Planning a Flexible Framework for Intermediate Mathematics

This is an overview of what we will be discussing during our curriculum framework professional development session on August 25th. We will be at the DDSB Education Centre, room 2015 from 8:30 AM to about 2:00 PM. I hope you will all be finished your initial planning before 2:00 PM, but I put some extra time as padding, to avoid cutting short any great discussions.

Why I Enjoy Math

This is one of the recent math problems that I've enjoyed solving. Fill in 4 cells to give products shown in each row/column, sum of 4 cells must be 80.

There are two main experiences that I can point to that really drove me to want to teach mathematics. The first was my grade 12 calculus course. What I enjoyed the most about that class was when we were given time to work on tough problems. I was lucky enough to be in a class where most (if not all) of the students really wanted to be there, and really wanted to learn. The teacher didn't pull any punches, and gave us really tough problems to work on. I remember working on those problems both individually and with other students for hours. The sheer difficulty of the problems made them intensely satisfying to solve.

The second really positive experience I had was one of my university math courses about the process of problem-solving. The course was entirely based on the book, "Thinking Mathematically", by Mason, Burton and Stacey. In that book, I found that the authors had summed up the essence of what is satisfying about mathematics: solving intricate and difficult puzzles by finding the simple logic that underlies them. The book showed where the art and the science of mathematics meet. (It sometimes feels like many of the new ideas of how to engage students in math are pulled from this book. The problems are simple enough that they can be understood and attempted by grade 7 students, but are complex enough that they can lead to university-level solutions.)

So, I do math because I enjoy solving intricate puzzles, and I find it extremely satisfying when seemingly impossible problems can be solved simply by looking at them in the right way. And the reason why I teach math is that I enjoy seeing students get the same satisfaction out of these puzzles.

The Perfect Lesson Plan: a Hypothesis

Unfortunately, there is a disconnect between the way I enjoy learning mathematics and the way that I felt I was expected to teach mathematics. I get excited about learning math when I'm faced with a challenging problem that I can generate good hypotheses for, but that takes a while to figure out the underlying structure. I enjoy teaching math when I see my students going through the same process: they have a tough problem that is easy to understand, but which is tough to solve. Take, for example, the first problem in "Thinking Mathematically".


This problem is simple enough that students can understand the question easily, but complex enough that it takes some time to figure out. That's precisely what makes it so enjoyable to work on. But when you go in to the Ontario Math Curriculum (Grade 8 Number Sense is on page 111), it's not immediately clear whether this question really teaches kids to "solve problems involving percents expressed to one decimal place (e.g., 12.5%) and whole-number percents greater than 100 (e.g., 115%)". It becomes easy to start asking myself if I've really taught that students what I was supposed to teach. Did I go over the concept too superficially? Should I have gone into more detail? What about the kids who struggle with the concept of decimals, fractions or percents--how do I help them succeed? And if I try to focus on the students who are struggling and fill the gaps in their understanding, what should the kids who find this easy be doing?

(You may have already heard an easy answer to this, such as, "You just work with the struggling students in small groups while the independent workers work independently"; unfortunately, the reality of working with students in the classroom often precludes easy answers. I have a couple elements of solutions and hypotheses to propose. I also have a couple good questions. But I can confess right now that I don't have any easy answers or solutions that will 100% work in your class for 100% of your students 100% of the time.)

The problem-solving rubric proposed in the book, "Thinking Mathematically"

Where I Got Stuck

Last September, I taught Chapter 5: Measurement of Circles in September. And I did Chapter 1: Number Relationships in October. And after that, I realized that I had no interest in continuing. I couldn't figure out why. It didn't make any sense to me--I love doing math. I love teaching math. I love the rigour involved in preparing lessons that are pedagogically sound, and that explore the nuance of the mathematics curriculum. Why wasn't I looking forward to the next unit?

So I did what I usually do: I started hypothesizing. Maybe I didn't change the program enough from last year (I don't like teaching the same thing the same way twice). Maybe I wasn't precise enough in my examples or definitions (perhaps students weren't fully understanding). Maybe I didn't want to write notes on the board (in retrospect, that was certainly part of the answer). Maybe I should ditch the textbook entirely and plan my year based only on the curriculum. No matter how much I personalize the program and add in my own activities, by starting with a textbook, I'm automatically following someone else's plan, and that limits my creativity. (I looked at my long-range plans, and they looked dreadfully boring.) Maybe I'm experienced enough now that I don't need to curriculum as a guide (I have a pretty good feel for what a grade 8-level problem should be). Maybe if I taught using just the curriculum, I could cover the entire thing in one year. Well, anyway, as much of the curriculum as anyone can cover, since there are so many curriculum expectations, how can anyone cover them all anyway?

