Reflection:
- I learned as a teacher of math that there is more than one way to solve a math problem. Not every math equation only has way one. My students all learned with different perspectives and they had different ideas that lead them to the same answer. I leanred that I should not force my students to think of math in only one perspective and instead teach them to look at all the different ways in which we can look at a problem.
Final: Analysis of Instruction and Engagement
Amber, Carlee, Ekaterina, Marie, and Tabitha,
BEDUC 419 -Knowing, Teaching, and Assessing Mathematics -Spring 2017
1. Description of Focal Clip
Describe the focal clip:
- Starts with us asking the students to think about the problem in their mind without sharing out loud.
- After the students solve the problem mentally we ask them to share with a partner what they got and how they solved the math string to get the answer.
- Then we asked if any of the students would like to share with their peers how they solved 5 + 20.
- Asher shared how he got his answer by adding 20 + 5 he just know from previous knowledge that the answer was 25.
- Areli then shared that she solved it by counting on from 20 on her fingers 5 more to get to 25.
- After hearing Areli strategy we told the students to keep in mind the 5 + 20 problem as we move on to the next problem of 5 + 23.
- We again asked them to solve the problem in their mind without sharing the answer out loud.
- Once all the students gave the thumbs up sign to say that they had an answer in mind for the problem of 5 + 23, we asked if anyone would like to share how to solve the problem.
- Asher shared how he know that 5 + 3 = 8 and that he added the 8 to the 20 to get 28. Marie asked him where he got the 5 and the 3 from and Asher said that he got the 5 from the 5 in the problem and the 3 from the 23 in the problem of 5 + 23.
- After Asher shared Noah described how he solved the problem. Noah said that he split the 5 into 2 and a 3. He then took the 2 from the 5 and added it to 23 to get 25. After that he added the 3 that he had taken from the 5 before and added it to 25 to get 28.
- June was the last to share her strategy for the 5 + 23 problem. She said that she used the problem above of 5 + 20 to solve the second problem by adding the 3 from 23 to the 25 to get 28.
- We then had a quick debrief about how June used the first problem to help her solve the second problem and how we were able to use many different strategies to get the same answer.
General Timestamps of the Video:
00:07 Start
00:07-00:21 Explanation of what students are to do
00:21-00:50 Wrote problem on paper and students solved the problem in their heads
00:50-01:42 Turn and talk (talk to your partner and tell them how you solved the problem)
01:42-02:25 Asher shares his strategy
02:25-03:26 Areli shares her strategy
03:26-04:20 Quick debrief of both strategies and what is going to happen next
04:20-04:44 Students solved problem in their heads
04:44-06:03 Asher shares his strategy
06:03-07:50 Noah shares his strategy
07:50-09:19 June shares her strategy
09:19-09:33 Quick debrief
09:33 End
What was the math activity:
Our math activity was two mental math number strings. A mental math number string is two number sentences where the first problem (5+20) influences how the students solve the following number sentence (5+23). We wrote the number sentences on the board and had the students solve in their heads instead of with paper and a pencil. We had the students who volunteered to share their strategies to solve the two number sentences and recorded them on the board.
What mathematical ideas were you focusing on in the lesson?
Our goal was to encourage students to reflect on how problems could be related to each other. To encourage students to use what they already know from (20+5) to help them solve new and more challenging problem (5+23).
What were the targeted strategies (if any)?
Our targeted strategy was for the students to use derived facts in order to use what was done in the first number string and apply it to solve the second number string.
2. Promoting a Positive Learning Environment
Describe how you demonstrate mutual respect for, rapport with, and responsiveness to students.
- We had four students sharing their ideas instead of just one or two. We wanted to hear from as many students as possible because we feel each students thoughts are important. We also did not say or treat as though one students’ strategy was better than the rest of the students strategies. We did this by giving students wait time after asking students a question to help them process their ideas and having them signal to us that they were ready or needed more time (Intentional Talk, pg. 21). The wait time gave students confidence in their idea as well as trust that the teacher won’t move on while they were still thinking. Wait time gives students confidence that the teacher will also not call on random but allow the students will a clear idea to talk first.
