First step was brainstorming a progression:
I flicked though the add/sub book to do this. Here.
This was to get me thinking about:
This was to get me thinking about:
- What types of responses I might expect from students.
- How I might categorise student thinking.
- How I might progress students through subtraction.
Diagnostic
A question that I just made up:
Mary had $54 dollars. She spent $6 on a drink and twice as much as the drink on a movie ticket. How much did she have left?
(Simple but not straight forward - double calc involved)
Things I saw in it: 54 - 6 should be doable by all levels - counting back and part whole. Could show if they could take off 10 as a strat if they were taking off 10 and 8 or even 6 and 12. Could be solved by taking off all the parts or by adding them and subtracting them. It also lent itsself reasonably easily to a rounding and compensating strat as (54-20)+2 if someone wanted too.
Levels (for the cards):
1: No idea
2: Count back in ones to solve
3: One strategy other than counting back in ones
4: Two strategies proved with number lines
Mary had $54 dollars. She spent $6 on a drink and twice as much as the drink on a movie ticket. How much did she have left?
(Simple but not straight forward - double calc involved)
Things I saw in it: 54 - 6 should be doable by all levels - counting back and part whole. Could show if they could take off 10 as a strat if they were taking off 10 and 8 or even 6 and 12. Could be solved by taking off all the parts or by adding them and subtracting them. It also lent itsself reasonably easily to a rounding and compensating strat as (54-20)+2 if someone wanted too.
Levels (for the cards):
1: No idea
2: Count back in ones to solve
3: One strategy other than counting back in ones
4: Two strategies proved with number lines
During This Lesson
Because this was a diagnostic I paired the students so it was easier for me to see who was doing what. Usually they would be in threes.
It was interesting - doing the mult/div stuff they had stopped asking me if they were allowed materials and were just getting them. However, with this change of context they reverted to asking me if they were allowed to get them.
I spent the first 6-7 min listening and then intervened with a pair who were not working together well.
I then spent about 5 min with each group teaching them how to record their thinking of a number line. Some had solved with a hundreds beads board (real name?) with part whole thinking: Make 54, take off 4, then 10 then another 4 and end up at 36, but could not record it.
I took note of and decided on a order that I wanted them to present some of the ideas.
I also noticed on student giving a mini lesson to others on how she "knew" that 40 - 4 was 36.
It was interesting - doing the mult/div stuff they had stopped asking me if they were allowed materials and were just getting them. However, with this change of context they reverted to asking me if they were allowed to get them.
I spent the first 6-7 min listening and then intervened with a pair who were not working together well.
I then spent about 5 min with each group teaching them how to record their thinking of a number line. Some had solved with a hundreds beads board (real name?) with part whole thinking: Make 54, take off 4, then 10 then another 4 and end up at 36, but could not record it.
I took note of and decided on a order that I wanted them to present some of the ideas.
I also noticed on student giving a mini lesson to others on how she "knew" that 40 - 4 was 36.
Took a snap of this because I wanted to revisit this at some stage...but was not focus for today. Perhaps a warm up. If you know that 10 - 4 = 6 ...... what else do you know is absolutely true?
I went around the groups and took the pictures that I needed and the students presented them in the order below. The discussion lasted 15 min at the end of the session and also took up 10 min after lunch to finish it off.
I went around the groups and took the pictures that I needed and the students presented them in the order below. The discussion lasted 15 min at the end of the session and also took up 10 min after lunch to finish it off.
First Group Responses
These are not all the responses, just the ones that I selected to be discussed at the end of the lesson.
This first one was selected because I wanted students to link this working out back to the question.
I primed the kids who did the work to ask the other students about what each of the numbers meant in the story. I modeled for them and they used that interaction to open the class discussion by running the same discussion with the class. "What is the 12 in the story?", "Where did the six come from?". "What is the 18 made up from?" -
This first one was selected because I wanted students to link this working out back to the question.
I primed the kids who did the work to ask the other students about what each of the numbers meant in the story. I modeled for them and they used that interaction to open the class discussion by running the same discussion with the class. "What is the 12 in the story?", "Where did the six come from?". "What is the 18 made up from?" -
The two pictures below were shown and explained by the group.
I asked:
Where can I see the part that was the drink being taken off?
Where can I see the part that is the movie ticket being taken off?
(nowhere really - we could have unpacked it and looked at the first 6 counted back and then the last 12, but I was short of time and wanted the focus to be on the latter ones)
What level was this? Got students to link back to our progressions.
I want the students to be continually linking their working out back to the actual problem. This is so that the sense making that the interpretation of the question requires follows though into the mathematical working out and recording.
I asked:
Where can I see the part that was the drink being taken off?
Where can I see the part that is the movie ticket being taken off?
(nowhere really - we could have unpacked it and looked at the first 6 counted back and then the last 12, but I was short of time and wanted the focus to be on the latter ones)
What level was this? Got students to link back to our progressions.
