Algebraic Reasoning Interview

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.