“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:

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The Types of Questions we Ask: which categories of questions should we focus on?

I think we can all agree that there are many different ways for our students to show what they know or understand, and that some problems ask for deeper understanding than others. In fact, many standardized math assessments, like PISA, aim to ask students questions at varying difficult levels (PISA uses 6 difficulty levels) to assess the same concept/skill. If we can learn one thing from assessments like these hopefully it is how to expect more of our students by going deeper… and in math class, this means asking better questions.

Robert Kaplinsky is a great example of an educator who has helped us better understand how to ask better questions. His work on Depth of Knowledge (DOK) has helped many teachers reflect on the questions they ask and has offered teachers examples of what higher DOK questions/problems look like.

In Ontario though we actually have an achievement chart that is aimed to help us think more about the types of questions/problems we expect our students be able to do. Basically, it is a rubric showing 4 levels of achievement across 4 categories. In Ontario it is expected that every teacher evaluate their students based on each the these categories. Many teachers, however, struggle to see the differences between these categories. Marian Small recently was the keynote speaker at OAME where she helped us think more about the categories by showing us how to delineate between the different categories of questions/problems:

  • Knowledge
  • Understanding
  • Application
  • Thinking

Knowledge vs. Understanding

Below are a few of Marian Small’s examples of questions that are designed to help us see the difference between questions aimed at knowledge and questions aimed at understanding:

As you can see from the above examples, each of the knowledge questions ask students to provide a correct answer. However, each of the understanding questions require students to both get a correct answer AND be able to show that they understand some of the key relationships involved. Marian’s point in showing us these comparisons was to tell us that we need to spend much more time and attention making sure our students understand the math they are learning.

Each of the questions that asks students to show their understanding also help us see what knowledge our students have, but the other way around is not true!

Hopefully you can see the potential benefits of striving for understanding, but I do believe these shifts need to be deliberate. My recommendation to help us aim for understanding is to ask more questions that ask students to:

  • Draw a visual representation to show why something works
  • Provide an example that fits given criteria
  • Explain when examples will or won’t work
  • Make choices (i.e., which numbers, visual representations… will be best to show proof)
  • show their understanding of key “Big Ideas” and relationships

Application vs. Thinking

Below are a few examples that can help us delineate the differences between application and thinking:

These examples might be particularly important for us to think about. To begin with, application questions often use some or all of the following:

  • use a context
  • require students to use things they already should know
  • provide a picture(s) or example(s) for students to see
  • provide almost all of the information and ask the student to find what is missing

Thinking questions, on the other hand, are the basis for what Stein et. al called “Doing Mathematics“. In Marian’s presentation, she discussed with us that these types of questions are why those who enjoy mathematics like doing mathematics. Thinking and reasoning are at the heart of what mathematics is all about! Thinking questions typically require the student to:

  • use non-algorithmic thinking
  • make sense of the problem
  • use relevant knowledge
  • notice important features of the problem
  • choose a possible solution path and possibly adjust if needed
  • persevere to monitor their own progress

Let’s take a minute to compare questions aimed at application and questions aimed at thinking. Application questions, while quite helpful in learning mathematics concepts (contexts should be used AS students learn), they typically offer less depth than thinking questions. In each of the above application questions, a student could easily ignore the context and fall back on learned procedures. On the other hand, each of the thinking questions might require the student to make and test conjectures, using the same procedures repeatedly to find a possible solution.

Ideally, we need to spend more time where our students are thinking… more time discussing thinking questions… and focus more on the important relationships/connections that will arise through working on these problems.

Final Thoughts

Somehow we need to find the right balance between using the 4 types of questions above, however, we need to recognize that most textbooks, most teacher-made assessments, and most online resources focus heavily (if not exclusively) on knowledge and occasionally application. The balance is way off!

Focusing on being able to monitor our own types of questions isn’t enough though. We need to recognize that relationships/connections between concepts/representations are at the heart of expecting more from our students. We need to know that thinking and reasoning are HOW our students should be learning. We need to confront practices that stand in the way of us moving toward understanding and thinking, and set aside resources that focus mainly on knowledge or application. If we want to make strides forward, we need to find resources that will help US understand the material deeper and provide us with good examples.

Questions to Reflect on:

  • What did your last quiz or test or exit card look like? What is your current balance of question types?
  • What resources do you use? What balance do they have?
  • Where do you go to find better Understanding or Thinking questions?
  • What was the last problem you did that made you interested in solving it? What was it about that problem that made you interested? Likely it was a Thinking question. What was it about that problem that made it interesting?
  • Much of the work related to filling gaps, intervention, assessment driving learning… points teachers toward students’ missing knowledge. How can we focus our attention more toward understanding and thinking given this reality?
  • How can we better define “mastery” given the 4 categories above? Mastery must be seen as more than getting a bunch of simple knowledge questions correct!
  • Who do you turn to to help you think more about the questions you ask? What professional relationships might be helpful for you?

