Posts by Deborah McCallum

How do we design our Math Class?


I have been doing a lot of reflecting about math. As a student myself, I have also been doing some research about types of math class situations, and considering what kinds of questions we need to ask to help students develop conceptual knowledge.


In our math classes it is important to provide students with cognitively demanding tasks. Rich tasks are great examples of this. Rich tasks provide multiple entry points AND multiple solution paths for students.


When we are planning our math classes, we can consider the following situations that we may be providing for our students. Next, we can think about what feedback we need to make the student learning more conceptual.


Situation #1 – Students are engaged in lessons that focus on basic knowledge, and procedures. Students need to get to a correct answer vs gaining a conceptual understanding of the strategies used. Students therefore are unable to make connections to deeper math concepts.


Key Questions:

  • What do we do to help students develop conceptual knowledge?
  • What supports need to be in place to help make this happen?


Situation #2 – students engage in more complex tasks, but then the tasks turn into procedural tasks, thus the students don’t get experience with increasingly difficult and complex tasks


Key Questions

  • How do we keep students engaged with a task, and allow them to experience increased cognitive demands and go through the ‘fits and starts’ that learners go through?


Situation #3 – Students may engage in rich tasks, with multiple entry points and multiple solution paths, however, they are unable to engage in a whole class discussion about different types of strategies.  


Key Questions

  • What steps can we take to conduct a whole class discussion?
  • How do we structure the strategies in a way that helps students to make the best connections?
  • What do we want students to get out of our whole class discussions?


Situation #4 – Students are engaged in rich tasks, with multiple entry points, multiple solution paths. Students are choosing from a variety of strategies and honing in new ones. They are able to explain why they chose a certain strategy. They are given time to explore their various strategies, and time to explain their reasoning. Whole class discussions allow for students to explore different strategies, and consider the efficiency of the strategies. Students make deeper connections, learn from peers and apply the new learning to a new task or ‘exit ticket’.


Key Questions:

  • How do we push our learning and the learning of our students in this type of a situation?


The questions can really help to guide our  thinking into how we are going to design our math class.

I would love to hear your ideas,


Deborah McCallum

Copyright, 2017




Educational Administration Quarterly

Vol 53, Issue 3, pp. 475 – 516

First Published January 31, 2017

Reflections on #TheMathPod with Cathy Fosnot: The Meaning of Context


I recently listened to The Math Pod recording with Cathy Fosnot and Stephen Hurley from VoiceEd Radio.

This podcast really helped me to think about the different lenses that we use to teach math. This is where my thinking went.


This caused me to connect with ‘lenses’ I have heard, particularly out in the media, include thinking about math as needing to be ‘back to the basics’. I personally assume that this lens implies that math is a ‘pure’ subject. A subject with right and wrong answers, set algorithms that need to be memorized and strategies that are inflexible and rigid.

I also think about the ‘lens’ that I have traditionally used, that includes a) figuring out what needs to be covered in the curriculum; b) finding out where the students are at; c) developing a rich task that allows to enter from their own developmental level; d) providing opportunities for students to build rich math talk and increase the discourse; e) share strategies and learn from one another, assess, consolidate and so on.

But where do you go from there?

This podcast was very enlightening because it was a strong reminder that we need to sequence multiple rich tasks to allow for the progressive development of math strategies and conceptual understanding, along important pathways of learning.


It also gave me great pause to think about the significance of ‘context’, and the importance of designing the sequence of rich tasks within meaningful contexts. This blew my mind, because I realised that I was focused on context as a day-to-day construct, not as embedded within sequences of tasks. Providing a meaningful contexts in this sense, helps students to not get lost in the abstraction- a very common occurrence for students still operating at more concrete levels. Context also enables multiple pathways of growth toward becoming efficiency.

I thought about culture, I thought about student identity and decolonizing the curriculum with contexts that include Indigenous Perspectives.

