# Task: Concrete to Abstract Multiplicative Comparison

While the fourth-grade students were still enjoying their lunch recess, the teacher, the student teacher, and I chatted about what the students could do during math that day.

Struggles with Models. The teacher mentioned that students still seemed to struggle when they drew a model (see examples images below: yellow, green, and red).

Suppose the context of a situation mentioned one person had 3 Pokémon cards (because those have proven to be immensely popular as example!) and a second person had four times as many cards. Then one student might draw the yellow model which represents the person with 3 cards in the first row, and the person with four times as many cards in the second row. Together, they have 15 cards.

Another student might draw a similar model to the green, but the box for the first person is as long as the four boxes of the second person combined. By itself, that’s fine. But it might lead to confusion about who has more, or confusion later about fractions, etc.

Finally, many students would draw something similar to the red model, with three groups of five cards each instead of five groups of three cards each, which represents one person with twice as many cards as the other. This model is especially problematic when the students are asked how many more cards the second person has than the first.

Struggles with Moving Between Additive and Multiplicative Comparison. The teacher also mentioned that students seemed to struggle to answer an additive comparison question after thinking about multiplicative comparison. (For example, in the question above, the second person has 9 more cards than the first person, but four times as many.)

We brainstormed about tasks the students could do that might help them draw the models correctly, but – most importantly – understand and use the models to make sense of a situation. We decided to return to a more concrete representation of the model, so we created two tasks: one for the Teacher Time center and the other for the Hands-On center. The student teacher and I agreed to split the students at Hands-On and each take half of a group.

In the introduction to the math lesson, and during the Teacher Time Center, the teacher had students represent a situation using foam cubes and small baskets.

For example, using the example above, students would put three foam cubes in one basket to represent the 3 Pokémon cards for the first person. Then they would put 3 foam cubes in each of four other baskets to represent the second person with 4 times as many Pokémon cards.

The teacher then drew or had them draw the diagram modeling the context of the situation. She asked them: “How many Pokémon cards do they have all together? How many more Pokémon cards does the second person have? etc.”

She then would ask a question starting with the whole amount: “If two people have 30 Pokémon cards together, and one person has 5 times as many as the second, then how many cards does the second person have? How many more cards does the first person have?”

We hoped to bridge from the very concrete “foam cubes in baskets” activity in the introduction of the lesson, to be slightly more abstract by using foam number cubes. (Although, as it turned out, it may have been better to use cubes with “pips” instead of “digits” because students did not seem to really connect the objects in a basket with numbers on a die.)

The student teacher and I each took three students per group in the Hands-On Center. Each fourth-grader had a set of six foam dice (with numerals). I had one foam die (with numerals). I asked students to think of an object they collected, and then I used that for the context. For example, in the first group of three, one student wanted ice cubes, another wanted robots, and the third wanted puppies. So, we compromised with “Icy, robot puppies” (as suggested by one of the students!). My question then was: “If I have 3 icy, robot puppies, and each of you have 2 times as many as me. How many more icy, robot puppies do you have than me?” I asked them to show me using their dice, and then to draw the model or show the model on the markerboard. I then asked them a few other questions comparing the amounts. I gave each of the students a turn to choose a context and then make up numbers that the rest of us could model.

# Week 6: Reflection

I was able to be in the class four days this week: every day but Wednesday (because they had a field trip that afternoon).

Much of what students worked on this week continued to be review of multiplicative comparison, and on Friday they moved on to “multiplying by 10s, 100s, and 1000s.

On Tuesday, I led the Teacher Time center, based on a multiplicative comparison worksheet. My teacher mentioned she would like it if we had a chance to use “n” in place of the unknown number. The first two questions on the worksheet gave students a multiplicative comparison situation, that ended with students doing an additive comparison. This shift seemed to be difficult for many students! It was interesting to see that many students “got” that the first two questions were multiplication – but most also struggled to flexibly move between multiplicative comparison and additive comparison.

