I started modifying and using some of the ideas with my students last year, and this year I actually have a small group of 7th graders that work on just word problems with me once a week and I am relying heavily on the information I learned about CGI.
While I didn't get to put all the kids into that group that I would have liked (because scheduling is STUPID), I do have some kiddos who were really having a hard time making sense of what a word problem was even saying when they started, and after 6 sessions, they just blew me away this week. Another teacher asked me to share what I'm doing in a blog, so this is the result.
Please realize, I am not some kind of CGI expert - I haven't even seen it done other than the videos from the book, and teachers at my school that I've convinced to try it with me but I like what I read, and am doing my best to figure out how to apply the core ideals to my teaching with junior high kids.
Here's what I've done so far:
My focus turned pretty quickly toward writing equations because it is such a difficult concept for students to master, especially when there are multiple variables. I have been AMAZED at how much they have grown over our few sessions:
First Session:
- Gave the kids multi-step problems with low numbers that could be directly modeled
- This day was mostly for me to assess what they really know/are able to do right now
- Students were able to solve the problems, but multiple kids mentioned that it was hard for them to know what happened in what order, so I could tell they were still struggling with comprehension, and some of them weren't connecting mathematical symbols with the real life circumstances that create them.
Second Session:
- Presented one step addition and subtraction word problems with NO NUMBERS (blanks instead).
- For example: "Justin had ___ cookies, then Becca gave him ___ more. Now he has ___ cupcakes"
- I asked students to discuss then prove what was happening between quantities (they had to identify what the blank was talking about, and prove why they thought it was that operation by citing evidence from the problem and giving examples).
- Students created numbers for the problems and wrote and solved equations representing the problems. Many students also drew pictures. Lots of sharing and explaining.
- At the end, students got to make up their own problems - they LOVED doing this and trying to stump each other by making them with extra information or complicated story lines.
- I did do names of problem types with kids this day, but wound up deciding that wasn't the most helpful thing so I dropped that off after this session
Third Session:
- Introduced multiplication and division problems. We talked about what types of things in real life would create multiplication and what types of things would create division.
- We tried a similar approach with blanks instead of numbers and students reasoning about what operation made sense and why.
- There was still a lot of confusion about how to tell what is getting divided and by what in problems that created a fractional answer, and also with rate problems. This makes sense because these problems are harder for students to model.
- Students were still choosing numbers and picking large numbers that were hard for them to draw, so I started switching to variables so we could focus on the operations instead of the numbers.
- This was actually a REALLY great challenge for them and totally made the discussions way better.
- Students also started creating cases to make their point and would plug in smaller numbers to try to justify reasoning within the story (it has always been hard for me to "teach" kids to break an argument into cases and they just did it themselves because they were working to make sense!).
- This was the first day we had visitors and the kids kind of freaked out and shut down while they were there. So we had a heart to heart at the end about that.
Fourth Session:
- Started with more multiplication and division with variables because once they shut down while the visitors were there, I wanted to give them a chance to get comfortable again. So this day was basically the same as session 3.
- Students are starting to write multiple equations with the inverse operations because they are understanding the relationships between quantities.
- Students mentioned wanting more challenge, so I asked if they'd like to try a multi-step problem next week - they are all in
- Discussions were WAY more confident today, so I know they are ready
- We had one visitor who was a teacher form a prior year so students were more comfortable
Fifth Session:
- 3 visitors. We did an ice-breaker to make sure the kids were comfortable this time.
- Multi-step problem (AND not the easiest to model directly):
- Calvin paints pictures and sells them at art shows
- He charges ___ for a large painting
- He charges ___ for a small painting
- Last month he sold ___ large paintings and ___ small paintings
- I showed only one sentence of the problem at a time, then asked students to give me facts they know so far
- When we would get to a sentence with a blank, I asked them what the blank was and made them prove it by citing evidence from the words in the problem
- I wrote what they said on the whiteboard then asked them what variable would make sense to represent that quantity (we talked about what the word quantity means)
- We had lots of good discussions about things like not using variables that look like numbers, not using the same variable for different quantities, and not using multiple variables for one quantity. We also talked about how writing multiplication as x is confusing, especially if x is a variable.
- Then I mentioned that there was no question, and asked them if they could think of a way to turn this into a word problem. We found how much money Calvin got for all his large paintings last month, then they really wanted to find the full total he made last month, so they set to work working on making an equation to represent that.
- I had lots of mini discussions with individual kids while they were working about what different parts of their equations meant and how they knew what operation to use.
