Four Fall Reflections

What a crazy couple months it has been! It began mid-September when I was asked to give a talk about the new mathematics curriculum in BC through the college. From here, I attended researchED New York early in October, where I got to connect with some awesome educators and present about connecting interleaved practice with teaching with the amazing Yana Weinstein. After this, I gave a math workshop to some future BC education assistants. We talked about math anxiety, early numeracy, and cognitive science, all while working on interesting math problems together! Yesterday I was in Saskatoon at the SUM Conference to present on non-routine cognitive tasks – synthesizing some information I gathered from Dan Meyer and Steve Leinwand. Finally, I have a small break before the next two workshops to hopefully eke out a post on some reflections I have had from traversing between these different worlds.

#1) Cumulative Review — Why Isn’t Everyone Doing It?

I have recently read “Accessible Mathematics” by Steve Leinwand, in which he outlines 10 instructional shifts to help raise student achievement. One of those shifts is to shift toward giving ongoing cumulative practice at the beginning of your math lessons. It does not have to be terribly extensive – perhaps just four or five short recall-type questions to ensure that students are not forgetting past concepts. It seems obvious that we should be doing this – but many of us are not!

Why should we be doing it? Well, this was somewhat tied to the presentation that Yana and I gave at researchED. It seems that interleaved and spaced practice are highly effective strategies to increase long-term learning in our students. For instance, I saw a 10% increase in the discrimination of problem type when I used interleaved practice in my integral calculus class last year. However, there are some things that we don’t know about interleaving that warrant future studies – like how many problem-types should we include, or how interleaving affects attention in our students.

Why are we not doing it? Efrat discussed some of the practical limitations of using interleaved and spaced practice at researchED New York. Teachers typically list time investment, lack of support, or an incompatible system as reasons for not utilizing spaced practice. What might change their minds? It seems that teachers are interested in ongoing professional development in cognitive science, and time to work with colleagues in order to help ease them into implementation of such tasks. As this is an area of interest to me, please contact me if you or your school is interested in ongoing professional development in cognitive science – I would be happy to help!

#2) Depressing — Why Aren’t We Collaborating?

Continuing on the conversation, we could ask why aren’t we collaborating more as a community? Let’s take a look at an example from my life. I had a student come into my calculus class with a TI calculator stating that his teachers at high school said they would absolutely require a TI calculator for college calculus. Literally, what?! With tools like Desmos at our fingertips, why is there a need to drag around a $200 brick? In addition to this, my department doesn’t allow graphical display calculators on major tests anyway. So it looks like I will need to reach out to the local community and try to spread the Desmos love. Why? Let’s look at it form the alternate viewpoint: If I teach Desmos to my students this year, but when they move on, the next teacher doesn’t know how (or doesn’t want to know how) to use Desmos, these students are now potentially disadvantaged. In essence, a teaching tool is greater when we share it with others in the profession and we develop long-term learning goals using similar tools.

#3) Planning — Using Space, Not Time

In a presentation by Nat Banting and Ilona Vashchyshyn, we were asked to consider planning a lesson using quadrants labelled as “Teaching Actions”, “Teaching Spaces”, “Anticipation”, and “Improvisation.” In other words, when it comes to planning, we need to consider our space (the room, manipulatives, desk arrangement) and our actions (modelling, watching, telling). And Nat and Ilona see our actions and spaces situated on a continuum between anticipation and improvisation. In fact, there has to be some improvisation within our classrooms, since it would technically be impossible for us to plan all the possible divergence that may happen in any given lesson.

Of interest to me was their belief that false dichotomies arise when we believe an individual spends all their time within one of the half-planes. For instance, if we believe an educator continuously anticipates and does not improvise in the class, then they are defined as a traditional teacher. On the other hand, those who are thought to improvise all the time are branded as reform or progressive teachers.

This also works for the horizontal half-planes. If an educator is too focused on the teaching spaces, the lesson might be branded as a differentiated instruction type of lesson; and if an educator is too focused on the teaching actions, the lesson might be branded an inquiry type lesson. There is probably more to this conversation, but I am still trying to think more on these two particular diagrams.


