2.+Early+Observations

= Superior Greenstone District School Board = = Learning Supposition for 2012-2013 = = Written by Kathleen Schram = = = **Early Observations**
 * Student Work Study Initiative **

In the beginning, many observations were made around students having a hard time reasoning and communicating in mathematics, and when they tried to use the symbolic form of math by applying a procedure/algorithm there were often errors in their communication.

//** November 13th - Gr. 8 class School B **// - evidence from ONAP stated that these students still struggles with grade 7 concepts, so teacher went back to see where they were with some grade 6 measurement expectations - some work samples collected that day:

8.78 + 12.6 = 110.628





Only 5 out of 18 students were able to find the correct response


 * ???Wonderings??? Are students able to visualize the math that is necessary? **

//** Nov. 16th - Gr. 4/5 Class School B **//

3 Gr. 5 students working on multiplication: When one student questions another about how he knew 6 x 31, his response is just that "I know" I interject and question how you know - when someone asks it's not 'cause they don't believe you, they often want to learn how you figured things out....

Student explains 6 x 31 = 186 he lines them up algorithmically and states, "6 x 3 = 18 and 6 x 1 = 6 & you just put them together...."

so I gave him 7 x 69 to try....I thought he might end up with 4263 but he did

7 x 9 = 63 and 7 x 6 = 42, then added 63 + 42 and stated the answer to be 105

I worked with the students for a short time with some explicit intervention around 6 is not 6 but 60 and worked with him on how to properly break apart using 10 and he was able to meet with success.

Later one of the other students came to ask me why the following wasn't working: When trying 2-digit by 2-digit (14 x 26) he did the following, but knew the answer of 160 did not make sense



A group of grade 4 students were working on the following:

The following video captures some of the student misconceptions in thinking. The video itself is quite lengthy, but below it I outline some of the key points. I came across their answer to 815 – 623 being 277 and the video begins after I asked them how they knew it was 277, and they tried to prove then that 815 – 277 would be 623. They have already moved the 1 ten to be 15 ones and placed a 0 in the tens column.

media type="custom" key="23333160" Right away in the beginning we see the need to represent all when subtracting 7 from 15, and when partner says “it’s 8” she just accepts answer without checking for herself. by representing all she realizes that it is 15 and by counting on on fingers is able to show others. But when asked “What’s the answer you decided?” Student responds “15” Then asked “15 is the answer to what?” and they weren’t able to make sense of it They then wrote 815 – 608....and many of the inefficient strategies were repeated //**at 5:17**// Students didn’t see the relationship that if 815 – 277 resulted in 608; then 815 – 608 should’ve resulted in 277; instead they ended up with 207 to which the student replied “you guys are getting me all confused” She then adds 207 to 623 to get 830 and then goes on to ask to minus 15 from the 207 to which they get 202. Subtracting the ones gets 2, then she moves to the hundreds and writes 2 then writes 0 in the tens...... the video continues with them struggling to make sense of why they are changing the numbers and what is happening when they do make the changes.... //**at 6:45**// she does write it as 623 + = 815, but again changes numbers to make them work without compensating in the end //**at 9:24**// I interject as I am curious as to whether or not an open number line would’ve helped and she is able to make some understanding, but doesn’t really see how this would help in solving a subtraction questions, and in the end even when she is adding numbers written horizontally as opposed to vertically, she seems to rotely try to follow a procedure by using connecting lines and placing a 0 in the hundreds place in front of the 77. //**at 12:1**//9 she realizes that 192 + 623 should give her 815; but when writes the numbers horizontally again struggles. She says 20 + 90 is 110, but then records it with the 0 in the tens and wants to write an 11 in the hundreds....then says “I’m very, very confused....” //**at 13:22**// when I prompt her to write it differently (writing horizontally) of which she replies “oh, now I get it” but then begins to follow a procedure of connecting with lines and writing + symbols (//**see 13:40**//), when she gets to 700 + 110 + 15, she still re-writes using connecting lines, instead of just making meaning of the numbers....
 * //at 0:58//** we here “0 – 7 is 7” partner says “0” and answer is accepted; “8 – 2 is 6” resulting in an answer of 608, but when checks to see if 608 is the same number as 623 they realize it isn’t and they then go on to “608 plus _ will equal to 623”


