Interview Report: Algebraic Thinking

Due to some technical issues, I was not able to videotape the interview.

Background Information

            The student I interviewed is currently a 10^th grader who is taking pre-calculus. Her name is Lisa and she is my cousin who lives in Oakland. From what I know about her, she has a really strong background in mathematics. Her pre-calculus teacher is Ms. Abernethy. I asked her the four questions on Page 37 on the Fostering Algebraic Thinking (FAT) book. For problem 3 & 4, I only went over a, b, c instead of all 5 parts with her because I thought that it was too redundant to ask her all the questions. However, the theory and activity remained the same to a certain extent. Due to some technical issues, I was not able to videotape the interview, but the interview lasted for about 30 minutes.

Algebraic Thinking

                According to Fostering Algebraic Thinking (FAT) by Driscoll (1999), “[people] characterize algebraic thinking as the ability to operate on an unknown quantity as if the quantity was known, in contrast to arithmetic reasoning which involves operations on known quantities (p. 1).  In other words, algebra thinking focused a lot about solving variables, but sometimes we cannot clearly define these variables. For example, solve for x, when the given function is x + 5 = 6. But, what does this really mean? This is when Algebra thinking steps in to help us think about the functions and how they work in our mind. To be more concise, no definition can fully define algebraic thinking because algebraic thinking can also be defined as generalizing through a given example, which is what we did for this interview report.

            If the student is building rules to represent the function, then this would be considered algebraic thinking because the student is forming their own conjectures to help him or her understand more about the problem. This is one of the key ideas in algebraic thinking, where students start to form patterns, make conjecture, and generalize the problem to help them comprehend the problem.

            The student used algebraic thinking when the student was discovering the patterns. The student told me that, “when you go in order, like from 1+2 to 2+3, the sum of the consecutive number jumps up by 2.” This showed me that the student had some sort of additive thinking in their mind when she was establishing the discoveries. Furthermore, this is a form of abstracting from computation because the student was generalizing using the relationships among addition.

            Furthermore, at the end of the interview, the student was also able to build rules to represent the given function. The student said, “see you can add 22+23 but by knowing the pattern of adding 3 consecutive numbers each time [it goes up by 3 each time], then you realize 45 can be made with 3 consecutive numbers too because 45 is a multiple of 3.” In the beginning of the interview, the student was not able to tell me about multiple of 3 for adding 3 consecutive numbers, but the student was able to develop a general rule for the problem at the end to help me confirm that the student has some sort of conceptual understanding in algebraic thinking.

Next Step

            As mentioned by Driscoll, students learn from three habits of mind: Doing-Undoing, building rules to represent functions, and abstracting from computation (p. 36). In order for students to do these thinking in their mind, the questions have to concise and clear, so that they can conceptualize the problem.

            A leading activity can be a group activity/investigation. Since we are currently learning consecutive numbers, I can ask the students to find the sum of the first 100 consecutive numbers without adding the 100 numbers by hand or calculator. I expect the students to make predictions with their group mates. Once they have some idea on how to do the problem, the students should then try to write a formula for this problem. The students are basically building rules to represent the given problem, and by doing so, the students are starting to see more patterns about consecutive numbers.

Reactions

            I learned that some of these questions can be really board. For example, Problem 2 stated, “For each number from 1 to 35, find all the ways to write it as a sum of two or more consecutive numbers? Explore and record three discoveries that can share with the class.” My student was just stating tons of patterns that may not be relevant to the problem. However, at the end of problem 4, the student was able to explain the pattern explicitly when the question was stated explicitly. The questions have to be clear in order for students to develop algebraic thinking with the given example.

            One thing that surprised me the most was that my student has a little trouble comprehending problem #1. I had to clarify it to her or ask her to reread the question in order for her to give me a good response. I found out that this interview was more abstract than the last interview we did. The proportional reasoning interview was straight forward, but in this interview, the student can form many conjectures or discoveries that are hard to use to measure their understanding of the content.

Works Cited

Mark Driscoll (1999). Fostering Algebraic Thinking: A Guide for Teachers Grades 6-10. Portsmouth, NH: Heinemann.