#4) Synthesis — Finding Your Balance

In Saskatoon I tried to synthesize some reading that I have been doing as of late. The first bit of information was regarding non-routine cognitive tasks I originally heard of from Dan Meyer at OAME 2017. The main premise is that a mathematical task can either have a real-world context or not. In addition to this, a mathematical task can involve “real work” or “fake work.” There are certain verb choices that we make in a math class that lead to real work (question, predict, analyze, debate) or to fake work (evaluate, simplify). Finally, doing fake work in a real world context is overrated; that is, dressing up a routine task with the air of real worldness is overused in math education. However, pushing students to do real work not in a real world context is underrated; that is, we often fall short of allowing students to use meaningful verbs like question, predict or analyze outside of real world contexts. Think “Calculate when the phone will be charged given the model.” (routine, plug ‘n’ chug, dressed up in real world clothing) versus “Predict the y-value given the data.” (non-routine, analyzing data to predict, non-dressed up mathy question).

In addition to Dan’s thoughts on non-routine tasks, I embedded Steve Leinwands idea to lead lessons with data. My thoughts were that if we are interested in moving toward doing real work, data can help drive questioning, noticing and predicting. Provided things go well with the lesson, we can follow up with verbs that allow us to extend, such as generalize or debate. If you are interested in seeing a bit more, my slides from the conference can be found here.

Realistically, I think it would be quite the challenge to create every lesson as a non-routine cognitive task. To me, it feels unrealistic. Also, I firmly believe that the verbs recall, calculate and simplify have a place in mathematics classes and that they should be respected. For instance, John Mighton of JUMP Mathematics consistently reminds me that cognitive load is important – that is, our students require some skill in order to begin a rich-task such as data analysis. This skill comes with practice, which can easily be acquired via spaced practice involving recalling facts. However, on the other side, Bjork reminds me of desirable difficulties. Could non-routine cognitive tasks be shaped in such a way to support learning and long-term retention?

As I continue to navigate the large divide of what feels like a fake world of mathematics and a real world of mathematics education, I can’t help but wonder how we might all be able to help shift the collective from fake work to real work.

2 Comments on “Four Fall Reflections

  1. “Does anyone actually think this holds?”

    I certainly don’t speak for everyone, but I do think that at one point I held this viewpoint: that traditional teachers tend not to improvise and that progressive teachers tend to improvise. Of course, I think your argument is true – I also view myself as a more traditional teacher who uses improvisation quite a bit.

    Funnily enough, I tend to think that traditional teachers tend not to anticipate as much as progressive teachers do, so maybe these two half-planes need to be reframed more.

    “There’s opportunity costs involved.”

    Yes. Absolutely agree with you. Personally, I feel like non-routine tasks are important though, and I want to think about ways to include them in my courses at post-secondary that don’t affect the needed routine practice that students do need. I also wonder if, the more non-routine tasks students see, the more routine they become, requiring less costs.


  2. On the other hand, those who are thought to improvise all the time are branded as reform or progressive teachers.

    Does anyone actually think this holds?

    I’m a thoroughly traditional teacher, using explicit methods almost exclusively, yet my lessons are often completely improvised. I frequently enter a class knowing what I am going to be teaching and pretty much nothing else (actually, I sometimes wander in not even knowing what I am going to teach, and wing it after I’ve done the review of yesterday’s work, based on how they go). I generally work with (previously unprepared) questions on the board based on how well they did the previous question, and even when I use textbooks I wander around them, taking bits from here and there.

    I’ve seen thoroughly progressive teachers who are wedded to worksheets, and who pre-plan everything.

    Could non-routine cognitive tasks be shaped in such a way to support learning and long-term retention?

    There’s opportunity costs involved. Non-routine tasks are necessarily slow, because we have no routine. If we give lots of such tasks we simply don’t have time to do the routine practice we also need. (It’s also exhausting, both for student and teacher.)

    We’re better just to accept that we can’t square the circle that is trying to get non-routine tasks to do other things.


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