I have realised that it is not about finding the perfect problem, or designing that great problem to be solved. It is about crafting a sequence of problems where students are able to access key strategies, but also able to invent their own strategies – within important contexts that reach across and between problems.

Rather than thinking about this through the lens of moving students along a linear path from the concrete to the abstract, with contexts that change daily, we can use sequencing to enabling multiple paths that foster deep understanding about patterns, relationships and properties about numbers.



Therefore, I would have to say that using a ‘lens’ of looking at math as a pure subject, would also be to assume that students do not need context, and only learn along linear paths from concrete to abstract.  But we know that students do not learn along a linear path, using the same strategies, and we know that context, if harnessed effectively, can produce very meaningful math learning.

I also think that the deepening understandings that emerge from this sequencing and contexts will lead to greater memorization of basic facts as well, for those of us who do see the importance of freeing up that working memory to do more complex tasks. This understanding is what helps to build more efficient mathematicians.

When it comes to math, there are no easy ways out. The reasoning, skills, procedures, concepts, strategies are all challenging! Does this mean that it is too hard to learn? Never. In fact, I think that it is perhaps the most important things to learn as it helps with thinking in all other areas of life. If we can really get at the heart of helping students to become more efficient in math, then we will have students that can understand relationships, patterns, and ways to figure out what we don’t know in many contexts within our lives.

As for my next step, I am going to really delve in to looking at how to find sequences of rich tasks, and supporting contexts that incorporate Indigenous perspectives.

I will also strive to understand student development and how to help them to become more efficient and develop deeper strategies. This is important to use as a lens for future math work, and moving beyond the lens of ‘this is what I need to cover in the unit, this is where my students are at, and this is the problem they need today’.

What is your next step?


Deborah McCallum


Math & Identity

I just read this great piece by Karin Brodie:

Entitled: Yes, mathematics can be decolonised. Here’s how to begin


When we think about math, we often think about the content – but what about the way we think about it, the way it is taught? If think about math in these ways, we are able to consider how identity plays a role in how we teach, understand, and apply math.

What is identity? It is connected to the groups that we affiliate with, the language we use, and who we learned the language from. I believe that we all have different identities depending upon the different groups that we belong to, and that this has implications in terms of the languages and discourses we use.

What is important is that I recognize the intersection of my identity with identities embodied within the Ontario Public School system, my school board, schools, and students that I will be working with next year. In identifying this intersection, can I truly facilitate math learning, and promote higher achievement for students? Especially if my identity is stark compared to the identities that exist within classrooms across Ontario schools?

But this is not comfortable. One of the ways we as educators try to deal with this discomfort is to think of math as a ‘purist’ subject. This is but one way that we can strive to reconcile the dissonance we can feel about dealing with multiple identities in math.

But it is important to hear the identities and cultures of our students, in order to ask better questions about how math can be learned, versus merely finding the ‘right’ answer.

What can we do?

I think that Culturally Responsive Pedagogy (CRP) is one way to begin to address this question and forge a path forward. Inherent in CRP is the idea that I as an educator would continue to use student culture to transcend the negative effects of dominant culture. It becomes a tool to explain the ways in which I will develop deeper cultural knowledge of students, and thus use cultural referents to increase opportunities for student learning.

Here is a great piece to learn more about CRP: Framework for a Culturally Responsive and Relevant Pedagogy:

I do have many questions however.

How do I know that I am actively supporting a safe school environment, and not just thinking that I am because it fits with my own identity and dominant culture in society?

How can teachers situate their own privilege and oppression of themselves, and that of others? It is through this that we can start to understand identity, and understand how diverse our experiences surrounding math can actually be.

When we consider the multiple identities of teachers and students, we can understand that a standardized test is just one type of outcome for student learning. There are so many additional ways that we can capitalize on to enhance student achievement in math, to help us move beyond the spaces where we simply consume knowledge, into spaces where we can critically examine mathematical knowledge and how it plays out in our lives, and with our own identities.

