Hi. My name is Tammy from Forever in First. If you're like me, I'm sure you love hearing all about teaching from across the Atlantic, but Kelli asked if this little teacher from Idaho would be her guest. Idaho isn't quite as thrilling as England, but hopefully what I have to say will hold your attention. Thanks Kelli for letting me drop by!
I'm fully aware that it's summer and your brain is probably on vacation just like mine, but would you do me a favor? I double dog dare you to compute the answer to this problem without using an algorithm. In other words, carrying, borrowing, and all the tricks that you most likely learned as a student are illegal. Grab a piece of paper if you need to and play along pretty please. Here goes.
847 + 256
How'd it go? Was it hard to ignore the rules of an algorithm or did you find it easy to think flexibly about computation? If we were in my classroom, I'd ask several of you to draw your strategies on the board and ask you explain your thinking to the rest of us. You would definitely notice the absence of any algorithms. You would see several different strategies that originated from the students instead of the teacher. You would hear the reasoning behind each strategy. You would hopefully walk away with several new computational tools and a more complete concept of how addition works. You would definitely know that there's never just one way to solve a math problem.
"Flexibility with a variety of computational strategies is an important tool for successful daily living. It is time to broaden our perspective of what it means to compute." John A. Van de Walle, Teaching Student-Centered Mathematics 158
You see, I've got math on my brain this summer. I'm in the middle of this thought-provoking book. (Get yourself a copy!) It definitely stretches my thinking and at times confirms what I had always thought. I love books like that. It covers about everything, but today I have just enough time to hit on the somewhat controversial topic of computation. Who knew computation could be so controversial?
Like most of us, I was taught algorithms in school, but as a teacher, I've simply never believed in them. The book states that they're not evil, but Van de Walle also says,
Does any of that sound familiar? The premise of this book and of the math classes I've taken in the past few years is that students learn to compute more efficiently and understand what they're doing more completely when they invent their own strategies. Teaching them to carry and borrow does not allow them the freedom to do this, even in classrooms where algorithms are conceptually taught well. The book lays out several other reasons why invented strategies are more beneficial than prescribed algorithms but I'm afraid this post would get way too long and you'd never forgive me, so I'll let you get the book and read it for yourself. When I allow my kids to think independently and flexibly about math, this is a taste of what they come up with. (These strategies originated from contextual tasks. I apologize that you don't know the contextual problem behind the strategy. Also, notice that many equations are written in horizontal form. This is on purpose. It's harder for algorithms to sneak their way into the mix when the problems aren't written vertically.)
Stepping away from the safety of an algorithm might create a little fear in the hearts of some teachers out there and understandably so. If that's you, you're in good company. Look here to read about some of the leaps of faith I've taken on my journey as a math teacher. If you're not ready to leap, then maybe this post has raised some questions or sparked some unexpected thinking. I believe Van de Walle would be pleased with that, and his book would definitely fill in the holes that I've left wide open. If you can get your hands on a copy, it will broaden your view of what it means to compute, and in my opinion, our students will be better mathematicians because of it.
"Far too many students learn them as meaningless procedures, develop error patterns, and require an excessive amount of reteaching or remediation." (162)
Does any of that sound familiar? The premise of this book and of the math classes I've taken in the past few years is that students learn to compute more efficiently and understand what they're doing more completely when they invent their own strategies. Teaching them to carry and borrow does not allow them the freedom to do this, even in classrooms where algorithms are conceptually taught well. The book lays out several other reasons why invented strategies are more beneficial than prescribed algorithms but I'm afraid this post would get way too long and you'd never forgive me, so I'll let you get the book and read it for yourself. When I allow my kids to think independently and flexibly about math, this is a taste of what they come up with. (These strategies originated from contextual tasks. I apologize that you don't know the contextual problem behind the strategy. Also, notice that many equations are written in horizontal form. This is on purpose. It's harder for algorithms to sneak their way into the mix when the problems aren't written vertically.)
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| (Yep, this is my writing, but the kids told me all the steps. I was simply their secretary.) |
13 comments:
I can see how this way of computing will definitely help students to understand numbers and how they work! It takes a shift in thinking to teach this way, but very worth it!
