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.

 

Memo #5

Brenner, M. (1994). A communication framework for mathematics classrooms: Exemplary instruction for culturally and linguistically diverse students.

Summary:
          Brenner's chapter addresses some of the concerns in teaching mathematics to linguistically diverse students.  One of the key problems is that many classrooms spend too much time on independent work and teacher lecture.  Unfortunately, academic discourse skill require more time to develop and more needs to be done in order to foster its learning.  Many females and minorities' attitudes towards math decline as they reach high school.  Aspects of school, context, curriculum, instruction, and institutional support can help support diverse students and improve mathematics communication.  Many students are not adequately prepared in math to pursue technical professions.  As a result, the forms of communication in the classroom need to change in order to develop useful mathematical understanding in students.  Students need to communicate about, in, and with mathematics.  Communicating about mathematics means talking about math from one's own perspective, reognizing when specific methods can be used and why.  Communicating in mathematics means using the academic language and key terms.  Communicating with mathematics means using math as a tool or utility.  Improving mathematics communication and the academic achievement of diverse learners in the curricular context involves giving students access to: problem solving, reasoning, estimation, and of course, communication.  Issues in the instructional context consist of matching instructional group arrangements to student levels, the language of instruction, and the role of technology.  Inssues involved in the institutional contexts include the environment of the school or district, and parent involvement.  This reading will be further discussed in the readings below.
 

Questions

1. What issues/questions about teaching English learners do the readings raise?
   
   Some of the issues brought up from Brenner is how although many student are benefiting from traditional teaching, the increasing number of diverse students in U.S. school demands a change in instruction in order to accomodate all students.  Many minorities are being tracked into lower level classes in mathematics diving many students to failure.  Futhermore, many students are not being adequately prepared in mathematics to pursue technical professions.  This needs to change to adapt to society's change towards technology.  It was also found that many students were uncomfortable in situtations in which they had to individually speak in front of their peers.  
   
   
2. How does the reading suggest you can promote these in your classroom:

(a) intellectual growth/academic excellence in mathematics

(b) equity

Both equity and intellectual grown/academic excellence in mathematics can be promoted through a variety of methods in mathematics classrooms.  Using a wide range of instructional strategies can help accomodate students with differing needs.  One instance of improving communication about mathematics involved using daily journals to write about what was learned in class.  Incorporating student experiences into instruction can further improve students' intellectual growth.  Because communicating mathematics is a key idea, interactive activities are highly important in terms of learning the material and academic language.  This might include the use of pair problem solving, cooperative groups, large group discussions, individual  student journals, and written assignments.  The use of technology can also help students fill in gaps.  Instruction needs to be changes in order to involve all student participation to promote equity because communication is a part of the learning process.  Also, the use of manipulatives and having student write their own problems can foster mastery.

Generalizing Structure and Function Relationships in Algebra

Powerpoint!

https://docs.google.com/open?id=0B0MTyL2FEVkqbDlzaE9tVVJrNWs

Interview Report I: Proportional Reasoning

         I interviewed an 11-year old 6th grader at Branciforte Middle School.  She is a student in my placement classroom studying basic math with Mr. Lammerding.  In this class, she has learned about integers, decimals, and has just been introduced to solving one step algebraic equations.  This class has not gone over ratios and proportional reasoning yet.  During this interview, I introduced the problem to her and gave her the paperclips to work with for Mr. Short.  She did not write down much work to find out Mr. Tall's height in small paper clips.  She got an answer of 10 paper clips.

Me: We're going to do a problem involving Mr. Short and Mr. Tall.  So here, we have Mr. Short.  I'm going to measure how tall Mr. Short is in paperclips.  How many paper clips tall is Mr. Short?
Student: 4 paper clips.
Me: Okay.  Write that down.
Student: (writes it down)
Me: Okay, Mr. Tall, whose picture I don't have with me, he is 6 of these same paper clips tall.  So I'm going to take these away.
Me: I want you to do 3 things.  I want you to measure how tall Mr. Short is with these small paper clips. (hands student paper clips)
Student: 6
Me: Okay, write that down.  Next, I want you to predict the height of Mr. Tall with small paper clips.  I told you earlier that we know Mr. Tall is 6 big paper clips tall.  And now we want to know tall Mr. Tall would be if we used small paper clips.
Student: 10
Me: Now, tell me, how did you solve the problem?
Student:  Because if this is one (Mr. Tall) is 6 (big paper clips), and these ones (small paper clips) are like half of it.  If you measure 6, and then 7, 8, 9, 10. um that tall. (Stacks small paper clips on top of each other)
Me:  Can you explain how you know your solution is complete?
Student: We were trying to find out Mr. Tall’s height using the small ones.  He is 10 small paper clips tall, so um, we are done.

1.      Proportional reasoning:

a) Define proportional reasoning.

Proportional thinking is defined to be the ability to construct and algebraically solve proportions.  In other words, it is "the comparative relation of one thing to another in part or whole and is expressed in terms of magnitude, quantity, or degree" (Lowery 1974).  It involves the strategies that are used to tackle a number of problems in mathematics and plays an important role in students' mathematical development.  Proportional reasoning is used in fractions, word problems, rates, functions, etc. In addition, proportional reasoning is not merely when someone can set up "cross-multiply and divide" algorithm for solving proportions.  It means understanding why the algorithm works, and the mathematical and multiplicative relationships that occur between 2 or more measure spaces in proportional situations.  It also means being able to recognize which situations are proportional and which are not.  It means being able to solve a variety of problem types, including missing value problems, numerical comparison problems, and two types of qualitative situations.   Students with proportional reasoning should also be able to overcome unfamiliar settings and non-integer numbers such as factions and decimals.

b) Describe what you would consider evidence that a student is using proportional reasoning.

