As a final assessment, students are asked to write a story interrelating slope, speed and direction of travel, starting position, y-intercept, and the equation of a linear function. Tara uses the cooperative learning technique, Think-Pair-Share, to get students to think and participate in writing original stories about the graphs. She gives them approximately one minute of quiet think time. Next, they are asked to share their story verbally with their partner. This takes about 30 seconds for each story. A lively discussion ensues as students share their stories with their partners.

Discussion between Colleagues

Tara, what is your philosophy of teaching mathematics?

Students need to be taught using various methods, mainly those that help them investigate how and why things work as they do. They should be encouraged to find answers to questions through opportunities for discovering ideas or rules and testing them out. They need to be asked many questions throughout lessons so that they truly understand the concept behind the material. By asking them thought-provoking questions, we help them to develop self-confidence in their math abilities. Basic skills cannot be forgotten. Tying them into daily lessons is best.

What are the preferred learning styles of your students? How do you accommodate for these styles?

Most of my students prefer to learn using hands-on activities and visual aids. I use many activities ranging from discovery lessons to cooperative group lessons to get all students involved. Students are taught early how to work in pairs and small groups. I create a comfortable environment, so all students can participate in whole-class discussions and not feel embarrassed or ashamed if they make a mistake. When talking about a given figure or situation, I always have a model or draw a picture. I also allow students to show and explain their work to the entire class on the overhead or whiteboard. This helps them to better understand the material, and it also helps those who are still struggling with the given concept to hear it from a student.

Explain what you do in the lesson in light of the cultural makeup of your students? Are there any teaching practices that you consistently apply based on the culture of your students? Please describe these, and give an example of each.

Most students have heard the fable. It seems to be a context that applies universally to all students. To accommodate as many students as possible, we vary activities so that students’ various learning styles are utilized: visual, kinesthetic, auditory, and reading/writing. We make certain that students use and translate among various representations: tables, graphs, equations, and stories.

What would you keep and/or change the next time this lesson is taught?

We found students having difficulty completing the tables in Figures 11.3 , 11.4, and 11.5. To help the students, we asked questions to help them understand, such as, “When the Hare woke up, how fast did he travel?” “How far did he go in 0.5 minutes?” “How does 0.5 minutes compare to a whole minute?” In the future, for The Hopping Hare we would like to lead the students through graphing and have them fill in the table for the Hare’s race simultaneously.

Another change was based on our observations of student behaviors. They were much more reserved the second day and not as focused on the lesson as on day one. We attributed the lack of participation to a schedule change that shortened classes. But we also decided that part of what was missing was physical activity. In seeing how linear functions are created and using the race scenarios of The Tortoise and the Hare to do so, students seemed to be better able to sensibly discuss various speeds and directions of travel.

What would you do differently if you taught this lesson to students of a different culture?

Nothing. The activities are culturally unbiased. The mix of the activity, visualization, and oral and written communication gives all students the opportunity to learn.

What recommendations would you make to a teacher who is about to begin teaching?

Teach students cooperative learning early Expect them to write often and do homework daily Encourage and expect students to actively participate in class, and always model the behavior and communication, both written and oral, that are acceptable. Stay organized, be open-minded to new ideas and methods, and keep all students involved. Never allow students to leave your classroom totally frustrated or overwhelmed. Give them something they are confident with to start, and then push further into the concept. Review often, and tie ideas together that have previously been taught.

Any other comments?

Char: The Tortoise and the Hare introduces the concepts of slope and intercept and sets the stage for further development of linear functions, which arise throughout school and college mathematics. We recommend that the activities be followed by a variety of activities that apply linear functions in other contexts, as found in Phillips (1991). In addition, we recommend that students be encouraged to apply linear functions in contexts found in newspapers, almanacs, menus, and so on. Linear functions arise in a variety of contexts such as grocery ads (buying multiple units), sales tax, hourly wages, percentage-off discounts, pizza menus (price compared with number of toppings),

conversions between certain units of measure, conversions between monetary units, and calories used per minute of activity. Students should be encouraged to complete experiments such as stacking same-sized items, tying knots in cord, hanging same-sized weights from a spring, and the like, to experience constant rates of change.

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