You Say You Want a Revolution

With two Saturday workshops left in our summer program we’re looking back on what we’ve learned so far, looking ahead at our two final workshops. We’re looking at how we can best use the technology to engage with the students, as well as reflecting on the feedback we’ve gotten so far.

Many of the activities have students use Desmos to build understanding of geometric relationships and graphical transformations. We’ve been trying to put students in situations where learning emerges organically from play. Some students have been telling us that they’d like more problem-oriented activities––things that feel more like the math they’ve learned in school, so we’re thinking about how to integrate that into this week’s workshop.

Last week’s activity was a combination of playing with rotations in Desmos, compiling what we’ve worked on to make something new, and review from the previous week for students who hadn’t made it to every breakout room. The creative breakout rooms involved students in using Desmos to animate their names or design a landscape image. The “revolutions” breakout room culminated with a small wheel rolling around the perimeter of a larger circle.

We’re looking forward to the year ahead and what our next workshops will look like. One of the obstacles we anticipate to getting our Desmos activities into classrooms, is teachers not wanting to learn how to use another online platform. Our workshops this summer have been focussed on reaching students, but, come fall, we’re thinking about potentially running a Desmos workshop for teachers.

Let us know what you think!

Peter Taylor Quote of the Week: “Cutting through the complexity to see the inner structure, that’s what mathematicians do.”

Putting a New Spin on Things

There’s a tendency in math education to teach trigonometric functions late and in fragmentary ways. Typically, students first meet sine and cosine in “triangle trig” and don’t encounter their relationship to the unit circle until grade 11. In last week’s workshop our younger students had their first meeting with sin and cos in driving a point around the unit circle. And they seemed to catch on to that quickly. That’s part of our “just show me how to do it” philosophy.

We started with a simple animation of a spinning stick, then added path tracing and movement of either the end points or the center points of the stick to create different kinds of images. Students did a remarkable job of using trig functions to create imaginative animations.

We’ve continued learning over the past few weeks about how best to organize breakout rooms in Zoom, and how to make sure that even in larger groups students are getting the support that they need. To organize breakout rooms in our “wandering” format, we need to make sure that the activities in each breakout room are distinct enough that they can be done in any order without losing their appeal.

The chat in Zoom calls is a tool we’re working with to try and make sure that student needs are being met as well as to build community within the RabbitMath summer workshops. Many students either don’t have a microphone and camera or are simply more comfortable with those things off. The chat is a valuable tool for us to use to engage students who might not want to speak up on mic. We want students to feel engaged, and for them to be involved in more than just the math, and that can be a challenge in an online environment. We’ve had breakout rooms at the start of each workshop for the past few weeks where students can chat and check in with each other, and this week we left some time at the end of the workshop for students to chat (on mic or in text) in the larger group.

This past week has also involved some thinking about the focus of our website and how we can improve its function. The website has, so far, been geared primarily to teachers looking for resources. We’re thinking about how we can add interactive modules that students can work through on their own, either as part of a class activity or for their own interest, more easily. We already have one such module here, and we’re going to add more in the coming weeks.

There’s also likely to be some reorganizing of the website by topic rather that just by grade, since many of our problems are low-floor high-ceiling and may be suitable for multiple grades. The workshops have given us some perspective about how well students can rise to a learning occasion!

Peter Taylor Quote of the Week: “They’re not coming to learn, they’re coming to play…and the learning happens.”

Quin There Done That

During last week’s workshop, we took a leap and changed the format. In past weeks we had made groups based on grade, skill level, or random groups assigned to be the right size. Last week we didn’t assign groups at all. We opened breakout rooms with different activities, and let students move through them at whatever pace and in whatever order suited their pace.

The activity was another one with Desmos animation, but it was a bit more artistically oriented. We introduced The Quintessential Human Traveler, or Quin for short, who students could animate to move through space after designing their own faces in Desmos. We saw lots of creativity from our students in how they designed their Quins and the ways they chose to animate them.

We created four different breakout rooms, each with its own focus. We had two breakout rooms for designing faces; one of the rooms was to design the face quickly and move on, while the other was meant for a slower pace. The latter room was chosen by students who either wanted more time and support to build their faces, or who wanted to take the time to add lots of details to their faces. The other two rooms were focused on different ways to animate Quin, either to bounce along a parabolic path or to squish and bounce up and down.

With more flexibility, students created a wide range of animations and pictures with a simple group of shapes, functions, and transformations. Some students worked through one of the Quin building breakout rooms as well as both of the animations, while others stayed on one of the Quin breakout rooms for most of the session to customize their faces or make multiple of them. It gave the students the freedom to work at their own paces, and helped the RabbitMath team make sure that students were getting the support they needed.

