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?

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!

Week 4: Reflection (Review / Cross Class Math Groups)

This week, I was in the math classroom on Tuesday, Thursday, and Friday. On Wednesday, the students took their first math test of the year.
On Tuesday, we reviewed. It was a bit chaotic, but I took some groups of students to review rounding, addition, and subtraction – it ended up being three groups or maybe four that I worked with. I gave students a few exercises and then asked if they wanted to challenge each other, so they wrote numbers to round or add. Each had a turn and seemed to enjoy the power / responsibility.
On Thursday, I led the hands-on game during centers. Many students had struggled with translating between standard, expanded, and verbal forms of numbers. The teacher, made a sorting game for them with three forms for each number. I found that the game worked best if three or four students laid out all of the cards face up and just went at it – trying to find as many matches as they could. They were pretty good about sharing and helping each other as well as being competitive. Then after a few rounds, they started dragging, so I asked if they wanted to make it into a “Memory” game and that upped the challenge a bit but seemed still fun for them. (Starting out in “memory” mode did not work well!)
On Friday, the fourth grade classes did a crossclass remediation. 22 students from two other classes came to my teacher’s classroom, and all of her students left for other classrooms. We worked on place value. I led a dice game where partners or triples would roll three ten-sided dice twice and try to write a number that was closest to a goal. The goals were written on their “score sheets” as: “closest to 0,” “closest to 1,000,000,” “closest to 500,000,” “closest to 250,000, and “closest to 750,000.” The game sort of worked but because I had 11 students at a time and because I don’t know them, it was fairly chaotic. (see more description of this game: Task: Place Value Dice Game)
Looking back, I should have shared clear expectations by asking students: “What should I hear? What should I see?” The teacher has a poster about “hands-on expectations” so I could have gone over that. But, the first group was way too wild. I tried to compensate with the second group by telling them I would take the dice away if they were throwing or playing with them. That led to about five minutes of answering questions like “What if I shake the dice and one falls out?” “what if I roll the dice and one rolls too far and falls on the floor?” Poor kids!

This task (game) was given to me by my teacher: I did not plan it, but only implemented it.

The goal of the task was for students to think about place value and “periods” (in the number 365, 728, “365” is a period in the thousands place and “728” is a period in the ones place. Then the students can read each period the same, but appending the appropriate period name: “Three hundred sixty-five thousand, seven hundred twenty-eight (ones).”)

Students were given three ten-sided dice and a worksheet for record-keeping.

Students would roll the set of three dice twice, writing down the three-digit number using the place value on each die.

Then they would decide which period would make more sense to be first, depending on the goal from the worksheet.

For example, if they rolled 365 with one roll and then 728 with a second roll, they could choose to have the number “365,728” or “728,365”.

The worksheet had about 12 “find a number closest to…” goals available:

• 3 goals were: “Closest to zero”
• 3 goals were: “Closest to 1,000,000”
• 2 goals were: “Closest to 750,000”
• 2 goals were: “Closest to 500,000”
• 2 goals were: “Closest to 250,000”

Students were in pairs and whoever was closest to a number “won” that round and got a point.

We started out on the carpet at the front of the room, but some students in the first group asked to move to desks to make it easier to roll the dice and record responses. I write more about the struggle I faced for my own communication of expectations in Week 4: Reflection (Review / Cross Class Math Groups).

In each group, some pairs of students were very focused on rolling the dice and finding the closest number. But at least one pair in each group struggled to stay focused – either one student would not take turns with the dice or would roll them wildly multiple times. I wondered if one solution would be that each student had their own set of dice and they could work together or individually. I also wondered if it would have been different if I had known the students and been able to shuffle partnerships to be more balanced.

In the moment, I decided that I had been unclear with my expectations with the first group, assuming that they would know what I expected regarding use of resources. For the second group, then, I briefly told them my expectations, ending with “if the dice don’t stay on the desks, I will take them away and you will have to complete a different activity.” At that point, hands went up and students asked about boundary cases: e.g., “if I roll the dice, and one falls out accidentally…”

For the partners that were on task, the game seemed fairly straightforward and even easy. I don’t remember if there were any “Choose your own goal” slots on the sheet, but I think it would have been interesting for students if they could choose their own goal and try to achieve it.