This is especially important with Indigenous students. Canada has a history of experience with colonizing Indigenous communities. Because Indigenous peoples were on this land first, it stands to reason that the diverse cultures of Indigenous peoples are allowed to be welcomed and understood in our classrooms, as a way to promote and enhance the identities of Indigenous individuals, cultures, and incorporate their diverse experiences with math.

It causes me to ponder the importance and power of language. Language is part of our identity, it forms how we know the world – thus how we understand and know math. We need to learn the languages and narratives of our student identities, and check out our own, in order to co-create the necessary mathematical experiences that will lead toward higher math achievement.

Perhaps it is important to use CRP to help co-create new languages of math in our unique environments of unique identities and cultures – that can help us shape our understandings of different cultures, contexts and sensitive issues. It will be important to have agreed upon norms, and exercise them in ways that help us to foster truth and respect. It will also be important for me to frame this as discourses of education, and not discourses of the individual.

It is also important to facilitate the creation of math opportunities that allow students to discuss their own aspirations for the future.  Noting how students solve problems, and sharing the different ways that problems are solved. I can strive to move away from relying on my own identity and personal experiences to make sense of how math should be solved in the classroom. In this way, I recognize that math is culturally defined, and that I can change the narrative that I learned from dominant culture that math is a pure subject that has the correct answers, and is culturally neutral.

It is time to get really uncomfortable with math.


Deborah McCallum

Joseph Boyden

This morning I read the following new article by Joseph Boyden in MacLeans. It gave me a lot to think about.

Here is the article.

This article really brings into question for me about what being Indigenous really means, and what elements create your own identity. On one hand, it has really struck me that so many ‘others’ are questioning what his identity should or shouldn’t be. However, I have come to know that identity is not all about DNA or blood quantum. It is also about the ‘intimate’ conversations and language you share with the people who shape who you are. It is the language that moves through them and with them that make you who you are – perhaps more than Blood Quantum. I fear that Boyden is benefiting from claiming Indigenous identity, without belonging to an Indigenous community in this way. He states that he identifies most of his life as an anglo-white male, growing up in a mainly anglo-christian household. I am curious about which community or culture is he giving back to? Working to make better?

This is more than just about Joseph Boyden and the fact that he has some Indigenous DNA. What does his story mean for others who define their identity as Indigenous? All of the others who also do not ‘look’ Indigenous’, but are, yet continually asked about what percentage of Indigenous they are – as if there is a magic number that decides.

I am curious as well, did he ever have to live with what it feels like to ‘Live in the hyphen’? Did he experience racism by both white and Indigenous cultures? Was he questioned and rejected by both anglo and his Indigenous cultures – I would infer no, as it appears that he has not been part of such a community his whole life. It feels like he has just decided to ‘live in the hyphen’ now, but only to reap the benefits of both worlds. Not to help heal the traumas, or contribute personally – as one would do with those in our lives who helped to shape us.

After much careful thought, and more reading, I have come to the conclusion that identity is very much about belonging. Whose community do you belong to, and who belongs to you? Blood quantum cannot be looked at in isolation. If you have no community that you can truly claim, and who claims you in return, then how can you truly identify with it? How can you give it a voice? And finally, if you do have a community, how are you using your gift and fame to make your community better, and not just yourself?

This brings up a lot of questions about identity, what community is, and about whose identities deserve privileges, and whose do not. Why do we decide this? We do it without even realising as well.

If you haven’t already heard it, this is a great Podcast to listen to as well: Ep. 73: White Settler Revisionism and Making Métis Everywhere 

I will continue to think about my aporia, and my personal discomfort surrounding this situation with Joseph Boyden.


Deborah McCallum

Language, Culture & Math

I just spent the last 3 days at a Summer Academy for Purposeful Math Planning. I was very intrigued when we were discussing number sense and the need to become more flexible with numbers and how we use them in our world. Only one person brought up the issue of culture and how numbers are perceived. It really gave me pause to deeply consider the impact our culture has on how we perceive math as well. Particularly in the areas of spatial sense.

