Viewing mathematics education from an international perspective, we’re doing pretty well. Where do we stand?
Consider equity versus quality on a graph (from Pasi Sahlberg presentation at ICSE 2016). We often measure equity (horizontal axis) by how narrow the gap is between high achievers and low achievers. Quality of student achievement can come from scores (as PISA). Where would a country like to be? Where is Canada?
Chris notes she was “in Scotland, listening to someone from Finland” and up in the cloud, high in both measures, was Canada (along with South Korea, Japan, Estonia, Finland, Hong Kong). Those countries are “Well known” for high achievement in assessments but also high equity.
We don’t hear about this, we often get painted with the same brush that the US gets painted with. While others, like the US, are still climbing the “stairway to heaven”. If we look at PISA scores for 2012 (international assessment of 15 year olds), Ontario is 514, while International Average is 494. So we’re well above, while the US is below that average. Yet as their stories are perhaps not as positive, we get drawn across the southern border.
Chris says that a researcher came to visit recently. A fellowship to Canada to see what we’re doing. They could “find out what’s in Singapore or Hong Kong” but that’s a very different context, Canada has more similarities to the USA. “You should all pat yourselves on the back”, because instead, you get beat up over the kinds of things that are happening. Yes, PISA scores dropped slightly in the last assessment, but it was exaggerated in the media.
Canada is fairly well respected in terms of our math education system. And we have high standards. We’re just not showing it. So what is it that makes us great? Chris doesn’t have all the answers, but wants to use a framework based on what she’s done in Ontario with teachers, to do some analysis. (ASIDE to Blog readers: Canada’s education system varies by province; Ontario isn’t the same as Quebec, which isn’t the same as the Western or Eastern provinces.)
The framework is based on the guiding principles for school mathematics. Consider “Principles to Action”, NCTM 2014. We’ll go through these points: (1) Access & Equity. (2) Tools & Technology. (3) Curriculum. (4) Teaching & Learning. We’ll save “Assessment” and “Professionalism” for another talk. Chris will be drawing on her recent research over the past 10 years. Which includes “Curriculum Implementation in Intermediate Mathematics” (CIIM), a focus on Gr 7-10 teachers, “What Counts in Math”, communities of practice studied over 2 yrs, “Pan-Canadian Pedagogy Project”, which looks at what things are like in BC, Quebec, etc, and the “Grade 9 Applied Project”, also studied over 2 yrs.
(1) Access & Equity. “I think we’d all nod to that”, and we want to be sure that’s not just for our gifted students, but also for our struggling learners. All students have the need - and Chris would say the right - to engage in rich mathematical activity. All students need (and have) the right to these things: To Engage, To Have Opportunities, To Time, To Feel Valued. It’s more than just worksheets.
(2) Tools & Tech. “Some of you had me when you were teacher candidates”, thus know Chris refers to manipulatives and mathematical thinking tools when she instructs. But it’s not just about manipulatives. Look at the kinds of technology that many of you use: Kahoot, Nearpod, Plickers, Desmos, Geogebra, Geometer’s Sketchpad. “All of these things I observe you using in your classrooms.” And they’re used in interesting ways.
Chris shared one way she saw a teacher use EQAO questions (from the Gr 9 mathematical equivalency test in Ontario). At the time it was plickers, could easily be kahoot or other. These were Gr 9. Applied level students, looking at a summer gardening job question: $10 flat plus $8 per hour, which graph matches (A, B, C, D). The teacher gave them a couple of minutes, they keyed in their response, and then she displayed responses... a pie graph about evenly split four ways.
Chris wonders where she’ll go with this. Teacher: “Turn to your partner, and tell them why you think your answer was right.” After some time for this, she put it up again, everyone individually keyed in responses. Now almost ALL of the responses were the same, and on the correct answer. It was a wonderful example - and this teacher did many more - of using the technology in a very collaborative kind of way. It was formative assessment for her, in a collaborative environment, more than a timed competition (which can be how some use those devices). The students can talk about why to pick this one over that one.
(3) Curriculum. Phrases like “Develops important..." are defined in that “Principles to Action” NCTM document. Chris claims these are satisfied in Ontario. But even though our current curriculum has been around since 2005, over 10 years, why are people still talking about it as NEW math?
Consider, what does the research tell us. What about learning progressions (what to teach first), about a network of learning, about new pedagogies? And what are other countries doing and how do we compare? ”When I visit other places”, they’re surprised that there would be consultation with education researchers or teachers. It’s not as though the curriculum comes out and we totally disagree, so we’re lucky to have that consultation progress.
Actions that students should do in a mathematics class: Identify, Compare, Describe, Pose problems, Evaluate, Gather, Connect, Expand, Investigate, Sketch. Chris adds her beginning teacher candidates are often doing those things themselves too. It’s not about the number of manipulatives on the table. Manipulatives don’t always mean it’s good lesson.
Chris shows an image from Singapore, their framework. (Note: Singapore was an outlier in the country graphic above, high quality despite middling equity.) The graphic shows math problem solving at the heart, and pod areas around for: Attitudes, Metacognition, Processes, Concepts, Skills. “You’ll see words that pop out from our curriculum.” Then there’s also talk of mathematical competencies with South Koreans, French, and Germans, similar to our mathematical processes. (You can just look on the internet and see other curriculums.)