- At 00:50 we asked students to think about their strategy and then told them to turn to their partner and talk about how they solved the problem. This teaching strategy is called ‘turn and talk’ and it’s a way to get students more comfortable sharing their ideas. “Turn and Talks” also give the teacher a way to listen to key words, miscues, in this case we were looking for students to make the larger numbers into manageable numbers like 5’s and valid strategy (Intentional talk, pg 21). From the turn and talk we had two students share their strategies, one using recalled facts, and the other using a counting-on strategy.
- At 5:10 we had students share their ideas on how they solved the problem. For the three strategies shared we used revoicing as one of the main talking moves. Revoicing is a common talk move in teaching that has the teacher, “repeat some or all of what the student has said, then ask the student to respond and verify whether or not the revoicing is correct” (Intentional talk, pg. 21). A specific example of revoicing was at 6:25, when our student was explaining how he solved 5+23. His idea was splitting the 5 into 3 and 2, at first he didn’t explain this and we tried to make sense by revoicing, “So you took the 2 from the 5 and added it to the 23?” From the revoice our student understood that his explanations had to be more concise for us to understand, and his explanations included were he got his numbers for his strategy.
3. Describing Student Mathematical Thinking
Provide explanations for the children’s mathematical thinking.
- For the first number string, 5+20
- Asher’s strategy (number fact/recall)
- When explaining his strategy, he said that he just knew that 5+20 is 25.
- This is an example of the number fact strategy. In the book Children’s Mathematics (Cognitively Guided Instruction) by Thomas P. Carpenter he explains that “through experience solving problems and reflecting on their strategies and the strategies of their classmates, children begin to know many number facts from memory” (Carpenter, p.40). Asher was able to know the number fact 5+20=25 from his memory.
- Areli’s (counting)
- She started at 20 and counted on using her 5 fingers on one hand to get 25.
- This shows us that on the CGI trajectory Arelie is on the counting on stage.
- In the book Children’s Mathematics the author Thomas P. Carpenter explains that “counting strategies represents more than efficient procedures to solve addition or subtraction problems. They indicate a level of understanding of number concepts and an ability to reflect on numbers as abstract entities” ( Carpenter, p.37)
- For the second number string, 5+23
- Asher’s strategy
- He started by explaining that he knew 3+5=8. He got the 5 from the 5 in the number sentence, and he got the 3 from splitting 23 into 20 and 3. He then added the 8 and the leftover 20 to end up with 28.
- He used derived facts in order to make the problem easier by splitting up 23 into 20 and 3 in order to create number sentences that he knew or that were easier.
- Noah’s strategy
- Derived facts, he split the 5 into 2 and 3. Then he added 2 to 23 making it 25. And finally he added the 3 to the 25.
- In the book Children’s Mathematics the author Thomas P. Carpenter explains that “ derived facts strategies support children to look at relations among numbers and operations in ways that support learning to know number facts from memory” (Carpenter, p.40)
- From working with Noah on previous lessons we could see that the derived fact strategies has lead him to knowing more facts from memory.
- June’s strategy
- She started by explaining that she knew 20+5 was 25 by using derived facts from the first number sentence. She got 20 from splitting the 23 into 20 and 3. She ended with adding the 25 and 3 to get 28.
- This shows that she is using what we did from the first number string to help her solve the harder second number string. This was our targeted strategy which we mentioned earlier.
4. Engaging Students in Learning and Deepening Student Learning during Instruction
- Describe your strategies to elicit students’ thinking and understanding
- 05:01-05:33 Asher shares his strategy
- “How did you find that? Did you count up? Did you just know that?”
- “And you pulled the five from?”
- “Where did you go from here? You have 8.”
- “20 to..?”
- 06:03-07:50 Noah shares his strategy
- Noah’s explanation of his strategy was a little hard to follow, so there was a lot of revoicing. Marie asked him, “So you took the 2 from the 5 and added it to 23? (06:48)” Originally she wrote it as 5 - 2 = 3, followed bye 23 + 2 = 25. The 5 - 2 = 3 confused Noah, and made it a little difficult for him to continue on explaining his strategy. It was up until Marie made a carrot under 5 and put a 2 and 3, that he said he carried on the 2 to add to 23 to make 25. From that point on, Noah was clear on where he got his numbers and where he put them due to the questions he was asked in the beginning.