In the article ‘Does Your Language Shape How You Think’ by Guy Deutscher, I was really drawn in when I read that speakers of geographic languages appear to have almost superhuman senses of orientation, and simply ‘feel’ where the directions are. I couldn’t help but consider how language has deep connections to visual and spatial sense and how we ultimately perform – especially with English when used in our Eurocentric, settler based curriculum.

As the article said:

The convention of communicating with geographic coordiates compels speakers from the youngest age to pay attention to the clues from the physical environment (the position of the sun, wind and so on) every second of their lives, and to develop an accurate memory of their own changing orientations at any given moment”.

The language we use compels our students to pay attention to different cues in the environment. Our language thus shapes our habits in ways that make our spatial understandings feel like second nature.

I was struck by the fact that different languages lend themselves to different languages of space. Some languages explore directions from a more egocentric point of view – ie., directions given in relation to ourselves, whereas others are more geographically oriented. This may not sound like a big deal, until you consider how deeply language shapes our realities and how we perceive and learn about the world around us depending upon the language we have learned.

More questions I have include:

What ‘habits of mind’ form due to the spatial language that we use?

How is our ability to succeed in math class affected by our language?

What if the instructions we give in say a math class is what is preventing a student from understanding instructions?

What about our English Language Learners who may be confused based on instructions that are more egocentric or more geographic?

Do we assume that the student has learning difficulties?

Also, what happens when we are trained via language to ignore directional rotations when we commit information to memory?


This is another example in the article that was very powerful to me – basically, if I walked into an adjoining hotel room that is opposite of mine, I might see an exact replica of my own room. However, if my friend who spoke a more geographical language walked into my room, they would not see an exact replica – rather they really would see that everything is reversed, and would have the language to describe that. This has big implications therefore in how we commit events to memory, recall them, solve problems, and critically think about the world around us.

The language we use compels our students to pay attention to different cues in the environment Our language thus shapes our habits in ways that make our spatial understandings feel like second nature.  It therefore will compel our students to think differently about math.

We make so many decisions each and every day about the world around us – so much of this is spatial. We just simply don’t know our language and habits impact our ability to succeed in math.

We really are at the center of our own worlds. If we determine that subjects like math are linear and one-dimensional, with set algorithms and languages to describe, know and understand, then we are absolutely missing the worlds of many of our students. To dismiss language, culture and our identities of our students could very well mean the difference of success and achievement vs failure.


Deborah McCallum


Teaching and Assessment with Math Processes


Teaching and assessment in math go hand in hand. What ties them together are the mathematical processes. Our job as teachers is to help students build mathematical knowledge and skills of the curriculum through the 7 mathematical processes. They  include:

  • problem solving
  • reasoning and proving
  • reflecting
  • selecting tools and computational strategies
  • connecting
  • representing
  • communicating

For instance, here are the math processes for Grade 4 from the Ontario Curriculum:


In order to begin to assess what our students have learned through the expectations and processes, we must set some learning goals to help our students learn.

A Learning Goal must:

  • Have a sense of purpose
  • Build on student ideas about math
  • Engage students
  • Help students develop mathematical ideas
  • Help teachers to assess student progress
  • Connect with the classroom activities
  • Connect with math processes


The kinds of activities that we engage in during math class that embody math processes may include the following:

* something to ponder: can you think about what math processes can be embodied in each of the following? Are there any more we can add?


It is important to ask the right questions. The questions help us to facilitate the discussion that will follow. Questions are also used to raise issues and problems


Inquiry Based Learning.

As students solve problems, they will develop their ability to ask questions and plan investigations to answer those questions and solve related problems. The goal is to invite student entry into the math problem, and facilitate their exploration of the math.


Gallery Walk

The focus of a Gallery Walk is on the student work and interactive discussion shared around the classroom. Students have the ability to read different solutions and provide written and verbal feedback to each other, communicate, and solve problems together.



