Monthly Archives: October 2013

Things to Focus On

A couple of weeks ago we ran a workshop with all of our secondary schools and talked about some things that can help us help kids learn math.  Here’s a summary of some of the points we discussed along with some relevant articles/videos/etc.

1)  Closing the Gender Gap

In general, our boys are doing better than our girls, particularly in the applied stream.  Jo Boaler’s book, “What’s Math Got to Do With It?” raises some good points on why girls sometimes fall behind boys in mathematics.  Her research has shown that while boys tend to be content following a set of rules, girls need to know the “why” behind the mathematics.  She advocates for “classes in which students discuss concepts, giving them access to a deep and connected understanding of math [which] are good for girls and for boys.”

Jo Boaler also has a video (from the course she offered this summer) talking about stereotypes and growth mindset. 

The research she quotes was done by Carol Dweck and can be found in this article: Is Math a Gift? Beliefs That Put Females at Risk

2) Creating Opportunities for Discussion

Discussion of math concepts is important in helping students develop a deeper understanding of the mathematics.  Malcolm Swan’s Standards Unit offers some great suggestions for incorporating and encouraging discussion in the classroom.

Cathy Bruce has a video about Math Talk on the Ontario Ministry of Education’s “Webcasts For Educators” site. There is also a talk by Dan Meyer about inquiry, the power of predicting and the importance of having a clear goal in mind.

One way to encourage discussion in math class is by giving students problems that encourage discussion.  The Grade 9 Writing Team that got together in the summer came up with some great problems that are available in First Class.  There are also some great blogs that offer some other suggestions for “perplexing problems.” A few to check out are Dan Meyer, Fawn Nguyen, Robert Kaplinsky, Sam Shah, and of course our very own What Did You Do in Math Today blog.

See OAME 2013 Presentation – Talk is Cheap so Let’s Make the Most of it

3)  The Importance of Mistakes

Mistakes are imperative for learning.  This video, again from the course Jo Boaler ran this summer, explains the science behind mistakes.

There is actually a blog about making mistakes! It is appropriately named Math Mistakes and contains pictures of errors students make in their work.

4) From Patterns to Algebra

This resource is invaluable for teaching grade 9 math in either the academic or applied stream.  It not only connects what students already know about linear relations to allow them to go deeper in grade 9, but also allows them to develop really flexible thinking around the multiple representations.  See You gotta try this!

5) Growth vs Fixed Mindset

The other piece we talked about is student (and teacher) mindset.  This is a new idea that we were introduced to during Jo Boaler’s summer course that has a lot of implications for teaching and learning.  This is the video we showed explaining the science behind brain plasticity and what that means for the teaching and learning of mathematics.

The tasks that we give students can be designed to encourage a growth mindset in our students.  Tasks of this nature include: (taken from Jo Boaler’s summer course)
1. Openness
2. Different ways of seeing
3. Multiple entry points
4. Multiple paths strategies
5. Clear learning goals and opportunities for feedback

6) EQAO Strategies Across Schools

We also spent some time talking about strategies that schools across the board are having success with when it comes to EQAO. These included:
1. Logistics – writing during one day, time between booklets, etc.
2. Support from across the school – peer tutors, LRT, greater awareness to school and parents
3. Using exemplars – give students exemplars (without the codes) and have them try to determine the coes, have students complete an open problem and compare their solution to the scoring guides to determine their score, etc.
4. Familiarity with EQAO-type questions – use previous EQAO questions on tests, reviews, diagnostics, as problems to investigate during class; have conversations about multiple choice distractors
5.  Extra support for students – EQAO midterm, MSIP, after-hours numeracy

 

The Border Problem

I have a new favourite problem!!  It looks like this:

Without counting one by one, determine how many shaded squares are in this 10×10 grid.

border-problem

This is not a new problem.  I think it’s appeared in many books and possibly even textbooks throughout the years.  My first exposure to it was actually in Jo Boaler and Cathy Humphreys’ book “Connecting Mathematical Ideas”.  I loved watching the video of middle school students explaining the way they saw the problem and then eventually generalizing the methods for an nxn square, but I always wondered what it would look like in a high school class.  Luckily one of my good friends allowed me to come in and try this problem with her class.  It was awesome!  Like exactly what I was hoping it would be!  Maybe even a bit better!  Here’s how it went:

