By Maria Steffero
As I look back now, I feel fortunate that I was picked for the study. I didn't realize it at the time, but the Rutgers study helped me develop a strong foundation for learning, team work and communication. The way we were taught really worked for me personally, and it's the way I've chosen to learn in my adult life.
- Romina, 2010
Introduction
What does mathematical reasoning look like? How do we listen for a convincing argument? As a middle school teacher in New Jersey, I am immersed daily in the development of my students' mathematical ideas. At the beginning of my career, I believed the more information I could explain and annotate at the chalkboard, the better my teaching would be. However, after I began to reflect on my pedagogy and practice, first as a masters' student and then as a doctoral candidate at Rutgers University, I realized learning begins with a teacher listening to her students, rather than the other way around. I started to understand the proverb, "If speaking is silver, then listening is gold." But how could I better attune my ear to students' thinking? Video recordings allowed me the wonderful opportunity to rewind, pause, and replay the intricacies of children's mathematical minds in action.
As both a student and researcher at Rutgers University, I have experienced the transformative power of video data analysis for my own teaching and understanding. I remember the first time I watched a video recording of a student named Romina as she described an elegant combinatorial proof to her fellow tenth graders. Where did her ideas come from and how did they develop after this session? It was as if I had just read a page from a single chapter in a beautifully written novel - I had to read the whole book! The video data archive at Rutgers allowed me to trace Romina's ideas over seventeen years as she encountered problem solving tasks and reflected on her thinking in clinical interviews. Imagine being able to see so far and wide into our own students' ideas over time. I could watch Romina consider the same problem in both 1992 as a fourth grader and in 2009 as a successful M.B.A. graduate.
I invite you to consider the selection of video clips included here featuring Romina's thoughtful engagement in mathematics over seventeen years. You will find a student working through a range of behaviors: questioning others' ideas, interacting with teacher-researchers, working through frustration, and performing multiple group roles. Consider how Romina constructed conceptual knowledge in collaborative learning environments where problems were shared, questioned, and argued. Listen to Romina reflect on her own growth from the elementary grades through college and into the workplace. In June of 2010 Romina was asked to comment on her participation in mathematical research studies. She described her experience with the many problem solving task sessions as "empowering" and elaborated "I was amazed with what a group with different strengths that worked well together could accomplish in a few hours." As you listen to Romina in the video clips, imagine how empowering it would be for all students if we could bring such careful attention to every student's voice!
Background
Romina was among the original participants in a longitudinal study resulting from a research partnership between the Rutgers University Graduate School of Education and a public school district in a working class community. Initiated in 1989 with a class of eighteen first graders, the study has continued for over twenty-one years with a smaller subset of the original class of students in addition to a few other students who joined later in the program. Problem-solving tasks occurred in elementary, middle, and high school. For the elementary and middle school sessions, a problem-solving task was introduced during an extended period of the regular school day to a large group of students who were working together in smaller partnerships. Romina's first problem-solving task within the longitudinal study occurred when she and Brian worked on the combinatorial "Towers 5-High" on February 6, 1992 in 4th grade for 40 minutes. Romina and Brian worked together again in 6th grade during the algebraic "Guess My Rule" task on October 1, 1993 for 45 minutes.
Whereas the elementary and middle grades the interventions took place with the whole classroom during an extended period of the normal school day, the high school years of the longitudinal study were conducted on a voluntary basis after school, often on Fridays, and in much smaller groups - only four or five students at a time. In what became known as the "Ankur's Challenge" session from 10th grade on January 9, 1998, the participants were Ankur, Michael, Jeff, Romina, and Brian and the session lasted 90 minutes. In the Taxicab Geometry session from 12th grade on Friday, May 5, 2000, the participants were Michael, Romina, Jeff, and Brian and the session lasted 100 minutes.