Here, the Chalkboard becomes a record of the entire lesson. This really helps us to model effective organization to our students. It also includes cooperative learning strategies including Think-Pair-Share, Think-Talk-Write & Placemat.


Math Congress

Here, the purpose is to support development of mathematicians in classroom learning community vs fixing mistakes in student work. We focus the whole-class discussion on 2-3 student solutions that are selected strategically by myself, the teacher. Students also share work with one another, check answers and strategies, ask questions to provoke clarification & elaboration, and defend and support mathematical thinking.


Assessments we use:

Assessments will include rubrics, performance tasks, formative and summative tasks, observations, portfolios, journals, interviews and products. Assessment will be based on Learning Goals, expectations, processes and the following Achievement Categories:

Knowledge and Understanding. Subject-specific content acquired in each grade (knowledge), and the comprehension of its meaning and significance (understanding).

Thinking. The use of critical and creative thinking skills and/or processes,3 as follows: – planning skills (e.g., understanding the problem, making a plan for solving the problem) – processing skills (e.g., carrying out a plan, looking back at the solution) – critical/creative thinking processes (e.g., inquiry, problem solving)

Communication. communicating mathematical ideas and solutions in writing, using numbers and algebraic symbols, and visually, using pictures, diagrams, charts, tables, graphs, and concrete materials).

Application. The use of knowledge and skills to make connections within and between various contexts.


All of the instructional and assessment practices can be interconnected with the Math Processes as defined in the Ontario Math Curriculum:



Professional Learning: Does it work?


I have been doing some research lately into Training Evaluation, and quite unexpectedly have become intrigued at how we measure professional development and whether it really works.

A lot of time, money, effort, resources, blood, sweat, and tears goes into PD. We as educators provide and receive PD regularly, but does it change our learning stances? A learning stance could be viewed as our own theory of learning, which impacts how we will continue to develop professionally. These stances cannot help but impact how we choose to change, or not make changes in our own practice.

Sometimes, educators might believe that we have the ‘right’ way, or that ‘we know what works in education’, or ‘we alone understand what the students need’. I do think that these stances can become problematic, in that they can prevent us from learning, growing and evolving with our students. If we are thinking about student learning, in addition to justifying money spent on PD, then we need to think about this uncomfortable area.

Also, in education we may focus more on the design of the Professional Learning, including learning principles, sequencing of training material, and job relevance. However, one area where we may be able to improve includes an increased emphasis on trainee characteristics including ability, skill, motivation and personality factors. In addition to work-environment characteristics including supervisory and peer support. All of which have tremendous impacts on learning, and perhaps this is a reason why schools tend to maintain their ‘culture’ over time. It becomes more of a situation where the learning gets changed to fit in with the culture, versus the culture changing to retain new learning. I think that this embodies a ‘transfer problem’. Can we truly transfer our learning from our professional development, and if so, how would we measure that?

Some interesting information that I have processed include 3 prevailing strategies that can be used used that could prevent us from making substantial changes to learning. (I will need to re-evaluate where I found similar information).

I have re-applied them with my own questions about how we as educators possibly deal with new information.

3 Strategies to avoid Change:

  1. Finding ways to reject the new content we are being presented with
  2. Modifying any new content to make the changes less demanding. This includes modifying the content as close as possible to current practice so that we can say we already teach that way, and
  3. Pinpointing only the content that we can easily implement. This means that we teachers will use elements of the content that we can easily apply to our teaching without changing it fundamentally.

I can’t help but wonder what this all means for education. Myself, I can see #2 and #3 happening quite unconsciously. After all, learning is very hard. Learning new things is uncomfortable. It can be very easy to look at a new professional development opportunity assume that it is already quite similar to what we already do – thereby missing key information that could be important.

I have many questions regarding the 3 strategies as well.

