“The More Strategies, the Better?

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:

See VisualPatterns.org for more visual patterns

…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:

See VisualPatterns.org for more visual patterns

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:

Decomposing & Recomposing – How we subtract

Throughout mathematics, the idea that objects and numbers can be decomposed and recomposed can be found almost everywhere. I plan on writing a few articles in the next while to discuss a few of these areas. In this post, I’d like to help us think about how and why we use visual representations and contexts to help our students make sense of the numbers they are using.

Decomposing and Recomposing

Foundational to almost every aspect of mathematics is the idea that things can be broken down into pieces or units in a variety of ways, and be then recomposed again. For example, the number 10 can be thought of as 2 groups of 5, or 5 groups of 2, or a 7 and a 3, or two-and-one-half and seven-and-one-half…

Understanding how numbers are decomposed and recomposed can help us make sense of subtraction when we consider 52-19 as being 52-10-9 or 52-20+1 or (40-10)+(12-9) or 49-19+3 (or many other possibilities)… Let’s take a look at how each of these might be used:

The traditional algorithm suggests that we decompose 52-19 based on the value of each column, making sure that each column can be subtracted 1 digit at a time… In this case, the question would be recomposed into (40-10)+(12-9). Take a look:

52 is decomposed into 40+10+2
19 is decomposed into 10+9
The problem is recomposed into (40-10) + (12-9)

While this above strategy makes sense when calculating via paper-and-pencil, it might not be helpful for our students to develop number sense, or in this case, maintain magnitude. That is, students might be getting the correct answer, but completely unaware that they have actually decomposed and recomposed the numbers they are using at all.

Other strategies for decomposing and recomposing the same question could look like:

Maintain 52
Decompose 19 into 10+9
Subtract 52-10 (landing on 42), then 42-9
Some students will further decompose 9 as 2+7 and recompose the problem as 42-2-7
Maintain 52
Decompose 19 as 20-1
Recompose the problem as 52-20+1
Decompose 52 as 49-3
Recompose the problem as 49-19+3

The first problem at the beginning was aimed at helping students see how to “regroup” or decompose/recompose via a standardized method. However, the second and third examples were far more likely used strategies for students/adults to use if using mental math. The last example pictured above, illustrates the notion of “constant difference” which is a key strategy to help students see subtraction as more than just removal (but as the difference). Constant difference could have been thought of as 52-19 = 53-20 or as 52-19 = 50-17, a similar problem that maintains the same difference between the larger and smaller values. Others still, could have shown a counting-on strategy (not shown above) to represent the relationship between addition and subtraction (19+____=53).

Why “Decompose” and “Recompose”?

The language we use along with the representations we want from our students matters a lot. Using terms like “borrowing” for subtraction does not share what is actually happening (we aren’t lending things expecting to receive something back later), nor does it help students maintain a sense of the numbers being used. Liping Ma’s research, shared in her book Knowing and Teaching Elementary Mathematics, shows a comparison between US and Chinese teachers in how they teach subtraction. Below you can see that the idea of regrouping, or as I am calling decomposing and recomposing, is not the norm in the US.

Visualizing the Math

There seems to be conflicting ideas about how visuals might be helpful for our students. To some, worksheets are handed out where students are expected to draw out base 10 blocks or number lines the way their teacher has required. To others, number talks are used to discuss strategies kids have used to answer the same question, with steps written out by their teachers.

In both of these situations, visuals might not be used effectively. For teachers who are expecting every student to follow a set of procedures to visually represent each question, I think they might be missing an important reason behind using visuals. Visuals are meant to help our students see others’ ideas to learn new strategies! The visuals help us see What is being discussed, Why it works, and How to use the strategy in the future.

Teachers who might be sharing number talks without visuals might also be missing this point. The number talk below is a great example of explaining each of the types of strategies, but it is missing a visual component that would help others see how the numbers are actually being decomposed and recomposed spatially.

If we were to think developmentally for a moment (see Dr. Alex Lawson’s continuum below), we should notice that the specific strategies we are aiming for, might actually be promoted with specific visuals. Those in the “Working with the Numbers” phase, should be spending more time with visuals that help us SEE the strategies listed.

Aiming for Fluency

While we all want our students to be fluent when using mathematics, I think it might be helpful to look specifically at what the term “procedural fluency” means here. Below is NCTM’s definition of “procedural fluency” (verbs highlighted by Tracy Zager):

Which of the above verbs might relate to our students being able to “decompose” and “recompose”?

Some things to think about:

  • How well do your students understand how numbers can be decomposed and recomposed? Can they see that 134 can be thought of as 1 group of 100, 3 groups of 10, and 4 ones AS WELL AS 13 groups of 10, and 4 ones, OR 1 group of 100, 2 groups of 10, and 14 ones…….? To decompose and recompose requires more than an understanding of digit values!!!
  • How do the contexts you choose and the visual representations you and your students use help your students make connections? Are they calculating subtraction questions, or are they thinking about which strategy is best based on the numbers given?
  • What developmental continuum do you use to help you know what to listen for?
  • How much time do your students spend calculating by hand? Mentally figuring out an answer? Using technology (a calculator)? What is your balance?
  • How might the ideas of decomposing and recomposing relate to other topics your students have learned and will learn in the future?
  • Are you teaching your students how to get an answer, or how to think?

If you are interested in learning more, I would recommend:

I’d love to continue the conversation about assessment in mathematics.  Leave a comment here or on Twitter @MarkChubb3