When I prepare my units, I try to start with the end in mind. I look around for any great resources (after all, why create something from scratch when I can take something good and make it great with the same amount of effort?), and I try to match them up against the curriculum and organize them in a way that should make sense to students. You can sort of see my process when I compiled the resources, "Nelson Math Grade 8 Long-Range Plans", "Jump Math Grade 8 Long-Range Plans", and "Ontario Curriculum Long-Range Plans". I tried to take the best parts of all three of those documents, and structure them in a way that would make everything as clear as possible to the students. As far as I was concerned at the time, a perfect lesson should cover all the curriculum expectations in an organized and structured manner, and should define all terms, hypotheses and theorems along the way. A perfect lesson should be rigorous, precise, and provide excellent examples for students to follow. You can see the result of that hypothesis in "Chapter 1: Number Relationships".

A Possible Solution

So here is the working hypothesis I tested last year: if I can find a way to teach mathematics so that the kids learn the same way that I like to learn, then they will have more fun and I will have more fun. (Oh, and also, we will all learn more and other stuff too.)

With that in mind, I started to re-design my curriculum so that, instead of teaching a bunch of theory, taking notes, and then trying out some problems, we tackled the big problems first.

The advantage to doing it this way is that the students started asking better questions, generating better hypotheses, and becoming more independent. The short-term downside was that many of the students felt destabilized. The structure I was using to teach had changed, and they justifiably felt unsure of what to expect. To be fair, I felt a bit destabilized myself. I like having structure.

And that brings us to the reason why we are all here today. How can we teach the way we want to teach while still retaining structure? How can we avoid the extremes of "too much flexibility" (which makes it difficult to plan, and to make sure that we are going to teach what needs to be taught) and "not enough flexibility" (where we can't adapt what we are teaching to the needs of the students)?

Well, there's a resource that helps tremendously with that problem.

The "Curriculum and Connections" Resource

Continuum and Connections

This collection of documents, created by the Ontario Ministry of Education, is a fantastic way to organize a flexible framework for mathematics. The documents cover grade 7 through grade 10, and distinguish between applied and academic at the high school level. The best part is that each of the documents is broken up into the strands for each grade.

For example, grade 7 and grade 8 mathematics have a section devoted to each of the following: (1) Fractions; (2) Integers; (3) Patterning and Algebraic Modelling; (4) Perimeter, Area and Volume; (5) Proportional Reasoning; and (6) Solving Equations. Furthermore, each of those sections are broken down into the 5 strands we are supposed to evaluate in those grades: Number Sense and Numeration; Measurement; Geometry and Spatial Sense; Patterning and Algebra; and Data Management and Probability.

The way the "Curriculum and Connections" resource is put together on the EduGAINS website, is unfortunately difficult to work with on a daily basis. I needed something that I could just put on the wall, and check off as I went, something that I could refer to quickly and easily. So, last year, I took some time to compile all the units (by grade level) into a series of 1-page documents containing all of the important information. Click here for the full "Curriculum and Connections" resource.

Covering All the Curriculum Expectations

One of the big advantages to teaching this way is that I no longer worry about "covering the curriculum". There are only six units in the entire year, and I can break them up however I want. In addition, I know I'm going to cover each strand once in every unit, so I'm never going to be in a position where, "Oh, no! I have to teach fractions sometime in the next three weeks so that I can get a grade for the report card!"

Using the Curriculum and Connections framework, I'm going to cover every strand multiple times each semester, so I really just need to make a judicious choice of what I'm evaluating.

Here are two examples of units that I've taught this past year, using the "Curriculum and Connections" framework.

1. Understanding the Relationship Between Percents, Fractions and Decimal Numbers

This is an overview of our strategy wall, after doing one project together.

This unit was based on the Curriculum and Connections grade 8 "Fractions" section. I co-constructed the learning goal and success criteria with the students, but I already had a good idea going in what I wanted to do. The big idea was to understand that "fractions, decimals and percents are just different ways of representing numbers" (you can see that on the top-right, in yellow), and that the same number can always be represented (at least) three different ways.

