Cantaloupe, Marian Small, and O.A.M.E.

“Honeydew you want to get married?” “Sorry, but I canteloupe.” -Anon

A few months ago, several teachers and administrators whom I follow on Twitter convinced me that, since I was visiting Ontario already, I should extend my visit to attend the Ontario Association for Mathematics Education (OAME) yearly conference.

After securing a place to stay with some friends, I signed up and had the utmost joy of having to select my sessions from this massive list. Fortunately, I had some idea of what I wanted to see and think about, so it only took me two hours to select my sessions. I made a few excellent selections, but perhaps I will discuss those on another day. For now, there is one sessions and one Ignite talk that I want to focus on for the time being.

I did something a little wild.

I signed up for a Marian Small talk.

For those of you who don’t know of Small’s work, I will let you go explore here. In short, I had predispositions about some of the philosophies that she stood for, but I wanted to go to see what the hype was about. I liken wanting to see her session to my relationship with cantaloupe – I think cantaloupe is gross, but I insist on trying it every few years to ensure that I still dislike it.

So here I am, sitting mid-row with a teacher whom I absolutely adore, and I am trying to keep my cool: (1) because this teacher literally has pet a tiger, and (2) because Marian Small’s talk is starting to be hit or miss for me, with a larger portion of the swings being misses. Aside from the misspelling of Pythagorus (always spell check your slides), I jotted down a few notes and questions from the session:

  1. Is there any evidence for visuals leading to increased test scores and fact recall? I don’t disagree that visuals/manipulatives are excellent tools to start the learning process; however, this was a big claim that didn’t seem to have any articles attached to it – I want to read the sources!
  2. I thought it was interesting to hear that students may move the visuals, manipulatives from the concrete world into their minds. I think there is some weight to this, as I use the number line to visualize addition and subtraction of integers in order to determine the overall sign. A few interesting visuals were given that I hadn’t played around with before, such as one for n^2 – 1 = (n+1)(n-1) shown below. Personally, I find these visuals appealing, as they give opportunity to introduce the concept in a tangible way and facilitate the discussion of the symbolic. Of course, my philosophy is that the visual serves the symbolic – the visual is never a means to an end.


    Visual displaying 4 x 2 = (3+1)(3-1) = 3 x 3 – 1 can be used to help motivate the general formula n^2 – 1 = (n+1)(n-1).

  3. Is one conversation enough for the ideas to “stick”? In my opinion, no – it might be enough to open up the conversation, but more consolidation is needed for movement to long term memory. This is part of a larger conversation I was having with many attendees that skills practice needs to be an integral part of our teaching. The bigger questions for us to think about are “What does this practice look like?” and “How much practice is sufficient?” Hopefully, I will write more on spaced and mixed practice soon, as I think this gives teachers a concrete way of thinking about what this might look like in practice. A few teachers from the Halton Board presented on their journey with these ideas, and it was an excellent and insightful presentation. If you are looking for more information on these ideas, I recommend following The Learning Scientists (@AceThatTestblog, as they often discuss these ideas.
  4.  In relation to the third point, there was a time in the session in which Marian Small mentioned that there should be more learning questions as opposed to more assessment questions. I’m not sure if this was a hint to more assessment for learning as opposed to more assessment of learning? Either way, let’s say we do agree that we should have more questioning in the classroom (in the “progressive” sense of the word). I am not against this, as we need to believe in the judgement of our teachers to select the most appropriate tools for classroom instruction – and some teachers will be excellent with more progressive types of instruction. To restrict them by telling them how they should teach is unfair. This said, I would argue that teachers using this style of instruction require vast mathematics content knowledge, and quite a bit of pedagogical content knowledge as well. (Here, pedagogical content knowledge refers to items such as knowing/understanding the multiple representations your students have, knowledge of where the learning outcomes are coming from and where they are going, common misconceptions, ability to guide students towards common goal despite various strategies, and so on.) Are we absolutely sure that teachers at the elementary level are obtaining this knowledge base during teacher training or while in-service?

Marian Small then really lost me. A few things were said that I was really at odds with. First, she mentioned that some students find the mechanical operations difficult and that open questioning can be used as a way for these students to see success without looking at the mechanical. I do agree with this statement, and I do agree that using open questions does allow students who struggle to be a bit more active and engaged. Provided we create good questions, it also has the potential benefit of reducing working memory load and ensuring that we are observing/critiquing a student’s knowledge on a particular skill where the mechanical may convolute our observations. For example, observe the completion problem:


Part (a) is a completion problem in which the first and last steps are known and students must complete the middle steps.

I don’t want my students to be punished in part (b) because they were unable to add/subtract rational expressions. If their answer in part (a) is incorrect, then this has potential to convolute my assessment of their skills in part (b). So I understand why we might be interested in using particular types of questions.

What I do worry about is the over-use of this open question methodology. If we constantly are using questions that remove the computational portions of mathematics, then when are students going to learn the computational part of mathematics? Do we value computational skill? If so, we should definitely reflect on our methodology – are we building in time for students to consolidate the mechanical as well as the conceptual? To me, this seems to harken back to points 3. and 4. above.

The next thing that I didn’t agree with was the idea that pictures are more elegant than algebra. While I do value an elegant picture, I can’t look past beautiful advancements in our mathematics history. Algebra is perhaps one of the most elegant recent advancements in mathematics. There is a reason why the language of mathematics rests on the pillar of algebra! It allows us to succinctly write our ideas in a universal language, so that anyone who knows the language can understand our message. Seriously, how cool is that?! Aside from my fan-girling about algebra, the message Marian Small may want to be sending here is that a visual is often helpful in allowing students to see connections and opening up the initial conversations about the workings of algebra. To me, this harkens back to point 2., where I stated that I see the visual serving the symbolic, not being a means to an end in of itself.


Here is where things got a bit interesting for me. One thing I have really been trying to do on my journeys is to hear everyone’s stories. My philosophy on this is that by understanding and sharing our stories we foster connection. Putting ourselves in a place of vulnerability, and seeing that vulnerability reciprocated leads to empathy, a part of our humanity that is often missing within the context of the “math wars”. I have been so honored to share in many of your stories, and for this I am truly grateful.

If you are unsure of the format of an Ignite talk, the premise is that the presenter has five minutes and twenty slides, each slide progressing after fifteen seconds. It is fast and furious and awesome. Marian Small was a part of the Ignite session and did something that I absolutely have to throw respect at. Speaking from the viewpoint of a parent, she laid out her beliefs of what an excellent education system might look like.

It was honest.

It was passionate.

It was the starting point for important conversations.

And that’s when it all clicked for me. Marian Small is respected in the educational community because of the way she is involved. She talks with teachers, not to teachers. She collaborates with teachers in classrooms instead of demanding to teachers about classrooms. Rather than connecting to teachers, she connects with teachers. And while I don’t agree with every message that she speaks, I respect this.

I wonder if this means it is time I give cantaloupe another shot?


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