Task: Math is Everywhere… in Logan

I adapted this task based on pages 120-121 in Van de Walle.1
Math is everywhere. People use it all the time in the world and at work and at home. Part of math is just being curious and asking questions and looking for patterns that help you make sense of the world. Yesterday, while you were at school, I took some pictures of different places where I think people do math.

1. Which of these do you recognize?
1. Willow Park Playground
2. Willow Park Wildlife Center
3. Logan Aquatic Center
4. Logan Library – Books
5. Logan Library – Programs
2. Which have you visited?
3. Can you think of any other places where you do math that isn’t your house and isn’t school?
4. What are some other places where you think you might do math?

Decide as a group: Which picture / place do you want to use for math today?

• What math could you do here – I mean, what math questions could you ask?
• I asked myself some math questions and chose one. Before you try to answer it, I want you to Analyze the question and Make Sense of it, so I’m going to ask you some questions.
• What does analyze mean?
• What is happening in the question?
• What will the answer tell us?
• Why is it interesting to know? Or who could the answer help?
• Do you think the answer will be a big number or a small number?
• Can you estimate the answer?

Willow Park Playground:

When you plan a playground, what do you have to think about?
The Sway Fun glider costs \$34,628 and a Swingset costs \$3,298. Which costs more? How much more? What does that tell the planning group?

Willow Park Wildlife Center

The Wildlife Learning Center needs to know how many kids will go to Story Time each month.
In 2008, there were 943 kids that visited Story Time the whole year. In 2013, there were 497 kids that visited. Which year did more kids visit? How many more? What does that tell the Center?

Logan Aquatic Center

The Aquatic Center needs to know if it’s using too much water.

There’s a park in Draper, Utah called Cowabunga Bay that uses 250,000 gallons. Logan Aquatic Center uses 800,000 gallons. Seven Peak Water Park uses 34,746,096 gallons2 . Which uses the most? How much more? What does that tell the Aquatic Center?

Logan Library – Books

The Logan Library tries to get more books every year so more children can use them.

In 2011, kids checked out 633,330 children’s books. In 2015, kids checked out 434,815 children’s books. Which year did kids check out more books? How many more? What does that tell the library?

Logan Library – Programs

The Logan Library has children’s programs and want more kids to use them. In 2011, there were 2015 kids that came to programs. And in 2015, there were 11,171. Which year did more kids come? How many more? What does that tell the library?

