Algebra Case Studies

  1. In case 2, lines 166 to 179, the teacher explains the approaches of Stanley, Burt, Jeff, and Barry. Analyze each of these three approaches. What understanding about mathematics might each approach indicate?

    1. Through examples, Stanly has shown that he understands that the number of cues in each tower in the sequence was three more than the one before it, he also moved quickly from the visual representation and wrote multiples of three from 1. Ex. Stanley made the first two tower then wrote “7, 10, 13, 16, 19, 22, 25”.
    2. Stanly understand that you can create sequences by adding the same number to get a new number, although Burt writes out number sentences in order to make sense of the number sequence while creating the towers. Burt understands he does not have to start from one each time, instead he can add three to the last tower height.
    3. From Barry and Jeff’s work, you can see that although they have an understanding of the number line which represents the tower sequence, they are more comfortable and prefer building each town out to discover the height. These boys are not yet at the level of comprehension where they can use multiplication rather than a number line.

2.   What do you see in the thinking of Anna in Case 3? What experiences might have contributed to her understanding?

  1. In case 3 I see that Anna has an understanding on how a bar graph works and is comfortable representing data in this form. I believe this is because of the experience she has each morning when she walking in to her classroom, her teacher has the students begin on bar graphing questions to build their skill level. Due to this, she is able to think about a series or sequence of information as ‘towers’ or bars on a graph. She does this with each tower representing a different day or element of a sequence.

3.  Consider the teacher’s question in Case 4, line 330. What does that question highlight? Discuss student responses after that question was posed.

  1. In Case 4, I feel that the teacher’s question “All of them?” encourages the children to make a generalization on how to work with patterns at any length. When Babette explains that she starts at the first cube of the color, she then keeps adding four until she gets to where she needs to be. The teacher then asks her “why”? She responds by saying that “the pattern is 4 cubes long and just keeps repeating in the same way and if the pattern was 6 cubes long she would count by sixes instead”. The teacher then uses this example to test and see if the strategy works for all numbers.

4.  What is the impact of the question that Catherine poses in Case 1, line 41? What do students need to understand in order to determine what will be in a later position in a repeating pattern?

  1. n Case 1, line 41, in response to a students saying that to find what was in a certain position in a pattern he counted “cat, dog, cat, dog”, the teacher asks “What do you mean you counted?”  This questions causes the student to rethink what he said about counting “cat, dog, cat, dog” and the fact that this is an unconventional way to count. In order for children to understand what comes next in a repeating pattern, children need to understand that patterns are a specific length and repeat in the same way forever. It also helps for students to skip from counting out every individual item in a pattern to skip counting by the number of items in that pattern.

5. In Case 4, lines 295 to 300, explain Dora’s method of finding color for the 16th cube. What does her method indicate about her understanding? Explain Juanita’s method in lines 305 to 314. What does her method indicate about her understanding? How is Dora’s method the same or different from Juanita’s?

  1.  To find the color of the 16th cube Dora used the original four cubes that represent one repetition of the pattern. She then repeated counting through the patterns using the one repetition of cubes she had to represent a long strand of the pattern until 16 that she didn’t have. Dora’s method is evidence  that she understands patterns, along with understanding the first four cubes can be used over again since they don’t strictly represent the numbers 1, 2, 3, and 4. Although meanwhile Juanita uses a longer drawn out cube train in order to find out the 16th cube. She understands the blue cubes are every other multiple of 2. Juanita also understands you can isolate a segment in a pattern because it repeats the same forever. She also goes on to show that she is aware  you don’t have to show every individual block to find a color of a certain cube, but that you can start at the first cube and skip count.

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