Some professional (and artistic) development
I must admit that I signed up for the QLSMA Math MiniConference on a whim, with the added knowledge that it would be a great excuse to visit Anneke. However, I was pleasantly surprised. And now I'm going to attempt to consolidate all that I learned so I can eventually put it to use!
The key note speakers had done research into bridging the divide between patterning and algebra. Now, their study focused more on elementary school students, but I could immediately see a connection to high school mathematics.
This is what I took away from their methodology:
- introduce a patterning concept by using an "input" "output" robot approach --> which is really the definition of a function!
- stay away from table of values until students get the connection between a pattern, equation and the graph representation (it is meaningless and confusing otherwise)
- get students to create patterns using pattern blocks, using the terms 1, 2, and 3 and then get other students to guess those patterns
- you can explore what happens when you are at position 0 (this is the y-intercept)
- originally get students to write the equations for these block patterns in the form "output = input x 2 + constant term"
- the robot exercise will get students away from thinking about the recursive formula of the pattern "add 2 each time" and thinking more in terms of general equations
In a high school classroom, I think this would be a brilliant way to talk about functions, and to introduce slope, y intercept, and the graph of a line. Once the "equation to graph" connection has been made, you can introduce x and y as "shortcut notation", and then start thinking of other applications for the multiplier (or slope).
There are so many possibilities for this! I know I am getting excited about little square plastic blocks, but I am always looking for simple math manipulatives, and the great thing is that there is absolutely nothing saying that the patterning material needs to be squares. It could be bingo chips, lego, or whatever else I can find (which I appreciate).
I can see great applications to modelling all types of functions -- especially transformations of them, and introducing things like exponential functions. I also think that the sequence and series unit of grade 11 math could seriously benefit from some hands on patterning. My students in the summer benefitted greatly from group work and whiteboards, but I think the patterning blocks would be a great tool to add to my arsenal.
The other session that I attended at the conference, was about the concept of a "bump up wall". The teacher presenting on the topic introduced it saying that she and her department had been having a lot of trouble with a grade nine summative assessment, and used the wall to improve student performance drastically within 2 class periods.
The wall gives exemplars for each level performance of a certain type of problem, along with "look fors" that the students can use to "bump up" their level. Students fill out a checklist before handing their work in which has a lot of these look fors on it. The next time they do a similar problem, they can look at the wall for pointers as well. It was impressive how fast the kids changed their tactics for solving a problem. And while I don't agree with using this practice for every single type of question -- I think that it has a lot of merit for gently teaching students how to structure their answer so that it is easy to follow. Honestly, that ability to lay out the solution to a problem is one of the more practical things that a student can learn from a math class, so it is definitely worth the time. Applications of this method can really be made for any subject, and really can be used in different ways. I can see a "bump up binder" being more useful in my alternative setting.
We made a sample "bump up wall" for a grade 10 applied math word problem. It was a good exercise, and lead to a lot of us questioning "what amount of thinking needs to happen for a Level 1?"
After this really great conference, I became an art student again, and spent the next 4 or 5 hours working on my second ever oil painting. What a great way to spend a Saturday!
The key note speakers had done research into bridging the divide between patterning and algebra. Now, their study focused more on elementary school students, but I could immediately see a connection to high school mathematics.
This is what I took away from their methodology:
- introduce a patterning concept by using an "input" "output" robot approach --> which is really the definition of a function!
- stay away from table of values until students get the connection between a pattern, equation and the graph representation (it is meaningless and confusing otherwise)
- get students to create patterns using pattern blocks, using the terms 1, 2, and 3 and then get other students to guess those patterns
- you can explore what happens when you are at position 0 (this is the y-intercept)
- originally get students to write the equations for these block patterns in the form "output = input x 2 + constant term"
- the robot exercise will get students away from thinking about the recursive formula of the pattern "add 2 each time" and thinking more in terms of general equations
In a high school classroom, I think this would be a brilliant way to talk about functions, and to introduce slope, y intercept, and the graph of a line. Once the "equation to graph" connection has been made, you can introduce x and y as "shortcut notation", and then start thinking of other applications for the multiplier (or slope).
There are so many possibilities for this! I know I am getting excited about little square plastic blocks, but I am always looking for simple math manipulatives, and the great thing is that there is absolutely nothing saying that the patterning material needs to be squares. It could be bingo chips, lego, or whatever else I can find (which I appreciate).
I can see great applications to modelling all types of functions -- especially transformations of them, and introducing things like exponential functions. I also think that the sequence and series unit of grade 11 math could seriously benefit from some hands on patterning. My students in the summer benefitted greatly from group work and whiteboards, but I think the patterning blocks would be a great tool to add to my arsenal.
The other session that I attended at the conference, was about the concept of a "bump up wall". The teacher presenting on the topic introduced it saying that she and her department had been having a lot of trouble with a grade nine summative assessment, and used the wall to improve student performance drastically within 2 class periods.
The wall gives exemplars for each level performance of a certain type of problem, along with "look fors" that the students can use to "bump up" their level. Students fill out a checklist before handing their work in which has a lot of these look fors on it. The next time they do a similar problem, they can look at the wall for pointers as well. It was impressive how fast the kids changed their tactics for solving a problem. And while I don't agree with using this practice for every single type of question -- I think that it has a lot of merit for gently teaching students how to structure their answer so that it is easy to follow. Honestly, that ability to lay out the solution to a problem is one of the more practical things that a student can learn from a math class, so it is definitely worth the time. Applications of this method can really be made for any subject, and really can be used in different ways. I can see a "bump up binder" being more useful in my alternative setting.
After this really great conference, I became an art student again, and spent the next 4 or 5 hours working on my second ever oil painting. What a great way to spend a Saturday!
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