(4) Teaching & Learning. We need to engage students in meaningful learning. So they’re not just cutting and pasting, they’re doing some math. Big ideas that get broken down, what does that look like. “Productive struggle”, support this, even if “we want to rescue those kids, to save them”.
Look at implementing tasks that promote reasoning and problem solving. “Rich Tasks.” It’s like a buzzword, but we’re looking for varied opportunities with multiple entry points - not a simple task, more one with high cognitive demand. Everyone at tables had the following problem on a sheet, and were asked to work on a solution “that you think a Gr 5-8 student might produce”.
PROBLEM: Tug of War 1: 4 frogs tie with 5 fairy godmothers. Tug of War 2: 1 dragon ties with 2 fairy godmothers and a frog. Who wins the third Tug of War: 1 dragon and 3 fairy godmothers versus 4 frogs?
We had 5 minutes to play. I think I had seen something like this before, as my immediate inclination was as follows. Focus on the 3rd scenario, swap out the 4 frogs for 5 fairies. Then, remove equal fairies until I had 1 dragon versus 2 fairies. The dragon will win; the fairies are missing a frog. Others at my table were looking at unit rates. We spoke briefly. Should the scenario be modified to avoid such a direct swap on my part? How deliberate was this option? If someone finds this solution, should they be encouraged to look for other ways?
Chris warns sometimes we give a task and “scaffolding too early can reduce the cognitive demand”, taking away opportunities for students to explore and build confidence. Support productive struggles.
Grade 7 teachers in a conversation on October 9, 2013: “I like the struggle, and I do that too.” ... It’s “part of my own struggle as a teacher, because you’re so wanting to just get in there and help them and save them. But the standing back and giving them time, and realizing that often they WILL get it, and get more out of it when you let them do that - when you let them struggle.” Which, Chris says, is what she knows Ontario teachers do.
We looked at some different kinds of solutions from a class. “Did you draw pictures like this?” One student explained reasoning as “we started with 4 frogs and we decided to value them as 10, because 10 is a friendly number”. It was a value of their “strength”. Why strength? “Well, I thought about Pokemon.” And from there to rates. Another student valued 20 and 20 (each side of the rope), so in a way, they’re doing a similar thing. “Once you assign a value to one, it influences the others” and that’s huge, realizing that.
Chris said it was very interesting, the diversity of solutions. Not necessarily using algebraic reasoning. (See mine, above.) In the end, the teacher pulled the students together and had them compare, to see a richness in the kinds of solutions. That deals with another teacher practice. Use and connect mathematical representations, to deepen understanding of concepts and procedures.
Also, facilitate meaningful mathematical discourse. Used to be teacher to student, student to teacher, but students hearing teacher talk is like in Charlie Brown. Understanding comes from student to student discussion, need to facilitate those. Give the opportunity for debate. “I teach with my door closed all the time because we are loud.” It’s okay not to agree. See shared thinking that almost sounds like one voice, back and forth discussion, while working on a problem.
Another practice is: Elicit and use evidence of student thinking. Such as having whiteboards around the whole room, shoutout to Peter Liljedahl (who was in the audience). It’s about making student thinking explicit, and eliciting student thinking, making it so the teacher can also see their thinking. Otherwise there’s no way to see what’s going on inside their head.
Also, pose purposeful questions. The teacher in “Ferris Bueller” (Anyone, anyone...) is not asking purposeful questions, ones that help me get a sense of understanding. Ask “What makes this a linear relation”, then give wait time, and listen, not just waiting for someone to say y=mx+b. “Teacher questions in action.” At the end, “Why does that make sense?” If there was one question to tattoo on your arm, that could be it.
And those are the kinds of things happening in Ontario classrooms. It’s that list (above) that Chris often sees as she wanders around classrooms and engages with teachers. And that’s what the principle document says too. We’re building classroom environments to promote math understanding and thinking.
What are the things that are common elements? (“Oh I do that too”.) Chris says they tried a concept map. What often came up was the goal of teaching, that idea of “Developing students’ math thinking and understanding”, from which components of a lesson come out. But a lot was on the environment. Knowing when to step in, making connections, choosing groups... and valuing student voice. “We saw that as huge.”
So Chris thinks we’re doing a great job in Ontario. There’s ways that you can engage in student learning, and it helps us stand out internationally. If there were any questions, feel free to ask her, firstname.lastname@example.org
|For more on Peter, see this post.|
Peter quipped that “every time I do something, I get 10 years worth of data”. Since it takes 10 years to do proper analysis, he’ll likely be busy for about 160 yrs now.
And with respect to the PISA study, referenced above in the measure of quality of student achievement? The last questions are: “How hard did you try.” and “How hard would you have tried if this was worth marks.” They don’t publish this in Canada, but Peter had recently tweeted the information out. (He got this data from Sweden.) The biggest differential between those numbers? Is in Sweden. The second biggest? Canada. Is it the students that “really try” who have the higher scores?
Thanks for reading this post! Feel free to comment below on any questions you might have, or anything that stood out for you. If you teach in Ontario, consider joining the OAME, and if you’re in Ottawa, your chapter is COMA.