## Monday, October 21, 2013

### Solving problems or problem solving?

This week I attended a four day training  (YES--4 days of sub plans!!!!) that was geared toward developing our understanding of how children learn mathematics and how to troubleshoot when they are struggling.  The content was fantastic.  The instructor was amazing.  Now my brain is absolutely churning.  Are you like that after professional development?  You wish there was one more day tacked on so you could actually PROCESS the information so you could be more ready to apply it?

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!

or find me on Twitter at @FourthGrStudio

Heather
2 Brainy Apples

1. Thanks for your thoughtful reply, Heather. I can tell you and I are cut from the same cloth! It is so important that we really change our perception about what the intent of math class is!

2. Ditto, Meg. I've been changing things a lot in my classroom since starting to read more and more about problem solving. It certainly makes the lessons interesting. :) A couple of weeks ago we were looking at number lines - just putting numbers on them - and I drew a few on the board and invited children to come up and put numbers where they thought they should go. Every hand in the room was waving until I added "and explain why you put it there." Hands dropped instantly. I'm glad to say that by the end of the first lesson I had lots of volunteers to explain their thinking - and they did a great job. It was fascinating to hear their reasoning and often it wasn't something I would have thought of but it was right! I spent quite a lot of time on number lines and I think it was worth it just for the "thinking" aspect. :) It certainly takes a lot more planning on my part but oh, the benefits!! Totally amazing!!

Lynn

1. Thanks, Lynn! It takes time, but it sure is worth it, isn't it? The students really benefit from the ideas of others, and it is fun to watch them start to gain confidence.

3. Here in New Zealand, we don't teach standard algorithms until year 7 (Grade 6). In fact unless they are able to use at least 2 part-whole strategies then we steer well away from them. (Part whole is like being able to know that 7 is made up of 5 and 2) Before that we are ALL about problem solving, and using strategies to work out number problems. At the school I teach at we are big on providing a huge chunk of time towards developing their Place Value knowledge at every level - and we have seen huge progress over the last few years! Great to see others developing in similar ways!!
Erin
The E-Z Class

1. Thanks for your perspective, Erin! I do NOT think spending huge amounts of time on place value can do anything but good for mathematical understanding! We talk about the part-whole strategies as well and also refer to the idea of "composing" and "decomposing" numbers. I hope you'll stop by often with your insight!

4. As usual, your post was timely and thought-provoking. We are at the tail end of a unit on subtraction right now and so many students can't understand what they are doing because the algorithm means nothing to them.
Have you seen the series of books by Michael Battista on cognition-based assessment and teaching? They are a lifesaver! http://www.amazon.com/Cognition-Based-Assessment-Teaching-Addition-Subtraction/dp/0325012717/ref=sr_1_1?s=books&ie=UTF8&qid=1382438777&sr=1-1&keywords=cognition-based+assessment+and+teaching+of+addition+and+subtraction+building+on+students%27+reasoning

5. Caroline--I have seen them but do not own them. I will check them out! Thanks so much for your kind words . . . I hope you can continue to build number sense with your class as you move forward. It takes time, but I believe it pays off!