The course actually took me way back to my college days where I was very involved with the university and the people working with CGI (Cognitively Guided Instruction). If you are unfamiliar with the work of Thomas Carpenter, I highly recommend you do some reading! In fact, a big part of the mathematical understanding elements in the CCSS are based on his research.
The piece of information wiggling around in my brain is a quotation that I wrote down from our presenter. I'm not sure if they are HER words or the words of someone else, but they are powerful and relate to the idea that we are teaching children traditional algorithms in math WAY too soon. I've been following a little discussion about this on Facebook, and all of these pieces have me thinking a LOT about what I believe to be true. Ready for the quotation? Read it a few times. I needed to.
"A written algorithm is meant to SHOW how you think, not to TEACH you how to think."
As I sat their thinking, my light bulb went on BIG time. I have always introduced multiple ways for students to solve problems. I love to hear how they think about math...but when it gets right down to it, at some point I do TELL them how to do it. There are some fantastic articles and books written about how much damage can be done when we interrupt children's thinking and "sense making" to fit their learning into what the adults feel is the right way. How many times have you seen a student do something goofy with an algorithm that makes no sense? Or when you ask them how they solved it they reply, "I crossed out the 1 and made it a 0." Or they shrug and can't even START to tell you! We really need to stop and think as we push ourselves to do more and more with students that we don't forget that how they learn is more important than getting through the workbook! Check out a few of these problems that have been eye openers for me over the last year. How do students need to APPLY math understanding rather than simply solve a math problem? Sometimes I feel we are "training" them to solve problems on the paper instead of coaching them to figure things out on their own.
|Hardly any students in my class could solve this problem accurately last spring. Guess what they put on the blue line? 2,500. It's halfway, right?|
|How about this one? You should have SEEN the ideas my students put on their sticky notes! What a "red flag" for me!|
|Or this one? I simply asked students to "Use your ruler to divide your paper exactly in half." Wow.|
|Or this one! My students had been DRILLED with the idea that fractions are equal parts. Where did we go wrong?|
So . . . I know this is a lot of rambling, but I would love to hear your thoughts! Do you think we are conditioning children to try to just solve the problems on the page instead of being problem solvers? As we move into the next generation of career opportunities, don't we want students with amazing number sense and problem solving--not just students who can solve it the way we (or the book!) teach it? This is where I absolutely LOVE the Standards for Mathematical Practice--the content should be taught through the lens of mathematical thinking. So...now that I have gone on and on and possibly not made much sense, what do you think? I have worked hard to incorporate more constructivist work in my classroom, but I sure have a long way to go!
The fraction examples above are actually a part of a full month-long unit I wrote when I discovered how much my students really struggled with constructing meaning about fractions. They could solve the problems on the paper, but I quickly learned that their understanding was marginal! Check it out if you are interested...it's full of activities to help students develop their own number sense about fractions. I've also included one of my freebies of questioning prompts that can help YOU help THEM to help THEMSELVES! Have a great week, everyone!
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