Transitioning to Mental Models for Thinking

I had another interesting experience while listening to a student thinking about math today.  This time I had the opportunity to listen to a grade 4 student’s thinking on the following question:

“I am reading a book with 207 pages in it.  I have read 188 pages so far.  How many more pages do I need to read to finish the book?”

This question is a part of our board-created interview assessment that provides teachers with an opportunity to dig into student thinking with questions that address the Quantity and Operational Sense Big ideas in our Number Sense and Numeration curriculum.  Missing addend questions, like the one above, seem to pose a particular challenge for our students, according to our EQAO data (and confirmed by our teachers).  So, these questions are a mainstay across the grades in our assessment.

One particular grade 4 boy was working on this problem and he asked to use the hundreds math rack.  I was curious about his choice of tool and so rather than giving him one, I asked him to explain to me his plan for solving the problem.  This is what he very clearly articulated to me:

“First I would make 188 on the math rack” (which I found interesting, since he had asked for only one 100 math rack).  “Then I would add ten to get 198 and then I would add 10 more to get 208” (with no hesitation on 208).  “But I only want 207, so I would take 1 away.  Then I could see what the answer is.”  WOW!

I was very excited and troubled at the same time in hearing this.  What a fabulous strategy, but why did he feel he needed the math rack to figure out what he had already figured out?  I recorded his words on his paper (afterwards I reflected that I could have perhaps modelled his thinking on the number line, but considering this was an assessment of his thinking and he hadn’t gone to the number line himself, perhaps this could go down as a next step for him).  I had him read back his words and I asked him if he could figure out his answer now without the math rack.  He thought out loud “10, 20, 19 – 19 more pages!”  I celebrated with him that he had used such an efficient strategy – and used a mental model to do it!

I’ve thought a lot about what might have happened if this student had immediate access to the math rack.  Would he have found himself getting lost in trying to model 188 on this tool, possibly forgetting the purpose of doing it in the first place?  Would there have been so many steps that he might have made an error.  The solution he articulated to me was so efficient that there practically wasn’t room for error.  It also highlighted his understanding of counting on by 10’s from any number (such an important understanding/skill for number line use).  To me, this student already has created a useful mental model that he can use for addition and subtraction, yet he is still closely attached to his concrete math rack.

I think some next steps for this student would include more experience with similar contextual situations, where his thinking could be modelled (by a teacher at first) to an open number line.  In this question, the numbers could be held onto mentally, but what would happen with larger numbers or numbers with a greater difference?  Encouraging a more flexible, mental tool may empower him to work with more challenging numbers and various contexts.

This experience is causing me to wonder again about the intentional use of concrete tools and the transition to graphic and mental models.  I do believe that concrete models/manipulatives are very important in building conceptual understanding and exploring strategies and big ideas.  However, we need to know how to identify when these concrete tools become crutches and may actually prevent students from further building flexible, efficient and accurate ways of solving mathematical problems.  We need to be prepared with a deep understanding of a variety of models, including flexible graphic models (like open number lines, arrays, rectangles for partitioning, etc.).  We need to know how to support students to see their thinking on different models and make connections between them, with the intention of creating powerful mental tools for thinking over time and with varied experiences.

Using concrete models for computational thinking is not the end goal, but is an important step in the journey.   Becoming fluent thinkers with a variety of strategies and tools based on conceptual understanding is the end goal.  I am looking forward to continuing to learn more about how to support students in making this transition towards flexible mental models.


“6 take away 6 is 5” – Listening for Understanding

I had the opportunity to be in several classes during the first week of class as teachers from NNDSB were implementing a program my colleague Jane Rutledge (@MathJane) and I wrote called the “The First X Days”.  It provides teachers with 9 lessons for the first 2 weeks of school to support the development of a non-threatening classroom environment, along with important norms and routines to set the stage for a great year of math learning.  Throughout the 9 lessons, students have an opportunity to engage in a variety of hands-on, creative and visual mathematics, inspired greatly by Jo Boaler’s work and teachers have the opportunity to get to begin to get to know their students as they observe their students working through these engaging problems and discussions.

I was in one particular grade 1 class during the conversation “What is math?” which extended into “When do you use math?”.  Students shared ideas that included numbers, shapes, patterns, cooking, addition, subtraction, etc.  Our math walk uncovered lots of math all around us that we enjoyed discussing.  One particular conversation really intrigued me as a student shared her understanding of subtraction and how she knew that 6 take away 6 is 5. Hmmm… Instead of correcting the student, the classroom teacher wisely asked the student if she could show us what she meant.  She proudly counted out 6 of her fingers (5 on one hand plus her thumb on the other hand) and then she put down her thumb and again repeated “6 take away 6 is 5”.  I was starting to understand her thinking around this!  The teacher was still curious about what she meant, so she asked her if she could draw for us how 6 take away 6 is 5.  (I was appreciating this educator’s curiosity and focus on visual representations!)  The young student gladly came up to the chart paper and wrote out the numbers 1, 2, 3, 4, 5, 6 and then she carefully crossed out the 6.  She clearly represented her thinking as she pointed to the 5.  Of course!  After crossing out the 6, there was 5 left!


