A tool for math educators that opens up possibilities

Spiralling…no, not out of control as one tries to complete reports, finish up that last little bit of marking or prep for the next few days…or the drain as many felt 2020 had been. Rather spiralling through the curriculum in Grade 9 (and more recently Grade 10) applied mathematics.

Unfortunately, when I think of spiralling it usually comes with a negative memory association—spiralling out of control, or, more colourfully, spiralling the toilet drain. But a spiralling curriculum, also referred to as interleaving, is a powerful tool which can have very positive effects on students’ ability to perform well and to retain information over a much longer period of time.

So what is spiralling? To borrow from Kyle Pearce of tapintoteenminds.com spiralling is: “…organizing topics that might traditionally be taught in blocks, chapters, or units of study over a short period of time and we are introducing topics in smaller chunks and spreading them out over a longer period of time.”

When I taught my first ever section of Grade 9 applied math (many standardized tests ago) I was handed a textbook by my department head and told to start at the beginning and do about a section a day. This seemed to make sense, and over the years I tweaked and changed things, but still generally followed the organizational paradigm set out by the textbook. Invariably at the end of the course, I would spend time reviewing topics that hadn’t been directly touched upon for the better part of a semester and, as my own familiarity with the curriculum was increasing, I found it increasingly difficult to understand how the students couldn’t retain concepts only a few months old.

What I was failing to realize at the time, was that I was falling victim to a number of psychological paradigms.

First is the illusion of understanding, which states that people feel they understand complex phenomena with far greater precision, coherence, and depth than they really do; an illusion that is far stronger for explanatory knowledge than many other kinds of knowledge, such as that for facts, procedures or narratives (Rozenblit and Keil, 2002). So, the students felt that they were more able to explain to me how different ideas in mathematics worked than they were actually able to. Rozenblit and Keil also found, that when the inadequacies of this illusion are made obvious to a person, they will adjust their attitude more moderately. They will better be able to self-identify their knowledge gaps. Unfortunately, for my students, the only time they were presented with the fact that they may not have understood as much as they thought they did was upon receiving their unit test. And by then we were usually a few days into the new unit.

Next is the idea of the forgetting curve, first proposed by Hermann Ebbinghaus in the late 19th century. While his methodology was suspect (he used himself as a subject) the idea that memories fade over time with a certain degree of measurability is one that has held up well. There is still debate about the exact shape of the forgetting curve and of how certain memories (those paired with significant emotional connection for instance) are forgotten at different rates (Averell and Heathcoate, 2011). However, there is enough reliable evidence to suggest that the more time that passes between learning a new skill, fact, or process, the more difficulty accurate recall becomes. So, to counteract the effects of the forgetting curve, active recall is required repeatedly over time, with breaks in between. Executed properly, the forgetting curve becomes less and less steep, leading to long-term acquisition of skills and greater post-learning performance.

The third psychological paradigm we fall prey to is a notion explored by Bjork (1994). When learning new tasks in a blocked fashion (that’s to say one unit at a time) versus interleaved (spiralled), participants reported feeling that they learned best when they were allowed to practice by blocking. What’s rather astonishing is that even when presented with the fact that the interleaved participants performed better, most participants still felt that blocking was the best way to learn.

I was faced with these paradigms and wanted to make a change. So, how did I start the process of adapting what I was doing? I was fortunate to have an opportunity a few years ago to spend a year as a school board-level math consultant. This gave me time to explore what was out there, attend more professional development than I would have been able to otherwise and, most importantly, allowed me to talk with and observe many math teachers. Jon Orr, Kyle Pearce, Alex Overwijk, and Peter Liljedahl—their ideas, theories and practical examples—changed how I would teach math forever.

I spent the better part of the next year completely rebuilding the Grade 9 applied math course, and at the core of this new course—spiralling!

My first hurdle was how to best logically go over everything, but at increasingly more complex levels. My solution was to attach a theme to each spiral. This had two added benefits. First, it gave a contextually relevant reference point to which I would often refer, further aiding memory retention. Second, it allowed for a much greater application of the concepts, which seemed prudent given that the course is in fact called Applied Mathematics.

Next came the daily structure, which I borrowed from Sylvia Wentland—my co-conspirator in this endeavour and an invaluable asset off whom I could bounce ideas and gain insight and motivation. The approach is to split the learning week into three pieces: Tuesday through Thursday are problem days, which I will elaborate on further in a moment; Friday is our consolidation and practice day; and Monday is a performance quiz.

Problem days see the students placed into groups randomly (visible random groupings) using technology (flipity.net) or cards. The groups are never preconceived and they change every day. There is some soft-skill development here, but that’s a discussion for another day. The students work at whiteboards spread around the perimeter of the classroom (vertical non-permanent surfaces) so their process and thinking are visible to myself as I circulate around the groups and to the other groups. A different strand is chosen each day and the problems are selected so that topics are covered over and over again, but each time at a deeper layer. As we move through spirals there is greater overlap between strands or references to older problems, but with a new twist.

Consolidation occurs by reviewing the daily problems, practicing skills (I use Knowledgehook and Desmos activities) and working with small groups to shore up any identified gaps. This is followed by the performance quiz; each one consisting of three questions (similar to the daily problems, but not identical). Each question is graded on a four-point scale: attempted, but not getting it; starting to get it; got it; got it and could teach it to others.

Student feedback has been positive. Students report decreased anxiety related to mathematics and to test-taking. They are more receptive to feedback and are increasingly using the feedback to develop their skills.

Furthermore, the students’ time to task has diminished, while their time on task has increased significantly. Math-talk has improved and the students’ willingness to attempt problems (especially more complex problems) has increased. No longer are questions left blank if the student doesn’t understand or have the solution immediately.

With the proposed destreaming of the Grade 9 math curriculum, it seems that the idea of spiraling would be particularly useful. In my experience, many of the students in my applied class, while grumbly at first, are quite capable of rising to the challenge of reading, understanding and planning a strategy and having a mathematical conversation with their peers and myself. By spiralling the curriculum, students with gaps are given more time for remediation and students with processing issues have more time to assimilate their learning. By starting slowly and gradually building, these students are given multiple opportunities to join the conversation. Furthermore, students demonstrating strength would have many occasions to support their peers and could also be given opportunities for extension and enrichment within the context of the current spiral. Given that destreaming results in a mixed-ability classroom it makes sense that spiralling would be a good fit, one that would help address some of the inequities and inequalities that destreaming is targeting.

I would strongly urge anyone considering the idea of spiralling to take the plunge. It’s been an adventure, but one well worth it for me and my students.

Great resources from teachers in Ontario: Jon Orr from Lambton-Kent DSB; Kyle Pearce from Greater Essex County DSB; just about everyone in Ottawa-Carleton DSB, but I’ve specifically enjoyed learning from Laura Wheeler, Alex Overwijk, Jimmy Pai, and Mary Bourassa; and of course the great people I see on a daily basis in my home board of Kawartha Pine Ridge DSB, Sylvia Wentland, Stacey Willis, and Ben Medd.

About Marc Paxton
Marc Paxton is a teacher in District 14, Kawartha Pine Ridge.

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