If you haven’t already, please take a look at Marian Small’s entire presentation where she labels understanding and thinking as the “fundamentals of mathematics”

I’d love to continue the conversation about the questions we ask of our students.  Leave a comment here or on Twitter @MarkChubb3

The role of “practice” in mathematics class

A few weeks ago a NYTimes published an article titled, Make Your Daughter Practice Math. She’ll Thank You Later, an opinion piece that, basically, asserts that girls would benefit from “extra required practice”.  I took a few minutes to look through the comments (which there are over 600) and noticed a polarizing set of personal comments related to what has worked or hasn’t worked for each person, or their own children.  Some sharing how practicing was an essential component for making them/their kids successful at mathematics, and others discussing stories related to frustration, humiliation and the need for children to enjoy and be interested in the subject.

Instead of picking apart the article and sharing the various issues I have with it (like the notion of “extra practice” should be given based on gender), or simply stating my own opinions, I think it would be far more productive to consider why practice might be important and specifically consider some key elements of what might make practice beneficial to more students.


To many, the term “practice” brings about childhood memories of completing pages of repeated random questions, or drills sheets where the same algorithm is used over and over again.  Students who successfully completed the first few questions typically had no issues completing each and every question.  For those who were successful, the belief is that the repetition helped.  For those who were less successful, the belief is that repeating an algorithm that didn’t make sense in the first place wasn’t helpful…  even if they can get an answer, they might still not understand (*Defining 2 opposing definitions of “understanding” here).

“Practice” for both of the views above is often thought of as rote tasks that are devoid of thinking, choices or sense making.  Before I share with you an alternative view of practice, I’d like to first consider how we have tackled “practice” for students who are developing as readers.

If we were to consider reading instruction for a moment, everyone would agree that it would be important to practice reading, however, most of us wouldn’t have thoughts of reading pages of random words on a page, we would likely think about picture books.  Books offer many important factors for young readers.  Pictures might help give clues to difficult words, the storyline offers interest and motivation to continue, and the messages within the book might bring about rich discussions related to the purpose of the book.  This kind of practice is both encourages students to continue reading, and helps them continue to get better at the same time.  However, this is very different from what we view as math “practice”.

In Dan Finkel’s Ted Talk (Five Principles of Extraordinary Math Teaching) he has attempted to help teachers and parents see the equivalent kind of practice for mathematics:

Finkel Quote


Below is a chart explaining the role of practice as it relates to what Dan Finkel calls play:

practice2

Take a look at the “Process” row for a moment.  Here you can see the difference between a repetitive drill kind of practice and the “playful experiences” kind of  practice Dan had called for.  Let’s take a quick example of how practice can be playful.


Students learning to add 2-digit numbers were asked to “practice” their understanding of addition by playing a game called “How Close to 100?”.  The rules:

  • Roll 2 dice to create a 2-digit number (your choice of 41 or 14)
  • Use base-10 materials as appropriate
  • Try to get as close to 100 as possible
  • 4th role you are allowed eliminating any 1 number IF you want

close to 100b

What choice would you make???  Some students might want to keep all 4 roles and use the 14 to get close to 100, while other students might take the 41 and try to eliminate one of the roles to see if they can get closer.


When practice involves active thinking and reasoning, our students get the practice they need and the motivation to sustain learning!  When practice allows students to gain a deeper understanding (in this case the visual of the base-10 materials) or make connections between concepts, our students are doing more than passive rule following – they are engaging in thinking mathematically!


In the end, we need to take greater care in making sure that the experiences we provide our students are aimed at the 5 strands shown below:

strands of mathematical proficency.png
Adding It Up: Helping Children Learn Mathematics


You might also be interested in thinking about how we might practice Geometrical terms/properties, or spatial reasoning, or exponents, or Bisectors


So I will leave you with some final thoughts:

  • What does “practice” look like in your classroom?  Does it involve thinking or decisions?  Would it be more engaging for your students to make practice involve more thinking?
  • How does this topic relate to the topic of “engagement”?  Is engagement about making tasks more fun or about making tasks require more thought?  Which view of engagement do you and your students subscribe to?
  • What does practice look like for your students outside of school?  Is there a place for practice at home?
  • Which of the 5 strands (shown above) are regularly present in your “practice” activities?  Are there strands you would like to make sure are embedded more regularly?

