Background
Information
The student I interviewed is
currently a 10th 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.
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