We also started sending a survey link to our students at the end of the workshop to collect feedback on how they felt about it. This is a tool that we’re hoping to use more in the coming weeks, including by adding a section for students to share their creations with us if they want us to share them on our website. Hopefully we’ll be able to add a gallery of student work in the coming week.

Peter Taylor Quote of the Week: “You’ll try things and fail, and then get it right and figure it out.”

Working Out Workshops

This past week at RabbitMath has included lots of planning for our summer workshops, starting this weekend. We have a big group of students signed up already and are working to give them the best experience possible in the online format.

One of the things we’ve given a lot of thought to is the implementation of activities that focus on using games to build mathematical thinking skills and intuition. This can work for a wider variety of students than problems that focus strictly on technical skills. The idea behind the focus on games is that the more students work with structure, the better they’ll understand it. Peter often uses the example of a smartphone; most of us don’t know how our phones work, but that doesn’t stop us from learning what they do and how we can use them. The same can be true of concepts in math and understanding of how they work often follows intuition built by working with them.

We have enough students signed up so far that we’d like to do lots of work in smaller groups where students can collaborate on problems together. For some activities it will make sense to break off into groups by grade, because different problem-solving strategies will work with different skills. Some of the problems we’ve looked at can be solved easily with calculus that a grade 12 student might know but can equally be solved using algebra or geometry. For other activities it will make more sense to divide students into groups of mixed skill levels so that they can learn from each other. In a physical classroom, students could look around the room and put themselves into a group, and we would mill around to help guide people towards solutions. In an online learning environment, we need to be more mindful in advance about how we’ll manage that to make sure that people are in groups best suited to their level of confidence with the concepts and the activity at hand.

Our goal for the Saturday workshops is to have students working with concepts and sophisticated narratives, and to be building things. We’re still working on the final details, but we’ve got lots planned for our summer program students. We’re looking forward to meeting them this weekend!

Peter Taylor Quote of the Week: “The dream is to have kids walk out of my lectures saying, ‘that was an experience.’”

De-Streaming In Ontario

In this past week we’ve talked a lot about Ontario’s de-streaming of grade nine math classrooms. This change is in line with advice from Organization for Economic Co-Operation and Development (OECD) to de-stream classrooms in grades nine and ten, though there haven’t been changes announced in other subjects or grades yet. This blog post is going to get into why those changes have been made, the advantages for students, and how the RabbitMath curriculum aligns with the objectives of a de-streamed classroom.

The OECD is a multinational organization made up of 37 member nations, including Canada. Their goal is to improve quality of life for sustainable economic growth on a global scale. Their three main tenets are soft law for international best practices, peer review to help member nations “examine and compare experience,” and public policy research. The public policy research end of things involves huge amounts of statistical data from member countries used for both new and existing research initiatives with the goal of creating “better policies for better lives.”

Their advice to de-stream grade nine and ten classes is based in data showing that streaming in early grades disadvantages students and exacerbates existing racial and socio-political inequalities. The Education Quality and Accountability office has found that with similar ingoing academic records, students are more likely to succeed in academic stream courses than applied ones. This becomes even more problematic when paired with a higher chance of students of colour, students from low income families, and students from other systemically disadvantaged communities being steered towards applied level courses.

Ontario is the only province in Canada to start streaming as early as grade nine. The decision to take academic or applied courses can be a major education and life choice. In theory, academic stream courses prepare students for university while applied courses are geared for students going to college. In practice, students in academic courses are more than twice as likely as those in applied to courses to pursue post secondary education out of high school. Grade eight students needing to make that kind of decision about their education is a big ask.

De-streaming grade nine math classes is far from a quick fix. Teachers will need strong teaching resources to help them adjust to a new kind of classroom. There’s a lot of discussion about whether the curriculum should move up or down in difficulty. At RabbitMath we prefer to think of the curriculum shifting sideways. Richness can be added to the curriculum in the form of low floor, high ceiling activities that focus on mathematical thinking rather than pure technical proficiency. There’s a lot of work ahead to shift to de-streamed classrooms. There’s also great benefit to be gained if it can be done thoughtfully, carefully, and with a commitment to equity in education.

Peter Taylor Quote of the Week: “As part of de-streaming we need to change our course objectives.”

Workshop Announcement and Escher Sketch

We’ve got exciting news this week; we’re opening registration for our inaugural summer math program! It will consist of free workshops for high school students (entering 9-12) to work with the math activities we’ve been developing. It will run from ten to noon on Saturday mornings starting in July and running through August. Keep an eye on our website for the “Summer Program” tab, going up later today, where you’ll be able to find the registration form.