This was an excellent example of the importance of teacher content knowledge when listening and observing students with the purpose of uncovering student starting points.  In this case, the teacher having an understanding the components of counting and quantity as it relates to student development was essential for identifying this student’s starting point and learning needs.  This young girl wasn’t completely incorrect in her understanding of subtraction.  She knew that if she took that single digit 6 or that one thumb away, then she’d be left with 5.  She had an understanding of “take away” and she was able to easily count and represent the digits to 6, demonstrating one-to-one correspondence and stable order.  I wondered if her learning need was around the concept of cardinality.

Cardinality:  the idea that the last count of a group of objects represents the total number of objects in the group. (Guide to Effective Instruction, Grade 1-3 Number Sense and Numeration, 2016) 

To her that single thumb represented 6, not 1!  If an educator listened to this without probing further and having an understanding of cardinality, that educator might be very concerned that this student was exhibiting some serious errors in her thinking, rather than realizing she was in a certain place on her landscape or trajectory of learning.  Once we understood that she was in need of experiences to learn about cardinality, our planning could be very precise to celebrate and acknowledge her starting points and then include a variety of games and tasks to support the development of this big idea.

The more I understand the important big ideas, strategies and models for students to develop conceptual and procedural knowledge, the clearer my listening, wonderings and interpretations become as I observe students at work.  We know that meeting students on the cusp of their understanding is essential to helping them build connections to new learning.  In order to do this, we need to continue to develop our content knowledge for teaching math.  These are concepts that we likely didn’t learn in our own schooling, so this is challenging work that lies ahead for us; work that is critical to best support our students in becoming skilled mathematicians.

So, the next time I hear a student say something that clearly sounds incorrect to me, I will pause and think of this student who was confident in saying that “6 take away 6 is 5”.   Just like with her, there may be understanding buried within the statement, along with hints for a next step.  It’s up to us to uncover that understanding and respond appropriately.

Common Ground

New School Year Resolution – Get going with my blog!  I’ve been intending to begin a website for a few years now, with only one entry to show for it.  There have been many times I’ve written most of a blog in my head, only to not get it onto my website.  This week that happened again, however whether it was because of how strongly I was feeling about the topic, or because I could carve out an hour this morning, I actually am putting the words down and sharing some of my thoughts on an emotionally charged subject – the statements made by the conservative party this week regarding changes in the way we teach math.

I am a common ground person.  By this I mean I regularly try to listen to what people are saying (whether or not I agree with them) to see if there’s something I can learn and to determine if there is common ground between us; a place where we can start from and build both of our understandings.  I don’t feel the media often takes this stance.  Every opinion is polarized which results in a very conflictive mentality in this, and many other debates.  For awhile I’ve been considering the common ground in the memorization vs conceptual understanding debate that is ongoing.  I have a few thoughts on this that I would like to share.

Misunderstood Common Ground 1:  Students need to be automatic with their facts.  Believe it or not, Ontario educators do believe this!  It’s in our curriculum, and it’s foundational for later success in math.  However, HOW we do this is likely an area for discussion.  Students who build automaticity through rote drill have very little to fall back on if their memory fails them – they have efficiency, without conceptual knowledge.  On the other hand, students who have many ways to determine a fact, but without efficiency will spend so much time figuring out 7 x 8 that they forget why they were calculating it in the first place – they have conceptual knowledge without efficiency.  So, can we find common ground here?  I’d like to think it’s in a place where students are provided lots of opportunities to build conceptual understanding of number relationships (i.e. inquiry problems, number talks and strings, math discourse, etc.), and as they build an understanding of strategies to work with numbers, they also have the opportunity for purposeful practice.  This may look like drills, but the key word here is PURPOSEFUL. If students are considering how to use their understanding of 10 times to know their 9 times facts, then they engage in games and drills to practice this UNDERSTOOD STRATEGY.  This is very different than a drill to memorize 9x without conceptual understanding.

