EDIT2: This post has turned into quite a monster. It connects several different concepts together, but is certainly long... Take your time and enjoy :)
*EDIT3: It seems that I am unable to follow through on my initial ambitious idea of recapping absolutely every session... and so I will focus on a few sessions that really made me think. Also, I have decided to split it into 2 parts.
The Canadian Mathematics Education Forum (CMEF) meets approximately once every 4 years. It is part of a larger organization called the Canadian Mathematical Society (CMS), but the focus is on creating a dialogue across different spectrum of individuals involved in mathematics education: elementary teachers, high school teachers, school board coordinators, instructional coaches, college and university mathematicians, graduate students, math education professors, and representatives from government and publishing. Participants came together from across Canada -- from the west coast to the east coast.
CMEF 2014 took place about 2 months ago (May 1st to May 4th). I am going to summarize some key ideas from the various sessions I went to, as well as provide some thoughts on each of the topics. Any mistakes are my own.
Thursday
France Caron: Reconnecting the curriculum - beyond tensions, myths, and paradoxes
The mathematics curriculum is often conceived as accumulation of knowledge for which a “vertical integration” is expected to ensure some continuity from one educational level to the next. We propose the idea that a “horizontal integration”, that would take into consideration the multiplicity of today’s mathematical practices, could help us go beyond resilient myths and reduce some of the tensions that currently exist in the teaching of mathematics. This vision does not entail the sacrifice of mathematics quintessential characteristics; on the contrary, it could help bring them forward. To substantiate the proposal and illustrate it with examples, we will make use of experiences, observations, documents and discussions. And we will not hesitate to stand on the shoulders of giants.
My summary:
It is a myth that mathematics are built one floor at a time. A better analogy would be that multiple floors can be built at a time, and that mathematics cannot be represented as a linear process. France showed us very nice pictures depicting this. I took a picture, but it was relatively blurry, and so I made an attempt to recreate it here:
The idea is that there are multiple buildings. It is colourful, plural, with connections, and is alive. When connecting university and secondary math, often several topics are considered, as well as connections between them. "Real world" mathematics, however, is often the elephant in the room.
Should students learn to compute
It encompasses a large number of different techniques and is time-consuming. Professors responses are sometimes: "because it's hard" or "you will know later, just endure it for now." The prevailing myth is that working in the abstract develops abstractions and reasoning." Another myth that comes out of that, is that calculus is not particularly useful, but it is a great filter of students. France believes that the value of a technique (like solving for the integral shown above), can be assessed from two perspectives: epistemic and pragmatic.
The epistemic value contributes to:
- understanding the concepts and their properties
- helping acquire future knowledge
- developing higher order thinking skills
- efficiency
- scope of applicability
We believe that the epistemic values are the most important ones. So why do we teach mathematics? There are so many different fields of mathematics, and they all have (naturally) different opinions on what mathematical field is important. So what is mathematics really about?
According to Brown & Porter (1995), mathematics is the study of pattern and structure and the logical analysis and calculation with patterns and structures. Within these, 3 areas emerge as 1) Modelling, 2) computations, and 3) proofs.
But what about problem solving? Problem solving is inherent in all three. France then discussed several individuals who had different thoughts about mathematics as a discipline (including Poincare). Modelling is not just applying mathematics. We need to think of it as "involving mathematics." There are many different ways to model instead of just curve fitting. Our desire to build towards calculus in the curriculum creates a disillusioned version of mathematics. It's not that calculus isn't useful - but that calculus isn't the only way to model, and isn't even the most intuitive for some problems. In our current society, mathematics is used more than ever. But it is also hidden more than ever. technology has been driven by the concept of the black box (magic occurring inside a box that does what you want it to, without us understanding it). We need to claim back math ownership of technology -- or at least part of it!
Proofs was the last area that France explored. It is perceived as boring because we have reduced it to a two column proof. When in fact it provides continuity across different concepts of mathematics.
Thoughts:
There are a lot of different ideas in here. A ton of things to think about. I will list some ideas that came to mind:- While France does touch on the topic of real world/non-real, this is a bit different than a previous post on real/fake world, as well as Dan's posts on real/fake world. We focused on purposes of mathematics that engages student. Some notable points were that 1) real-world mathematics does not guarantee interest; 2) definitions of "real" may vary across different individuals, classrooms, context. France, instead, looks at the curriculum level. It does intersect at the interest of the student.
- The vision of mathematics as one single building from the ground up is a devastating one. I liked the analogy that France made to a landscape as opposed to a single skyscraper. Calculus is at the top of a pyramid that students climb without understanding why. Calculus certainly is beautiful. But this beauty is obscured by abstraction that we assume all students are ready for - simply because they are similar in age. This beauty is also strangely emphasized over the other fields of mathematics, which paints a poor picture of mathematics in general.