I want the students to be continually linking their working out back to the actual problem. This is so that the sense making that the interpretation of the question requires follows though into the mathematical working out and recording.
This was the final work that we examined on that day:
My questions were:
What does the $54 mean?
What does that $36 mean?
Were is the drink?
Where is the movie ticket?
What did this group do to make the subtraction easy?
What does the $54 mean?
What does that $36 mean?
Were is the drink?
Where is the movie ticket?
What did this group do to make the subtraction easy?
What would I do differently?
I would have put one more example of student work as part of the final discussion. It was the number line which showed 54 - 4 - 10 - 4 = 36. This would have been so great to ask where is the drink and where is the movie ticket! This will really take some thinking to upack for the kids because in this example they added them up to 18 and then split the 18 into 4,10 and 4.
It would actually make a really good warm up on another day!
It would actually make a really good warm up on another day!
Second Group
Same diagnostic given to second mixed ability of group.
Only selected two bits of working out for the class to see.
This was a particularly clear example of counting back (stage 4).
Only selected two bits of working out for the class to see.
This was a particularly clear example of counting back (stage 4).
Below is an example of some stage 5 thinking (heavily scaffolded by me) that they were able to rehearse and perform. They had worked it out on the beads and showed them how to take each step and show it on a number line.
I used this movie to model how to do a number line for the entire class.
Extension and Moving Forward
Two kids wanted some math to do at lunch and one wanted to take some extra homework home. I changed the problem to $154 starting money and left the rest of the problem the way it was. I also said that "someone" solved the first problem like this: 54 - 20 = 34, and then 34 + 2 = 36 (and gave an example number line). Could they work out what they were doing and use that same strat?
Hopefully the seeds of some of these ideas will be planted in 1/4 of the group (and two of them were not the most competent). Interesting to see if these resurface and are taught to the other students.
Hopefully the seeds of some of these ideas will be planted in 1/4 of the group (and two of them were not the most competent). Interesting to see if these resurface and are taught to the other students.
Things I learned from the Diagnostic (from both groups)
- No one knew how to record on a number line well.
- Some could copy one out completely but not sketch it with just the important bits.
- Taking off 10 instantly was a problem for a few.
- 10-x = was a problem for a few.
- Breaking up the numbers into anything other than 10 and x, and doubles, was a really new idea for many.
- Drawing many dots and counting back was the refuge of a few kids who I expected more of (after seeing thinking in other domains
- Most kids were stage 4 (counting back)
- The word "twice" was not a problem for the first group but it stumped the hell out of the second mixed ability group.
Sooooo.......
- Will keep these things in mind when designing questions....
- Will do a lot of work on linking the question, materials and working on a number line together
- Will schedule some maths buddy stuff to hopefully target the -10 and the 10-x stuff
- Warm up for counting back in 10's from three digit number.
A new reflection question
A new reflection question....well more a scaffolding phrase....seemed that it might be really helpful near the beginning of a unit when kids were making more drastic shifts (hopefully) in their thinking:
I used to think . . . Now I think . . . This routine provides teachers and students with an opportunity to formally recognize a change in their thinking from the beginning of an Investigation to the end or from the start of a Problem to its conclusion. Students confront their original thinking about a concept and reflect on what has changed about that thinking. Acknowledging a change in thinking is powerful for all learners. This is easily done with sticky notes.
Ritchhart, R., Church, M., and Morrison, K. (2011). Making Thinking Visible: How to Promote Engagement, Understanding, and Independence for All Learners. Hoboken, NJ: John Wiley & Sons, Inc.
I used to think . . . Now I think . . . This routine provides teachers and students with an opportunity to formally recognize a change in their thinking from the beginning of an Investigation to the end or from the start of a Problem to its conclusion. Students confront their original thinking about a concept and reflect on what has changed about that thinking. Acknowledging a change in thinking is powerful for all learners. This is easily done with sticky notes.
Ritchhart, R., Church, M., and Morrison, K. (2011). Making Thinking Visible: How to Promote Engagement, Understanding, and Independence for All Learners. Hoboken, NJ: John Wiley & Sons, Inc.
Something Interesting
Something I had noticed (over the last two weeks) was that some students were finding an answer to the problem and then generating equations that gave the same answer even if they made no sense to the context. Just to generate extra examples. Some of the earler work we did might have given them the impression that this was good maths?
It happeded again today - one students gave the answer - $36 and his working out read - "I start with 100 and then take off 7tens and 4 ones and it makes $36" (the fact that this was wrong was beside the point)! It had nothing to do with spending 6 and 12. I asked where the drink was in teh working out and where the movie ticket were to get him back on track. Going to need to add a check - Make sense suggestion somewhere in the process.....
It happeded again today - one students gave the answer - $36 and his working out read - "I start with 100 and then take off 7tens and 4 ones and it makes $36" (the fact that this was wrong was beside the point)! It had nothing to do with spending 6 and 12. I asked where the drink was in teh working out and where the movie ticket were to get him back on track. Going to need to add a check - Make sense suggestion somewhere in the process.....