As many teachers implement number talks/math strings and lessons where students are learning through problem solving, the idea that there are many ways to answer a question or problem becomes more important. However, I think we need to unpack the beliefs and practices surrounding what it means for our students to have different “strategies”. A few common beliefs and practices include:
Really, there are benefits and issues with each of these thoughts…. and the right answer is actually really much more complicated than any of these. To help us consider where our own decisions lie, let’s start by considering an actual example. If students were given a pattern with the first 4 terms like this:
…and asked how many shapes there would be on the 24th design (how many squares and circles in total). Students could tackle this in many ways:
- Draw out the 24th step by building on and keeping track of each step number
- Build the 24th step by adding on and keeping track of the step number
- Make a T-table and use skip counting to find each new step (5, 9, 13, 17…).
- Find the explicit rule from the first few images’ data placed on a T-table (“I see the pattern is 5, 9, 13, 17. each new image uses 4 new shapes, so the pattern is a multiplied by 4 pattern…. and I think the rule should be ‘number of images = step number x4+1’. Let me double check…”).
- Notice the “constant” and multiplicative aspects of the visual, then find the explicit rule (I see that each image increases by 4 new shapes on the right, so the multiplicative aspect of this pattern is x4, and term 0 might just be 1 circle. So the pattern must be x4+1″).
- Create a graph, then find the explicit rule based on starting point and growth (“When I graph this, my line hits the y-axis at 1, and increases by 4 each time, so the pattern rule must be x4+1”).
While each of these might offer a correct answer, we as the teacher need to assess (figure out what our students are doing/thinking) and then decide on how to react accordingly. If a student is using an additive strategy (building each step, or creating a t-table with every line recorded using skip counting), their strategy is a very early model of understanding here and we might want to challenge this/these students to find or use other methods that use multiplicative reasoning. Saying “do it another way” might be helpful here, but it might not be helpful for other students. If on the other hand, a student DID use multiplicative reasoning, and we suggest “do it another way”, then they fill out a t-table with every line indicated, we might actually be promoting the use of less sophisticated reasoning.
On the other hand, if we tell/show students exactly how to find the multiplicative rule, and everyone is doing it well, then I would worry that students would struggle with future learning. For example, if everyone is told to make a t-table, and find the recursive pattern (above would be a recursive pattern of+4 for total shapes), then use that as the multiplicative basis for the explicit rule x4 to make x4+1), then students are likely just following steps, and are not internalizing what specifically in the visual pattern here is +4 or x4… or where the constant of +1 is. I would expect these students to really struggle with figuring out patterns like the following that is non-linear:
Students told to start with a t-table and find the explicit pattern rule are likely not even paying attention to what in the visual is growing, how it is growing or what is constant between each figure. So, potentially, moving students too quickly to the most sophisticated models will likely miss out on the development necessary for them to be successful later.
While multiple strategies are helpful to know, it is important for US to know which strategies are early understandings, and which are more sophisticated. WE need to know which students to push and when to allow everyone to do it THEIR way, then hold a math congress together to discuss relationships between strategies, and which strategies might be more beneficial in which circumstance. It is the relationships between strategies that is the MOST important thing for us to consider!
Focusing on OUR Understanding:
In order for us to know which sequence of learning is best for our students, and be able to respond to our students’ current understandings, we need to be aware of how any particular math concepts develops over time. Let’s be clear, understanding and using a progression like this takes time and experience for US to understand and become comfortable with.
While most educational resources are filled with lessons and assessment opportunities, very few offer ideas for us as teachers about what to look for as students are working, and how to respond to different students based on their current thinking. This is what Deborah Ball calls “Math Knowledge for Teaching”:
If any teacher wants to improve their practice, I believe this is the space that will have the most impact! If schools are interested in improving math instruction, helping teachers know what to look for, and how to respond is likely the best place to tackle. If districts are aiming for ways to improve, helping each teacher learn more about these progressions will likely be what’s going to make the biggest impacts!
Where to Start?
If you want to deepen our understanding of the math we teach, including better understanding how math develops over time, I would suggest:
- Providing more open questions, and looking at student samples as a team of teachers
- Using math resources that have been specifically designed with progressions in mind (Cathy Fosnot’s Contexts for Learning and minilessons, Cathy Bruce & Ruth Beatty’s From Patterns to Algebra, Alex Lawson’s What to Look For…), and monitoring student strategies over time
- Anticipating possible student strategies, and using a continuum or landscape (Cathy Fosnot’s Landscapes, Lawson’s Continua, Clement’s Trajectories, Van Hiele’s levels of geometric thought…) as a guide to help you see how your students are progressing
- Collaborate with other educators using resources designed for teachers to deepen their understanding and provide examples for us to use with kids (Marian Small’s Understanding the Math we Teach, Van de Walle’s Teaching Student Centered Mathematics, Alex Lawsons’s What to Look For, Doug Clements’ Learning and Teaching Early Math…)
- Have discussions with other math educators about the math you teach and how students develop over time.
Questions to Reflect on:
- How do you typically respond to your students when you give them opportunities to share their thinking? Which of the 3 beliefs/practices is most common for you? How might this post help you consider other beliefs/practices?
- How can you both honour students’ current understandings, yet still help students progress toward more sophisticated understandings?
- Given that your students’ understandings at the beginning of any new learning differ greatly, how do you both learn about your students’ thoughts and respond to them in ways that are productive? (This is different than testing kids prior knowledge or sorting students by ability. See Daro’s video)
- Who do you turn to to help you think more about the math you teach, or they ways you respond to students? What professional relationships might be helpful for you?
- What resources do you consult to help you develop your own understanding?
I’d love to continue the conversation about how we respond to our students’ thinking. Leave a comment here or on Twitter @MarkChubb3
If interested in this topic, you might be interested in reading:
Thinking Mathematically 