- The example France gave with the integral is an interesting one. Why DO we put our students through difficult problems without context? We throw problems after problems at students that may be difficult, but are uninteresting. Context and freedom helps this. By context I mean providing potential reasons for why we may be interested in a problem (through 3acts, through fascinating problems...etc). By freedom I mean the freedom for students to explore questions, problems, processes, and answers. Much of the current issues stem out of a fundamental misunderstanding of what mathematics is. Across Canada we've been having difficulty moving forward because of the portrayal of mathematics as a singular entity grown out of computation. People are unable to see mathematics as a vehicle for better thinking, and they are unable to imagine mathematics as a multitude of different ideas bound together by logical proofs.
Friday
Chris Suurtamm: Assessments that elicits and supports mathematical thinking
The second day began with Chris Suurtamm from University of Ottawa. Her talk was titled: Assessment that elicits and supports mathematical thinking, and the following is the summary provided by the CMEF website:
Current thinking and research in assessment and in mathematics education recognizes that mathematics is a complex process that is inadequately assessed using merely a paper-and-pencil test. But what does an assessment program look like that elicits, values and supports students’ mathematical thinking? This presentation will challenge the audience with taking a critical look at assessment from the perspective of what mathematics is valued and supported. It will also draw on research to share a variety of strategies and practices that support and value the development of students’ mathematical thinking and deeper understanding.
My Summary:
Assessments have shifted along with the paradigm shift in mathematics education. With respect to assessment, we have moved from assessment as an event to measure the acquisition of knowledge to the ongoing attention to support student learning. With respect to mathematics education, we have moved from mathematics as a set of rules and procedures to the social practice of investigating mathematical ideas. The implications are that:
1. assessment should be ongoing and embedded in instruction
2. the use of a variety of forms of assessment is needed to address the complex nature of learning and the range of mathematical activity occurring in classrooms.
3. assessment informs teacher practice and promotes student learning.
Chris talks briefly about the NCTM (1995) assessment standards as an example. The importance of assessment is paramount. Because, ultimately, what we say we assess is what we tell students we value. Of the 6 NCTM assessment standards (mathematics, learning, equity, openness, inferences, coherence), she discussed the two of them: the Mathematics standard and the Learning standard.
The mathematics standard: assessment should reflect the mathematics that all students need to know and be able to do. In other words, we should not focus on the mathematics that is most easily measured (hand written work, calculations, computations). Instead, assessments should reflect the full range of mathematical content and mathematical process.
The learning standard: assessment should enhance mathematics learning. Assessments are learning opportunities as well as opportunities for students to show what they know and can do. In other words, it is not only assessment of learning, but also for/as learning. Assessment that enhances student learning becomes a routine part of ongoing classroom activity rather than an interruptive event. Assessment activities are consistent with activities used in instruction (e.g. learning in groups, assessed in groups).
Chris lists several questions to keep in mind when creating and facilitating assessments, but this one sticks out the most: How do your assessment activities enhance student learning?
She carried on to two student video examples from the work of Carpenter, Franke, and Levi (2003). The two students each had a different problem, and had different responses. The first student thought out loud, and the second one paused for a few moments, and arrived at the answer. The second student was asked to elaborate, and also showed good insight and understanding into the problem. A notable point that came up during the discussions that Chris facilitated was that the second student did not use inverse operations. Does this mean that he doesn't understand inverse operations? No, it merely means that we do not have evidence of what he doesn't understand.
What teachers would like from students are deep understanding. What are the teacher practices that elicit this type of thinking? We went on to look at a different problem that we were all given some time to do. There were a large variety of different types of solutions. We then looked at transcripts of student/student and student/teacher conversations while the students are solving problems, and there were effective teacher moves and questions that facilitated students' learning progress.
Chris ended with two important slides.
1. Assessments to make student thinking visible:
- questioning, listening, and responding
- conferencing
- journals
- listening to student conversations
- sharing student solutions - making student thinking public
- consider what mathematics is being assessed - is this the mathematics that is valued?
- consider the opportunities for students to learn from the assessment
- consider whether the assessment strategy mirrors instructional strategies (coherence)
- consider how the assessment elicits, values, and develops students' mathematical thinking
Thoughts
There is so much to think about from this session. I am interested in assessments for at least the 3 different reasons below: 1) it plays such a pivotal role in developing student learning and student thinking; 2) it is largely under-developed and misunderstood by teachers in my experience; 3) it is a continual challenge to match the current development in pedagogy.
One of the few things that struck me is the difficult in being able to do certain things that are theoretically sound and in alignment with current thinking in pedagogy. For example, the idea of group assessments in an individualized reward system of grades; the difficulty of the teacher facilitator changing roles to become the teacher assessor...etc.
