Everyone has an Opinion

As a math educator there is nothing I dread more than the wrath of opinions sent my way when standardized tests scores are released (not to mention the fact the it seems to get worse every year...somehow the media gets hold of results before they are public). Everyone has an opinion. And they are entitled to one.

But it does not make them an expert.

It is easy to sit at the sidelines and put blame on someone else - teachers, the curriculum, and the like. Currently most of them blame is on the curriculum and/or so-called "discovery math" instruction.

The Curriculum

Standardized expectations set by the ministry that is publicly available here.

Now I do not ever call myself a math ed expert. My experience is limited to Ontario in grade 9-12 with a bit of tutoring at the grade 6 level. This is the only experience I can speak from. But I am sure I could find many colleagues (elementary and secondary) that would agree that there is too much emphasis on math content. Every grade level is loaded with content that is needed and content has always been in the drivers seat. Elementary grades have to cover content in 5 areas each with a multitude of specific expectations to get to.

Flip to the front matter of either sets of curricula and you will find 7 Mathematical Processes. These are supposed to be the lens through which math is taught - and we need to remind each other of this. Perhaps a redesign of curriculum to make this front matter the meat of each grade would be beneficial. Check out BC's new curriculum - it is competency focused. I am sure it is not perfect, but frankly, it is genius.

You won't catch me claiming that the curriculum does not need to be revisited.

"Discovery Math"

I use the phrase discovery learning in quotations because it is misunderstood by most people. It is over-simplified into this little box definition and believed by many to mean that we give students a problem and then never help them. This is a myth.

I am sure that there is a lot of support needed for teachers to embrace the research behind "purposeful struggle" and I am also willing to go out on a limb and guess that there are many teachers teaching math who are not comfortable doing it (I have friends who would attest to this) and are probably even less comfortable with embracing a different way of teaching. But discovery math is not to blame.

We absolutely need learners to play with numbers and learn ways that they are related. They need to find the number sense within them. Math is not as simple as we make it out to be. It is not a bunch of facts that we memorize and use without understanding (lack of understanding leads to mistakes!) - hey you might be good at arithmetic, but this does not a mathematician make. We need to develop a future filled with people who can problem solve, use logic, reflect and communicate - we do not need a future of human calculators.

Using traditional teaching forced me to always tell kids what to do and how to do it. I could tell them why, but to them why was not important. Students got a lesson and then practiced that work. They only worked on that one skill and never made connections to other ideas. Traditional teaching forced me to teach to the middle of the class and made differentiation nearly impossible.

Discovery math allows me to do a variety of things and to differentiate my classroom. Research supports learning as a complex structure that requires much more than memorizing (and even the parts you do need to "remember" need to be forgotten and recalled many times to bring them into long-term memory). In fact, learning is one of the most counter-intuitive things I know. I bet most people can come up with at least one example of something they thought they had learned only later to realize they had tricked themselves into it - only their short term memory had any idea at the time. By using a discovery approach students have to struggle with ideas (makes learning seem like it takes more effort to achieve - because it does - but lasts a lot longer and leads to making connections to ideas already understood) but they get to do it in an environment with 29 of their peers and a learning coach to help out when the "struggle" moves from purposeful to frustrated.
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Don't get me wrong. I don't have an answer. And there isn't a simple one. I cannot tell you why only 50% of last year's grade 6 students met the provincial expectations (but I can tell you that these tests are not straight-forward and disadvantage many groups of students).

I have rambled at this point and could keep going. I could justify every decision that I made in deciding how I would be running my classroom this year. We don't make these decisions on a whim. We do not do them just because someone tells us to.

One opinion writer this week wrote that the one thing that has not changed are kids - that kids are still kids, capable of learning. This is an oversimplification of a very complex discussion - I could easily disagree with her. But one thing is for sure.

We do it for the kids. Every day.

MHF Cycle 2 Reflection

I have previously posted a reflection of our first cycle here.

Here is the plan that we used for our first cycle.
(Most of the activity credits go to Jamie Mitchell and Steph Girvan in the Halton Disctrict School Board - Thank you for sharing your resources - including your blood, sweat and tears guys!)

