Spiralled MPM2D Update

This semester I taught two sections of grade 10 academic math, both of which I spiralled. Although I have taught this course for many years, this was my second time spiralling it. I blogged my way though last year's journey, starting here. I thought it might be worth sharing what I did this time around as the changes resulted from my reflections the first time through the course.

The one big change was doing more quadratics earlier in the course. About half the course is quadratics so I tried to devote about half of each cycle to quadratics. We factored starting in cycle 1; the rationale being that factoring doesn't always stick so multiple exposures to it (and lots of practice) should help students better retain how to factor.

Cycle 1:

Cycle 2:

Cycle 3:

Not-really-a-cycle Cycle 4:

We managed to finish the content on December 22 so when we returned from the break there was a sufficient amount of time to review each strand and have an optional test for each strand. These tests covered 9 of the 10 curriculum expectations and each expectation was optional. Each student could choose which expectation they wanted to show - some did none while others did all 9.

I modified some of the homework sets from last year, but continued to spiral the homework as well. I tailored it to include questions that I know many students needed to practice, but always kept it to one page.

I'm sure there were other differences that I cannot recall at the moment. If you have questions I would be happy to answer them in the comments.

My Spiralling Process

Before I actually begin this post, I will link to some resources about spiralling as my intent here is not to explain what it is or its benefits. Suffice it to say that I am really enjoying teaching this way. If you have no idea what spiralling is, you could read my blog post recapping a session that I co-presented at TMC14, found here. There are also some interesting articles/blog posts found here, here, here and here.

I am often asked how I spiral (interleave) a course. Thus far, I have not had particularly insightful answers, as the only course I had spiralled prior to this semester was grade 10 applied (MFM2P). I stole everything my first time spiralling that course from Alex Overwijk, with whom I collaborated throughout that semester. I cannot thank him enough for taking the time to work with me and Sheri Walker, and for generously sharing all his resources. 

What has changed? This semester I am spiralling the grade 10 academic course (MPM2D) for the first time. I am doing this alone, but feel confident having spiralled MFM2P for the past two years (and continuing to do so), and having taught MPM2D dozens of times (so I know the content inside and out). By the end of TMC15 this past summer, I had envisioned how I could make spiralling this course work for me. Although I found it very difficult to do a lot of planning before actually being in the trenches of teaching the course, I worked out a draft of my plan. I first stated the overall curriculum expectations for the course. I then worked out what I wanted the first cycle to look like and got less and less specific with each subsequent cycle. It looked like this:

I then made a gdoc for my first cycle. Choosing which activities fit in with my goals takes a little time, but I try to make each day as meaningful as possible. I also want my students to enjoy math class so choosing good activities that will engage them is key.

Soon after the semester began, I added columns for what I actually did each day as things always take longer than planned and my plans are always flexible. I adjust what I do based on how the previous day went. Here is what that doc looks like now:

I am recording what I do each day, along with what homework I assigned. Speaking of homework, I also decided to create my own homework sets. I am doing lagging homework some days. For example, homework set 11 had one question where students had to find the equation of a line given two points, one question where they had to sketch a parabola given the equation in factored form, one question with binomial multiplications and one question where they had to solve systems of linear equations using substitution. This way the content is always fresh and students who struggled with a concept will get more opportunities to practice and improve. I try to give feedback on homework (it never counts toward the grade) several times a week, and check only for completion occasionally. I feel that this has helped me really know where each student is in the course. Lagging homework also doesn't let students "escape" any of the material. If they didn't understand something they soon learn that it's going to keep coming back and that they really do need to take advantage of some extra help. I post full solutions to all the homework after it has been handed in, allowing students to figure out some of their own errors and how to correct them.

In addition to the gdoc above, I have gone through the curriculum document and highlighted all the specific expectations that I will hit during cycle 1, and pencilled in cycle 2, 3 or 4 for each of the others. I hope to create a digital version of this at some point.

Evaluations in a spiralled course look a little different. They have multiple curriculum expectations, making them more like a little exam than a standard unit test. I mark on levels (R, 1, 2, 3, 4) by expectation. This makes it clear to each student how they are doing on each expectation. Should they get a level R, they will be given an opportunity to reassess before the next evaluation, after getting some help from me. I record results using a locally-developed program called MaMa which allows me to see progress throughout the semester. Here is a screenshot from a previous year:

The curriculum expectations are along the left and each test is a different colour, getting darker as the semester progresses.

I am also blogging my way through this course. I write a post each day where I explain what I did and reflect on how things went. I also post link to all the handouts. If you are interested in reading more, my first post for this semester can be found here. I also blogged my way through the MFM2P course last year and the one before.

Do you have questions about spiralling that I have not addressed? Please fire them my way in the comments.

