Video Analysis 2: "Number Operations: Multiplication & Division"

“Number Operations: Multiplication & Division” Video-4^th Grade

This was a math video called “Number Operations: Multiplication & Division.” Becca Sherman was conducting lessons in a 4^th grade classroom about multiplication and division. These clips, much like the video clips we had to watch from the first video analysis, were valuable to watch because it was as if I was able to jump into these students’ minds and see their thought process. I was also able to observe the different teaching strategies the teacher used (i.e. the bell to gain students’ attention, partner sharing, etc.). Additionally, it was interesting to see the various math approaches the teacher exhibited (i.e. bar model, array of dots, etc.) and how well the students were able to comprehend these approaches. There were no video clips of the teachers discussing pre-planning or the goals of the lessons. The lesson simply began with the teacher introducing the lesson. There was an introduction (Part A and B) presented to the students, problems (the lessons) that were split into four parts (Problem 1-Part A and B, Problem 2-Part A and B, Problem 3-Part A and B, and Problem 4-Part A, B, C, D, E, and F), and a closure (then followed up by a faculty debriefing video clip).

Introduction:

           

            Ms. Sherman made the students aware that they were working with multiplication and division. She also pointed out the quote that was written on the board, “A picture is worth a thousand words.” After asking students what they thought this quote meant, she indicated that they will try to get to know what that quote means after practicing with today’s lesson, multiplication and division. Ms. Sherman next asked the students to think about something they know about multiplication. They then shared their thoughts about multiplication with a partner. It was interesting to watch the “Introduction-Part B” video clip of the male student trying to define multiplication. It was obvious that he struggled to devise a definition of multiplication. He may have known how to do multiplication (knows examples; 2x2=4) but doesn’t know what it actually means to multiply, as I think is the case with many students. Or maybe he didn’t really understand any aspect of multiplication and therefore, was completely confused all together. I think a common misconception when teaching mathematics is that many educators do not ask “why” enough and/or assume their students actually know what the meaning is behind what they are doing. However, if students and teachers do not explore the “why,” the students aren’t going to be able to truly gain an understanding and make sense of the concept they are working with. This, in turn, results to students likely confronting more struggles with the math concept and less success.

The Lesson:

Problem 1-Part A and B & Problem 2-Part A and B

           

            In the video clips, Problem 1-Part A and B and Problem 2-Part A and B, the students were accessing their prior knowledge about multiplication and division. The teacher was able to assess their understanding and confidence levels about those math concepts. Ms. Sherman had “Multiplication” and “Division” written on the whiteboard. In Problem 1, students were looking at the term “Multiplication.” After sharing thoughts about multiplication with their partners, responses were shared as a whole class. Some students agreed that multiplication was like adding (i.e. 2x8=16 is the same as 8+8). Other students provided similar examples (i.e. 5x2=10 is the same as 5+5). Students seemed to be stuck on timing by 2’s and struggled to think of other examples (Is what you are saying true with times something else?). The students brought up the idea of “groups” (i.e. 9x2 is the same thing as two groups of nine and 9+9). Ms. Sherman then emphasized this idea of “groups” saying that they needed to create equal groups and could use pictures to help them create these groups. She drew a bar model of two groups of 3 and had a student come up to the whiteboard to add another group of three to the bar model.

After discussing multiplication with partners and the whole class, students were asked to discuss “division” with their partner (What do they know about division?). After sharing thoughts about division with their partners, responses were shared as a whole class. Some students believed division was like subtraction while other students believed division was like multiplication (i.e. “It’s like multiplication but you divide it. Like 9 divided by 3 equals 3 and 3x3=9; “It’s like multiplication but you switch it. Like 5x3=15 and 15 divided by 3 equals 5.”) Ms. Sherman asked the students to elaborate on their thoughts about whether they thought division was like subtracting or multiplication. After briefly drawing a fact family triangle (which the students seemed to be familiar with), Ms. Sherman was trying to get the students to aim their responses towards the idea of equal groups. She threw the number “100” on the whiteboard and wanted the students to think about how they could divide it into some equal groups. One student mentioned that 100 could be divided into two equal groups of 50 (100 divided by 50 equals 2 and 2x50=100). I think Ms. Sherman realized that she probably started with too large of a number for the students to break into groups, and therefore, she decided to have the students work with the number 12 (instead of 100). She drew 12 dots on the board (a 3 by 4 array) and wanted students to try and make equal groups out of that. Many students were able to successfully make equal groups (i.e. 3 groups of 4, 4 groups of 3, 2 groups of 6). I think many students were starting to see the connection between multiplication and division.

