Turning Math Upside Down
Cathy Seeley is the former President of the National Council for Teaching Mathematics. She has spent the last several years talking about Turning Teaching Upside Down in mathematics and this article reviews what she is saying. Her phrase, Upside Down, depending where you are on the teaching spectrum will look a lot like problem based learning, or student-centered learning, or just the way we teach math. The name doesn’t matter. The base of it is what Seeley calls the Big 3:
- Understanding mathematics (making sense of it)
- Doing mathematics (skills, facts, procedures)
- Using mathematics (thinking, reasoning, applying, solving a range of problems)
In one sense, it’s very simple. “Students learn more when we let them wrestle with a math problem before we teach them how to solve it.” But that’s not intuitive. Doing before teaching doesn’t seem very clever at all. Making sense of it surely asks me to tell the student what and how to do something before attempting it! How can the student apply mathematical reasoning if she hasn’t been taught what mathematical reasoning is?
So the typical mathematics class is actually similar to the way most classes are structured:
- Present the concept (from a well-prepared lesson plan that follows a clear scope and sequence)
- Work through some examples
- Give individual feedback
- Respond to questions
- Assign homework
- Take up the homework the next day
In many mathematics textbooks, this process is followed in the same way. Notably, very rarely is extraneous information included in the examples, and the problems merely require the student to substitute. It has been clearly shown that the “able” student can actually do extremely well in mathematics without more than surface understanding. Summative assessments certainly do not typically discriminate those who truly understand from those who can apply the right formula. It is also true that in this system, those who were identified as “good” math students early on lived up to that prediction and so did those who “struggled”. There is actually a site providing clear instructions as to how to fake a pass including this glorious advice: “Learn formulas and get an idea of how to apply them. It always looks impressive if you have the formulas spot on and make some confused steps towards the outcome. It’s subjective what a good step and a poor/clueless step is in mathematics, and that definitely may earn you a pass, if you know how to do your thing….. That’s what you want – loads and loads of stuff in your temporary memory.”
Seeley suggests that you start in a different place with your math students:
- The students tackle a problem they may not know how to solve yet.
- We (students and teacher) talk together about your thinking and your work.
- Teacher helps connect the class discussion to the goal of the lesson.
This process builds on the knowledge that the brain ‘grows’ when it struggles productively with something difficult. Incidentally, Disney knew this already in the 1950s when it created Donald Duck in Mathmagicland. This video also identifies another key ingredient that Seeley identifies: making it relevant! ‘Why do we have to learn this’ is an entirely valid question. Mathematics can be relevant but often is taught in silos of knowing that shuts down interest altogether. How do you shut students’ thinking down, always unintentionally? Seeley offers a compelling list:
Focus on covering material • Teach bits and pieces, instead of chunks and clusters (of the curriculum, standards, test speciﬁcations) • Show them exactly what to do • Ask one too many questions • Answer all of their questions • Tell them if they’re right or wrong
The more effective way to open students’ thinking is, Seeley says, is through mathematical communication, students engaging, and constructively struggling with good problems. There are some helpful videos on this site illustrating this different way of approaching the teaching process using the students’ own thinking as the starting point. Dan Meyer calls this the 3 Act lesson.
- Introduce the central conflict of your story/task clearly, visually, viscerally, using as few words as possible (a video or image really helps)
- The protagonist/student overcomes obstacles, looks for resources, and develops new tools.
- Resolve the conflict and set up a sequel/extension.
An example at a higher level of mathematics is the following example:
A pre-calculus teacher puts a graph on the board with some coordinates labeled in two different colors. The teacher tells students there might be an error in the coordinates shown in red. Students work in pairs to discuss the posted work, considering whether there is a mistake and determining how they will make their case to the rest of the class. The teacher then convenes the class for a large-group discussion in which the students present their thinking to their peers, eventually coming to agreement about the correct solution. PBS has this example and many other resources as well.
When looking at these kinds of resources, Seeley suggests some questions that you might pose in your mathematics professional learning community:
- What kind of problem or task does the teacher use to start the lesson?
- How does the teacher encourage students’ thinking and stimulate student discourse?
- What kinds of questions does the teacher ask?
- What do you notice about the roles of the teacher and student?
- How does the teacher sequence students’ presentation of their work?
- How does the teacher connect the class discussion to the mathematical outcome of the lesson?
- How is this classroom similar to or different from your classroom or the other classrooms you see?
Whatever you call it, transforming mathematics teaching by making it engaging for the student is no longer a mystery and there are many fine resources to help you do it.