The first question asked something along the lines of: A fire dragon burned 8 times as many huts as a fire beetle. The fire beetle only burned 3 huts. How many more huts did the fire dragon burn than the fire beetle?

First Thought. A few struggled with understanding that the first question started with 9 groups of 3. These students struggled to set the model up correctly and so I tried to give them opportunities to use tiles as physical manipulatives to understand the situation. Some students interchanged 9 groups of 3 with 3 groups of 9, which makes sense because multiplication is commutative. However, in the situation, (1 group of 3 + 8 groups of 3) will result in a different final answer (24 huts – 3 huts = 21 huts) than (1 group of 9 + 2 groups of 9) which results in (18 huts – 9 huts = 9 huts).
Second Thought. Most students found their answer by multiplying 9×3 = 27 huts. Only a few students really noticed or understood that 27 huts was not the answer, but that, instead, “how many more huts did the dragon burn than the beetle” meant to subtract. It seemed that shifting from multiplicative comparison to additive comparison was difficult. I tried to support the students by focusing their attention on the final question and asking them what they thought it meant, but many students seemed to really be mentally blocked from using subtraction.
Third Thought. I tried to work in the “n” with the first two groups, but I think even in a small group there is a wide range of understanding of what the tiles meant or what the squares in the model meant, so moving to calling the number “n” seemed to be just one other thing at the moment. Only one student among the twelve in those first two groups really seemed to respond to that.
Fourth Thought. I will keep working to communicate manipulative expectations. The groups really varied in their reactions to the availability of physical tiles. Some students immediately began grabbing tiles without using them as mathematical tools, but instead began creating patterns with them. I tried to communicate clearly, but again this communication is something I will have to continue working on.

#### Multiplication (Equal Groups):

• V has some pockets.  He puts a few pennies in each pocket.  How many pennies does V have?
• Alternate numbers:    Easier: (2, 3); Hard: (9, 4); Harder: (15, 8); Hardest: (12, 28; 32, 44)
• Questions to Ask: What is happening in the story? How many pockets do you have? What information you need to find an answer?

#### Comparison (Product Unknown):

• L picked up some rocks. S picked up a few times as many rocks as L. How many rocks did L pick up?
• Alternate numbers:    Easier: (2, 4) or (6, 2) or (3, 4); Hard: (9, 6); Harder: (13, 7 OR 14, 8); Hardest: (23, 24)
• Questions to Ask: What is happening in the story? Do you collect rocks? What are things that you collect? What information you need to find an answer?

#### Comparison (Group Size Unknown – Partition Division):

• N has some fish. N has a few times as many as A. How many fish does A have?
• Alternate numbers:    Easier: (10, 2) or (6, 3); Hard: (25, 5) or (42, 7); Harder: (110, 11) or (132, 12); Hardest: (304, 16)
• Questions to Ask: What is happening in the story? Do you have fish or other pets or stuffed animals? What information you need to find an answer?

#### Comparison (Number of Groups Unknown — Measurement Division):

• P has a few pieces of chocolate. V has even fewer pieces of chocolate. How many times as many pieces of chocolate did P have compared to V?
• Alternate numbers:    Easier: (10, 2) or (6, 3); Hard: (25, 5) or (42, 7); Harder: (110, 11) or (132, 12); Hardest: (304, 16)
• Questions to Ask: What is happening in the story? Do you like chocolate or other candies? What are candies that you like? What information you need to find an answer?