The student work you see on the board is for two problems, "Nana's Lemonade" and "Dueling Discounts", created by Dan Meyer. For an overview of how I structured this unit, see the post in its original context. At

2. From Patterning to Algebra

Finished Strategy Wall (1). This is what our strategy wall looked like at the end of this problem.

This unit was based on the grade 8 Curriculum and Connections section, "Patterning and Algebraic Modelling". For the students, I named it "From Patterning to Algebra" because that was more meaningful to them. The big idea was to "Understand that pattern rules, tables of values and graphs are simply different ways of representing patterns" (in yellow at the top left). We created our success criteria together, and we added definitions as the unit progressed, to help students learn the terminology that goes with the ideas they were expressing. You can see how I structured that unit in this post. I also included the following evaluation in that unit.

2016-03-04 From Patterning to Algebra - Pattern Rule

Distinguishing Between Skills (Computing) and Reason (Problem-Solving)

Now that we have a framework upon which to base our teaching, the next problem is, "What do we put in that framework?". This is where we get more into the art of teaching, rather than the science. I'll do my best to show you what I do, and hopefully some of that will be useful for you, but most probably, at least some of this won't work with your style of teaching, so please look at this with a grain of salt.

The Ontario mathematics curriculum identifies 7 important mathematical processes: (1) problem solving; (2) reasoning and proving; (3) reflecting; (4) selecting tools and computational strategies; (5) connecting; (6) representing; and (7) communicating. I find it difficult to address all of these processes using just one teaching strategy. I suspect that it would be difficult to reach every student in the classroom using just one teaching strategy as well.

There are two main types of activities that I use to structure student learning. The first is called "Number Talks", and is based on the book "Making Number Talks Matter" by Humphreys and Parker. The book is available on Amazon, but if you bug your school's numeracy coach enough, they might be able to find a copy for you. The second type of activity I like to use are "Three-Act Math" problems, which is (sort of) Dan Meyer's take on three-part math lessons.

Number Talks are 20-minute lessons that cover the skills or computing aspect of solving relatively straightforward problems. Essentially, I chose a problem that I hope will get students to come up with good strategies. Each successive Number Talk builds on the one before it, until students have developed a variety of effective and efficient ways to solve "skill" problems. I will do one or two examples with you during our workshop, but in the meantime, this is how I introduced Number Talks to my class this past year.

Number Talks have replaced the "exercises" portion of my math class. I used to spend a lot of time worrying about the exact best way to structure exercises so that students develop a variety of strategies, and are challenged but not overwhelmed (here is an example of a series exercises for exploring square roots and squares). Now, with Number Talks, I instead use what students have learned so far, and where I want them to go, to choose a problem that I think will help them develop the type of strategies that will help them with a particular skill.

Three-act math lessons are where students get to work in groups to solve complicated problems and develop reasoning, proving and reflecting skills. With good three-act math problems, students form a hypothesis, test the hypothesis, maybe solve the problem, and then reflect on their solution. Often, at the end, they will realize that their solution was inaccurate, partially accurate or reasonably precise, at which point they can reflect on the usefulness of their mathematical models. Over time, they often realize for themselves the statistics aphorism:

All mathematical models are wrong. Some are useful.

I'm going to do an example of a three-act math problem with you as well, and we'll talk about how to make them as effective as possible for your students.

Trials and Errors

In no particular order, here are a couple of the questions that I am struggling with right now. I have elements of answers, but I don't have a coherent global theory that answers all of the problems.

  • How to emphasize group work vs. individual work? (Should one count more than the other? How much time should students spend working individually or in teams?)
  • How do you make sure that every student has learned from their work? (What kinds of activities help you determine what a student has, in fact, learned?)
  • How do you evaluate students when you are constantly giving them feedback? (If you give them questions or next steps, how does that affect your grading of their work?)

Overview and Comments

Here is what I hope to cover during this in-service:

  • Introduction to "Curriculum and Connections" (large group)
  • Choose an order in which to do the 6 units proposed in Curriculum and Connections (randomly generated groups of 2-3)
  • Go through the planning process for one of those units:
    • Do a Number Talk to teach the skills section of the problem (large group)
    • Do a 3-act math problem to teach the problem-solving side of it (randomly generated groups of 2-3)
    • Do an evaluation (exit ticket or math journal entry) to evaluate summatively (plan in groups of 2-3, do the problem individually)

I hope to add pictures of the work we did together to this post, so that we can refer back to it later. If you have questions or comments, please add them below.

UPDATE: Pictures from the Workshop

Thanks for coming out, everyone! This was a lot of fun, and I learned a lot.