One of the things we have found as a department is that students often struggle with the algebra portions of this course. Because of this I offered my students an "algebra crash course". I attempted to remind them the core of algebra and solving equations through manipulatives and a clear reminder of what inverse means (i.e. that log is a function, so has an inverse). These should be ingrained ideas that these students have and I find myself often wondering how to best help students at the high school level with these skills. If anyone reading this has any ideas please share!

As you can see in our plan we had two traditional tests in this cycle. We split the algebra portion up into two sections, polynomial & rational functions and logarithmic & trigonometric functions. The last part of the cycle has students explore combinations of functions through investigation of graphs and getting students to do their best to generalize rules for different types of combinations. As a final evaluation in this unit we had student-teacher conferences.

Students had a conference like this one during cycle one as practice (for all of them I was using Google Forms to track and DocAppender to give student immediate access to feedback). For this conference students were given two functions in small groups. They were asked to identify the characteristics of those two functions and then to as a group predict the superposition characteristics of those two functions. On the day of their conferences students rolled a die to get a random second combination. Students were given 5 minutes to prepare and then had 5 minutes to share as much as they could about that combined function. The key was that they were to explain why they believed those were the resulting characteristics, not just to list them.

I found this evaluation very insightful into student reasoning and understanding of characteristics as a whole. It also provided insight into the emphasis that I should consider putting onto the graphical representation of functions in earlier courses. I have started to think that we take for granted what students take away from graphs.

I really enjoyed the experience with conference with these classes and definitely plan to continue working on using them in other courses. Getting students to explain things verbally has an ability to show student learning that reading a written response just cannot do. The power of triangulation of evidence.

Planning a Spiraled Course

This year I am embarking on a new journey - I am working at a different school and have more math in my schedule than I have had since my first year of teaching. As a part of that journey our MHF 4U (Advanced Functions) course team is taking a crack at "spiralling" the course.

Over the summer I spent some time laying out the course to begin to plan. I started with the skeleton Overarching Learning Goals (OLGs) that were created last year for math to come up with OLGs for the course (I wrote about these skeleton OLGs here) so that I would have already wrapped my head around the overall themes of the course.

Then I created a new document to start the actual planning. I pulled the overall expectations (OEs) for the course and the front matter of the math curriculum (math processes (MP)) into the chart by strand and then created a new column where I put in only key words (content & skills) from those OEs and MP. From those words I looked for common themes in the skills/content that I noticed and colour coded them.

Through this process I noticed a major theme in recognizing characteristics of functions and making connections between representations of functions (numerical, graphical, and algebraic). This seemed to be the backbone of a large portion of the course so it made sense to make this into a group of expectations - and cycle 1 was born.

Here are images of that document (they are a work in progress, evolving as we work our way through the course):

Creating the other cycles became largely about noticing the layers involved in the course. I wanted to build the remainder of the course by adding on layers of difficulty, which would allow us to revisit the same concepts. You may have noticed that the second cycle adds on algebraic techniques but is still focused on the same things introduced in cycle 1. This is the purpose of spiralling - students are able to see the same things over multiple exposures to better build their understanding of the material.

Studies are showing that the use of spiralling techniques will help with long-term retention for learners. It is not necessarily about improved results within a particular course, but will help with the foundations moving forward for longer-term success. My hope is that this type of pedagogy can also help with engagement and mindset for learning in the mathematics classroom.

The planning process was somewhat time consuming but was worthwhile for moving forward into the course as I knew what the purpose of the first cycle was and could see the long-term goals. The difficulty was not being able to co-plan with my course team (complicated by it being summer, going into a new school, etc). Now that we are a few weeks into the semester the team is more on the same page and is starting to be able to see the long term plan more easily.

One of the ideas we added to this plan was to have students start and maintain a portfolio where they would put information as it is learned organized by type of function. We are also going to do part of our final 30% as a conference with students - so students have been told that maintaining their portfolio is to aid them with this conference at the end of the semester. The possibilities are exciting.