## Reflection on Task and Implementation

The implementation was good, but a little rushed! I’m still learning how to keep them on track. I’ll figure this out eventually. It seemed that communicating my expectations about how they used the resources still needs work.
I felt like it was going better by the last group – starting out with brief expectations and getting them focused. I worked with four groups of three students each. The second-to-last asked if they could have one minute of free play before we began and that did seem to help somewhat, but it was difficult to keep them on task, so I didn’t do that again with the last group.
Well – I thought I had left out numbers, but forgot that “few” is fairly well defined as “3 or 4” and students in each group pointed that out. So I will know better next time! But that means they are good at looking for the key words.
For the first question, I asked students to think about the story and to count their pockets to see what a good number would be for V’s pockets, and then I told them some number of pennies. Students in every group automatically multiplied, so I asked them how they knew it was multiplication and could they show me using the squares. One student literally put squares in each pocket and then counted them up, others grouped the squares flat on the desk or in stacks or other designs.
I learned that the students overall had a pretty good sense of multiplication and comparison. One student, and a few others, mentioned on the third question that it felt tricky – it felt like A should have more fish than N, even though they said they could see N had the most fish.
On the other hand, some students did struggle with using blocks to represent the story, hesitating to figure out why they could use squares in groups to find the answer. I tried in the third or fourth question to encourage them to write numbers on the squares instead of counting out every square that they needed, because their teacher had shown them they could do that in the intro to the lesson. It didn’t seem that any of the students felt comfortable with that – although one student drew squares on her desk with the number in the square to represent each group and so her group members also tried that.  I’m not entirely sure if the other students’ discomfort was about writing on the squares or whether they just liked to use the squares or if they really were struggling to see the representation. So, that was interesting!
I also learned that the students really responded when I left the questions open – for each question, I asked which numbers made sense to use and thought they might always choose simple numbers. I had them take turns, and some were simple – like 6 times 6 but others were a little harder like 11 times 4, 24 times 6, or 500 times 5.  One student suggested 11 times 4 – he and one of the other team members knew it was 44 right away, but the third member didn’t. But she did really well to find a strategy – she told me she could just make it 4 times 10 instead and then add 4 to get 44.
They also really liked having their names in the problems and they enjoyed suggesting other things they might pick up or own.
I feel that I probably did not get exactly what I should have from the activity – mainly because of the lack of time. Also, I think the students are not yet completely comfortable with me in the classroom.

# Week 5: Reflection

Because of my schedule, a reassessment (of ideas from the first exam), and a Professional Day at the school on Friday, I only was able to go to school on Wednesday this week.

On Wednesday, I led the Task: Multiplicative Comparison that I had prepared. I worked with small groups of 3, rather than full groups of 5 or 6. Unfortunately, I saw them during the Hands-On center rather than during Teacher Time (which is normally when I will do a task). Hands-On center time is usually when students choose (or are given) a math game to play with each other, so the students I chose to work with were anxious to finish and play a game. Paradoxically, that made it difficult to focus on the questions and we took 15 minutes each time. In the future for a task like this one, I would try to make the task more like a game, only work with one or two students, or request to work with only two groups of two students each across the whole centers time.

Something that I noticed, about which I can sympathize with undergraduate student teachers, is that I see how busy the class always is and how my teacher has organized it to run smoothly and I don’t want to disrupt that. I could have asked for the extra time with one group of students (as I wrote above) – I was aware I would need it – but I chickened out because I wanted the task to support the teacher and students in their current context. It was difficult to ask even for half of a group rather than a whole group.

Another, somewhat unrelated observation, is that some students have asked me to call them by nicknames (e.g., “Peanut Butter Sandwich,” “Pretty, pretty princess,” “J.J.,” and “Renaldo” or “The Great Renaldo”). Partly, they suggested the nicknames because I was learning their names, and partly because I think they feel special. I think it’s fun, but I have noticed some students ask me, “Why do they have nicknames?” I try to answer: “They asked me if I would use that name.” But I can see that it privileges some students over others – even though each student could ask, there are only some students that will ask. Some students would like to have a nickname, but will not ask for it. Some students would like to have a nickname, but will not ask for it. Other students don’t care one way or the other. I’m not sure how to make that “fair” other than to let them know I’m happy to call them nicknames if they’d like one!