I interviewed a 14 year old freshman
at Watsonville High School. She is a
student in my placement classroom studying Algebra 1 with Joy Winterlyn. She has been exposed to some Algebra in
middle school, but this is the first time she has taking Algebra 1. At this point, the class has covered solving
multistep and word problems involving linear equations and inequalities in one
variable. They have also gone over
graphing linear equations by looking at x and y-intercepts as well as sketch
the region defined by a linear inequality.
They are currently learning how to find solutions to systems of linear
equations. During this interview, I introduced
the problem to the student, but she was unfortunately unable to finish it in
the allotted 20 minutes. She was only
able to complete most of part (a) and (b).
I was merely able to find out what she could notice about sums of
consecutive numbers.
Do you notice anything? When I try to put them together, I can’t always get 35.
Do you notice anything else?
When I add them, I have to skip a
number.
Are you noticing anything about the sums?
Most of them are odd. I can’t find any that add up to 1 or 2. And you can’t find 2 consecutive numbers that add up at an even number without a tens place.
(continues writing out sums)
Most of them are odd. I can’t find any that add up to 1 or 2. And you can’t find 2 consecutive numbers that add up at an even number without a tens place.
(continues writing out sums)
Okay, now, I want you to write down three
things you noticed while you were doing this.
They can be something you have already said, or something new that you
noticed as well.
(writes it down).
1. Algebraic thinking:
a) Define algebraic thinking.
There is no simple way to define Algebraic thinking. This is because Algebraic thinking involves a
being able to think in a wide range of areas.
Driscoll (1999) defines algebraic thinking as “the capacity to represent
quantitative situations so that relations among variables become apparent” or
“how problem solvers model problems” (pg. 1).
Furthermore, Driscoll emphasizes three aspects of thinking:
Doing-Undoing, Building rules to represent Functions, and Abstracting from
Computation. The latter two refer to
generalizing, a key idea in Algebraic thinking.
Doing-Undoing refers to the idea the Algebraic thinking generally
involves reversibility, in other words, being able to “undo mathematical
processes as well as do them”. For
example, this might include solving for a variable x in the equation x + 2 = 5.
One could subtract 2 on both sides for find the value of x = 3. On the other hand, we
can also substitute 3 into x for the original equation. Building rules to represent functions means
“the capacity to recognize patterns and organize data to represent situations
in which input is related to output by well-defined functional rules” (pg
2). A case of this can be represented in
the given problem: Take an input number, divide it by 2 and add 5. Lastly, abstracting from computation is “the
capacity to think about computations independently of particular numbers that
are used” (pg. 2). This might mean being
able to compute the sum, 1 + 2 + 3 +…+100,
and noticing that pairs of numbers add up to 101 (i.e., 1+100, 2+99,
3+98..etc). In general, Algebraic
thinking can be described as being able to generalize or look past particular
examples within a given mathematical situation and draw conclusions.
b) Describe what you would consider evidence that a student is using
algebraic thinking.
Within the context of algebra,
evidence of this kind of thinking can be shown through student explanations and
their use of generalization. Driscoll
(1999 Pg. 87-87) lists some examples of students showing Algebraic
thinking. Students might generate some
cases for a function and compare each result to look for a pattern. They might be able to verbally describe a
rule they noticed in the pattern. The
ability to represent the rule as a graph, table, equation, or expression is
also evidence of algebraic thinking.
Furthermore, using the ideas of doing-undoing and seeing that the
particular rule can go from input to output and vice versa shows that the
student is able to recognize reversibility.
If the student was able to test cases of consecutive numbers as well as notice
commonalities between the sums and construct a rule from it either verbally or
algebraically, I would consider that to be algebraic thinking.
c) How did this student use algebraic thinking? Look for “building
rules to represent functions” and “abstracting from computation”
Unfortunately, the given the 20
minutes allotted for the interview, the student was unable to show much true
algebraic thinking. She was able to do
some of part (a) and (b). At this
point, she had tried out different combinations of consecutive numbers in order
to get sums between 1 and 35. She did
not get to the point where she was able to notice any rules so represent a
function nor did she abstract from computation.
2. Evidence: What evidence do you have in what the student said or did that he/she
was using algebraic thinking? Look for evidence of a) Building rules to
represent functions (see page 16) and b) Abstracting from computation (see page
17)
This student did not show evidence
of Algebraic thinking. Again, she was
able to follow the instructions and add up consecutive number to get most of
the sums between 1 and 35. She noticed
that she could not get 2 consecutive numbers to equal a single digit
number. She also noticed most of her
sums were odd. Rephrasing the
instructions, she also noticed that with consecutive numbers, one could add up
to numbers between 1 and 35. According
to Driscoll, she seems to be generating “cases and compares them
systematically, searching for a rule” (Driscoll 1999, pg 86). Although she is able to compare some of her
findings, she does not actually find the rule for this sequence of
numbers. Therefore, she does not show
any algebraic thinking.
3. Next Steps: If you were teaching this student’s class,
how could you use what you learned while interviewing this child in a lesson
plan? What might be a good lesson to
follow up on this interview?
This student did not get a chance to
show me any algebraic thinking, so my follow up lesson will assume that she
does not have any algebraic thinking skills in terms of generalizing. Generalizing branches off into two aspects:
globalizing and extending (Driscoll Pg 94).
Since it appears the student needs to work on her globalization skills,
this lesson will focus on building those skills. She will be given a problem called Toothpick
Squares (Driscoll 1999 Pg 95). This
problem presents a pattern of growing squares made from toothpicks. In this problem, students can work with
manipulatives (toothpicks) to visually see the pattern. They can make table and write down what they
might notice about the number of toothpicks in each square. After I’ve noticed students have a good
amount of data, I might ask
her what rule we might be able to construct for this sequence of numbers using
a variety of questioning techniques. For
example, I might ask “Is there information here that lets me predict what’s
going to happen?” or “Am I doing the same steps over and over? What are they?”
(Driscoll 1999, pg 92). These questions
will motivate the student to think about the patterns that are occurring in the
list of numbers and help her slowly build up a rule for the sequence of
numbers.
1. Reactions: What did you learn? What surprised you?
What are you wondering about?
From this interview, I learned that doing this particular type of
problem generally requires more than 20 minutes. Within that time, a student may only be able
to generate cases, but not necessarily show any actual algebraic thinking. In other ward, they may not be able to have
enough time to draw conclusions about the pattern and generalize it into an
algebraic rule in that time. It seems
that many students might have trouble with identifying a pattern when it is
both additive and multiplicative.
References
Mark Driscoll (1999). Fostering
Algebraic Thinking: A Guide for Teachers Grades 6-10. Portsmouth, NH:
Heinemann. Chapters 1, 2, 4, 5, 6, and 7.
