Archive for May, 2011

Homework exercises assigned to students should reinforce the big ideas consistent with the professor’s teaching strategy. Homework exercises should not be selected haphazardly.  A primary goal of assigning homework is to help students become more enthusiastic and proficient learners. Instructors may implement a variety of strategies surrounding homework.

Homework does not need to be assigned every day. Instructors must be sensitive to the student process of learning and strategically assign homework at the appropriate time. Homework should be assigned when students have basic understanding of the ideas contained in the homework. Giving practice problems to students who lack understanding can have several negative effects (Kohn, 2006).

  • The student may feel stupid. Over and over again, she is reminded of what she can’t do.
  • The student may become accustomed to doing things the wrong way because he repeatedly applies mistaken assumptions.
  • The student may fake that she understands the assigned homework by asking someone else for the correct answers to help to conceal her ignorance.
  • The student concludes that math is something people aren’t expected to understand.

At the appropriate time, homework should be assigned. But what homework? What is the purpose of assigning that particular homework assignment at that particular time? The author team uses several strategies to select and assign meaningful, purposeful homework assignments.

1. Have students read the textbook.

Homework can involve reading the textbook.  The author team designed the textbook to be read by students so textbook explanations are communicated in a conversational tone. Examples and solutions, pictures and graphs, and the written discussion work together to assist students in making sense of important concepts. To further engage students, the Peer into the Past feature accentuates the historical context of mathematical ideas. The Just in Time feature provides prerequisite review material at the exact time it is needed.

2. Give short assignments when focusing on concepts

Homework assignments can be short. By keeping the homework assignments short, students know that they have the time to dig deep and make sense of the mathematical ideas. Instead of spending an hour repeating a memorized procedure, students may spend an hour working to make sense of just a couple of problems.

3. Give longer assignments when practicing procedures

When students have made sense of the underlying ideas of a particular concept, it is appropriate to have them practice procedures. The Concepts and Skills portion of the homework exercises includes many procedural exercises.

4. Assign challenging exercises to stretch learner understanding.

Homework assignments can be challenging. The Show You Know portion of the homework exercises provide students the opportunity to write, explain, describe, and explore core concepts. The Stretch Your Mind portion of the homework exercises are designed to challenge the strongest students.  These exercises may be assigned to some students and not to others based on the student’s current mathematical understanding and learning needs.

5. Be flexible in determining due dates

Homework assignments may take time to complete. If the purpose of the assignment is to have students develop convincing arguments or well articulated explanations, allow additional time for students to produce their best work. For example, the end-of-chapter Make It Real Project typically requires students to collect and analyze data from an area of personal interest. An instructor may choose to allow up to a week or more for students to complete these projects.

6. Don’t grade homework assignments

Homework assignments do not need to be graded. Research has shown that homework in the best classrooms is not checked – it is shared. In fact, some educators argue that to grade homework is especially destructive because this tells students that the point of the exercise isn’t to help them learn: it’s to evaluate them on whether or not they’ve already succeeded. Consider having students present selected homework problems on the board and give them homework credit for presenting regardless of the accuracy of the mathematical work, recognizing that the students may learn more from learning how to correct a common mistake than from watching a student present a perfectly executed problem. In a supportive classroom environment, the instructor and students can discuss on-board errors comfortably without causing the presenter to feel inferior to his classmates. Alternatively, have the students discuss the homework exercises in small groups and reach consensus on strategies that worked.

7. Assign homework to be done as a group

Homework does not have to be done individually. Once a classroom culture of collaboration and teamwork has been created, it may be helpful to continue this culture outside of the classroom through the assignment of group homework. Admittedly, there are difficulties in assembling teams of students outside of class; however, the benefits of a fully functioning team working together to complete a meaningful task are great. Be flexible in considering student needs and schedules while simultaneously seeking ways to help students build effective teams.

8. Use a variety of strategies to assess homework

There are many different actions that an instructor may take when reviewing a homework assignment. Consider the following ideas and select the strategies that make the most sense for each assignment.

  • Classroom conversation. Ask students to share their ideas in small groups. Students may then be randomly selected to share and explain something that was discussed in the small group
  • Collect and grade. Homework should only be collected and graded if students will receive meaningful feedback from the instructor. This may be done a handful of times during the semester.
  • Paired board work. Students may work together with a partner to write solutions to problems on the board for discussion with the whole class.
  • Check for effort. Depending on the purpose of the assignment, an instructor may check only to see if students have worked on making sense of the ideas.

As students begin to feel comfortable with the classroom environment and begin to engage in mathematical thinking, the instructor helps focus the students on the most critical mathematical concepts (the “big ideas”). In the textbook, the big ideas are identified at the start of the section as learning objectives. By including problems and activities aligned with these learning objectives as part of the classroom instruction, instructors reinforce the importance of these core concepts. Empowering resources for the instructor include exercises not assigned for homework, activities from the Classroom Activity Guide, worked-out examples from the text, and activities from the interactive lecture materials. The instructor may use these resources for on-board examples or small-group activities.

