So I have been thinking a lot about this chart found in PRIME:
There is a lot to take in here, but I want to point your attention to the “Goal” and “Roles” rows. Take a look again at these two rows. I think a lot of the differences we see between classrooms, between lessons, between the beliefs we hear online… comes back to what our GOALs are… and therefore, our ROLES.
For some, their goal is to fill gaps, make sure our students can do a skill or group of skills. When this is our goal, likely we are explicitly teaching. The role of the student is to listen, or as the chart says, “passively” listen. The focus of the mathematics is typically procedural and symbolic. Learning happens here by the teacher showing, students trying to imitate the teachers’ procedures. With this approach, “learning” comes from the teacher, is passed on to the student, then time is given so a student can master what the teacher just showed them. The belief system at play here is that math is about remembering and following steps/rules!
For others, their goal is to guide students carefully into a deep understanding of the math… to connect learning together… to build conceptual understanding. The teacher’s role here is to guide not tell, and the students role is to make things make sense. There is likely more discourse happening here because the students are taking on more of an active role as they are expected to develop meaning. Learning happens through lessons, but in a different way than a skills approach. Here learning isn’t transmitted from the teacher to the student, students take an active role in understanding. The teacher might allow time for students to talk with their neighbor, or work independently for a moment, or explore visual representations, or share their thinking… The goal of this lesson is conceptual understanding, but also the development of procedures that make sense. The belief system here is that visual representations and contexts can help students make sense. Students are capable of developing a conceptual understanding which leads to a bridge between concepts and procedures…
Still, for others, their goal is to develop mathematical thinkers. Students being able to follow rules isn’t good enough for these teachers. They want students who can make connections between concepts, and develop reasoning skills. Relational understanding and mathematical reasoning are their goals! Using the process expectations are how these students learn. Learning starts with low floor/ high ceiling problems where students can answer in ways that make sense to them. From there students share their thinking and learn WITH and FROM each other. The teacher’s role is more complicated here, since it relies on the students’ thinking. The 5 Practices are used to make sure the learning is deep and meaningful. The belief system here is that students learn best when they have had enough time to grapple with the thinking themselves first… and that learning HOW to think mathematically will help you with future learning!
I don’t want to make it sound like there are 3 different types of teachers… or that one of these is a better way to teach or learn. In fact, I am not sure any teacher is in any one of these columns every day. Rather, I believe that we are likely moving in and out of these different teaching approaches regularly.
I do, however, believe that there are two different ways of thinking about these approaches though:
- Moving from Skills, to Concepts, to Problems
- Moving from Problems, to Concepts and Procedures
This might seem like a subtle difference, but I think it is something that we need to reflect on, and better understand others’ thinking.
Moving from skills to concepts and then finally to problem solving can be thought of as a Teaching FOR Problem Solving approach. Taking this stance seems to align with the goals and beliefs of the Skills Approach in that learning happens FROM the teacher and only when students are ready can they solve problems.
On the other hand, moving from problems to concepts and procedures can be thought of as a Teaching THROUGH Problem Solving approach. Taking this stance seems to align with the goals and beliefs of the Conceptual/Constructivist approach.
This is a really powerful caption from Cathy Seeley’s book Making Sense of Math: How to Help Every Student Become a Mathematical Thinker and Problem Solver. But again it shows me what the goal of the author is, “developing mathematical proficiency that includes the development of thinking skills and the ability to tackle problems that may not fit a particular format.” Let me say that last part again… “…tackle problems that may not fit a particular format.” A skills approach does not have this as the goal… rather the goal is typically simple… master this one thing today.
Take a look further into her thinking about how we can develop mathematically proficient students:
“Upside-down teaching.” Hmmm. It sounds like she is telling us that a Gradual Release of Responsibility model doesn’t work in mathematics. However, I think it has more to do with our goals. If my goal is to develop skills, then we are likely going to start with direct instruction. The problem with this is, I will be building students reliant on MY thinking. Taking the “upside-down” approach listed here, or the teaching THROUGH problem solving approach I mentioned earlier should help us build mathematicians who can reason mathematically, make sense of concepts and ultimately have a relational understanding! But this leaves us with the question, what order and for how long should I be using each approach?
I really want you to think about what day 1 looks like for any topic you teach… and what day 2 looks like… and day 3… and day X. Here is a sketch a friend shared with me (they found it on twitter). What would your graphic look like? What might be similar? Or different?
The reason I’m writing this blog post is for us to really consider if we are reaching our goals… and to reflect on our own beliefs.
Does the role you take match your goals?
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