But all these difficulties in assessments are reasons why I believe it is worth exploring.
But all these difficulties in assessments are reasons why I believe it is worth exploring.
Patrick Reynolds: Adrift in a sea of video tutorials
My next session was onto Patrick Reynolds from University of New Brunswick
My summary:
I didn't take as many notes during this one. If you are reading this blog, you are likely familiar with the #mtt2k discussion last year. Patrick began with some introduction of the video phenomenon, as well as citing a few studies that have looked at the approach. Among the videos that he showed, this Veritasium was a good one that caught my eye (you may have seen this before)
We ended with a few talking points.
1. the online video medium has potential way beyond "let me show you how to do [technique]"
2. hundreds of math educators creating "chain rule example" videos is an ineffective use of our time and talents
3. the comment "I learned more watching 10 minutes of Khan than in 1 hr of lecture" was shown to be ineffective in some studies (retention is actually low, and students don't actually have full understanding) -- but positive sentiments from students shouldn't be dismissed easily.
Thoughts:
This is a topic that I've thought about before. Plenty of other people across #MTBoS has discussed and thought about this as well - and it can be a divisive topic (goes to show that while we share a desire to innovate, think, and learn, we are far from conforming or having the same opinion on everything).
Michael Pruner: Sometimes the best high tech is low tech
The next session was from Michael Pruner from British Columbia Association of Mathematics Teachers. While I didn't really know what to expect from this session, this turned out to be the beginning of several earth shattering sessions. It even warranted:
Best conference I have EVER attended. The ideas coming out will revolutionize my teaching till retirement. Well done #CMEFFCEM14
— Alex Overwijk (@AlexOverwijk) May 3, 2014
This was from Al Overwijk who is currently doing some pretty amazing stuff with respect to pedagogy.My Summary:
Michael began with his personal story. He always had the lingering feeling that his classrooms are not interactive enough. He feels like a performer, the class is too quiet, but this didn't bring everything together for him that things need to change. Michael differentiates between what he called "gaming" and "studenting." Where "gaming" is when students are just trying to beat the system instead of actually learning - which is what "studenting" is defined as.
Michael then talked about several points that changed for him.
First is the "now you try one" technique. He indicated, with some slides of research that I will go into in the second part, that no thought or learning went through the students mind with this type of pedagogical approach.
Second he tackled -- probably a more controversial point even among teachers who are currently trying new stuff -- the idea of note taking. In this new pedagogical move, students don't take notes! Instead, students write on the white boards collaboratively, and Michael takes pictures afterwards and share them on his website. He cited research (likely from Peter Liljedahl, whose sessions I will discuss in the second part) here about notes
Third he tackled another (likely controversial) topic -- homework. He shared how he assigned no homework. This was an academic streamed class, and there was no homework! Kids were significantly more engaged, had a lot more fun, and interacted with the mathematics instead of passively letting others (including peers and teachers) take on the tasks.
What do we do about this? We change the game.
He then went on to describing three things that he did in his classroom:
1. Visibly random groups
2. Vertical non-permanent surfaces
3. Students do not take notes (evidences are posted as pictures instead)
The third one is actually not one of the three major pillars that Peter Liljedahl includes as the three pillars... but that will come later (The third pillar is Engaging Tasks)
Michael made these changes, and it was powerful. Students were engaged, they were talking, they were working, they were thinking.
Thoughts
There are lots to think about here. This session aligns a lot with many different things I've been thinking about (and reading about). I've used vertical non-permanent surfaces in the past (simply because the classrooms that I worked in had lots of blackboards), and I did notice engagement, but I definitely did not do that every day.Vertical: The majority of the time I had students work on chart paper, which means they worked on a flat surface. Or alternatively sometimes they worked on whiteboards that would lie on their desks. I have noticed in the past that this was sometimes ineffective for some groups.
Non-permanent: A student in the past actually asked why we didn't just use a giant white board instead of wasting chart paper all the time. I suppose her point was about saving trees... but there is a more powerful point here about non-permanence of written work. With permanent markers on chart paper, students have a harder time putting their markers on the chart paper to produce work. I've had several instances where students would ask for a new piece of paper because they've messed up one letter in their explanations. Then they would carry on wasting a lot of time recreating what they've already written.
Notes and homework. All of these were good starting points of things to think about while I was in Michael's session. I had more chances to think about this deeper when I got to Peter's session the next day. These two are definitely huge radical shifts for a lot of people, even for some of those who have strong buy-ins for activity-based learning and group work.
I will likely revisit all of these thoughts and ideas when more is elaborated in the later sessions.
Coming up in part 2 - some more thought provoking sessions...!
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