The Canadian Mathematics Education Forum (Forum Canadien sur l'Enseignement des Mathamatics) 2014 was held in my hometown of Ottawa this past weekend (actually from May 1-4). The CMEF meets every few years, it was previously in 2009 (Vancouver). It brings together university professors, public school teachers, and educators from the private sector, from all across Canada.
I'm going to give a quick summary of sessions in this post. It may be expanded on in the future, when I find the time, but my sporadic blogging about November has taught me that waiting is a bad plan.
1) Plenary Lecture: Reconnecting the Curriculum - beyond tensions, myths and paradoxes. (France Caron)
There is a Double Discontinuity: School math doesn't necessarily reflect on University math, either as an undergraduate, or as a teacher returning to the system. ("University studies become a pleasant memory.") Meanwhile, math in the rest of the world seems to exist outside this closed system.
|I believe Jim Pai found these images himself|
MYTH 2: Math is hard; as it should be, it's learning.
MYTH 3: Working in the abstract develops abstraction and reasoning.
MYTH 4: Calculus is not particularly useful.
MYTH 5: Calculus is a great [student] filter.
MYTH 6: Proofs are boring; students are not interested in proofs.
There is a Dangerous Paradox: Math has never been so present, yet never so hidden (inside black boxes). Can we recognize programming and algorithm development as mathematics? Are black boxes unavoidable with "technology"?
-Three goals for math. Modeling, Computation, Proof.
-Applied math is messy. Instead of applying mathematics, involve mathematics. (Appliquer v. Impliquer)
-The integral of the square root of 'tan x' is fiendishly difficult.
2) Plenary Lecture: Assessment that elicits and supports mathematical thinking (Chris Suurtamm)
There's a shift occurring from mathematical rules and procedures towards the social practice of investigating ideas. This session looked at some of NCTM's 6 standards for School Math. A question posed of Grade 8s was analyzed, as well as videos of students explaining how they solved problems. See www.pilab.ca
-Assessment should reflect what students need to know and do, not what is most easily measured.
3) Vignette: Adrift in a Sea of Video Tutorials (Patrick Reynolds)
What makes for a good video? A bad one? The "Mystery Teacher Theatre 2000" competition was referenced, as well as this Veritasium video: "Khan Academy and the Effectiveness of Science Videos". In that one, we see that videos must tackle misconceptions, otherwise students will not change their ideas, so incorrect beliefs remain - and worse, students will feel more confident despite their misconceptions!
If students "know" the answer, they won't search for a video, but if they don't know, they won't know what to search FOR. ("What is this called?" really means "How can I google this?") Coming to a video with questions is valid. Watching it to "learn" is not; we must spark a question first, possibly one about the viewer's preconceptions. Question: Is hundreds of math educators creating 'Chain Rule Example' videos an effective use of our time and talents? A five minute video can take hours to put together.
-Ideally, connect new material with something familiar. (This may be a familiar concept, or a familiar mistake?)
-Time delay answers, perhaps don't even give the answers in the same video.
ASIDE: The Four Points vignette was also during this slot, I spoke to Al Overwijk about it. Can you create four points such that there are only two different distances between them when they are all connected? A square is one way, what are others?
4) Vignette: Students and their Instructor Co-Developing the Final Exam (Tina Rapke)
There is a mismatch between classroom practices and closed book exams. To allow for student input: Six groups are made (4 ppl per group) and given 6 hours to create "practice" exams, including a solution key. (Course expectations are provided.) Groups then write each others' exams and assess themselves. All practice exam keys are posted online. At least 60% of the exam is taken from these questions, merely modifying the numbers.
Observations: If there is no set number of questions, students will make the exam too long. Doing this does not inflate marks to 100%, there will always be some things students "don't get". It's assessment OF learning... but does it lead to memorization? Who decides the legitimacy of a piece of math? This also may work best when problems are posed in class throughout the term, to breed familiarity.
-A good way to see what students see as high priority, and have them claim ownership of the material.
5) Vignette A: Sometimes the best high tech is low tech (Michael Pruner)
Research has shown the following methods to be ineffective:
-"Now You Try". Can lead to over 50% of people mimicking, rather than actually checking their understanding.
-"Note taking". Only a little over 33% of a class kept up with note taking, after which only 10% actually used notes to study.
-"Homework". The only visible difference between graded and ungraded homework is that in the former, more cheating/copying occurs.
A way to change your practice:
-Vertical non-permanent surfaces, to improve visibility. Non-permanent removes fear of writing until things are right.
-Don't give notes. Have one pen per group, can take pictures to track evidence. Put them in a PDF and place online as notes.
-Change how we answer questions. Don't answer proximity questions.
-Change homework to be more individualized.
-There was more (see image); for me this would be a radical shift. More discussion occurred in plenary for Saturday (see #9).
6) Vignette B: Problem Posing in Consumer Mathematics Classes (Jeff Irvine)
The resource for a College level class shouldn't be a text. Use the Newspaper. Every article in the Metro (free paper) can be the basis for a math problem. Allow students to choose the link they will study from an article. Promotes self-efficacy. Also watch language - don't give "word problems" give "missing information questions" (eg. blanking out part of an ad that could be reconstructed).