Evidence that a student is using proportional reasoning can come in a variety of forms, depending on the student’s stage.  Being able to see that there is a pattern between two measure spaces and relating the two together through a multiplicative relationship can be a form of proportional reasoning.  For example, if the student is introduced to a problem in which 2 miles are run in 18 minutes, they may be able to recognize that each mile is ran in 9 minutes.  From there, if the rate does not change, the student can then also find out how long it takes to run any number of miles.  In other words, the student is able to connect the two measure spaces together and see the multiplicative relationship between the two measure spaces.  In particular, for this problem, evidence of proportional reasoning would be shown after the student can identify that the ratio between the small paper clips and the big paper clips is 4 to 6.  With the given information that Mr. Tall is 6 big paper clips tall, the student should be able to recognize that multiplying 6 by 6/4 would give us height of Mr. Tall in small paper clips.   

I might also describe that the beginnings of proportional reasoning might involve being the ability to compare two different units.  Being able to see the comparison is an important part of building the relationship.  Although visually comparing may not be the most accurate, it demonstrates some understanding of proportions.  If a student is able to show the relationship between two measure spaces as a fraction or ratio, and then use that for find a missing value, then that can also demonstrate proportional reasoning.

c) How did this student use proportional reasoning? Look for evidence of unitizing, relative thinking, absolute thinking, multiplicative strategies, etc.

       My student mostly used visual strategies.  When she first wrote down the height of Mr. Short, she wrote 4 feet.  This shows she was looking at the big paper clips as a unit.  In terms of proportional reasoning, she did notice that the small paper clips were about half the size of the big paper clips, so she visually estimated that Mr. Tall would be 10 small paper clips tall.  In other words, she noticed that it would take about 10 small paper clips to be of equal size to 6 big paper clips.  Although she was able to visually compare the two paper clips are two separate units, she did not show proportional reasoning.  According to Lowery (1974), this student is in pre-stage.  Lowery points out that a student who is in pre-stage “uses proportionality, but is an estimation of length ratio instead of a computation of it from the data” (Lowery 1974).  As noted, this student estimated the relationship between the smallies and biggies.  She did not use computation to reach her answer; instead, she visually used the small paper clips to see how many of them would be equivalent to 6 biggies.

2.      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?

      After this activity, I learned that many students likely use similar visual reasoning strategies because they have not been introduces to proportional reasoning.  As a result, my follow up lesson would include an activity in which students will work with manipulatives in which they will need to compare two different units.  This activity will be similar to the Mr. Short and Mr. Tall activity.  Students will be introduced to a problem in which they will have 2 different units: quarters and dimes.

      Each pair of students will be supplied with 6 quarters, and 8 dimes.  They will be asked to line up the least amount of quarters and dimes so they are of equal height.  The aim of this is for them to notice that the length of 3 quarters is equal to the length of 4 dimes, in other words, they have a 3 to 4 relationship.  Next, they will have to line up 6 quarters to 8 dimes.  During this activity, the pairs will discuss with one another any patterns they are noticing.  After this, I will present a new problem and ask them how many dimes lengths are equivalent to 9 quarter lengths, 12 quarter lengths, 15 quarter lengths?  This data will be recorded into a chart.  By this point, students might discover that there is a multiplicative relationship in which if the quarters are multiplied, the dimes are as well.  Cramer, Post, & Currier (1993) say that all proportional situations have multiplicative relationships.  Therefore, it is incredibly important that students are able to investigate and come to this conclusion on their own.

       I will emphasize the use of relative thinking, which Lamon (1993) explains as the ability to see the ratio as a separate quantity from our data.  Next, I will have time tell me how many dimes lengths are equivalent to 48 quarter lengths.  This will require student to do some computation using the ratio in order to get to their solution.  Furthermore, I will also have them find how many dime lengths might be in 70 quarter lengths so they can see they we do not have to have whole number answers, and that this will work for numbers that are neither divisible by 3 nor 4.  Within this activity, student will be able to work with manipulatives and interact with a pair to explain the reason to their solutions.  After this activity, I will wrap up the activity and ask for what relationships student may have noticed.  I would further emphasize the multiplication relationships proportional problems have between measure spaces.

3.      Reactions: What did you learn? What surprised you? What are you wondering about?

      From this interview, I was able to get some insight on common misconceptions when faced with proportional reasoning situations.  I found that at the 6^th grade level, many students have not been formally taught ratios and proportions.  As a result, my student did not use proportional reasoning to find her solution.  A common strategy a 6^th grader might use is estimation based on what they see.  I found that she was pretty good at visually comparing the two objects to get to a good estimate.  This is a strength that could be used later to help her develop her proportional reasoning skills later on.  I am curious about what steps it takes for students to fully understand proportional reasoning from this stage.

Works Cited

Cramer, K., Post, T., Currier, S. (1993). Learning and Teaching Ratio and Proportion: Research Implications. In D. Owens (Ed.) Research ideas for the classroom: Middle grades mathematics. NY, NY: Macmillan Publishing Company.

Lamon (1993) Ratio and Proportion: Connecting content and children’s thinking. Journal for Research in Mathematics Education, Vol. 24(1), 41-61)

Lowery, L. (1974). Proportional Reasoning. In Learning About Learning Series. Berkeley, CA Univ. Of California, pages 17-20 and 37-38