# Commentary

The three-way collaboration among Char, Tara, and Kara produced an engaging unit that motivated students to solve algebraic problems from an investigative approach (SMP1). Through the activities, students discovered the concepts of slope and y-intercept from experiences with motion by modeling races between the Tortoise and the Hare. Students discovered that the slope of a line and the coefficient of the x-term in the equation y = mx + b indicates the rate at which a character is moving. They also discovered that the starting location of the character, the y-intercept of the graph, and the constant added to the x-term in the equation of a line all represent the same information. These activities led very nicely to the formal definitions of slope and y-intercept and their uses in other situations. The unit is a perfect example of how technology can help make abstract concepts very concrete, dynamic, and approachable for students.

The students’ attempts to read and create tables, to apply and correct their intuitions to interpreting and creating graphs, followed by the development of equations, are in line with research recommendations for the development of the function concept (SMP4). Thornton (2001) notes three approaches to algebra that are crucial to students’ conceptual understanding: developing patterns, using symbols, and applying functions. She writes:

The power of algebra lies in its capacity to develop and communicate insight by representing situations in alternative ways. Whether developed through alternative visualizations, symbolic, manipulation, or a functional approach, each of these alternative representations leads to new insights into mathematical relationships.

(392)

In Principles and Standards for School Mathematics (NCTM, 2000), the Connections Standard calls for making connections among mathematical ideas and for focusing on how those ideas build on one another to produce a coherent whole (64-66). In this unit, students used multiple approaches for looking at a situation. Throughout they used a functions-and-graph approach for interpreting results and creating their own stories; they used a patterns approach to construct tables and to develop the slope concept; they applied the formal symbolic approach when they determined equations from tables (SMP2). Most important is the fact that students were able to make connections among these representations as tools for explaining their thinking (SMP3).

The Common Core State Standards for Mathematics (CCSSI, 2010) recommends making meaningful connections between mathematical strands through the application of the mathematical practice standards of modeling, structure, and repeated reasoning (8-72), The unit is an excellent example of these practices because students are taught as college-intending students and afforded all the tools required to foster higher levels of thinking appropriate to their understanding of important mathematics. For educators who believe that calculators are useful only for checking work or are harmful to the development of basic skills, the unit shows a powerful application of technology in fostering connections to enhance conceptual development of algebraic and geometric concepts, as well as reinforcing basic skills.

During the learning process, the students made many mistakes. Their misconceptions were addressed through Char’s, Tara’s, and Kara’s skillful questioning and rephrasing, as well as through small group and full class discussion. Students enjoyed and understood the activities that were set in a scenario with which they were familiar. They connected with the context of walking and racing and appreciated the real-life aspect through which they were introduced to linear functions. They learned these powerful mathematical ideas through activities that challenged them to solve problems just as mathematicians do when no ready answer or algorithm is available: They explored and searched for patterns and had to make sense of the concepts by creating and defending their own real-life situations and models (SMP1, 3). This process is conducive to producing mathematically proficient students, according to CCSSM: “Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later” (CCSSI, 2010, p. 7). As students continue to be so engaged with mathematics, they will come to appreciate the modeling process as described by Meyer (2015):

The world will rarely cooperate completely with our mathematical models; there will be a few leftover slices. But seeing some documentation of their difference—with a photograph, video, personal testimony, or a live demonstration—will help students understand the true of power of mathematical modeling, one of the most powerful tools humans have for understanding their world, although not omnipotent.

(583)

Unlike the other profiles in this book, this profile makes no mention of student characteristics. Can readers tell from the unit, or from the students’ actions and responses, which dominant culture was represented? Looking to the lesson for clues, one could stretch a guess that the connections to the fables of the Greek philosopher, Aesop, would be of particular interest to Greek or European students. But Aesop’s tales are translated into many languages and have become part of many cultures.

I grew up thinking that Aesop’s tale of The Ant and the Grasshopper was a Haitian folk tale because I had heard it many times growing up in a Haitian community. Tara is correct in saying, “Most students have heard the fable. It seems to be a context that applies universally to all students.” And for those students who may not have heard it, the lesson begins with the reading of the fable so that all students start with a common understanding.

Assuming that this unit was based on cultural learning styles, do the styles addressed in the unit provide clues to the cultural makeup of Tara’s class? In previous chapters we read that kinesthetic or people-oriented activities work well with African Americans and Hispanics. Many Native Americans prefer learning by watching or doing, and many Haitians prefer an oral approach to learning. Cooperative learning is also the strategy most often recommended for diverse cultural groups. However, adhering to the caution not to stereotype groups, the unit incorporates all of these preferences and more. In effect, this unit would very likely motivate students because it begins at a level that students can master. As it proceeds through stages that move from the concrete to the abstract, it does so through a variety of methods. Among the many techniques applied are students’ thinking for themselves, getting help from others, using paper and pencil, applying technology, observing a student complete a task, sitting at desks, moving and running across the room to gather data, completing homework, explaining work, listening to the teacher, viewing results on the whiteboard projector or the board, creating graph stories to share, and working in groups. In short, within the period of this unit, there were aspects to appeal to multiple learning styles.

If we consider the students’ work, responses, and involvement in the lesson, then we might guess that the students were at least of average ability for achievement in mathematics. In actuality, the class was a lower-level math class consisting of 26% academic support, 19% special education, and 11% ESL (English as a second language) students. The socioeconomic level of students in this class was fairly low, with the majority receiving federal assistance for free or reduced-price lunches. Ethnically, the class was primarily Caucasian with two Hispanic students and one Asian student. The students were not among those who enjoyed mathematics. Indeed, as Tara explains:

The first activity of the school year completed by the class was to write four words relating to mathematics using the initial letters M, A, T, H. The overwhelming majority of students wrote these descriptors: Multiply, Add, Times, and Hate. The last descriptor made it evident that great energy and creativity would be needed to change this attitude. It was a challenging class that needed material presented in ways other than straight from a textbook. Attempts to teach straight from the text resulted in a repeat of teaching the material using hands-on materials the second time through.

The unit aimed to help students learn the concepts well the first time through activities that accommodated their need for intrinsic motivation and a large amount of academic and emotional support. The attention the teachers gave to understanding students’ needs by including a variety of engaging activities helped facilitate the process for students’ to identify themselves as part of a mathematics community. According to Anderson (2007):

Our identity includes our perception of our experiences with others as well as our aspirations. In this way, our identity—who we are—is formed in relationships with others, extending from the past and stretching into the future. As students move through school, they come to learn who they are as mathematics learners through their experiences in mathematics classrooms; in interactions with teachers, parents, and peers; and in relation to their anticipated futures.

(8)

Wager (2014) in her research equates this process to that of equitable opportunities necessary for children to learn mathematics. She sees this environment as conducive to the development of identities where children “develop their identities as learners and doers of mathematics by participating in practices that support agency Agency refers to the opportunity for children to take action with regard to their own learning” (313).

Finally, the three-way collaboration between Char, a university professor, and classroom teachers, Kara and Tara, exemplifies a successful professional relationship built by colleagues to share teaching and assessment strategies for the improvement of students’ learning. Such collaboration is at the heart of implementing a culturally responsive pedagogy in the curriculum.

For additional activities with technology coauthored or written by one of the teachers, see Graphs in Real Time, in Mathematics Teaching in the Middle School 5 (October 1999): 92-99; R. N. Rubenstein, C. Beckmann, and D. R. Thompson, Teaching and Learning Middle Grades Mathematics (Emeryville, CA: Key College Press, 2004).