Memo 6

            The article mentions that the typical math classroom is really teacher-centered. Brenner (1994) claimed, “Teacher lectures occurred in slightly more than half of the class days” (p. 234). The teacher lectures over half of the period and the class does not have time for group discussions guided by the teacher. In fact, students do not have time to communicate with others in a math classroom. Furthermore, seat arrangement in a typical classroom is usually in rows, and this prevents students from working together in groups. Thus, this can impede the students learning from others because of the lack of communication throughout the class.

            English Learners (ELs) started making transition in to learning English when they were at a very really early age. Brenner (1994) advised, “In a number of schools this transition occurs first in mathematics in the mistaken belief that mathematics is a universal language or entails minimal language use” (p.234). This can be an issue for English learners. Schools assumed that the students already have enough background to comprehend mathematics in English. Furthermore, they believed that math is a universal language, which can be taught to anyone, but they have forgotten that the language in mathematics is in ENGLISH. For example, if a student is given a math problem with just a figure and it asks the students to “arrange the letter from least to greatest order,” but what does this mean to them when they can’t even comprehend the problem because of lack of understanding in the English language. Hence, this can be a big issue because schools believe that mathematics is a universal language that can be understood with minimal understanding of the English Language, but this is wrong.

            The reading suggests that teacher should act as an orchestrator or a facilitator when they are teaching. By doing so, the teacher can provide scaffolds to help the students while they are working in groups or individually. Furthermore, by providing assistance to the students, they are more likely to attain the upper limit of the zone of proximal development (ZPD). Brenner (1994 claimed, “Particular emphasis is given to ways in which the teacher can enhance students’ expression of their ideas orally, in writing and in the course of peer discussion” (p. 240). By acting as a facilitator, the teacher can help the students enhance their performance in presenting themselves orally. Not just orally, the students will also be able to form their own conjectures, examples, and counter examples when they have proper scaffolds from the teacher. Therefore, it is more likely that the students will gain intellectual growth/academic excellence in mathematics if they receive scaffolds from the teacher.

            In order to provide equity to the students in learning mathematics, the schools must give instructions in a way that is assessable for all students. Brenner (1994) mentioned, “Hence, the research on bilingual education indicates that LEP students are likely to be better off receiving instruction in their native language” (p. 256). The schools can teach mathematics in the students’ second language, so that it’s more accessible for them because they might have troubles understanding mathematics with English instructions. This means that they cannot understand the math if they the language is confusing for them. Therefore, it’s better off providing students mathematics instruction in their native language.

           

            The schools can also implement projects like the Algebra Project (Implemented by Moses). The project included all kinds of preparation for high school algebra. For example, the projects included parent involvements for helping teachers raise the students’ expectations. Furthermore, the curriculum was more emphasis on the students’ cultures and the Spanish Language. It also made it culturally relevant to the students, which interested the students to learn the contents. This can be another way to promote equity in a classroom because it provides activities that are culturally relevant to the students. Thus, we need to implement activities that are culturally relevant to the students.

MEMO 5, Presentation Week 6. Fostering Algebraic Thinking (FAT) Chapter 6 & 7

My powerpoint for my presentation. Fostering Algebraic Thinking (FAT) Chapter 6 & 7

https://docs.google.com/open?id=0B-sXWx2Ez9SaVUFGWG9fU01CUHc

Interview Report I: Proportional Reasoning

http://www.youtube.com/watch?v=R3CBuLv72kw&feature=youtu.be

Background Information

            The student I interviewed is currently a 9^th grader who is taking geometry. The student is from my focus class, so from my observation, I know that he has a good background in mathematics. His math teacher is Ms. Mendez, which is my cooperating teacher (CT). The child is 14 years old. In the interview, I was supposed to chain up the paper clips when I do the demonstration of using the big paper clips to measure Mr. Short. However, I decided to draw a hat on top of Mr. Short to make sure that 4 big paper clips will be the size of him without being chained because I did not want the student to waste time to chain the paper clips. The theory and activity remained the same to a certain extent.