 * ???Wonderings??? Evidence that students are not able to make meaning of the procedures they are following, and that it is important to develop classroom culture around the importance of 'making our thinking visible'. That when someone asks, it's not because they don't believe you, or that you're wrong, but they often want to learn how you figured things out. That if we learn together we will be better able to develop a deeper understanding/meaning of the math we are doing. **

The resource Number Talks was shared and I went in to a few classes to model how number talks could be used to elicit student thinking and to begin to develop various models to represent student thinking. It was difficult to push student thinking to something other than the standard algorithm, however it was evident that some students could apply the standard algorithm correctly (but were truly just following a procedure) and that for ones who weren't quite able to follow the procedure and tried to make sense of being flexible with numbers initially struggled.

In some of the classrooms I was working in, teachers were already using Number Talks, and classroom culture had been established where ''making student thinking visible" was valued, but as evident in the video below, they would try to apply addition strategies to subtraction situations and would run into trouble.

//**Dec. 12th Grade 6/7 class School A**//

These students are working through a subtraction 'number talk' of 57 - 48 of which you will here the answers of 5, 9 and 11 being defended. Teachers try to model what the students are doing, and students try to provide explanation to misconceptions.

media type="custom" key="23334044"

The whole time there is one student trying to defend her answer of 5 and at the end you hear her say "I'm right", and tries to defend her answer of 5 in the following clip:

media type="custom" key="23334094"

She is overusing an addition strategy of compensation for subtraction. She could've very easily used a counting on strategy and arrived at the correct answer, but was struggling trying to see why this strategy wouldn't work in this situation. //**At 1:16**// one of the students interjects with a misconception that you can't round both numbers. He tries to say that we learned that yesterday, but the teacher interjects with "but did we though?" and notes that we didn't get into it. At that point he goes off on his own to try to make meaning of what he was onto yesterday when he was able to make sense of changing one number, but not both.

After playing with the number line for a bit, the students were able to make meaning of the following:



That if subtraction is difference than when looking at the difference between 48 and 57, and changing that difference to 50 and 55, that we needed to keep in mind the way the numbers changed and that the 2 on each side would have to be remembered.

The students began to see the importance of the value of the number line as a model for thinking.

//**Nov. 28th Gr.2 at School B**//

Students in this class were becoming quite proficient at counting forwards and backwards on an open number line, so when presented with this problem, I observed these two students. See video below.

Once again the video is quite lengthy, but what was noticed was that students were able to follow the procedure of solving on an open number line (although weren't very fluent with counting backwards), and they were a bit concerned about who would be able to record the jumps and who would record the numbers..... At 2:24 in the video students were asked to try another strategy...which resulted in some difficulty. Soloman mentions using ten rods, and then realizes he could do a 'take a way', but is only able to write the problem as a subtraction sentence. At 3:17 they begin to try to represent a different way on their own. Rachelle was able to make a representation and use a counting on strategy, at 6:11, but at 8:00 they try to make sense of it using the base 10 blocks and it helps her to make meaning of her representation. At 7:38 when pushed to use the actual manipulatives Soloman represents the 65, but when has to take away 28, is able to remove 25, then just removes from other piles to make the solution match. At 10:39 I ask him to show again, to talk it out loud...goes on to try, but then says "it's hers"...that he is not able to make meaning of it....at 11:54 Rachelle represents 65, and removes the 25 and also reaches up to remove from another pile that isn't within the 65 when I ask here "how come you're taking from these ones" she replies, "because there's not enough ones", then when checking (at 12:45) isn't able to make sense of it. At which time the block of time ends..... media type="custom" key="23334688"

After discussion with teacher, she tried in a large group setting to have a group represent the situation with the base 10 and came to the realization that although many were able to arrive at a solution on the open number line, they were not able to make sense using the base 10 - no one recognized that the 1 ten could also be represented as 10 ones (the trading aspect....)


 * ???Wonderings???the importance of students being able to compose and decompose numbers in a variety of ways to make sense of different representations **


 * It is at this time that the political situation impeded on my traveling, so we did not continue on with this work in the Grade 2 classroom in School B until early February. **

= Continue on to 'Supposition Develops'  =