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?

Please leave a comment

Could you share the reference for the chart (PRIME)? Your post is intriguing. The “I, We, You” method is used a lot in the school where I teach. I’d like to share some “new” ideas with my colleagues.

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PRIME kits were created from Marian Small’s research. There are 5 kits that can be used to help determine what is important across different aspects of math, diagnostic material, next steps…

The last kit though is for system leaders. There you will find this chart. Here’s the kit:

http://www.prime.nelson.com/teacher/schoolmath.html

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You know what I’m going to say – I know I’m like a broken record. But when do I have days 1, 2, 3 … x on a single concept? How do I bundle concepts so that I can have multiple days to explore them? I’m used to teaching a new skill each day, which leaves little time for concept development. Can I just choose half my standards to teach next year?

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1 standard per day? I don’t think we can leave stuff out… The hard part is making all those connections.

What you are speaking about is the issue with “mile wide, inch deep” that has traditionally plagued math instruction.

We need to slow down and get to the big ideas behind the math.

What is a big idea in your year that connects content together?

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I do leave some things out. I teach 7th grade students who struggle and I’d rather have students exposed to some of the concepts and really get to understand them as opposed to flying through all topics and have them frustrated.

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Good stuff… I always love a post that gets me to rethink the way I do things!

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I think the key to choosing a particular approach lies with what you feel is in the best interest of your students. You are not choosing an approach based on what is easiest or most comfortable for the teacher, you are using your ongoing assessment to drive your instructional approach. Student response, student activity and student output drive the bus. With that said, in most situations, a constructivist approach should be used first because I this approach will provide you with feedback that identifies both skill development and conceptual understanding.

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Great post, Mark. I really like how you laid this out and connected the 3 approaches referenced from PRIME to Cathy Seeley’s upside-down teaching.

I was very intrigued by the PRIME resources and classes, are they only offered in Canada? I’m looking for effective training for HS math teachers to get them to shift away from I Do, We Do, You Do, and cold-call and response with single students at a time.

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I don’t think it’s as easy as having people read a book about different approaches. In order for us to imagine something different we have to experience it ourselves. This means that we need to spend time getting to know the concepts we teach deeper. I would recommend opportunities where teachers can get together and plan specific lessons based on specific content. You can’t get better at everything at the same time. Start small. Build experiences that help others imagine a better vision of learning mathematics that actually helps more students.

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