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

Noticing and Wondering: A powerful tool for assessment

Last week I had the privilege of presenting with Nehlan Binfield at OAME on the topic of assessment in mathematics.  We aimed to position assessment as both a crucial aspect of teaching, yet simplify what it means for us to assess effectively and how we might use our assessments to help our students and class learn.  If interested, here is an abreviated version of our presentation:

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We started off by running through a Notice and Wonder with the group.  Given the image above, we noticed colours, sizes, patterns, symmetries (line symmetry and rotational symmetry), some pieces that looked like “trees” and other pieces that looked like “trees without stumps”…

Followed by us wondering about how many this image would be worth if a white was equal to 1, and what the next term in a pattern would look like if this was part of a growing pattern…

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We didn’t have time, but if you are interested you can see the whole exchange of how the images were originally created in Daniel Finkel’s quick video.

We then continued down the path of noticing and wondering about the image above.  After several minutes, we had come together to really understand the strategy called Notice and Wonder:

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As well as taking a quick look at how we can record our students’ thinking:

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Shared by Jamie Duncan


At this point in our session, we changed our focus from Noticing and Wondering about images of mathematics, to noticing and wondering about our students’ thinking.  To do this, we viewed the following video (click here to view) of a student attempting to find the answer of what eight, nine-cent stamps would be worth:

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The group noticed the student in the video counting, pausing before each new decade, using two hands to “track” her thinking…  The group noticed that she used most of a 10-frame to think about counting by ones into groups of 9.

We then asked the group to consider the wonders about this student or her thinking and use these wonders to think about what they would say or do next.

  • Would you show her a strategy?
  • Would you ask a question to help you understand their thinking better?
  • Would you suggest a tool?
    Would you give her a different question?

It seemed to us, that the most common next steps might not be the ones that were effectively using our assessment of what this child was actually doing.

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Looking through Fosnot’s landscape we noticed that this student was using a “counting by ones” strategy (at least when confronted with 9s), and that skip-counting and repeated addition were the next strategies on her horizon.

While many teachers might want to jump into helping and showing, we invited teachers to first consider whether or not we were paying attention to what she WAS actually doing, as opposed to what she wasn’t doing.


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This led nicely into a conversation about the difference between Assessment and Evaluation.  We noticed that we many talk to us about “assessment”, they actually are thinking about “evaluation”.  Yet, if we are to better understand teaching and learning of mathematics, assessment seems like a far better option!

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So, if we want to get better at listening interpretively, then we need to be noticing more:

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Yet still… it is far too common for schools to use evaluative comments.  The phrases below do not sit right with me… and together we need to find ways to change the current narrative in our schools!!!

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Evaluation practices, ranking kids, benchmarking tests… all seem to be aimed at perpetuating the narrative that some kids can’t do math… and distracts us from understanding our students’ current thinking.

So, we aimed our presentation at seeing other possibilities:

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To continue the presentation, we shared a few other videos of student in the processs of thinking (click here to view the video).  We paused the video directly after this student said “30ish” and asked the group again to notice and wonder… followed by thinking about what we would say/do next. b15

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Followed by another quick video (click here to view).  We watched the video up until she says “so it’s like 14…”.  Again, we noticed and wondered about this students’ thinking… and asked the group what they would say or do next.

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After watching the whole video, we discussed the kinds of questions we ask students:

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If we are truly aimed at “assessment”, which basically is the process of understanding our students’ thinking, then we need to be aware of the kinds of questions we ask, and our purpose for asking those questions!  (For more about this see link).


We finished our presentation off with a framework that is helpful for us to use when thinking about how our assessment data can move our class forward:

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We shared a selection of student work and asked the group to think about what they noticed… what they wonderered… then what they would do next.

For more about how the 5 Practices can be helpful to drive your instruction, see here.


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So, let’s remember what is really meant by “assessing” our students…

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…and be aware that this might be challenging for us…

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…but in the end, if we continue to listen to our students’ thinking, ask questions that will help us understand their thoughts, continue to press our students’ thinking, and bring the learning together in ways where our students are learning WITH and FROM each other, then we will be taking “a giant step toward becoming a master teacher”!


So I’ll leave you with some final thoughts:

  • What do comments sound like in your school(s)?  Are they asset based (examples of what your students ARE doing) or deficit based (“they can’t multiply”… “my low kids don’t get it…”)?
  • What do you do if you are interested in getting better at improving your assessment practices like we’ve discussed here, but your district is asking for data on spreadsheets that are designed to rank kids evaluatively?
  • What do we need to do to change the conversation from “level 2 kids” (evaluative statements that negatively impact our students) to conversations about what our students CAN do and ARE currently doing?
  •  What math knowledge is needed for us to be able to notice mathematicially important milestones in our students?  Can trajectories or landscapes or continua help us know what to notice better?

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


If you are interested in reading more on similar topics, might I suggest:

Or take a look at the whole slide show here