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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LOVED your post! I totally agree that we are teaching kids to solve problems. I actually like the new CCSS for math. Why? Because it focuses on teaching younger kids the fundamentals of math (place value) before adding the more difficult skills. One of my favorite changes? NOT teaching kids the standard algorithm for multi-digit addition and subtraction until 4th grade. I know I will probably get some flack for that statement, but when we teach kids how to regroup, we spend maybe a week using manipulatives, then we jump to the algorithm. Clearly they don't understand the concept of regrouping when they are making procedural errors and they don't realize their answer doesn't make sense. They don't understand place value when they think they can subtract 3 from 1 and get 2. Kids can solve addition and subtraction with regrouping without knowing the standard algorithm if they are taught the fundamentals of place value to a deeper understanding. Only when they have a deep and firm understanding of place value should they be taught the algorithm. This holds true for operations involving fractions. We are so quick to teach the algorithm so we can "move" on to the next skill, we forget that kids have to understand what is really happening with the numbers if they are to truly understand the concept. Teaching problem solving involves showing kids that there are multiple ways to solve math problems. That's the beauty of mathematics- more than one way to figure out the answer. I tell my students it isn't the answer I am interested in as much as the route they take to get the answer. Students have to realize there is no one way to solve a math problem. They need to understand that if one way doesn't make sense for them, there should be another way in their tool bag to use. Drawing models should be a part of every math class, every day, for every grade level. Last year I taught EIP math, and several of my students couldn't remember all the steps in the algorithms or they would get them mixed up, but if I asked them to draw a model, they were successful almost every single time. Same for solving word problems. Instead of teaching them solely key words, they need to be taught how to draw a model of the word problem. It makes the word problem more concrete for them and they can visually see what is happening.

ReplyDeleteLove your post!!!!!

Heather

2 Brainy Apples

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!

DeleteDitto, 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!!

ReplyDeleteLynn

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.

DeleteHere 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!!

ReplyDeleteErin

The E-Z Class

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!

DeleteAs 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.

ReplyDeleteHave 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

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!

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