Do need to have flexibility in curriculum, as never know where/when the elements will come from (through problem solving is in curriculum!). Also may need to go beyond curriculum (like intersection of exponentials). Usually manage 90-95% coverage.
-Students engage. Basis of a problem should be proportional to students' level.
7) Vignette: Beautiful Math Moves, Dancing the Transformations (Susan Robinson)
Being the functions creates a character viewpoint, as opposed to an external view. One student who returned from post secondary said moving in math class helped to create a sense of 3D space. Setting down a grid can also help create notion of scale (vertical may not be equal to horizontal, depending on room). Teacher moving between students can highlight patterns.
Particular uses: Expansions and Compressions. Help to see the stretch is not from a "vertex" but from your "core". More: When part of a class as a group is "y=x" and the other part is "y=1/x", students can add themselves to reach the same point, and you have now graphed "y=x+1/x".
-There is more to this than moving arms: has applications even in limits and vectors. Also, teachers themselves have trouble with common language when it comes to compression factors.
-y=x^(2/3) looks neat.
8) Vignette: Incorporating inquiry based learning in assessments (Jimmy Pai)
Inquiry-based as defined means: Learning through design; Project based; Problem based. The last is the most common. Given results can be unpredictable, not a conditioned response, how to assess and evaluate?
Begin with the Task/Problem/Activity, as decided by students. Can be more structured when first doing this, or a video may be used to narrow the focus. From a prompt, design something based on ministry expectations. Have groups on day 1, individuals on day 2, possibly add an extra element that day. Use an evidence record to evaluate more than just what is written, as math gets discussed.
-Difficult to see everything at once, but can focus on the individuals for which evidence is previously lacking.
9) Plenary Lecture: Environments to Occasion Problem Solving (Peter Liljedahl)
You can't just give a "problem to solve" to students... OR a "method to do it" to teachers. It gets filtered through existing norms. The results will accentuate/amplify what is already occurring in the classroom environment. The environment itself needs to be one that creates problem solving. Adjust with the BIG tool first, then fine tune.
|Filtered through existing norms|
There will be chaos. But it's a good chaos. It can remove social/cultural barriers, engagement and enthusiasm increases, knowledge mobilizes, and the kid who simply covers himself with chalk doesn't find that as interesting after a while. Elementary, where you do all subjects, should change up groups a couple times per day (as opposed to per month), and secondary subjects should change groups per day (as opposed to never). Groupings of less than 6 (5 is dangerous). It takes about two Mondays for students to realize that this is going to persist.
Fine tuning involves looking at how you ask questions, handle questions, handle groups (eg. the person with the marker can only write ideas of others, or switch it every few minutes), etc. When everyone in the class has noticed something, you can level the class to the bottom, rather than dragging knowledge up to where you want.
The "Hawthorne effect" (improvement is due to change, rather than the nature of the change) doesn't seem to be a large factor. In fact, thinking too much about the multitude of other factors can create a "move to paralysis" and so it's best to intentionally ignore them. Also noted that when you impose artificial constructs (eg. roles) on a dynamic process like this, you hold back progress.
-Everything. But key for me is that you need to buy into this, and for me that's going to take time.
10) Vignette: Climate Change in the mathematics classroom (Richard Barwell)
Measuring our climate requires math. It may not be that people don't understand the process - they may not understand any math was involved! How does one average the surface temperature for an entire planet? How are predictions made? There's also an element of democracy, as a societal response is needed. Can present unbiased facts in a math class, and data where YOU are may be more personal.
From the data, have students generate questions to pursue. Need to try and strike a balance between "what do we do with all this?" and "here's the step by step process".
-There is relevant data available and a multitude of related questions.
11) Vignette: Bootstrapping Thinking: Role of Engaging Tasks (Peter Liljedahl)
Bootstrapping: "Starting from nothing". The only way to make everyone comfortable is to make everyone uncomfortable. A self-differentiating task was presented involving ordering of playing cards. Noted that to achieve flow, you need to vary the challenge as skill level increases. Too fast leads to anxiety. Too slow leads to boredom. This is impossible to balance in a class of 30, but is achievable with 8 separate groups.
|First 3 are external|
MYTH: There is a resource somewhere, indexed with tasks to use. No. Moreover, you cannot create an engaging task that hits AN expectation, because to hit that point removes the possibility of solving using all other options. You can merely create a problem that goes in the right direction. Assessment must also look like what happens in the classroom.
"Everything I've just told you is guaranteed to fail." It's all in how you approach it, none of this is "teacher proof", and there will be resistance. What do you do next?
-Chaos is necessary for learning, but this is not something to jump into blindly. More thought is necessary (for me).
