Today, I want to touch on struggle space, misconceptions, and reasoning skills. It sounds like a lot, but it is all intertwined.
We are reaching the end of our explicit look into fractions, and are starting to explore measurement expectations. Yes, I do tie my "units" together, because all of the components of math are so important that I don't want students to feel that they should "close the book" once they've had a test. Hence, no tests - just constant, daily assessment.
Today I wanted to assess some of their fractioning skills (can they take a whole number like 1000 and split it into fractions, instead of looking at the "whole" in the fraction as "1"?) I also wanted to diagnose their prior knowledge of linear measurement, namely the conversion between metres to kilometres. So, I asked:
Mr. Patrick went on a trip this weekend.
He travelled 1km.
If he had only travelled a third of that, how far would he have gone in metres.
I wanted to know:
- do they know how many metres are in a kilometre?
- do they know how to use fractions as pieces of a whole, thereby splitting/dividing the number into the fractional amount, in this case being a third (1/3)?
I specifically wrote "how far would he have gone in metres" because I wanted to cue them to the conversions without explicitly telling them. I felt that if I hadn't, they would have taken 1km and split it into thirds as kilometres, which was not the point of the activity.
This was just the first part of the question.
To be honest, I was a little bit surprised that most had difficulty figuring out how many metres were in a kilometre. Fortunately, they saw that they needed to convert it (even though they didn't have the language to use, like "convert"). This was the intended struggle space.
Most struggled for a solid ten minutes. Most knew that it would be something like 100 metres. This was a great space for them to discuss and debate, and to work on their reasoning skills. Take a look at the video below, of one of my students explaining why she believed that the kilometre held one hundred metres. It is actually fairly convincing!
So, she wasn't correct. That's okay!! In fact, it's great, because she will remember her reasoning, and will correct the misconceptions. After a few more minutes of wondering, I directed some of the students who were struggling the most to grab one of the three text books that I keep on the resource shelf. They located the index, and found kilometres. They identified pretty quickly that 1km=1000m. Excellent!
I decided to use this opportunity to give them a little more confidence, as they often struggle more than others. I called the class's attention, as I felt the struggling had gone on long enough, and I wanted to see them actually working on the thick math of the question (there was a second part, which you'll see momentarily). I told the class that I appreciated all of the thinking and work they'd done so far, but that it was time to reveal how many metres were in a kilometre. I asked the kids with the text book to deliver the message, which they did proudly, and everyone said, "OH YEAH!" Then, they went back to work.
Some struggled to do the actual division. I encouraged them to struggle a bit through some long division, but allowed them to use calculators when they had shown me that they understood why they were dividing or splitting one of the numbers by another. This is so important - like any technology, it is useless if you don't understand the reasoning behind it. If you were to split 3, because of the thirds, into 1000 pieces, you would come out with an entirely wrong answer, and you would likely be baffled as to why it was so teeny-tiny. It could be a great learning opportunity, as well, which is why I stuck close by any groups using calculators.
There was another excellent opportunity built into this question. 1000 divided unequally, at 333.33333.... and so on, forever and ever, amen. We've been slowly looking at how to round the decimals, which is an expectation in Gr. 5, and this question allowed the kids a real opportunity to see it in action. Some struggled with the idea that 333x3=999 ... they wondered and were frustrated with the fact that it wasn't 1000, but that if they used 334, it would be way too many.
With the skills developed from part 1 of the question (fractional splitting and measurement conversions), they moved on to part 2:
On the way home, Mr. Patrick took the scenic route.
He travelled 12 kilometres.
What are some other ways to describe the distance?
How long would he have travelled if he only went a third of the route?
In this part of the question, I wanted to see:
- can they transfer the skills from part 1?
- can they convert 12km to 12,000m?
- will anyone be able to/think of converting it into cm?
With part 1 complete, most were able to struggle a bit and identify the conversion of 12 to 12,000, reasoning aloud to me that if 1 is equal to 1000, then 12 would obviously be 12000. Some of them knew it right away.
One group was able to find out how many centimetres it would be, and solved the fractional piece using both metres and centimetres. Then, they went on to determine how far I'd travelled in total, using every different measurement unit. In one lesson, those two students demonstrated that they had already achieved the curriculum expectation. For them, I can move on.
As we brought it all together at the end, I really had a sense that everyone was on the same page. I knew that I would be able to present tomorrow's explicit lesson on conversion and have them all grasp what I would be saying, because they had struggled, reasoned, and had many misconceptions repaired today.
Based on all of this, I feel like it is a strong explanation of the power of struggling and having a misconception; where explicit instruction and redirection can fit in; the importance of caution in the delivery of a correction; and the beauty in strong reasoning, whether or not the actual facts are correct.
This allows me to reflect on my teaching - am I allowing the kids to grow of their own accord, while still travelling in the right general direction? Are my questions allowing them to explore the intended curricular areas? Am I creating the right opportunities for struggle space that is reasonable and not beyond their zone of proximal development? Am I allowing the students with opportunities to discuss, explore and reason, for real and in depth?
Struggle space, misconceptions & reasoning - in my eyes, the cornerstones of a solid math program.
No comments:
Post a Comment