Through numerous interviews, Romina shared her reflections on her problem solving experiences. These clinical and semi-structured interviews ranged in length from about a half hour to approximately two hours each. The researchers questioned Romina about the following topics: her memories of the longitudinal study, her associations with particular years of classroom mathematics, her thoughts about learning, and her self-perception as a problem solver.
Towers Problem 5-High
Problem Statement: "Your group has two colors of Unifix® cubes. Work together and make as many different towers, 5 cubes tall, as is possible when selecting from two colors. Convince us that you have found them all."
In this clip the fourth grade partners Romina and Brian work to construct a solution to the "Towers Problem 5-High." At the beginning of this clip, Romina and Brian check for duplicates among the twenty-one towers they have already built and wonder if they have generated all possible towers that meet the problem criteria. Romina conjectures they have all the towers. Then the teacher-researcher asks Brian and Romina about their thinking. When Brian says that they are using a strategy of "do the opposite," the researcher asks what he means and then follows by asking if towers always have an opposite. The students investigate the new conjecture that every tower has an opposite by attempting to find a tower without an opposite. After verifying all of their existing towers can be put into opposite pairs, Romina tells Brian to stand the towers up together in groups of two which she calls "matches." One of their original towers does not have an opposite and after they generate its opposite, they have a total of twenty-two towers.
Towers five-high selecting from two colors: Developing local organization
Guess My Rule: October 1, 1993 (6th Grade)
Problem Statement: "Each problem has a table with two columns headed by the symbols: 
and 
. The objective of the activity is to create a "rule" for each problem which takes the given input values of the box column and results in the corresponding output values provided in the triangle column. In problem number three, the table provides: {(0,1), (1, 4), (2, 7), (3, 10)}. In problem number four, the table provides: {(0,7), (1, 17), (2, 27), (3, 37), (4, 47), (5, 57)}."
In this clip Romina and Brian work as partners. At this point they have moved to problem three on their worksheets and express to the researcher they are progressing well. However, when comparing rules for problem number three Romina says she thinks "it" (the y-intercept) is 1. When Brian challenges her conjecture, she works to convince him. For the next problem, Romina asserts the slope for the new linear function rule is "ten". Romina soon leans over and corrects the rule Brian is writing on his paper. She explains her method for identifying the y-intercept for the function rule: "All you have to do, Brian, is take the first number and add it." Brian then asks her if the rest of the rule is "blank times four" or "times seven." Romina describes she finds the slope by looking for "whatever is between seven and seventeen" - two of the entries in the triangle column. Then Brian writes the correct rule for problem number four as a linear function with a slope of ten and y-intercept seven.
Romina and Brian work on Guess My Rule problems 3 and 4
Romina's Proof: January 9, 1998 (10th Grade)
First Problem Statement: "Choosing from two colors of Unifix® cubes, red and yellow, how many total combinations exist for towers 5 tall, that each contains two red? Convince us that you have found them all."
Second Problem Statement (Ankur's Challenge): "How many towers can you build four tall, selecting from cubes available in three different colors of Unifix® cubes, so that the resulting towers contain at least one of each color?"
In this edited and narrated episode from PUP-MATH, five 10th grade students consider two different problems. During the session Ankur and Michael work together at one end of the table and Jeff, Romina, and Brian work at the other end. Ankur and Michael use binary notation to solve the first problem. While waiting for Jeff, Romina, and Brian to complete the first tower problem of five- tall with a choice of two colors, Ankur poses a new question about towers four tall with a selection of three colors: "How many with at least one of each color?" Romina develops a notation that she calls "ones, zeroes, and Xs" to represent the three colors in the problem. First at the table and then at the chalkboard, Romina presents her reasoning to the others as she works to convince them there are thirty-six total towers that meet Ankur's criteria. She shows them her representation to justify the six possible arrangements for two "1s" in the four positions of the tower. She fills the blank positions with "X0" or "0X" for the other two colors. She concludes with the calculation 6 x 2 x 3 = 36.
