Questions and Comments

Click on the links below to find the answers to some commonly asked questions about teaching this course. If your question is not listed here, or if you would like to share your comments, e-mail the authors at bgray@howardcc.edu.

What is collaborative learning?
How can we motivate students to learn in a collaborative setting?
What can we do to ensure every student understands the major concepts?
What can be done when a group falls behind or gets ahead?
What can be done when a group member does not actively participate or contribute?
How do graphing calculators assist students using this text?
Will students be totally dependent on the graphing calculator in using this text?
Are procedures for using the graphing calculator included in this text?

What is collaborative learning?

The basic idea of collaborative learning is that a group undertakes an assignment, brainstorms about the issues, works efficiently by using each individual's strengths in a division of tasks, and brings the assignment to closure in a suitable form. Collaborative learning is customarily done in small groups, usually of size three or four.

Back to top

How can we motivate students to learn in a collaborative setting?

You will find it easy to get to know your students in a collaborative classroom and learn what motivates them as individuals and as members of your class. We found the following suggestions to be helpful.

• Impress upon your students the need for teamwork in the professions for which they are preparing and that your classroom is a place for them to gain experience in collaborative work.
• In the beginning, your students will come to you for help every time they encounter a problem. Encourage them to discuss the problem, brainstorm, and share information within their groups. If they need further help, ask them to consult with other groups. If the difficulty persists, offer hints, stepping in to show a solution only after you judged they tried sufficiently on their own.
• Let some percentage of the grade depend on collaborative working skills. One way is to give one or more group exams (in addition to individual exams) during the semester. On a group exam, each group submits one set of answers for the whole group.

Back to top

What can we do to ensure every student understands the major concepts?

Each activity in the text is meant to demonstrate one or more mathematical concepts. Discussion questions are presented when sufficient problems are completed so that a pattern may be recognized or some conclusions may be drawn. Students are prompted to explain these verbally and write down their explanations. We often use a combination of small and large group discussions to share and reinforce these ideas.

We suggest you wait until all groups have the opportunity to answer the discussion question(s) before you review them in a whole class setting. If some groups have gone on to the next set of problems, pull them back for the whole class discussion.

For a class discussion, randomly select students from different groups to read their written explanations. After each reading, have the class discuss the accuracy of the explanation, using as criteria the following questions.

• Is the explanation pertinent?
• Does the explanation convey what the student had in mind?
• Are all the important points included in the explanation?
• Are any exceptions to the rule(s) or pattern(s) noted in the explanation?

By analyzing and critiquing each other's writing, students learn to differentiate good writing from poor and to improve their own. We suggest that you have as many readings as time allows so students see a variety of styles and quality.

A brief lecture at the close of an activity may be useful to ensure each student recognizes the important concepts. A 3 x 5 card written by each student stating the concepts learned and questions not understood will help you respond to your students' needs.

Back to top

What can be done when a group falls behind or gets ahead?

We usually prefer to have all groups work on the same activity during a class session. It is more efficient and productive when we have to stop to address concepts and problem-solving strategies causing difficulties for the entire class. Planning class lessons, tests, and so on, is much easier when groups are similarly paced. We found the following suggestions helpful.

• If most groups are similarly paced in the middle, with an extreme group at each end, ask each member of the fastest group (who finished an activity) to team up with one person from the slowest group and give that person individual help. Encourage the tutor to guide the tutored student toward the source of his confusion and the means for resolving it.
• If one or more groups complete an activity but the others are still working, send members of the faster groups to assist the other groups until the whole class finishes the activity.
• Similarly, if one group is falling seriously behind, send each member to faster performing groups to catch up.
• If some groups repeatedly fall behind, redistribute group membership to get more consistent pacing. Reserve the right to reorganize groups for the benefit of the class. Students accept and adjust to group changes if you establish good rapport with them and make them aware of your genuine interest in their learning and success. Encourage groups to interact with each other so students create bonds with other students in the class as well. Any necessary rearranging is then much easier.

Back to top

What can be done when a group member does not actively participate or contribute?

Group members are responsible for helping each other learn, clearing up misunderstandings, and analyzing conceptual difficulties experienced by individual members. A member shirks responsibilities by being habitually late to class, by erratic attendance, by coming to class unprepared, or by ignoring the group's work in class. Here are some approaches we try in response to this problem.

• Remind the class that groups chose their members and so, after giving sufficient warning, groups have the right to expel members.
• An expelled member cannot ask you to be placed in another group. As in real life, he has to try to convince another group to accept him. You need not act as a placement agency in the classroom.
• If a member is dissatisfied with her group, advise open communication within the group. If the group cannot resolve the issue, the unhappy member has to decide whether to resign and look for another group or to stay and make the best of it. Only if all else fails do we suggest that you take part in the dispute within groups who chose their own members.
• If you assigned the groups, then it is your responsibility to move students to other groups if the group becomes dysfunctional.

We do recognize that often there is no easy solution but we also believe students should be encouraged and helped to make these decisions for themselves.

Back to top

How do graphing calculators assist students using this text?

In Mathematics in Action the activities consist of problems from real life situations. Therefore, the outcomes in this text are normally not integers and the scaling used in graphing is normally not standard. In the past problems of this nature were often arduous and lengthy for students. Today the graphing calculator gives the student an alternative way to solve a problem. Because it reduces the need for lengthy manipulation, it allows time for addressing and understanding important mathematical concepts in real life situations.

Through visual and tabular representations, graphing calculators help students develop an intuitive understanding of mathematical concepts and learn the meanings of abstract algebraic concepts. Graphing calculators aid students in making connections between numerical, geometric and algebraic depictions of variables and functions.

With graphing calculators as tools students learn how to do independent mathematical exploration and discovery.

Back to top

Will students be totally dependent on the graphing calculator in using this text?

Total dependency on the graphing calculator is not wise at this stage of the student's mathematical development. Many grids are provided throughout the text for students to sketch graphs manually. The textbook also guides students to solve some problems graphically and algebraically without the graphing calculator and then to use the calculator to verify the results. Encourage students to estimate by hand the magnitude of expected results to guard against entry and other errors when using a calculator.

Back to top

Are procedures for using the graphing calculator included in this text?

To use the graphing calculator successfully students must know how to effectively operate it. While students entering your class will most likely have knowledge of the scientific calculator, the graphing calculator is a more sophisticated machine and will require more assistance from the instructor. You may want to prepare an introduction including some of the very basics. Some of the calculator steps are given within the text as they naturally assist in the solution of a problem. The Appendix contains a section on using the TI83 to learn specific operations related to the activities. Some procedures are the responsibility of the student and the instructor. Just as the skills in the text are introduced "just in time" the calculator procedures are also addressed "just when needed".

Back to top