Lori
Conversations in Literacy
Thanks Lori. It's definitely a shift, but like you said, a good one. Thanks Kelli for letting me join you here today!
❀ Tammy
Forever in First
I have been blog hopping for a little over a year and this post is by far my absolute favourite! I jut recently got my Masters in Elementary Education with a focus on math and science (in Alberta, Canada) and this book is one of my 'bibles'! This is absolutely how I teach and it makes such a difference to the success of my Gr. 2 students. We have about 3 weeks left of this school year and it won't be until next week that I will introduce algorithms. The algorithm itself is not bad...it is the belief that the ONLY way to solve the problem presented is through the algorithm. Obviously that is not true based on how many different strategies your students were successfully using to find a solution. The 'answer' is not the be-all and end-all, it is knowing HOW to manipulate those numbers (compose and decompose) that makes the difference. Once they know that, an algorithm can be introduced, and used successfully and they will understand why it works efficiently! (and please never say 'carry the 1' or you will undo all the hard work from the year.....that '1' is a 10 and it is so important that they know that. There is another author you might like to investigate, she is Marion Small and I think her book is called something like
'Big Ideas from Marion Small' COngratulations on being on the leading edge of what research is telling us makes little ones into mathematicians!
I just finished a workshop on this very thing. One of the books suggested was this one on your blog. The eye opener for me was more that the students were given an opportunity to show what they know in a different way. One of the teachers even said " I have always known how it was in my brain but couldn't figure out ow to explain it until now" It is a different way than what was drilled into our heads. Doing is learning. Understanding concepts much deeper is the key! I can't wait to begin my new school year with my new insight.
That's a great book and you explained things very well:) I love your students computations!
Thanks for sharing this, Tammy. You're always challenging my thinking (even though I always agree with you -- ha ha).
❀Barbara❀
Grade ONEderful
Ruby Slippers Blog Designs
Sherry, I'm greatly honored that this is your favorite post. That means a lot to me.
Heather, thank you for your comment. You'll have an exciting year putting your new insight into practice!
Barbara, I love that you and I think alike! (You make me laugh by the way.)
This is how we teach maths in New Zealand, no more algorithms here. I find it hard to teach this way, only because I learnt the 'old' way. I am hopeless at retaining numbers in my head, but the children today seem to have no problem doing it. I am just thankful that I teach year 2 (grade 1) and the highest stage I go to is adding 2 digits that don't involve renaming! (teacher next door to me does the next stage for the bright cookies) I would be screwed if I had to teach the next level! hahahaha
Carolyn
Sowing Seeds of Learning
Carolyn, I've wanted to visit New Zealand schools for a long time. You've given me another reason to do just that!
I had the privilege of having Tammy as my Instructional Coach my first years teaching in Idaho five years ago. She is amazing! I wish you could all come see her in action...Anyway, I've taught a below grade level math class the last two years in fifth grade and keep running into students who have tried to memorize meaningless algorithms and keep making simple miscalculations that have markedly slowed their progress. As they have time to explore these methods Tammy has shown us, they break free and become confident and wonderfully able mathematicians. Love this!
I had the privilege of having Tammy as my Instructional Coach my first years teaching in Idaho five years ago. She is amazing! I wish you could all come see her in action...Anyway, I've taught a below grade level math class the last two years in fifth grade and keep running into students who have tried to memorize meaningless algorithms and keep making simple miscalculations that have markedly slowed their progress. As they have time to explore these methods Tammy has shown us, they break free and become confident and wonderfully able mathematicians. Love this!
Lori J, thanks for the sweet comment. Hopefully you'll start to see more and more accomplished mathematicians coming your way as we continue to figure out what we're doing! I know they're in good hands with you. :)
Tammy,
Math is SO far out of my element, but you have certainly piqued my interest! I wish you could come to TX and be a math specialist for us . . . or at least consult with our teachers. It's an area we struggle in. Thanks for a great guest post. You do know that this means that now I want you to guest at the Corner, right???
Barbara
The Corner On Character
Barbara, too bad Texas and Idaho aren't close. I could just zip on over and chat about math. Oh, and I consider it a great compliment that you would consider me as a guest on your blog. Thank you for having such confidence in me!
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