- The visitors split up and talked with different kids so ALL of my students were forced to explain A LOT today - which was such good practice for them!
- The visitors were really impressed with how much more articulate the kids were than just two sessions ago. The kids were also so much more confident and excited to share their thinking.
- This was the first day where kids really shared their thinking with conviction, and you can see in one student's work where she plugged in numbers to help prove that her equation would represent the situation.
- They ATE UP this challenge - we talked about the fact that MANY visitors were planning to come the following week and I asked if they wanted to keep the difficulty the same - they said no way! Bring on the challenge!
- I have posted some pictures of their work below:
Sixth Session:
- MANY visitors scheduled to visit for "Pineapple Day" (10 teachers besides me total)
- Unfortunate absences - only 3 kids! They were nervous about all the teachers with so few of them but they said they were still up for the challenge (yay!)
- Here's the problem they worked on:
- Brayden and Gavin were playing touch foot ball against Cole and Freddy.
- Touchdowns were worth ___ points
- Brayden and Gavin scored ___ touchdowns
- Cole and Freddy's team scored ___ touchdowns
- What questions could we ask to make this a word problem?
- I picked this problem on purpose to address the following things: Two names for one quantity (because of the two-person teams), multiple quantities about touchdowns in some way so students would really have to keep track of points versus touchdowns, and I planned to ask a compare problem at the end if a student didn't come up with it.
- We had REALLY good discussions about what variables should represent things and how to tell the difference between to two teams, as well as whether the blanks represented points or touchdowns.
- Each student came up with a different word problem, so we solved them all. The students decided they wanted to do them in the following order:
- How many total touchdowns were in the whole game
- How many points Cole and Freddy scored
- The total points for the whole game
- The first two problems didn't take too long, but all the students were writing multiple equations and using inverse operations. We talked about how to prove that they are really saying the same thing and all representing the situation in different ways and we had good conversations about the commutative property and practiced saying inverse operations instead of just saying "backwards" and "opposite"
- THEN... there were there LOTS of interesting student discussions (between each other and with me individually) to figure out the last one.
- The COOLEST thing happened - every kid came up with a totally different equation. I was pretty excited because I knew when we came back together to summarize and connect we were gonna see a LOT of great connections.
- I am KICKING myself because I didn't remember to take pictures at the end :( Here is what I remember of their solutions:
- One student had the sum of each individual team's touchdowns in parenthesis then multiplied by the points per touchdown to equal the total points
- Another student had the points per touchdown multiplied times each of the team's touchdowns first, then found the sum of the team's points for the total
- The third student had separate equations for each team's points where she defined new variables, then summed those new variables to find the total
- We had SUCH a good conversation - we talked about how to know the first and second equation were connected and a student even remembered the terminology "distributive property," students were able to see how each equation was connected to the others and explained in multiple ways, including by plugging in values to help explain their thinking, and at the VERY end, one of the students (who started the year out as my least confident and most confused) asked if we could also say that the total points divided by the points per touchdown equals the sum of each team's touchdowns. He mentioned inverse operations and plugged in numbers to try to convince others. This was SO cool - this kid who has been practically silent in my regular math class, and who has STRUGGLED to make sense of concepts in 7th grade is starting move things around in literal equations, and be able to explain it. And with an audience of a bunch of teachers. I was seriously blown away - it was so cool :)
My Takeaways So Far:
- We might spend a whole hour on one problem, but holy cow we address a TON of stuff in that one problem
- Students are making huge leaps with symbolic notation because they are FINALLY making sense of the problem, then using the context as a tool to help them understand the mathematics
- I always pick the problem(s) I do for the day with certain goals in mind, especially in terms of common misconceptions, vocabulary, and mathematical structure, and EVERY day they find a way to surpass that in some way by bringing up something interesting, noticing something new, or asking great questions
- These little people have turned from shy, unsure, sad-about-math kids into confident, eloquent, deep-thinking mathematicians, and it has ONLY been 6 hours!
- I have these kids before lunch on a minimum day, and they go home after lunch, so I am literally the last class of the day, and before lunch, and instead of being tired and complaining, EVERY day they beg to stay longer because they just want to solve one more. I have to physically kick them out of my room to get them to stop doing word problems. They are showing me that math doesn't HAVE to be painful, and it also doesn't have to be a "show" to be fun for kids - if kids understand and get to make choices about how they solve problems, they actually can have fun.
- I have so much fun teaching this class - they teach me just as much as I teach them and I am so grateful I have had this opportunity this year.
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