 Problem 3-Part A and B, and Problem 4-Part A, B, C, D, E, and F

            For the next problem, Problem 3, students were to use mental math to think about the problem 26x4. I think Ms. Sherman wanted to see if students could apply the bar model and the idea of grouping. After students independently thought about this problem, there was a partner discussion and then a whole class discussion. Before discussing their methods, Ms. Sherman first had students share their answers they got from solving 26x4 mentally. One student mentioned they got 104 as their answer and another student mentioned he got 70 as his answer. Ms. Sherman also asked the students if this problem had only one correct answer or multiple correct answers, and many students said multiple correct answers (which was surprising). Ms. Sherman made sure to clarify that students knew that this problem only had one answer. It seemed many students did the traditional stacking multiplication strategy where the problem was set up vertically, and the students multiplied and “carried.” Next, Ms. Sherman wanted students to indicate how they could draw a picture of 26x4 (since "a picture is worth a thousand words"). She drew a bar model with 4 groups of 26.  One student shared that he did 26+26=52 and 26+26=52 and then added 52+52 (so he doubled 26 and then doubled it again). Lastly, another student shared a different approach. This student said, “I knew the 20 was in the 10s place so I did 20, 40, 60, 80 four times (20 times 4). Then did 6 four times, 6, 12, 18, 24 (6 times 4). Then got 80 and 24, added it, and got 104 (the correct answer)." I thought this was very advanced thinking and was surprised to see that this student thought of the problem in this way. Technically, this student broke apart 26, and he broke it into a 20 and a 6 (factor tree). I believe many students were trying to figure out different methods as to how to solve the problem, which was great, but they still seemed to struggle with grouping/bar model.

            The last problem that the students worked with, Problem 4, asked the students to take the words of a story and make a math picture to help solve the problem (now going from word form to picture form). The problem was as follows: “Maria saved $24. She saved 3 times as much as Wayne.” This problem was different in that no questions were posed. Students were asked to think about this story, what they know/what’s going on, and who is in it. The students then independently thought about this problem, drew a math picture at their desk of this problem, and then shared what they drew with their table. It was interesting to observe that at first, many students struggled drawing a picture to represent the problem. As they kept working at it, many students eventually expanded their numbers that they wrote down into more of a picture form. However, they still faced trouble when trying to explain what they wrote/drew. Furthermore, it was remarkable to see how split up the class was on the problem; about half the class thought $72 was the answer, and the other half thought $8 was the answer. This showed me that while many students can solve multiplication and division problems in number form, they have a misconception when reading story problems and deciding what concept to apply (multiplication or division). In other words, many students saw the word “times” in the story problem and assumed that meant to multiply (24x3=72). 

            Ms. Sherman then introduces a student’s strategy, Charlie’s model (bar model). His model showed Maria’s money with three boxes that totaled $24 and showed Wayne’s money with a question mark (need to figure out how much money Wayne has). These four boxes were all in green showing that they all have to be the same number (equal groups have to go in the boxes). The last question mark is the total of all their money. On a piece of paper, students copied Charlie’s model and filled in the missing numbers. Similar as to how students thought before Charlie’s model was introduced, many students either put 24 in the boxes or put 8 in the boxes. After students worked with Charlie’s model at their desks, Ms. Sherman presented them with two questions: How much did Wayne save? How much did Maria and Wayne save together? The students were then instructed to answer these questions underneath their pictures (needed to think about if there was a number that would be the same for all the green boxes that would work for this story). When the whole class discussed Charlie’s model, a student suggested putting 24 in Maria’s boxes. Ms. Sherman explained why 24 couldn’t be possible (Maria doesn’t save $72; she saves $24). Another student shared that Charlie put 8 in all the boxes (because 8x3=24 so three boxes of eight shows that in all Maria has $24). I don’t think many students wrapped their head around Charlie’s model. I believe many students had difficulty understanding why $72 was not correct as well as had trouble understanding why it should be $8 in all the boxes and why the total was $32.

Closure

            Ms. Sherman led students in the idea of putting $8 in all the boxes and showed the students that if they have 4 equal groups of $8 (4x8), they have $32 in total. The last task Ms. Sherman had her students participate in was to write approximately three sentences at the bottom of their paper about what they will remember about Charlie’s way (what they liked, didn’t like, something about his drawing, if it made sense to them, etc.). I think many students struggled to reflect on Charlie’s model and how he solved the problem.

