Sunday, 29 November 2015

PD: Math in Context

My other two blogs (math webcomic and time travel serial) are queued up early this week, so time for another “Good Professional Development but Had No Time To Blog” post. Today’s topic: COMA Social 2015, from October 1st, featuring Kyle Pearce (aka @MathletePearce). See his presentation, “Making Math Contextual, Visual, and Concrete!” at and/or read below.


While teachers may get excited about the correlation coefficient, students are often less enthusiastic. How can something like that be taught so that students won’t freak out or feel overwhelmed? Kyle says “I thought this is what good teaching looked like - very organized, structured notes”. Yet students were bored.

Maybe technology will change everything? SmartBoards? Well, students were behaviourally better (not talking while he wrote on the chalk board) but still not intellectually engaged. This because it was more of a substitution, not an augmentation... the latter is needed for functional improvement.

Here’s how we learned math: 1. Take up HW; 2. Definitions, formulae, procedures (to set student up for success); 3. Examples; 4. Homework. Yet the examples were abstract, removed of context, and then asking a week later, often the concept hadn’t stuck/was forgotten. This model creates two groups of students: “Good at Math” & “Not Good at Math”. Except “Good at Math” meant understanding terminology, and following procedures, aka “Good at Memorization”. Meanwhile, those “Not Good at Math” might not memorize, or might be capable of memorizing but were unwilling to play that game.

Even students “Good at Memorization” run into problems. There is a larger “Struggle with Unfamiliar Problems” group. (If a test question isn’t like the examples, it is seen as “unfair”.) When we take in new knowledge, we have to tie it to prior knowledge, but traditional methods (and textbooks in particular) will silo concepts into tiny blocks, removing the chance to see those connections.


Showing the connections (how math concepts in one chapter exist elsewhere), and making them more contextual, visual, and concrete, leads to more confidence. Student success and understanding can follow from that. Avoid the natural immunity to change (yes the algorithm is beautiful, yet we need to know when it applies), and a desire to aim for entertainment (showing “pi” as “pie” or inventing a “rap”) - put the student engagement where we need it.

Kyle showed a 7 step process posted next to an “instant brewer” coffee machine. (“Actually I just press this button.”) Do we need an algorithm to use one of the easiest objects in the universe? Must we be told to plug it in? He notes that process wasn’t created for the USER of the machine, but for the OWNER of the machine. Now the user has no reason to ask the owner anything! But what if there’s an upgrade? Such algorithms don’t equal engagement, or understanding. 

While technology can functionally improve a classroom, it’s the task that’s going to redefine the class. (Kyle showed a few apps quickly; don’t find an app specific to fractions, otherwise we’ll be switching from app to app for problems. And “Evergreen” apps can be used but they’re not math specific.) The fear is that people will ignore the effective teaching aspect, which has to happen before the transformational technology.

Kyle posed a question about stacking paper up a wall - and showed this with a photo. (Clipart doesn’t do much for him.) Ideally get a number of questions and settle on one in particular. Here if we ask “How many pack of paper to the ceiling?” we can now get predictions. Use “High/Low” strategy (Dan Meyer 3 acts) - what number would be too low? Too high? With a padlet for Google Docs, predictions can be put online - or jot on a whiteboard and take a picture. (Kyle got a class set of iPads funded.)

Don’t give the number yet. Request what other information is needed: How many papers in a stack? Is that an 8 foot or 10 foot ceiling? Figure out what is useful. Refine the predictions - and Kyle had us log into the playkh website with a PIN to play along like a game show.

Students can upload their own solutions. If Kyle wanted to look at exploring proportions (versus unit rates), he could decide which solution to show the class first. Also look for the best incorrect responses. Students can realize that by not showing work, it’s leading to simple mistakes - and you don’t have to reveal whose solution it is. (You can also rotate on the fly!)


Do teacher solutions look like student solutions? Sometimes it’s unnatural to do things the way we’re asking students to do it. The “game show” can then offer extension problems to check “what do students know, what do students not know” rather than asking. For instance, we may not have talked much about variables yet, but can mention it here to check prior knowledge.

Rather than looking up the solution, actually show it (Kyle enlisted the help of a custodian). His answer doesn’t match the math - students don’t think about this. Weight is compressing the stacks? Pushed up the ceiling a bit? Tower is leaning? Floor is slanted? We have made this more visual. Maybe not concrete in terms of physically holding the paper, but might not be needed here. (Will we remember doing math or merely baking cookies?)

Another extension: What would stacks on a table look like? (Linear?) Can a student identify that there is a relationship here between two variables, not merely a division and done? Which variable impacts the other? Proportional reasoning made more explicitly linear (direct/partial variations). Kyle noted that this wouldn’t all be addressed in a single day.

We might want to bump into the algebra, rather than make it explicit. What goal are we aiming for when we include a table? (Solving an equation.. first differences.. the y-intercept!) What we have as given information is slope and a point. Pull y=mx+b from students rather than have them copy an algorithm (as in “a note”). The only way students can do it if we strip away all this context and use simple numbers is if they mimic the teacher/process.

Move the “algorithm” to CONSOLIDATION after the activity. “What does the point (1,5) represent in the context of the stacking paper task?” (1 package gives us total height of 5 - including table) “What does slope/unit rate represent?” (height of a pack) “How tall would this table be with these numbers?” Scaffold toward tasks, moving towards the abstract, rather than the other way around. Giving procedures and then hoping for problem solving at the end won’t work - they’ll be scared by the end.


Executing things this way can shrink the “Not Good at Math” group down, but MORE, it will shrink the “Struggle with Unfamiliar Problems” group down. Continue digging deeper, when you’re ready to tackle another concept. If there are two package stacks of different heights, can we figure out the table underneath? What is the learning goal for students to bump into here? (Solving given two points. Intuitively uses the slope formula, in context.)

Scaffold out to: What happened here? Does it work all the time? “Our students are capable of doing all these things we want them to - we need them to find creative ways of getting there.” Then simplify what they’re doing.

“We don’t do math because it’s harder, we do math because it’s easier.”

Thanks for reading!
-For last year’s COMA Social recap, see "Public Math Relations" with Marian Small.
-For how I set up my classroom lately, see "Grouping Tagline".

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