A big question that we’ve worked with in the past week is what it means to decide what mathematics is or is not ‘developmentally appropriate’ for students at different grade levels. Teachers are given checklists to go through in the classroom each year of skills and concepts that students should be able to master. Bigger picture ideas are often avoided because they are admittedly harder to handle. In other academic subjects, ideas are often introduced that are well above the level of what could be produced or fully understood by the class.

A goal we’re working towards with our math activities is to bring the structural beauty and complexity of mathematics to the high school classroom. Computer programs are a useful tool in scaffolding concepts that may be more complicated mechanically than students could access on their own. Doing the mechanical calculations in the background allows for a more immediate connection to bigger picture ideas in math.

One such program is called Escher Sketch, available here. M.C. Escher was a Dutch artist well known for his optical illusions and tessellations and this program allows you to emulate his geometric tiling patterns with your own art. It makes it simple to create your own tessellations — patterns made up of geometric shapes that tile the plane without gaps or overlap.

The program allows you to pick the kind of symmetry you’d like to exhibit and at the same time copy it into a tiling. It has the potential to be a helpful tool in building understanding of transformation and symmetries. This is one of the ideas we’ll be working with more as we get further into the summer.

Peter Taylor Quote of the Week: “Asking the right questions is the hardest thing.”

Summer and Ideas and Structure (Oh My!)

There’s been a bit of a hiatus on the blog for the past year, but we’re back! Rest assured, we’ve been busy bunnies the whole time. Now that summer has started we’re hopping into high gear with some fresh faces on the team and lots of familiar ones. We’ve hit the ground running and are excited to share the work we’ve been doing. 

Our new main direction for this summer’s work is to direct our focus to the intermediate levels, grade nine and ten. The first problem that we looked at in our group meetings was one involving parabolas and tangent lines, with the idea that we might be able to get it into grade nine classrooms. That specific problem turned out to work with more ideas from the grade ten curriculum, but we’re still interested in getting more of the structures from the problem into other formats more suitable for introduction earlier on. One of our mantras is “start early.”

Some grade nine students may have difficulty with the algebraic concepts surrounding parabolic graphs and their transformations, but are there ways to introduce them geometrically first? The graphs that generally make it into grade nine classes are of straight lines, and parabolas offer a richer understanding of graphical transformations that will be valuable if they can be introduced in a manageable way. 

One of the ideas we’re exploring is that of using what we might call “apps” to enrich the curriculum. These are pieces of mathematical structure that students can work with even when they don’t fully understand how they work. This applies to things like giving students a formula for a tangent line to a curve at any point before they’ve encountered the ideas of calculus that would allow them to derive such a formula for themselves. Introducing concepts this way allows for problems to fit into the “low floor, high ceiling” approach, in which one problem can offer richness and interest for a wide range of skill levels. 

We’ll keep you updated as things progress!

Peter Taylor Quote of the Week: “Be prepared to modify your objectives.” 

The world is waiting

Suddenly the world has retreated and we are anxious, patient, caring.  We had been watching RabbitMath at work in a few classrooms, but now that it looks like formal school will not be appearing again this year. I’m thinking about making videos and developing resources that are easier to use at home, on your own, without a teacher at hand. Perhaps now is a time for tempering and reflection. I want to simplify a bit. 

Objectives

I am often struck by how different the structure of the curriculum is from what is typically in the schools.  At our teachers’ workshop in Ottawa in August there were many comments that the stories were certainly interesting, but they were also challenging—for both students and teachers.  However it was the kind of challenge we all needed. 

Of course most students already find math hard enough, but the challenge of RabbitMath is of a different kind.  And what I’m thinking is that a different kind of challenge requires a different set of objectives. 

The past few decades have been focused on objectives that were principally technical.  Increasingly it was found that student’s technical proficiency was poor, and so the technical component of the curriculum was ramped up and so was the amount of testing.  Guess what?––there was no significant improvement.  So more ramping up and (surprise surprise) still no real improvement. 

Now I certainly value technical proficiency, but the effect of focusing simply on that led to a curriculum that was, in a word, fragmentary. The stories were fragments. Oh there were stories alright, but question 13 is about Shreya’s bank account and question 14 is about rides in an amusement park, and 15 is about renting a car.  It’s like an English curriculum where the novels are all just a paragraph long. There’s really no “experience,” no sustained narrative thread. 

In the end, even more important than the fact that so many students have low technical proficiency, is the fact that when they get to university or even to their career, they can’t handle sophisticated narratives.  They live in a fragmented world.

That’s the objective we want to focus on—being able to work with whole complex structures.  Technical skill must follow organically from that. 

- Peter Taylor

Welcome to our free-flowing blog.

The main contributors to the blog will be Peter and Chris, but anyone in the RabbitMath community is welcome to share ideas or points of view--though we reserve the right to edit. [Of course we will always run edited text past you before publishing.] If you are interested please send your comments to Peter. 

Mimi Thian