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Hypothetical Learning Trajectory

4.3. Hypothetical Learning Trajectory II

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are very closely related and we though the candies will attract students’ attention.

Moreover, the candies are also very hands-on that students can touch, move, arrange and rearrange. We made some adjustments based on the result of the preliminary experiment. We will deliberately use only 20 candies with 2 different flavors to minimize the possibility of students using other structures than group of 10 or group of 5.

Activity 2: Group Presentation (refer to activity 2 HLT I) Activity 3: Flash card games (refer to activity 4 HLT I)

Activity 4: Double structure through singing (refer to activity 3 HLT I) Activity 5: Double structure using candy packing

At this moment, students should have been able to pick up some double sums from the double song. In this activity, our objective is to relate those double sums with structured visualization. To maintain the consistency, we will use the group of 10 candy packing.

Students will work on a worksheet in which they will find an empty candy packing. They will be asked to fill out some candies in the packing by coloring the worksheet. Students strategies of structuring the candies can be seen by the coloring pattern they made. We conjecture the song will influence students in coloring the worksheet that they would use double structure.

After coloring activity, students will have another worksheet. In this worksheet, they have to determine the number of candies given. Students will also be asked to explain their counting strategy. Here, we expect students used double strategy in their counting.

Activity 6: Number relation through “The sum I know” worksheet

From the previous activities we hope students have developed an understanding of double structure. In this activity, our main goal is to construct an addition framework that use students’ current knowledge of addition as a point of reference to explore more additions close to it. By this moment, students should have known some double sums; therefore we will use them as the points of reference. For example, if students know that 8 + 8 = 16, they can use it to determine 8 + 9, that is 8 + 8 + 1 = 16 + 1 = 17.

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Students will have a worksheet, in which they will be asked to circle all the sums they know by heart. We expect students will circle some double sums which were sung in the song. These double sums will be the point of reference for students to do “almost double” additions such as 6 + 7, 7 + 8, etc. Students will be given 5 minutes to circle as many as additions they know. After that, the teacher will ask them to stop, and write the result of the addition that they had circled. By writing the result, we hope student will discover a pattern of addition framework, that each time they move one step to the right; the addition gets bigger by one.

We anticipate that not all students could come up with the framework of addition, thus we will hold a class room discussion. We provide a large “The sum I know” worksheet and put it on the white board in front of the class. Some students will be asked to circle the worksheet, and the teacher will use the circled additions to start a discussion. Teacher would pose a question such as “You have circled 6 + 6, and who can circle 6 + 7?, how did you know that?”. We conjecture that some students will use the addition framework, 6 + 7 = 6 + 6 + 1 = 13. In this discussion, students will share their strategy and we hope more students would come to an understanding of using framework of addition to solve “almost double” problems.

Activity 7: Friends of 10 through egg box structures

In this activity we focus on the friends of 10 strategy. We will use egg box structures to develop students’ understanding of the friends of ten. First students will be asked to tell how many eggs are in the box and give a reasoning of their counting strategy. We conjecture students will use addition or subtraction to shorten the counting. For example, “there are 8 eggs, because 5 eggs are in the first row, and 3 eggs in the second row” or “there are 8 eggs because 2 eggs are missing”. In the first statement, students used groups of 5 while in the second, they used groups of 10.

Next, students will work on a worksheet in which they have to fill out the number of black dots (representing eggs) and the number of white dots (representing missing eggs). Each pair of blacks and whites will make 10 when added together. The teacher will drive a small discussion so that students will be able to conceive this idea.

From this worksheet, we hope students will come to an understanding that the number of black and white dots together makes 10.

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Activity 8: Friends of ten through finding friend games

By this moment, we conjecture that students should have been able to find the friends of 10, for example 6 and 4, 7 and 3, etc. Our goal in this activity is to strengthen students understanding of the friends of 10. We design a game where students have a number and they must find a friend so that together they would make 10. We conjecture that students would not have any difficulties doing this game.

Moreover, we predict some students will find their friends by doing counting on or some students might have known the friends of 10 by heart.

Students will not only find the friends of 10, but also other numbers less than 20. For example, when the teacher writes 17 on the white board, each student have to find one or two friends to make 17. During this game, students will be constantly doing additions; we hope this could be a good practice for them.

Activity 9: Addition up to 20 using math rack (refer to activity 9 HLT I)

4.4. Progressive Design of HLT I and HLT II: a Summary

To sum up, we describe the progressive design of HLT I and HLT II in the following table:

HLT I

Canceled

Canceled

Canceled

HLT II

Activity 1: Candy packing Activity 1: Candy packing

Adjusted by only use 20 candies

Activity 2: Group Presentation Activity 2: Group Presentation

Activity 3: Double song Activity 3: Flash card games

Activity 4: Flash card games Activity 4: Double song

Activity 5: Finger structures Activity 5: double structure in a

candy box

Improved with a worksheet Activity 6: The Sum I know

worksheet

Activity 6: The Sum I know worksheet

Adjusted by only giving 5 minutes for students to work on the worksheet

Activity 7: Friends of 10 in a candy box

Activity 7: Friends of 10 in an egg box

Activity 8: Throwing disc game Activity 8 : Friends of 10

through finding friend games Activity 9: Addition up to 20 by

using a math rack

Activity 9: Addition up to 20 by using a math rack

Table 4.1: the progressive design of HLT I and HLT II

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Chapter 5