Proportional Reasoning

                Proportional reasoning involved comparison of two or more extensive measures. For example, when Lamon (1993) talked about the associated sets, he talked about using pizza to teach the students about proportion reasoning. The pizza question asked, “The student is shown a picture of 7 girls with 3 pizzas and 3 boys with 1 pizza. Who gets more pizza, the girls or the boys?” (p. 5). These kinds of questions can help students think critically while figuring out an answer because it requires understanding of proportional reasoning. Lamon (1993) claimed, “Proportional reasoning was said to occur when a student could demonstrate understanding of the equivalence of appropriate scalar ratios and the invariance of the function ratio between to measures spaces, whether or not the student could represent these relationship symbolically.” When the pizza question was being asked to the girl, the girl provided a logistic explanation. However, this would not make sense if the student uses math symbols to symbolize her words. But the key point in proportional reasoning is that the students have to use clear explanation to show their understanding of the question instead of answering the question using procedural method.

            A student is using proportional reasoning when the student can provide accurate calculation to the problem. Furthermore, the student should also be able to explain what he or she is doing in the problem. By just computing the answer, the student shows that he has strong procedural skills, such as computing, seeing patterns, and etc. In addition, if the student is able to explain his or her answer with good reasoning and logic, then it shows that the student is having some sort of conceptual understanding in proportional reasoning. In proportional reasoning, conceptual understanding is far more important than procedural fluency because they have to be able to understand the concept in order to have good reasoning.

            Through the interview, I was not able to see that the student has much proportional reasoning because he simply did not use anything related to ratios. However, the student was at the pre-Proportional Reasoning Stage. Lowery (1974) claimed, “A student is at the pre-Proportional Reasoning Stage of development if his or her explanation refers to estimates, guesses, appearance, or extraneous factors without using the data of if he or she uses the data in an illogical way” (p.18). The student told me that the answer is “eight” because he simply added 2 small paper clips to find Mr. Tall’s height when Mr. Short is 6 small paper clips. The student is having his own guesses and reasoning to figure out the answer, but it shows no proficient level of proportional reasoning ability. Furthermore, this is an illogical way to use the data. However, the student does tend to use the most obvious solution to answer the question. Hence, the student used the additive property instead of using proportional reasoning to solve the problem.

Next Step 

            As the interview assignment mentioned, students usually do not master proportional reasoning until they are 16-18. My student was 14 years old, so it makes sense that he might be a little confused with problems like that. If I was teaching the student’s class, I believed that I would need to introduce the topic proportional reasoning starting from the beginning because it seemed like the student did not have much experience in proportional reasoning back in middle school. I would not use textbooks to teach the students proportional reasoning because it’s simply too procedural. Cramer et al. (1993) claimed, “Proportional reasoning abilities are more involved than textbooks would suggest” (p. 169). Textbook would not be able to teach students proportional reasoning ability, instead I would start off by providing hands-on activities to the class to get the students engaged, so they will be willing to learn proportional reasoning.

            My activity can simply consist of coins of dimes and quarters. Since most students use money in a daily basis, they would find this activity relates to them, so they are more likely to be engaged in this activity. They would also like to play with money, since everyone likes money. Using coins, the students will see the ratio 2:5, which means every 2 quarters is the same as 5 dimes in terms of values. By giving out this activity and providing proper scaffolds to the students, the students will have an early foundation of proportional reasoning.

Reactions

            I actually interviewed two students from the same classes at a different time, and I learned that they made the same mistakes. Both of my students gave me the answer 8 because they thought you are supposed to add 2 instead of using proportional reasoning. Surprisingly, they were also some of my top students in the class, so I thought they would be able to answer my question. This makes me wonder did they learn proportional reasoning back in middle school. Was the teacher in their middle school unqualified to teach proportional reasoning? Was this the reason why students did not master the topic proportional reasoning? Hence, I believe that it is necessary to reteach proportional reasoning in high school, so that students can become proficient in proportional reasoning.

Works Cited

Cramer, K., Post, T., Currier, S. (1993). Learning and teaching ratio and proportion: Research implications. In D. Owens (Ed.) Research ideas for the classroom: Middle grades mathematics. NY, NY: Macmillan Publishing Company.

Lamon (1993) Ratio and Proportion: Connecting content and children’s thinking. Journal for Research in Mathematics Education, Vol. 24(1), 41-61)

Lowery, L. (1974). Proportional Reasoning. In Learning About Learning Series. Berkeley, CA Univ. Of California, pages 17-20 and 37-38