12) Crowdmark Luncheon
Sponsored lunch, but worthy of an entry for a couple reasons. First, it was the founder and CEO of @Crowdmark himself who came out. Kudos. Second, it is an interesting technology - you scan in tests, each page with a QR code, grade online (with the ability to enter links/graphics!) and can email results to students. Can also flag entries for others to look at. Pointed out that at 2 hours of grading per week, with 28 million teachers in the world that's 2 billion hours of grading per year... a lot of which is paper shuffling and entry. Current pilot project at Universities, hoping for feedback from schools. Seems really good for contests or boardwide assessments, but if you need to look at multiple pages at once, that technology isn't available until summer.
13) Panel: Statistical Thinking in Schools
Individual teachers spoke, after which there was to be a panel discussion. Owing to time constraints, the panel was shunted to a vignette, but here's what each presentation entailed:
A) Michael Campbell
Spoke about the huge mix of people in the 4U Data course, and how it can be taught with limited computer time. Namely a lot of experimental work (even for confidence intervals) and individual data cards which had been made for sampling. A student said the concrete work here helped with later abstraction.
B) John Braun
A plug for "R", an online statistical tool with origins in "S". It's open source, see www.stats.uwo.ca/faculty/braun Quotable: "Things are hard if you don't have the time, energy and inclination", be that math or teaching itself. Also, in 2013 the minting of two dollar coins equalled the original 1996 minting, meaning a 50% chance of that date.
C) Georges Monette
Looking at the news, something not taught post-secondary. Notably the idea of "predictive" associations versus "causal" associations, and how statistical solutions usually involve asking questions, not finding answers. The difficult questions come when Causal Questions are answered (in media) using Observational Data, shown in a "Fundamental 2x2 table of statistics".
D) Len Rak
What is "Real Life" for college students? When categorizing data from an open ended question about 'cell phones' or 'technology', only 10% gave a robust response, trying to explain a large $0 occurrence. When the same data was framed as 'housing', over 50% gave such a robust response. Discovered by trial and error. They also are motivated by "knowing their mark", so he's reported it as a z-value, which increases engagement for that unit.
-Data is interesting for the reasons given. More insights like this need to be shared.
14) Vignette: Une serie qui chante (Anik Souliere & Melisande Fortin-Boivert)
I was hoping to get to at least one french talk, particularly after the opening plenary was bilingual. Here, the slides were (mostly) english and the talk was (mostly) french. Notably they had the audience vote at times for options using a sheet of paper which could be folded in half to show A/B/C/D. But no two papers were alike, so if you saw a "D" in front of you, that didn't mean "A" was necessarily on the other side. That was kind of brilliant.
Looked at music from a mathematical perspective - any complex pitch that is periodic will be seen as musical. Any unperiodic "noise" (clapping of hands) will not be heard that way. A voice is closer to an actual sine wave when graphed, owing to having fewer harmonics. The equation of a single harmonic of frequency was presented, fourier series was referenced. The openness of an assignment being "invent a complex note" threw the students off. I had to leave this one early... look for more here en francais:
-Mathematics can and does explain our perception of music.
15) Vignette: Ad hoc - Song Parody for Concept Retention
Over the weekend, two people said to me that I should consider doing my "Song Parody" bit that I've presented before. I had my laptop, so I decided I might as well sign up... if no one came, no one came. No one came. Well, until about 20 minutes into the time slot, so I gave a quick 15 minute overview to three people, half of that time being videos. I think they enjoyed it. A couple people also asked me about it outside of the session itself.
16) Plenary Lecture: Ontario's Stifling Mathematics Curriculum (Peter Harrison)
Saturday closed off with this. Point P(1,3) is transformed according to (y, (y+1)/x). There were many places to go with this problem:
a) Calculate coordinates until something "interesting" happens. (SPOILER: It cycles. Students can then be told to try the transformation on other points.)
b) Repeat the process with P(x, y), using algebra. Noted that some steps require factoring on two variables, and restrictions pop up.
c) Some seed points won't work, describe what's happening. (SPOILER: See restrictions, above.)
d) Find the point for which P0=P1 (ie - maps to itself). There's more than one, and it turns out to be interesting.
|But wait! There's more...|
The kicker is that NONE of that interesting mathematics can be found in the Ontario Curriculum anymore. Unless perhaps you look in the first 30 pages of the document, relating to problem solving and such. Instead, these days topics seem to have been selected to gear us towards an IB model - why?
There was some pushback from the audience in that if we don't have expectations, creating them ourselves is more work/more time, that people in the same course might end up with different approaches, and that parents would have trouble not knowing what's going on. Peter agreed that these were valid issues, but that "good PD" could help remove the conflict aspect while improving collaboration, and that the result would be beneficial.
-There is more to curriculum than specific courses we are teaching. Be aware of good problems, even if (like in 11 above) they can't be targeted to an expectation.
In the morning there was another Panel listed for "Does the curriculum need some fresh air?" and then Looking Towards CMEF 2018. But when all is said and done, I still need to teach this week, and I'm so behind in my marking that I could mark for 48 hours straight and probably still not be done. So I didn't go.
Oh well, hopefully you found reading this to be worthwhile! Feel free to indicate the most interesting thing in the comments below.