- 07:50-09:19 June shares her strategy
- “So you took 20 from 23. If you took 20 from 23, what is the other number that’s left? (08:10)”
- “She knew 20 + 5 = 25 from up here, where did you go from here? (08:25)”
- “So you did 25 + 3, and the three came from here or this? (08:50)”
- “And you said you knew these two are ones, that’s where 8.. (09:05)”
b. How did you respond to students’ sharings? What did it sound like?
- Throughout the lesson both of the instructors listened to the student’s strategies carefully and tried their best to write down exactly what the student has stated without interfering or adding their own thoughts or steps in the process.
- When concluding the lesson Marie pointed out that June used what she learned from the first problem to solve the second problem, she was also careful to point out that it was not the only strategy and that other approaches were just as accurate and important.
- In the book Children’s Mathematics the author Thomas P. Carpenter explains that sharing accomplishes two main goals. First, it helps students enhance their understanding of the problem because as a students verbalizes their ideas they can “make and deepen connections among mathematical ideas”. Second, not only do other students get to listen to their classmate strategies and learn new ways to solve problems, they also “learn what counts as a complete mathematical explanation” (Carpenter , p.142).
· c. How did you facilitate interactions amongst students? Or orient students to each other’s ideas?
What did it sound like?
- 7:59: Encouraged students to use strategies of other students from earlier problem (5+20) in order to solve a more challenging problem (23+5)
- “did anybody use any of these strategies (points to the first problem) to solve the problems down here (points to the second problem)”
d. Explain how you and the students used representations to support students’ understanding and use of mathematical concepts? How did you represent the students’ ideas? What did the representations look like? How did you use the representations to deepen students’ understanding of the mathematics?
- We put our students exact thoughts/explanations onto the big sheet of paper.
- The students ideas were written down word for word on how they described solving the problem mentally. We deepened their understanding by asking them where they got each number when they added and by asking them how they split the problem so that it was easier for them to add.
- 20 + 5 = 25 , represented Asher’s recall of what said he immediately knew
- 20 + (hand each finger counts up from 20, i.e. 21, 22, etc.) = 25
- Areli said that she started from 20, and used five of her fingers to count up to the answer. We drew the 20 she started with, and each finger that she used and its coordinating number.
- By drawing out the hand and labeling each finger 21,22,23,24,25 we were able to show the other students in the group how she approached the problem and to present the students with another strategy that they could use to solve future problems.
- 3 + 5 = 8 , 20 + 8 = 28
- Asher said that he took the ones (3 + 5), and added them together to get the 8 leaving 20. And then he added the 8 to 20, getting 28
- To help deepen the classes understanding, we showed them that Asher sorted out the problem by ones and tens to make it easier for him to solve. We showed them that sorting is another great strategy that can be used if they’re comfortable with it to solve future problems.
- Parted 23 into 20 and 3 , 20 + 5 = 25 (previous number sentence), 5 + 3 = 8, 25 + 3 (left over 3 from 23) = 28
- June said her strategy to find the answer was to part 23 into 20 and three, and then 20 + 5 = 25 which she knew from the previous number sentence that we worked on. From that point she added the ones knowing 3 + 5 = 8, so whens she added 25 + 3 = 28.
- June portrayed an example of finding a comfortable starting point to a number sentence that looks intimidating. By using her previous knowledge of 20 + 5 = 25, June thought it would be easier to know what three more would be. June’s example allowed the students to see that it’s okay to either go forward or backwards to find a comfortable starting point.
- Similarly to June’s example, Noah also split a number to find a comfortable starting point. He wanted to get the 23 to a number he felt comfortable with, which was 25. From that point on Noah felt more confident in being able to solve the problem, versus feeling intimidated in the beginning. It allowed us to show the importance of finding a point in a number sentence that enables one to feel confident that they can carry out the equation.
5. Analyzing Teaching
Refer to specific examples from the video clip in your explanations. (use timestamps)
a. In what other ways did your instruction support learning for the students in your group?