1.  We put the graphic above up for a moment and asked kids to give us a thumbs up when they knew the number of shaded squares.  I asked someone to explain how they were thinking and then wrote that method on the board.  “So is this how you thought about it?” I asked.  They agreed.  “Did anyone think about it differently?”  It kept going until we had a list like this:10x10

2.  I asked students to imagine they had a 6×6 square now.  “How would Jessica figure out how many shaded square there are?” I asked them.  I gave them a moment to talk about it with their partners.  When I did this again with Angelo’s method, the conversation got interesting.  The kids started doing funky things with the numbers “6 x 10 – 6” was one of the ones I heard.  I could tell that they weren’t thinking about the visual, and definitely not thinking about the expression as a function.  I asked them how we could check and they told me that they could add up the numbers and see if it equaled what Jessica’s equaled.  Not a bad method – definitely was a step towards getting us to talk about equivalent expressions which was one of my learning goals. I kept referring them back to the “imaginary 6×6 grid” and we eventually ended up with a list like this:

6x6

3.  I asked students if we could do this for any sized grid.  They all told me we could for any number… even for a 100241 x 100241.  “So instead of making a list for every single possible grid, could we just say that we have like a say, nxn, square and n could be any one of those numbers?”  I asked them to talk about that in their partners.  These students all have had lots of exposure to variables in the past, but I find they often have started to forget what they really mean by the beginning of grade 9.  They start to think they are some fixed thing and then start doing all sorts of weird things with them.  The students all agreed that that would be alright.  They also said that we could pick any letter as the variable.  I asked them what Jessica’s method would look like for an nxn grid and gave them some time to talk about it and write it on their whiteboards. As I circulated to see what students were doing I saw a bit of this ” n+n+(n-2)+(n-2)” but a lot of this “n+n+p+p” or something to that effect.  I was expecting this, but I still didn’t know what the best way to deal with it was.  I didn’t say anything to individual groups, but once I saw every group had something written down I asked them to put up their whiteboards and look around the room at what other people had written.  They did, but there weren’t really any lightbulbs flashing, so I started the discussion “I see two different types of answers (wrote them on the board). Are these both the same?” Most students were nodding their head.  “Okay, so in the n+n+p+p, what does n represent?” A student raised her hand and said it was the dimension of the grid.  “Okay, so what does the p represent?” No hands.  And then one, “the p is like the inside part of the other two sides that the n didn’t count”.  I asked students to talk about this – is this true?  how do we know that this is what p represents? Some good discussion started happening.  Some of the ones that had the (n-2)’s already written became more convinced, others who had written the n+n+p+p’s started looking over to the (n-2) people.  I brought them back, “what did you talk about?” One student told me that she had converted, “we need to write the n-2 so that we know that it is a row minus the 2 squares we already counted.” Most students seemed convinced about this, others seemed a bit shaky.  I erased the n+n+p+p and left the other expression on the board.  Then we moved on to Angelo’s method.  As you can imagine, similar conversations ensued.  After some good discussion and some synapses firing we ended up with a list like this:nxn

4.  We had already touched on the idea of equivalence with the 6×6 list (the fact that all the expressions should equal the same number), so I brought it up again with the nxn list.  “How do we know that these expressions all are equal?” No volunteers, but I was out of time.  I left them with that challenge for homework – to pick two of those expressions and show to me how you know they are equal.  My purpose was to hopefully get kids to start thinking about equivalent expressions and also like terms, the big idea being that even though expressions look different they actually mean the same thing, and we can manipulate them to make them look differently. 

Because I wasn’t able to be there the next day, I’m not sure what kids did to show that the expressions were equal.  The classroom teacher told me that the next day they pulled the algebra tiles out to start modelling these expressions with them and proving equality.  I think that is a great way to prove this, and also a great lead in to combining like terms, distributive property, etc.