Misunderstood Common Ground 2:  Memorization is one important component of learning.  For a number of years, I was not on board with this, until I revisited my definition of memorization.  I realized that I was defining memorization as something that must be based on rote practice only.  It wasn’t until I had the opportunity to ask Jo Boaler a question about the Black Cab drivers that she has referred to in her talks that this became clear for me.  She clarified that yes, the cab drivers have memorized the countless complicated streets and routes in London, but this was based on driving and experiencing them, thereby creating a mental map of the area.  This can’t be achieved through rote drill.  This was a moment of realization for me that memorization through experience, building conceptual understanding and strategies is critical to creating important pathways in our synapses.  As concepts are understood, there can be time for repeated practice – but it is purposeful and helps to build automaticity based on understanding.

It is my deep hope that as we embark on this journey of curriculum reconsideration under a new provincial party that we can keep in mind our common ground and ensure that we are basing important decisions on clear understandings and research.  We need to remember that our number one common ground is the goal of supporting the development of strong mathematicians to prepare them for a future that is sure to include problem solving, critical thinking, creativity, and definitely mathematics.

Math Representations

I’ve done a lot of thinking around mathematical representations over the last year.  What are they?  How do they show thinking?  How do they support thinking?  What are the most effective representations for a specific child for a given math concept or developmental level?

While exploring these wonderings, I’ve had the opportunity to read many articles and research on the role of representations in the journey of the development of mathematical understanding.  I’m most fascinated by the forms of representations (i.e. concrete, virtual, graphic, abstract), along with a concept known as “Concreteness Fading”.

This idea of Concreteness Fading discusses the benefits and drawbacks to both concrete and abstract representations of and for thinking.  In doing so, the model suggests that there be connections or links made between representations that will most often begin with concrete and then fade towards graphic and abstract representations.  Although we should always consider beginning with concrete, I think it’s important that we don’t generalize to say that these representations must always always be in a linear pathway.  (We know learning isn’t linear!)  I’m curious about the fluid interaction of these types of representations as students gradually build their Number Sense.

In discussing types of representations, we can talk about Concrete, Graphic (aka pictorial or iconic) and Abstract.  I’m thinking that there is a lot to consider in terms of how we interact through our senses with these different forms of tools. For example, concrete is a hands-on tool that can be touched, moved, manipulated and we generally rely on sight and touch to make meaning of it. Graphic/pictorial/iconic resembles the concrete models, but now relies on visual interpretations.  Symbolic no longer looks like the concrete or graphic model, but uses abstract symbols or thinking.  The meaning in these symbolic representations needs to link to some foundational understanding.  I’m also wondering where the idea of virtual representations fits into this.  Virtual tools can be manipulated, like concrete, but also have a graphic quality to them, where the sensory interaction with the tool is again limited to visual.

One representation I am particularly interested in is the Open Number Line.  As Dr. Fosnot mentions regularly in her work, there is much research out there to suggest that it is an incredibly powerful tool for the development of Number Sense.  It supports the sense of magnitude as one moves up and down the number line.  It also allows for flexibility in approaching computations in ways that are in line with the natural development in children.  The question then is, considering the idea of Concreteness Fading, how best can an educator support the development of the Open Number Line tool with students?  Dr. Fosnot talks about the models first being representative of the context (i.e. measuring tool of the cube number line in her unit “Measuring for the Art Show”), then as a way to show and communicate strategies and thinking, and then finally moving into being a tool that can be used for thinking.

In several schools I’ve been working in, we are starting to ask questions about the development of representations of and for student thinking.  In one particular school, educators are beginning to ask some great questions about the ways we can link between counting and 5/10 anchors in ELK-Grade 1, with concrete number lines (developed through the Art Show unit) and graphic cube lines/number path in grade 1-2, towards open number lines in grades 2-3.  I think, in many ways, the jump between counting on the concrete number line and flexibly using the open number line is big, and we need to carefully consider supporting students in this transition.

To explore these questions, we are beginning with using whiteboard versions of blank rekenreks, cube lines and open number lines in grades 1 and 2.  We are intentionally thinking about how we can support the linking between these representations, along with the concrete as well.  When students have access to these whiteboards and the concrete tools during mini-lessons, how are they using them?  How do we know when students are ready to be exploring the concrete tools in a graphic way? Are these whiteboards providing an important link between concrete, graphic, and symbolic understanding of number? What sorts of questions can we ask children to support this development with these tools?  How can we best use these whiteboard (i.e. non-permanent) tools, in conjunction with concrete and virtual tools ?  (All of our grade 1-4 students also have access to DreamBox Learning).

Here are examples of our blank rekenrek and Number Path/Cube Line whiteboards that we are beginning to explore with students:

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As we enter into the last month of the school year, I am excited to think about what we are already wondering about in terms of representations.  I think we are in a place to begin to learn a lot about the impact of graphic representations on the development of Number Sense.   Looking forward to seeing what learning surfaces!