My first goal was to follow the “Problem Analysis” from Van de Walle on pp. 120-121. The idea is to support students in making sense of the problem and the answer – so not really very different than what you showed them which was awesome! So I wanted to go through:
• What is happening in the question?
• What will the answer tell us?
• Why is it interesting to know? Or who could the answer help?
• Do you think the answer will be a big number or a small number?
• Can you estimate the answer?
My second goal (that I hoped would support the first!) was to support the “asking math questions” idea by bringing in the pictures from around Logan – mostly Willow Park and Logan Library. I had others but I talked myself down from preparing for so many problems and thought so many pictures would really be distracting. Part of the goal here is to support students use of their “funds of knowledge” which includes what they know from school and what they know from home and the other things that they do because research has shown that students compartmentalize and so don’t bring in their problem-solving skills, sense-making, or intuition, etc. that they might use outside of school.
This is what really happened…
The reality – I thought I could have kept them on track and moved faster, but I was pleased to here all of their ideas and enthusiasm.  What I actually did was:
1. Math is everywhere. The goal today is to think about math outside of the class and outside of your homework. Part of math is just being curious and asking math questions about things you notice or wonder about. Where are some place you think you could ask math questions?
1. I made sure each person told me at least one, although some repeated. Here are some of their responses: stores, library, video games, gummy bear factory (!?!), cookie factory, gas station, the world (how many people are there? how many gallons of water are in the ocean?), Fredrico’s Pizza, Tony’s Pizzeria, Macey’s, Tony’s Grove, LegoLand (how many Legos are in a big creation?), Wal-Mart, Bear Lake, Salt Lake, a road trip, Oregon.
2. Yesterday, while you were at school, I tried to guess where you fourth-graders might ask math questions when you aren’t in school. So I took pictures that I’m going to show you and I wonder if you have ever seen these places or been to them? (I passed out the pictures and they had no problem recognizing each – and pointed out that they had a field trip last year at the Wildlife Learning Center by Willow Park Zoo.  I think everyone had been to every place, except I think Isaac said he’s from Smithfield so doesn’t go the Logan Library.
3. As a group, choose which place we will ask our math questions about today. (They were reasonably quick to do this – every group wanted to talk about the Aquatic Center and the Logan Library – but only in only one group did Library win. I told them we would do the library if we had time.)
4. What are some math questions that you wonder about?
1. Logan Aquatic Center: How many people go there? (on Wednesday, in a week, in the summer, in the winter, between 2:30 and 3:30); How many people go on Wednesday compared to Friday? or go there vs. stay home? How many workers are there? How many lifeguards? How many lifeguards are there each week? How many people have been hurt on the slides? How many people have broken a leg on the slides? How many people have swallowed a gallon of water? How long is a slide? How long is a curly slide compared to a straight slide? How deep is the deepest pool? How tall is the tallest slide? How much water is in one pool? How many gallons are in a pool? How much does visiting cost? How much would lunch cost?
2. Logan Library: What is the number of books there? How many at each grade level? How many magazines are there? How many words on a page of a book? How many videos on a YouTube page? (or a YouTube search?) What street is it on? How many people are there?
5. Estimation:
1. Logan Aquatic Center: The question I chose to ask was: How many gallons of water are in all of the pools and slides? So, I found out how many gallons. What do you think the answer is? (I passed out markers and erasers here, and had them write down what they thought. That was interesting – I should have been prepared to write those answers down, but I didn’t. The answer according to the LAC website is 800,000 gallons. Maybe 4 students guessed at the magnitude – 123,000 or 250,000.  Several students guessed a two-digit number: 32, or between 20-30 gallons.  Most students assumed it was a huge number and so wrote something like 999,999,999,999,999,999 or 2,000,000,000,000,000,000,000… They all seemed surprised by 800,000.)
2. Logan Library: The question I chose to ask was: How many children’s books are checked out in a year? So, I found out how many books. What do you think the answer is? (Two students wrote numbers in the hundred thousand range – these really varied. The answer was 434,815 books in 2015. Time was up or I would have compared the books in 2011).
6. Subtraction question:
1. Logan Aquatic Center: I wanted to know how Logan Aquatic Center compared to other water parks. So I found two others: Cowabunga Bay (in Draper) and Seven Peaks (in Provo). Have any of you gone to those? (Some had – if they had, I made them guess if they thought it had more water or less than LAC.)  Cowabunga Bay uses 250,000 gallons and Seven Peaks uses 34,746,096 gallons2 . Write those numbers down. (Okay – that was my fault! Most struggled to write the 34 million down – but a few were okay. I know that’s bigger than they’ve practiced. By the time we had any two numbers down, time was up! That was okay!)
So – we didn’t really analyze any because of the other parts, but I felt like it was still productive even though we never actually subtracted. I really liked that they chose a place – but I think that took more time than made it useful. I think if I did this again, I would just bring one picture or I would have partners choose one and they would work on different problems as partners.

1  Pages 1120-121: Van de Walle, J. A., Karp, K. S., Lovin, L. A. H., & Bay-Williams, J. M. (2014). Teaching Student-centered Mathematics: Developmentally Appropriate Instruction for Grades 3-5 (Vol. 2). Pearson Higher Ed.

2  Note: I later realized that the difference in the amount of water is due to the difference in online resources that I used. For Seven Peaks, I found a document describing water usage in Salt Lake City – the 34 million number is probably per year, not its capacity!