When students are working in teams on an assigned task, the instructor facilitates the learning and mastering of the big ideas by doing the following:

  • Redirecting student thinking through effective questioning
  • Posing alternative ways of thinking and alternate strategies to be considered
  • Providing a mini-lecture to fill in gaps in student understanding
  • Asking a student to share his/her thinking with the class
  • Requesting that a student paraphrase, clarify, or extend another student’s thinking
  • Requiring a student to demonstrate her understanding of the problem using multiple representations
  • Allowing for the pursuit of a dead-end strategy recognizing that this helps to develop persistent problem-solving

At the conclusion of a group activity, student presentations should be used to motivate mental engagement and reinforce the newly acquired learning. Students are randomly chosen to represent their team’s work. Students can use a hand-held whiteboard or a document camera to display their work. While a student is presenting, others in the class actively listen and ask themselves, “Do I agree?” and “Does this make sense?” Students need to feel comfortable and encouraged to disagree and to challenge what is presented in a constructive and positive way. Some activities may extend over multiple class periods.

Student-centered classrooms are characterized by more thinking than remembering, In these classrooms, students do much of the thinking and help shape the day’s lesson consistent with the learning objectives. By being simultaneously flexible and focused, the instructor allows students to explore yet refocuses them as needed to ensure the big ideas are addressed. The instructor continually remembers that the purpose of the lesson is to make sense of mathematical ideas and use those ideas to set up and solve problems.

The development of an effective class culture begins on the first day of class but may take several class periods to develop. Students are typically uncomfortable working with people they don’t know. Team building activities are excellent ways to break down the barriers and begin to build relationships. During the first several class periods, students should work with as many different classmates in teams as possible. This can be achieved by assigning students to teams by their favorite foods, birthdates, number of siblings, shoe size, and so on. Once the day’s teams are assigned, conduct a 10-minute class activity with the intent to get students talking to one another in a discussion that may not be related to mathematics. By allowing students to interact on a level playing field in terms of knowledge and experience, students learn to share ideas, debate, challenge, and argue rather than worry about what they will look like to their classmates in terms of their mathematical “know-how”.

The goal of these team building activities is to build a culture that will transfer to mathematical learning. Time spent developing an effective classroom culture through team building pays dividends throughout the course. The deep learning that occurs when students are engaged in an empowered learning environment prevents having to re-teach material that would not have been learned in a less engaging, instructor-centered environment. An added benefit of the team building activities is that students learn how to work effectively in a team, negotiate conflict, and communicate clearly.

The physical setup of the classroom helps to define the classroom culture. Ideally, furniture should be arranged to allow for student-to-student interaction. The following diagram shows one classroom layout that facilitates student interaction and group work.This setup also facilitates the student use of hand-held whiteboards. These whiteboards make it easy to display student work for presentation and discussion. Although small whiteboards may be purchased commercially, they can also be hand-crafted from large sheets of the whiteboard material purchase at a local hardware store. A sketch of sample whiteboard is shown in the following figure.

Mathematics classrooms where students sit passively, working in isolation to copy notes, should be a thing of the past. Instead, the classroom should be alive with collaborative problem solving activity. Current professional standards (AMATYC and MAA) suggest that an active classroom learning environment should be the norm. Too often instructors and students spend much of their time focusing on basic computational skills rather than engaging in mathematically rich problem-solving experiences. Worthwhile mathematical thinking occurs when students struggle and learn to overcome initial frustrations to make sense of powerful ideas while solving realistic problems. Developing this type of thinking requires a classroom environment where students are comfortable sharing their correct or incorrect ideas in the public forum of a classroom setting. However, such an environment does not occur naturally: it must be developed by the instructor. The ideal classroom learning environment is characterized by the following attributes.

  • is positive and cooperative
  • piques student interest and provides the opportunity to enjoy learning
  • creates a strong personal sense of motivation and responsibility
  • encourages relevant questions and discussions
  • helps students learn that making mistakes is a natural and acceptable part of the learning process
  • develops student confidence in making sense of mathematics
  • produces an understanding that correctness is determined by the logic of mathematics, not by the instructor,  a “smart” student, or the back-of-the-book

College Algebra: A Make It Real Approach  and Precalculus: A Make It Real Approach use real-world scenarios to draw out mathematical concepts. This approach motivates students to learn and helps them see the relevance of mathematics in their personal lives. Using real-world contexts helps instructors focus on student understanding and sense making. As students make sense of foundational concepts, the procedural aspects of mathematics become relevant and emerge as an important part of the problem solving process.

With the rapid advances in technology, today’s learners need to be (1) willing to attack “tough” problems, (2) persistent, flexible, and creative in solving problems, and (3) effective in communicating. With these textbooks, students learn to creatively and effectively solve problems using numerical, graphical, and analytical methods. The examples and exercises inspire students to use novel and creative approaches. Students communicate their understanding by describing their thinking, challenging the thinking of others, and justifying their solutions. These textbooks promote student collaboration as well as student/instructor interaction.

Too often students view mathematics as a parade of isolated facts, skills, and calculations to be memorized and quickly recalled instead of a coherent and well-connected body of knowledge to be understood. These textbooks were purposefully designed so that students will think and solve rather than memorize and recite.

Creating an effective learning environment involves classroom culture, classroom layout, building teams, teaching big ideas, effective homework strategies,  and learning assessment strategies.