- At 00:50-01:42 we used the turn and talk strategy by having them talk with a partner about how they solved the number string. This helped us create a safe environment for our students to share with one another as well as for them to learn about their peers strategies from their peers.
- During the beginning of the debrief at 09:19, Maire explained how June used the first number string (5+20) to help her solve the last problem of 5 + 23. This was to help enforce the target strategy of using the first number string to solve the second number string. However, Marie also made sure to point out that it was not the only strategy that worked and that other approaches were just as accurate and important.
- In Noah’s strategy at 06:03-07:50 during the 5 + 23 problem, Marie used a revoiced talk move by asking where he got his numbers of 3 and 2 from instead of just explaining his process. This helped them make the connection to how he added 2 to 23 to get 25 and then added 3 to 25 to get 28. Having focused questions during the student’s process would’ve helped clarity for students and teachers.
b. What changes would you make to your instruction to better support student learning (e.g., missed opportunities)? How might you have elicited, responded, represented students thinking differently now that you can listen and imagine trying again? How might you orient students to each other differently? How do you think these changes would improve student learning?
- At the end of the lesson at 09:19 if we had more time we would change use pointing out what they students did the same in both problems to ask what patterns or similarities the students see between the two problems.
- Noah is a shy student who rarely to never speaks up whenever we do activities. During 06:03-07:50, Noah volunteered to share his strategy. His explanation was a little vague and choppy, which we wouldn’t say is inordinary. With that being said though, a way to help Noah feel more comfortable with sharing would be to allow him to explain his whole strategy before trying to put it down on paper or allowing him to come up and write it himself. Both of those opportunities could’ve helped Noah feel like his strategy is valuable and interested in sharing his thought processes. We saw that writing the strategy down first helped our quieter students become confident in their strategies and were more comfortable saying it out loud.
- Orienting the students in a different seating arrangement might have helped student engagement in this lesson. Our students are very quiet and in the row format we saw students who were not engaged or confident in speaking their ideas out loud to either other students or teachers. In other lessons we oriented the students differently by having the students sit in a circle instead of rows. In this configuration we saw student engagement in ‘me toos’ and sharing ideas back and forth went up as well as more hands raised and willingness to share ideas with other students as well as teachers.
6. Future Lessons
Describe what lesson you would like to do next with this particular group of students. Use course materials to support your answer.
With this group of students we would focus next on a compare and connect lesson created by the use of their strategies from this lesson. Compare and connect lesson is a lesson that compares two or more strategies from a student’s work from a previous lesson. As the teacher, we will be trying to anticipate what students will notice about the different strategies as well as encourage the students to begin to use, or broaden their thinking about their own strategies (Intentional talk, pg. 40). We chose a compare and connect lesson to help our students to have some baseline strategies for addition. Our group of students has a variety of strategies and methods, and with a compare and connect we want to highlight some stable strategies for our lower students to work with and for our higher students to memorize in order to move from the derived facts stage of CGI trajectory to the recall stage. We would start by going over what similarities that they see between 5 + 20 and 5 +23. We will be using the problem of 5+23 as the base problem for the compare and connect, and will use two of the student’s strategies by Asher and Noah. We can anticipate students to compare both strategies which use splitting numbers to combine in order to get their answers. As in Asher’s strategy, adding together the one’s place (3+5) and adding the ten’s place (20) to the one’s place. As in Noah’s strategy, splitting the 5 into 2 and 3 and adding that 2 to 23 to get 25 and adding the 3 to that 25 to get 28. We can also anticipate that the students will compare that both strategies ended with the same answer. In the two different strategies we can also anticipate that they will see a difference in the use of numbers being split in each of the problems.
References:
Carpenter, T., Fennema, E., Franke, M.L., Levi, L., Empson, S.B. (2014). Children’s mathematics:
Cognitively Guided Instruction. Portsmouth, NH: Heinemann.
Kazemi, E., & Hintz, A. (2014). Intentional Talk: How to Structure and Lead Productive Mathematical
Discussions. Portland, ME: Stenhouse Publishers