Algebra Interview

Interview Report: Algebraic Thinking

            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)

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,of toothpicks in each sqaure.es (toothpicks) to visually see the pattern.  They can make table and write 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.