I think we have all been there (or maybe it is wishful thinking that I am not alone:)…
The class is sharing strategies for solving a problem and all of a sudden, one student explains his/her “shortcut” or algorithm to the class. It’s true, it works, and you know you will get there, but my first thought is always “Do they know why? while my immediate second thought is, Oh no, now this will look faster to some of my students who will quickly grab onto it to save themselves some time, not caring why it even works.
I deal with this in many ways and it really depends on the situation. Is there time to go deeper into that idea at this point? Is it something that arises later and this could just be a nice “Interesting, that is something to think about”? Is it something that will lose more than half of the class and can be addressed with that student later to gauge understanding? Or is it something that will end up as a version of the Telephone Game?
If you have not had the pleasure of playing the Telephone Game, it goes something like this: One person whispers a sentence to another person, that person whispers the same thing (or as close as they can remember) to the next person in line, so on and so forth until it arrives back to the first person. Typically, when the initial sentence makes it to the last person, it is not the same.
This time, a decimal addition and subtraction strategy has fallen victim to the Telephone Game. During one of our number talks a week or so ago, a student, let’s call her Jane, mentioned that she just “adds and subtracts the numbers as if the decimals are not there and then puts the decimal back into her answer.” She explained it for the problem we were currently working on and we moved on. I made the decision to revisit this with her later in the class period to clarify her thinking and not make it a class discussion at this point. I knew in our upcoming lessons we would get to this “decimal movement” when we started multiplying decimals by powers of 10 and I thought this would be a perfect example to bring back up when we got into that lesson.
Evidently, I waited too long.
Over the course of a couple of days playing Fill Two, Empty Two, and Closest to 1, I started to hear a buzz in many students’ explanations that sounded much like Jane’s. However, it seemed as this idea made its way around my classroom, some very important pieces were missing. Some students were not taking place value into account while many others were losing any concept of sense-making about their answer. For example when adding 3.6 + 2.24, some were adding 36 + 224, arriving at 260 and having no idea where would be a sensible place for the decimal. While Jane was correct, and she understood she needed to put the numbers in the same place value, this part of her reasoning was lost in the Game. Not to mention the “why.”
Hmmm, now, what to do? I didn’t want to explain Jane’s process without going into what is actually happening to the addends when the decimal is “removed” and then “put it back in”, but I questioned whether I would be jumping too far, too quickly for some when we were just getting a handle on adding and subtracting decimals. I hate to ignore ideas or make them feel unimportant when I feel they truly are. I made the decision in my planning yesterday to attack this telephone game head on.
After our number talk today, I typed Jane’s claim on the SMARTBoard and asked the tables to prove if it was always true, and if it was, why can we do that?
Many students started “testing” a lot of problems to see if it worked every time. Some tested and tested but struggled with Jane’s “putting the decimal back in.” YEAH, because now we can talk about estimating and reasonableness!
Others re-emphasized the point of the same place values combining. This will be a nice discussion of how the base ten system works when combining place values, ie, ten in a place will always make one of the place to its left.
Two tables did start to talk about the addends multiplying by 10 and/or 100 and then dividing the answer by the same to adjust it. One table jotted down some work, but the others were still in the discussion phase. They did say 2.50 x 100 gives you 250. I asked how they knew that and they said that 2 x 100= 200 and 0.50 x 100 = 50 so it has to be 250.
Tomorrow’s conversation will be very interesting! I have so many thoughts about my goals for the convo, but here are my initial thoughts….
1- Reasonableness is SO important, estimate, estimate, estimate!
2 – How adding like place values acts similar across all place values.
3 – What is happening when we “take out the decimal”
4 – How adjusting the addends in the same way affects the sum. Really the bigger generalization for any addition problem.
Love this! Your kids do some great reasoning. Thank you for sharing your classroom so openly, we are all learning from you and your kids.
I always enjoy seeing how your students are thinking through the math process. Thanks for sharing!
Tricks are good only when you understand what’s going on, and best of all when you come up with them yourself. So great that you’re making them think through the whole process and experiment with lots of examples. Thanks for the story and your thoughts!