What are some best practices for teaching high school mathematics?

By Kevin Reins

This week I read the responses of a two part series in Ed Week by Larry Ferlazzo. The question of the week was, “What are some best practices for teaching high school mathematics?” This intrigued me as I was preparing the 18th revision of my secondary mathematics methods course here at USD.

“…there are a zillion different instructional strategies and practices that math teachers can use in high school.”

The focus of part 1 was on these ‘instructional strategies and practices,’

  • teach to big ideas (see image above), it allows students to have interconnected schema
  • focus on the processes and connections between different processes
  • use instructional routines (see Why instructional routines?)
  • keep a record of conversations when you orchestrate full group discussions
  • be selective and cautious in your use of technology
  • incorporate high leverage long-term strategies
    • -David Wees
  • embrace mistakes, normalizing mistakes, safe space for discussion and correction, utilize error analysis
  • formal error analysis through test corrections
    • -Jillian Henry
  •  provide relevance and contexts for the mathematics
  • engage students in a variety of practices and strategies
  • provide scaffolds for those who need the extra support when working with challenging content
  • plan intentionally and deliberately so your instruction is impactful, consistent, and effective
  • develop a community of learners where group participation and interaction is expected
  • employ student-centered teaching and learning
  • provide opportunities for students to develop and strengthen their skills of mathematical communication (including vocabulary)
  • make the development of a variety of problem-solving techniques a priority
  • eliminate the blank paper; require students to write (1) determine a strategy that could be used to solve the problem, (2) write a question that you have about the problem, (3) record everything you know about the content related to the problem.
  • develop their ability to ask good questions during problem solving phases, Entry (getting started), Moving (when stuck), Reflection (thinking about thinking), and Extension (deeper thinking).
  • utilize graphic organizers to help them employ processes independently
    • -Tammy Jones & Leslie Texas
  •  What works at elementary or middle level works for high school
  • pose interesting problems or set the stage for students to pose interesting questions/problems about the situation
  • encourage investigations, experiments, collaboration, and discourse as students explore problems
  • expect representations or models for the problems being investigated
  • engage students in discourse, creating mathematical arguments and critiquing the reasoning of others
  • proving their work with both formal and informal proofs 
    • Anne Collins

Part 2: Students must ‘engage in math problem-solving’ and not just ‘follow procedures.’

The acquisition of best practices for teaching high school mathematics is necessary for student academic success.

The focus of part 2 was on engaging students in problem-solving. The following was said by the experts interviewed,

  • you must have as your guiding philosophical principle the belief that all students can learn
  • you must provide opportunities for them to fall in love with learning
  • Standards for Mathematical Practices can serve as a guide for the ways students need to be engaged in mathematics
  • choose open-ended problems
  • focus more on the process rather than the correct answers
  • challenge them with mathematically rigorous tasks, choosing a Higher-Level Demand Task
  • learn how to anticipate student responses and misconceptions for tasks
  • ask students to find multiple strategies to the tasks you present
  • learn how to help students learn from mistakes
    • -Wendy Monroy, LA math coach
  • math learning should be developing conceptual understandings of the mathematics
  • focus on the conceptual relationships
  • create a synergy between the lower levels and higher levels of thinking through inquiry
  • create a social environment that promotes team work and collaboration
  • provide an open, secure environment that allows for mistakes as a part of the learning process
  •  use an inductive teaching approach (vs deductive)
  • reduce teacher talk time (increase productive mathematical discourse)
  • differentiate by content, product, and affect (Tomlinson)
  • use all types of assessment; visible thinking routines, “I use to think… Now I think…” (Harvard University’s Project Zero)
  • use a flexible fronts layout of the classroom which encourages more collaboration
    • -Jennifer Chang Wathall, educational consultant in concept-based mathematics/curriculum
  •  give challenging problems that build patience and persistence in their maturing problem solving skills
  • then spend ample time in joyful struggle
  • create rich mathematical dialogue that leaves the building
  • 12 challenging problems that 5 of which will appear on the final, and give them time in class to work on them (e.g., A point P, inside a square, was 3, 4, and 5 units away from three of the corners. Find the length of the side of the square.)
    • -Sunil Singh, author of Pi of Life: The Hidden Happiness of Mathematics
  • sufficient time to make sure that students know how to solve problems using different methods
  • look for opportunities for students to have multiple entry points or strategies for solving a problem
  • take time to discuss strategic choices
  • find flaws in short cuts and when certain methods won’t work
  • open their mind to new and different approaches
    • -Matthew Beyranevand, author of Teach Math Like This, Not Like That: Four Critical Areas to Improve Student Learning.

So after bolding all of the big ideas of the laundry list of instructional strategies and practices that were provided I compared it to the content that I normally would teach in my secondary mathematics methods course. The result was two ideas, one new, and one that could use a deeper focus. I would like to incorporate more ideas on how to utilize math mistakes in the classroom as learning opportunities. I also would like to explore instructional routines a bit more.

To think more deeply about embracing mistakes, normalizing mistakes, and creating a safe space for discussion and correction, I think it is important to start off knowing what some common math mistakes in high school are. I found Math Mistakes website that does just that. This should be a good start for a discussion on how to utilize some of these mistakes when they pop up as a learning opportunity.

With respect to instructional routines, I read Why instructional routines? It turns out I know what they are and how one should utilize tasks in teaching. One instructional routine that David Wees talks about is, Contemplate then Calculate, as a tool for learning how to use the 5 Practices for Orchestrating Productive Mathematical Discussions. The high level goals of Contemplate then Calculate are to support students in surfacing and naming mathematical structure, more broadly in pausing to think about the mathematics present in a task before attempting a solution strategy, and even more broadly in learning from other students’ different strategies for solving the same problem.

“Instructional [routines] are tasks enacted in classrooms that structure the relationship between the teacher and the students around content in ways that consistently maintain high expectations of student learning while adapting to the contingencies of particular instructional interactions.” Kazemi, E., Franke, M., & Lampert, M. (2009)

I’m looking forward to expanding my teaching and learning opportunities to include both instructional routines and normalizing mistakes.

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