over challenging content? Many common study habits like re-reading and highlighting text create illusions of mastery but are in fact completely ineffective. This books explores insights from research in cognitive science on learning, memory, and the brain, as well its implications on teaching and learning. 

​After the session, I couldn’t wait to bring these ideas back to the math team at my school in Suzhou, China so we could start putting them in action too! We began by looking at issues we noticed our students faced:

 

• Not knowing, understanding, or practicing math vocabulary
• Low retention rates of material from one year to the next (in some cases from one week to the next!)
• Lack of basic skills (algebra, numeracy)
• Low perseverance
• Low completion rates for homework

We made it our goal to address some of the issues above, taking many ideas directly from the session presented by Hill, Rawle, and Kueh.

Like Hill, Rawl, and Kueh, we removed unit tests, which freed up a significant amount time for other topics and activities. Instead, we moved to weekly cumulative quizzes that held students accountable to everything they have learned in class up to the Friday before quiz day (no skills expire!).

The weekly schedule looks as follows:

​Fast Fours. Four warm up questions printed front and back on half a sheet of A4 paper that’s ready for students as they walk into class. Each question relates to a different math topic that may be review from previous years, numeracy focused, or review of current material. By mixing up the problem types, we are introducing interleaving to students, the idea that we learn better when multiple topics or subjects are woven into the same learning session.

 

If you are interested in redesigning your course or looking for ideas on where to begin, I would say the Fast Fours are the easiest to implement. They work well because in all likelihood, every student is able to answer at least one question out of four, it is low stakes (not graded), gives you time to check in with students at the beginning of the class, check homework, answer questions, and it also gives students an opportunity to collaborate and help each other. (Scroll to the bottom of this post to see examples of these documents).

Weekly Quizzes. The quizzes themselves are one-page, double-sided documents comprised of three main sections. Part A focuses on vocabulary where we ask students to match key terms or fill in the blanks. Part B is review of previously learned material, and may include basic algebra questions from previous grade levels. Part C is new material that was covered the week before.

The quizzes only take up half a block and we drop the lowest two quizzes at the end of the year. If students are away for a quiz, they do not write the quiz and instead it counts as one of their dropped quizzes. 

Problem Solving/Project Days. At the start of the semester, we wanted to implement problem solving days that helped students dive deeper into the content they learned throughout the week and try some more challenging problems. Alternatively, our vision was to use these days as project days.

Fun Fridays. Our Friday blocks are shortened and many teachers find this time unproductive for teaching new material, which made having Fun Fridays built into our schedule a good fit. During this time, we may play a fun review game with students, have them explore an activity on Desmos, or one might even teach them something outside of the prescribed curricular content like coding.

Homework. In their original course redesign, Hill, Rawle and Kueh wrote customized homework assignments that introduced the ideas of interleaving and spaced practice to their students. That is, their homework assignments would begin a set of ten mandatory questions: five questions from previous material, and five questions from the lesson, as well as one or two challenge questions.

Unfortunately, our team was unable to implement so many changes at once, so we simply kept homework the same, and instead implemented randomized homework checks. Our hopes were to emphasize the importance of practice, and keep students accountable for it. 

Some of my takeaways from this semester:
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1. Fast Fours. I’m definitely keeping the fast fours in my classes. In their of year reflections, students mentioned that this was one of their favourite things to do because it helped them remember content they had not practiced for a while, and they were able to get immediate feedback on it. One student suggested having a balance of easy and more difficult questions for those who finish early (perhaps a 3:1 easy to challenging ratio).
2. Weekly quizzes. Since the weekly quizzes introduce interleaved and spaced practice, they reduce the need for large blocks of class time devoted to final exam review as we were continuously reviewing content throughout the entire semester. In terms of grading, however, it is important to keep up with it to ensure that students get feedback before the next quiz. An outcome we did not expect was that despite seeing several iterations of the same types of questions, students continued to struggle with the finance unit and were unable to identify the correct formula to use for the question.
3. Vocabulary. As a department, we found it extremely valuable to teach, review, repeat, and practice math specific vocabulary to help students increase fluency and be better equipped to answer difficult problems. Many Chinese students arrive in our classes already having much of the essential background knowledge in math but lack the English skills to succeed, so we have found this to be a fruitful endeavor. We plan to begin our Math 10 classes with a mini vocabulary unit to give students started with some common terminology and foundational knowledge for the upcoming semester.
4. Problem Solving/Project Days. Problem solving was a lot harder to implement, and we did not have a clear structure for it. As a result, Thursdays were mainly used for projects or as additional lesson days.

Overall, the teachers in my department felt the changes were worthwhile to implement and will continue with the same program for semester two, with a few new projects that we’ll be adding to some units that did not have one. In the future, I’d like to rethink how we might implement problem solving days in a more structured way. 

Finance Unit. I’m unhappy with the way finance is currently being taught to our students, and I think we can do better. I remember very little in the way of learning about finance when I was in high school. This was usually the topic my teachers skimmed over, and hence my dislike for it as a teacher now myself. In textbooks, it is usually presented as a series of formulas and how to apply those formulas, which is, I think, an area where we are doing our students a disservice. I’m a strong believer in getting students to first understanding the math behind the formulas, and some of these formulas (like the loan formula, for instance), does not lend itself well to building students’ conceptual understanding of it at the grade 10 level.

 

A useful analogy from Barbara Oakley’s course Learning How to Learn goes like this: a formula is like a summary, it describes several important ideas that mathematicians have packaged into a simple and elegant mathematical statement. Take Newton’s second law of motion for example, which is stated formally as, “The acceleration of an object as produced by a net force is directly proportional to the magnitude of the net force, in the same direction as the net force, and inversely proportional to the mass of the object” (physicsclassroom.com). Simply put, the relationship can be condensed into the mathematical formula f = ma.  As such, we must understand the meaning behind each symbol and look at how they work together to tell a story.

My plan is to put together a rough plan for how we can revamp the finance unit, pitch these ideas to my team before the start of semester two, and see if we can collectively find a way to improve the way we teach this unit to our students (more to come in a later post).  

Active Retrieval. Next semester, I plan to pause frequently during lessons to quiz my students on material. I’ll ask them to put their away notes, and engage in some simple recall exercises. A useful analogy to think about this is described in Make It Stick; Dr. Wenderoth, a biology professor at the University of Washington tells her students to “Think of your minds as a forest, and the answer is in there somewhere… The more times you make a path to find it, the stronger that path will become.” This is exactly what happens in your brain as you engage in active retrieval to strengthen new neural connections as you gain new knowledge or learn new skills. 

Elaboration. Once a week, I will ask students to complete a written reflection or summary of ideas learned throughout the week in their own words, with added connections and extensions if applicable. It will be a five sentence summary of concepts learned, with enough detail to help recall important ideas when it is read it at a later date, not too much detail that students end up reciting the entire lesson.