Algebraic Reasoning
The
student I interview is a geometry student, whose name is Alex and he is sixteen
years of age. He currently is a sophomore in Pajaro Valley High School and is
maintaining a C+ in Geometry class. The interview lasts twenty five minutes.
During the interview, Alex is able to list the sum of two consecutive numbers
from twenty to thirty five and make conclusions from his results. With extra
time left, I am able to get Alex to move on to the sum of three consecutive
numbers, four consecutive numbers, and even touch upon five consecutive
numbers. Let us not stray too far away from the point of the interview. I
believe any student is able to list out consecutive numbers which does not let
the student show much algebraic reasoning. To deviate to the point, the point
of this interview is to test the student’s algebraic reasoning by analyzing and
apply the results, generalizing, and assuming.
Alex
starts by listing the sum of two consecutive numbers. After creating a chart of
numbers from twenty to thirty five, I ask him a series of questions to test his
algebraic reasoning. One question I ask is if he can make a theory base off his
results. From looking at the list of two consecutive numbers on the list, Alex
immediately notices a pattern that the sum of two consecutive numbers gives
only odd numbers and not even numbers. I proceed to ask a follow up question
and the question I ask is “Are we able to make the number eighty using two
consecutive numbers?” The response he gives is no because if we show that we
are not able to make even numbers using the sum of two consecutive numbers,
then we are not able to make eighty with two consecutive numbers. Alex’s
response shows that he is able to apply his conclusion base off his results and
use it to predict what will happen with bigger even numbers. He draws
conclusions because there are certain things that he looks for such as commonalities
among numbers in the two consecutive sums, patterns in adding two consecutive
numbers, and numerical results of odd and even numbers.
The
final series of test comes down to Alex being able to provide a formula to find
the sum of two consecutive numbers of any numbers. As he thinks aloud, he tells
me he starts by dividing the number in half. If the number gives two whole
numbers, then subtract one from one number and add it to the second number. If
it the number gives a decimal number, then subtract half from one number and
add it to the second number. The wordy algorithm he writes down and shows me
works brilliantly and I think it is very creative. Alex shows me his algebraic
reasoning is strong because he is able to come up with his own algorithm for
figuring out the two consecutive numbers of any numbers. The author of
“Fostering Algebraic Thinking” by Driscoll describes the Alex’s thinking
process as recognizing regularities in the sum of consecutive number systems in
order to find shortcuts that help solve future issues. In this case, Alex
formalizes a theory base off results and verbally explains his thoughts as he
is solving the problem.
After
noticing I can get no more reasoning out of him from using the sum of two
consecutive numbers, I move on to questions about the sum of three consecutive
numbers. Alex lists the sums of numbers through twenty to thirty five. Upon
finishing, he notices that there are a lot more numbers that cannot be written
using the sum of three consecutive numbers. He supports his reasoning by saying
that adding more numbers limits his results from twenty to thirty five. What he
means by this is that he has to add three consecutive numbers and doing so,
gives us really large results which mean that there are bigger gaps in between
results from twenty to thirty five. I ask him if he can draw a conclusion using
his results. He is unable to generalize a conclusion. Other than that, he is
able to say that higher degree of consecutive numbers gives fewer results
(bigger gaps) between the numbers twenty to thirty five. At this point I ask
him to list the sum of four consecutive numbers and five consecutive numbers
for numbers between twenty to thirty five. I get ask the same question and
again I get the same response from Alex.
What
could have been done differently to get more reasoning out of Alex is to have
him organize his results. In the video, the results are all over the place and
it is to see patterns in the sum of consecutive numbers. Another change I have
is to work with smaller numbers, but not too small. Ideal numbers are one to
twenty five. I choose to use these numbers because they are big enough to be
made up of three or even four consecutive numbers, but not too big as to
confuse Alex on his calculations. Also, using smaller numbers means Alex has an
easier time calculating to divert more attention and concentration to thinking
rather than spending his efforts on calculating for the correct answer.