PUP Math Romina's proof to Ankur's challenge
Interview Reflections I: May 18, 1999 (11th Grade)
Interview Questions: What do you think would happen if all classes were conducted like those in the longitudinal study? Why do you say you're not confident in your abilities?
In this clip Romina describes what would happen "if people learned the way I did" in the longitudinal study. She conjectures that people would be able to "learn more and be able to do more" if they experienced such problem solving task sessions as she did through the years. Asserting you cannot "live your whole life being told what to do," Romina argues that when you "do it yourself," you will gain "more knowledge about everything." She describes that by doing, understanding follows. "Everything I do, I understand." Specifically reflecting on her own knowledge, Romina explains it is necessary for her to know in her "own way" and not someone else's. Romina says she is "not confident" because, although she knows she can "do a lot," she is not always able to explain what she knows to others, especially if they want the explanation in terms of a particular "rule" or established authority "guy." She describes how she can "know it in my way, but not their way." She provides an example of how she knows how to use Pascal's triangle, but she does not recall its formal name. Reiterating "I just know everything in my own way" she concludes that "everything has Romina's definition to it."
Romina interview reflections (11th grade): Everything has Romina's definition
Interview Reflections II: July 21, 1999 (12th Grade)
Interview Questions: Is there anything you would like to say that we have not been asking you about this experience? Would you feel more comfortable if people reassured you more about whether you were on the right track with a problem? How would you advise a teacher on how you would build up someone's self esteem in math? What is mathematics?
In reflecting on the two week Rutgers Summer Institute between her junior and senior high school year, Romina summarizes her time as a "good experience" where she and the others engaged in problem solving, thinking, talking, arguing, and proving. When asked whether she would have felt more comfortable to have been reassured more during tasks, Romina asserts "I don't like being reassured" in a problem. She defines "reassurance" as when "they're treating me like a little child," "holding our hand," and "walking us through." She says this process of instruction allows the students to "get the right answer." When "they" - the instructors - do not reassure them the students can then go in an unknown direction that "makes it all the better." In reference to self esteem in mathematics, Romina explains she believes there are "two big different areas of math" which she identifies as "thinking" and "spitting out numbers." She claims she was "never good" at the "spitting out numbers" part of mathematics, but she was "decent" at the "thinking about it." She says she would encourage teachers to incorporate the "thinking" part because, in her definition, mathematics is problem solving. Romina elaborates mathematics is "everywhere" and in "every situation you could possibly think of."
Romina interview reflections (12th grade): Mathematics is everywhere
Taxicab Geometry: May 5, 2000 (12th Grade)
Problem Statement: The problem was presented to the students with an accompanying representation on a single (fourth) quadrant of a coordinate grid of squares with the "taxi stand" located at (0,0) and the three "pick-up" points A (blue), B(red) and C(green) at (1,-4), (4,-3) and (5,-5) respectively, implying that movement could only occur horizontally or vertically toward a point. The problem states that: A taxi driver is given a specific territory of a town, as represented by the grid. All trips originate at the taxi stand. One very slow night, the driver is dispatched only three times; each time, she picks up passengers at one of the intersections indicated on the map. To pass the time, she considers all the possible routes she could have taken to each pick-up point and wonders if she could have chosen a shorter route. What is the shortest route from the taxi stand to each point? How do you know it is the shortest? Is there more than one shortest route to each point? If not, why not? If so, how many? Justify your answers.