A Summary of Spiraling through the Curriculum with Activities

At TMC14, Alex Overwijk and I shared our experience spiraling through the curriculum using activity-based teaching. I'm going to attempt to summarize what that entails. The PowerPoint is on the wiki.

Let me start by giving credit where it is due. Alex and Bruce McLaurin (@BDMcLaurin) started spiraling at their school 5 years ago. They are the experts on it now and are sharing the experience with teachers far and wide. I got involved this past school year when I spiraled my grade 10 applied math class, second semester. Alex wanted to collaborate on this with me and Sheri Walker so we met 3 times during the semester to figure out what we wanted to do, try some things out and plan each cycle. We took the lead from Alex on which activities to look at and added a few things of our own. This was truly jumping in the deep end of the pool, but I firmly believe it is the only way to do this successfully. At the end of our session Alex said that if you try to do it slowly and incrementally, the kids will drag you back down and you will go there because that is your comfort zone. I agree, so jump on in!

A bit about grade 10 applied math. The students in this course have generally been unsuccessful at some point in math and most do not like math. There are a lot of behavioural issues and I would guess that in the average class, over half of the students have an IEP. Asking these kids to sit and take notes then do homework is a recipe for frustration all around. Instead, we tried to get them up and moving, doing math in context, even if it was a contrived context, and assigned no homework. There was no point assigning homework - those who would do it are the ones who didn't need to and the ones who needed to wouldn't do it. So no homework, no textbook; we created worksheets for practice which some students finished and others didn't. They did what they could in 75 minutes each day, which was different for each student.

Here is how it was structured. Over the course of the semester we did 4 full cycles and a bit extra at the end. This means no more units. Each cycle covered each of the overall course expectations (they start on page 53) and we tested all of these at the end of each cycle. 

We uncovered the material through activities. Hands-on as much as possible, with manipulatives, graphing technology and whatever "stuff" we needed - spaghetti, pennies, linking cubes, algebra tiles, cups, etc. The activities started out very scaffolded and become more open-ended and richer as the course progresses. I blogged about it all starting here and I will link to Alex's blog posts below.

The first cycle started with the 26 Squares activity which took about 3 weeks. We hit all 3 strands of our curriculum. Next we did some toothpick activities to work on linear modeling and equations. We did Andrew Stadel's File Cabinet 3-act next to hit on some of the measurement strand. We used Smarties, Jujubes and pennies to solve systems of equations. That is where we ended cycle 1.

Please note that these are Smarties:

These are actually called rockets:

During the 2nd cycle we did Spaghetti Bridges for linear modeling/solving, a series of activities including High Fives and Frogs for linear and quadratics, found inaccessible heights for similar triangles and trig along with a roof truss task for the same expectations, then I did some work with surface area/volume of prisms and pyramids (I think Alex and Sheri did something different). Then came test #2.

The following chart, although hard to read, gives you an idea of the number of activities and which expectations they are hitting.

Alex also did a card tossing activity which we got participants to try out at TMC14. It is a lot of fun to do and hits a lot of curriculum expectations. I find that the more of this I do, the more I can see how to extend an activity to get more out of it.

(thanks to Nathan Kraft for the photo)

Throughout the course students saw the same concepts several times. For example, in cycle 1 they solved systems of equations only with manipulatives. They did not even write equations to represent the situation. In cycle 2, they did the same but also started to write something to represent the situation - some wrote equations, some used symbols or pictures. In cycle 3, they learned how to use elimination through a very logical progression of questions that made sense in context. And we reasoned through the answer. They may not have been able to present a beautiful algebraic solution, but they could solve the system and could explain to you how they did it. They were not following an algorithm - they understood what they were doing. In cycle 4 they solved systems to work through the cup stacking activities. We created a solid foundation and built upon it each cycle.

The benefits from the PowerPoint are shown below, but one of the big ones for me is time - you have time to get through the course - in fact they see most of the the curriculum in the first 6 weeks; you don't need to feel rushed to get an activity done - if students need an extra day, take it; if someone is away for an extended period of time, they miss an activity or maybe 2, but we will cycle back and see it again.

As a side note, I got comp books for my students because I enjoy structure a little more than Alex. (By "enjoy" I really mean "need".) For each topic I would print out an example or fill-in-the-blank type of sheet for students to glue in their comp book and fill in. This way they all had a resource to refer to. Some students took their comp books home to help them study for tests and they were allowed to use them for the end-of-year summative task.

I hope this helps shed a little light on what is involved in spiraling through the curriculum with activities. Please throw any questions my way in the comments. Perhaps the strongest endorsement I can give is that I will never teach this course in a "traditional" way again, and that I have volunteered to teach it (the course that no one wants to teach) both semesters next year.