4.+The+Work+Continues

= Superior Greenstone District School Board = = Student Work Study Initiative = = Learning Supposition for 2012-2013 =

Written by Kathleen Schram
=** The Work Continues **=

Now with an idea of where we were going, we began to watch and question students around multiple representations. What we noticed was that students would often just try to make another representation match - not seeing any relationships between them; not being flexible with their understandings of numbers.

Learning from the students was like playing with Russian nesting dolls.... as we worked at developing a layering of mathematical concepts we often uncovered another layer of needed understanding....and that all of them are needed to work together to build the foundation of mathematical understanding.



//**Gr. 2 School B**//

In the video below Matt tries to use multiple representations to demonstrate 48 - 16, but is not able to make meaning of them.

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Ashtyn & Isaiah try to model 52 - 24 (April 9th)

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These students are able to solve 52 - 24 on the open number line by writing 52 and 24 on the number line and finding the difference between them. When asked to represent 52 with base ten they did so using 5 tens and 2 ones, when asked to represent another way many of the students would just re-arrange their 5 tens and 2 ones. An interesting observation to note here is that (like Ashtyn did here) when students traded 1 ten in for 10 ones and recounted - they would revert to counting by 10's only with the tens, then going on to count on by ones (even though they had just made the pile of 10), instead of recognizing the 10 ones could still be counted orally by a jump of 10. In this video at the end, when they were asked where their answer of 28 was on the number line (hoping they would see it as their total in jumps or the difference between these numbers) the student just places where he thinks 28 would be on the number line -able to do so on the open number line (writing 52 and 24 on the number line, then finding the difference between)


 * Interesting in that the grade 6/7 students in School A (See Early Observations) also struggled with that concept of subtraction being the difference between the numbers.....

Multiple Representation of 69 (Soloman & Matt - April 16th)

media type="custom" key="23372900" Evidence of the students progressing in recognizing the multiple ways of representing a number, but seem to start with the standard representation then follow the pattern of trading 1 ten for 10 ones until the end when it it represented all with ones.

On carpet Representing 32 in 2 ways (April 17th) media type="custom" key="23372958"

The purpose was to push them to recognize that some representations of the number allow for an easier removal. As noticed many were still just rearranging the blocks for a different representation. Much discussion with the students was needed around this topic. Then when the teacher asked them which representation it would be easier to take 18 away from, they could see that with the representations of 32 with 3 tens and 2 ones that it wasn't as easy as their other representations....that they would have to trade their 1 ten for 10 ones to be able to do so.

This led us into using Cuisenaire rods to continue to reinforce the importance of multiple representations. Students investigated to see the relationship between the rods and then were asked to represent 17

media type="custom" key="23373158" Interesting to note that many students went to using the orange as a ten and the white as ones; others lined up 17 ones and then found rods that would work; and that when students would count, many would revert to counting by ones - example in video of touching the red rod twice when counting - not yet really unitizing or fluently able to count by other multiples.

The Cuisenaire rods helped to make the link back to the open number line, as students often arranged them in a linear model.



"One of the most important models for developing computational strategies is the open number line, which supports the development of some key big ideas in addition and subtraction." (Dolk & Fosnot; Fostering Children's Mathematical Development) The key being that it "supports the development" and is not just used as a tool to carry out a procedure.

This becomes evident as in the video below as this grade 7 student School A is pushed to try and represent the concept of working with integers on a number line.

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= Continue on to 'Where We Are Now'  =