This clip features the four 12th grade students Romina, Brian, Jeff, and Michael in the midst of working as a group to solve the taxicab problem. Earlier clips show the students developing their initial strategies to solve the problem, tracing the shortest distances to the three given points, and solving the number of shortest paths to point A. Having previously employed patterns, symbolic notations, and colored diagrams to count paths to B and C, Jeff explains their current notation on the taxicab grid. They conjecture there is a relationship between the taxicab problem and Pascal's Triangle. In this clip the researcher questions "why this pattern works" and how they would justify a connection. After rewriting Pascal's triangle on a separate sheet Romina begins to ask a series of questions about where each "block" comes from in the array. She suggests they "use Towers." She and Michael begin to question each other about how entries in Pascal's Triangle are related to the taxicab problem. Romina uses the letters "A" and "B," and later "x" and "y," to represent the two directions of horizontal or vertical direction, respectively, on the taxicab grid as well as the two colors in the Towers problem. Michael suggests they could also think of the two directions with toppings for pizzas.
Taxicab Problem, Clip 4 of 5: Explaining the Taxicab and Towers Isomorphism
Interview Reflections III: March 11, 2002 (College Sophomore)
Interview Questions: Situate your Rutgers experience in the context of where you are now - did it cause you to work differently? What were the strengths or weaknesses?
In this clip Romina remembers her experience in the Rutgers longitudinal study as one in which problem solving consisted of collaboration and questions. She describes herself as someone who does not "learn well" from a textbook and struggles in college because there "you have to read a book and then you're tested from what's in the book." Romina claims that she is "better at learning" when she engaged in "thinking about things, discussions, group work." She elaborates on her self-perception that she can "deal with groups," "work very well with groups," and do "some of my best work with other people." Romina states that working with groups has "helped" her in preparing for the future. Explaining she hopes to be "in some sort of leadership role" where she would need to "deal with people and delegate," Romina attributes her comfort with "discussing things" with people to why she got a job for which she did not at first seem qualified.
Romina interview reflections (college sophomore): Collaboration is strength
Interview Reflections IV: May 12, 2006 (Corporate Consultant)
Interview Questions: What do you think are important skills for young adults just entering the job market to have? In your workplace, what's the range you see of how people solve problems from best to worst?
In this clip Romina, Angela, and Magda describe their current jobs in the context of problem solving. Working as a business analyst for a global consulting firm, Romina explains the recruiting aspect of her current job. She observes college students' performances on "case competitions." When asked about types of problem-solving, Romina distinguishes between those people who have and have not been to business school. She observes the business school students usually begin a problem by looking for a relevant "framework" like Porter's Five Forces or "SWOT" analysis. She then speaks of those people who "just think" and then "come up with a better solution" than the business students. Romina remarks that business students "force all this knowledge" and become "constrained by this business mentality." After Angela mentions her business friends would often use formulas to solve problems, Romina describes how she knows certain concepts "inside and out" and yet she struggles with procedural computation. At work she encounters people who apply formulas but "couldn't understand the concept behind it." She describes how she explains concepts to others using an example like the Towers problem.
Romina interview reflections (corporate consultant): Problem solving in business
Interview Reflections V: July 15, 2009 (M.B.A. Graduate)
Interview Questions: What does it mean to you to know something really well? How would you define math now? What would be something you know really well?
For Romina, knowing something really well has three components: understanding the concept, recalling meaning after a long passage of time, and being able to explain the concept to others. She asserts that performing a procedure like calculating a derivative would not constitute knowledge of calculus. Romina states even today she would be able to "explain" for others both background and computation of "the fundamental theorem of calculus" - specifically, "how all these things happened and worked." Romina continues to distinguish between the conceptual "understanding how slope works" versus the procedural "actually figuring out the slope." She describes the "understanding" as "much more higher level," whereas "figuring out" computations is "basic" and "number crunching" with which tools like Excel help. Romina implies a lack of appreciation for knowing only a procedure - what she calls the "mechanics - just moving numbers around." She admits frustration in the past that she "couldn't do the mechanical part" of mathematics, but that since she does "understand" the ideas behind concepts, she will "get through life just fine." As a problem-solver, Romina states she is "pretty good" at a process that includes: "getting a lot of information," "organizing" the information to "see" the problem, and finally "working to find the solution."
Romina interview reflections (M.B.A. graduate): Understanding the ideas