First, are they merely proof of the human condition and how we want to learn in ways that help us to feel comfortable? If we remain comfortable, what are implications of this for our students?

What about our educational institutions? How can our schools actively create cultures where we teachers value this feeling of being uncomfortable with learning? Does this behoove educational institutions to create new organizational cultures? How can leaders work to shake up learning cultures that need to change? Who, or what variables, decides whether a learning culture needs to change anyway?

At what point can we take a step back, feel confident in what we are doing, and give ourselves that pat on the back for working so hard and having a competent learning stance? Can we do that? Should we do that?

How do motivation and prior experience impact whether we will allow ourselves to become uncomfortable with learning? And finally, how do we accurately measure the transfer of learning in the first place? Can our learning stances change?

Finally, if we knew the answers to these questions, would it change the way we provide Professional Development for educators?

Does PD work and how do we know?

Certainly a lot to think about. Much more than what can realistically be discussed in a small blog post.

What are your personal insights on this? 


Deborah McCallum

c 2016


Spatial Reasoning and Student Success



Spatial Reasoning

This year, I have had the privilege of designing a brand new makerspace for our school. In addition, I have been able to focus on visual-spatial reasoning as the thread that pulls together science, math and technology.

What is spatial reasoning?

According to the Ministry of Education, Spatial reasoning is the ability to engage in reasoning, and understand the location, rotation and movement of ourselves and other objects in space. It involves a number of processes and concepts. More information about this can be found here:


Why is Spatial Reasoning important?

There already exists a very strong body of research that spatial thinking correlates with later performance in math. In addition, research consistently demonstrates strong linkages between spatial ability and success in math and science — and those students with strong visual and spatial sense are more likely to succeed in STEAM careers.

It is absolutely clear that early exposure to visual-spatial reasoning is very important.

However, as educators, we traditionally have failed to recognize that our youngest students are actually able to perform way above the expected levels of spatial reasoning. We generally leave these tasks for older students. This has to change.

Not only is this a problem because we are neglecting our youngest students who already come to school with a high level of spatial-reasoning skills, but this also means that our youngest students are not having equal access to spatial reasoning activities that they are able to perform. This is a social justice issue. Especially when we consider that visual-spatial reasoning positively correlates with later performance in math (Mazzocco & Myers, 2003). If we know the research, and have the opportunity to employ high quality spatial reasoning activities for all students in Kindergarten, should we let older curriculum and older beliefs hold us back? Do we recognize when we are teaching in the ways that we used to be taught? What if we had the ability to ensure all of our youngest students engage in spatial reasoning? How would this impact their future?

In fact, students who experience issues with math, often have difficulties with geometry and visual spatial sense (Zhang, et al., 2012). This to me sounds like an amazing opportunity to understand mathematical achievement via spatial reasoning. The earlier we recognize this, the earlier we can respond.

Wouldn’t it be great if we gave all students the ability to access higher level learning associated with visual-spatial sense right from the get-go? Imagine the impact this could have in overall math achievement throughout our students entire school career, and beyond, in their STEAM based careers.

To me, I think this behooves us to ensure we have access to makerspaces – regardless of where they are located in our schools – to promote visual spatial reasoning skills.

What do you think?


Deborah McCallum

c 2016

Mazzocco, M. M. M., & Thompson, R. E. (2005). Kindergarten predictors of math learning disability. Learning Disablilities Research & Practice, 20(3), 142-155. doi:10.1111/j.1540-5826.2005.00129.x
Mazzocco, M. M. M., & Myers, G. F. (2003). Complexities in identifying and defining mathematics learning disability in the primary school age years. Annals of Dyslexia, 53, 218–253
Zhang, D., Ding, Y., Stegall, J., & Mo, L. (2012). The effect of Visual‐Chunking‐Representation accommodation on geometry testing for students with math disabilities. Learning Disabilities Research & Practice, 27(4), 167-177. doi:10.1111/j.1540-5826.2012.00364.x