Faculty Debriefing:

           

            Ms. Sherman was glad that the students were so eager to share their thoughts about multiplication and division. For multiplication, Ms. Sherman expressed to the other faculty members that it was hard to get the students to expand with the 2’s. Moreover, she expressed that the idea of equal groups didn’t come up naturally (she had to introduce and emphasize the idea of equal groups). Additionally, Ms. Sherman mentioned that the students said that multiplication is like addition and division is like subtraction. However, they couldn’t back this up (couldn’t explain this). Furthermore, Ms. Sherman decided to extend the warm-up longer so students could try to explore the idea of groups when dividing. However, she still felt she was almost forcing this idea upon them. When students were doing their drawings of the problem during the lesson, Ms. Sherman was happy that the students were engaged and thinking. She also appreciated that even though the questions weren’t presented at first, the students were starting to think of answers to the questions. Ms. Sherman was well aware that many students had the misconception of 24 x3, and when she put up the Charlie model students still struggled with the misconception of 24 x 3. She said that half of the students incorrectly wrote 24x3 to find out that Wayne had $72 (which was three times as much). This proves that despite her explanations of grouping and her pictures she displayed, many students still struggled to grasp the concept.

            Another faculty member shared with the group that he saw that a student had $8 for his answer but when everyone else at his group said that they had $24 for their answer, the boy switched his answer to $24 and the boy was unable to explain why he switched his answer. The idea of copying other students’ answers presents the important question, “How many children understood it and how many had copied the correct answer?” Other faculty members noticed that students struggled to explain their answers and/or did not understand what number was correct to put in the boxes.

            I think the faculty members realized the importance of getting students to understand multiplication and division and the connections to equal groups rather than just understanding math facts (students need to understand the “why”). As opposed to simply throwing out math problems/facts, students need to first build their algebraic and mathematical vocabulary to help them gain an understanding and deeper confidence level for the math concept they are working with.

Overall Thoughts:

            Watching these video clips was a great learning experience and provided me with some positive and negative thoughts. First, I loved that the students were so engaged in the lesson and that lots of questions were being asked to stimulate student thinking. Additionally, I liked how she asked one of the students to come up to the whiteboard and draw her answer. I also liked how students thought independently then shared with a partner and then shared with the whole-class. Furthermore, I enjoyed how Ms. Sherman used the bell to grab student attention, and also how she would say something like, “Show me you’re ready for mental math by putting your hands on your head.” I think these are great attention grabbers that cue the students in and regain their focus. I also enjoyed how Ms. Sherman took the time to write what the students said on the anchor charts (she didn’t just discuss their answers, but she also wrote them down). This provided students a nice visual as to what their peers were saying. While there were aspects I appreciated about the video clips, there are various negative aspects I would like to point out. First, while she guided and engaged the students by asking a lot of questions, I think she would ask too many questions at once. Many times Ms. Sherman would ask a question and then right away ask another question right after the one she just asked. I think this confused students and didn’t allow them enough process time. I think it would have been more beneficial if she asked one question at a time as well as provided enough wait time before saying anything else. Additionally, I liked the quote she displayed on the board (“A picture is worth a thousand words”). However, she never came back to this quote at the end. I think it would have been more meaningful if the students came back to this quote and discussed how this quote relates to multiplication and division. Similarly, Ms. Sherman often said throughout the lesson, “We’ll come back to this idea/or this picture later,” but she never did go back to it. Often times she would skip around, and it was frustrating because it was hard to follow her. If I were her student, I would have been overwhelmed and probably would have lost focus. Speaking of focus, I noticed a lot of students started losing focus towards the end of the lesson. I have a feeling that many students were so lost near the end that they just stopped paying attention. Sometimes I felt that Ms. Sherman was trying to force too many ideas upon her students at once, which caused for too much confusion, stress, and frustration. Maybe Ms. Sherman needed to split the days (one day for multiplication and another for division). I also think she needed to spend more time on developing vocabulary to lay a better foundation for this lesson. While I think it is beyond meaningful to let the students explore for themselves, the students also need to have the right amount of preparation for this exploration. Ms. Sherman had to clear up so much confusion in this lesson and asked so many guiding questions that the open exploration may have hurt more than helped. Once again, maybe if she worked on multiplication and division and built some vocabulary knowledge before today’s lesson, students may have been more successfully prepared to explore.

            I enjoyed watching these video clips, and not only did I learn some good tips to implement into my future classroom, but I also learned some strategies NOT to implement. I learned that it’s really important to think about your classroom, if your classroom is prepared for your teaching material, how to adapt your lessons to best meet the needs of your learners, and what tools/strategies you can use to help your learners to succeed the most. Hopefully, Ms. Sherman will go back and continue discussing multiplication and division with her students and build upon the vocabulary, because while they have a good introduction to these concepts, they could definitely use some more practice!