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## by Cindy Wyels, California State University Channel Islands |

Worksheets are an effective tool in ongoing efforts encouraging our students to engage their brains during class. Worksheets used in class can also help direct students' learning out-of-class. The following list, with links to discussion and illustrative examples, gives examples of goals that can be addressed by using worksheets.

- Helping students focus on an underlying ?big picture?
- Bridging the gap between watching and doing
- Focusing students? attention in class
- Delivering and/or summarizing content efficiently
- Encouraging students to communicate their mathematical ideas
- Teaching students how to learn from their textbooks
- Connecting new material to previously-covered material

One obvious disadvantage to incorporating worksheets into one?s teaching is the extra time that must go into creating them. However, I find that an imperfect worksheet often serves a particular purpose almost as well as a ?perfect? one ? a finding that relieves the pressure of finding sufficient time to create the perfect worksheet. And of course, worksheets can be revised and improved in subsequent semesters when repeating a particular class. But even better, worksheets ? as with many teaching innovations ? can be shared with and improved by colleagues. Mathematics faculty at my institution habitually share their course materials with colleagues, explicitly giving each other the right to revise and reuse worksheets. I often return to a course after an absence of a few years to find that my colleagues have substantially improved some worksheets that I initially created. Joint authorship, both simultaneous and asynchronous, brings additional benefits. Sharing worksheets often spurs conversation about teaching ? for instance, about course goals and how best to achieve them. Additionally, using another?s worksheet and attributing credit properly provides our students a subtle model of how to work together in an academic setting.

A less obvious drawback to using worksheets pertains to class time. It's all too easy to underestimate the time students will need to thoughtfully work their way through a worksheet ? and this is our valuable class time! Often the process reveals underlying problems: gaps in knowledge or skills that I assumed they already had. (Of course, identifying such knowledge/ skill gaps provides any instructor some useful information.) This time-related drawback is alleviated as I gain experience. Getting students comfortable with the idea that they won't always finish a worksheet also helps: I often have "straight-forward" and "more-involved" sections on one worksheet, and tell the students that I expect them to work through the former, then move on to the latter as time allows. This strategy also serves to keep the more advanced students from becoming disengaged while they wait for their classmates to finish. (Example: integration strategies)

These two costs of using worksheets are both mitigated over time, and are outweighed by the benefits. Students? benefits can be inferred from the list above. Students seem to appreciate the extra effort that goes into creating worksheets: our student evaluations typically mention worksheets in a highly positive manner. And faculty benefit too: from enhanced faculty interaction (as we share our worksheets), from becoming more informed as to what students are ?getting,? and from that same positive feedback on evaluations. Of course, our real benefit comes from the professional satisfaction in knowing we're doing all we can to help our students learn mathematics.

**Acknowledgment**

Many thanks go to Karrolyne Fogel of California Lutheran University for helpful conversations, fruitful worksheet swaps, and for always keeping the focus on student learning.

**Helping students focus on an underlying ?big picture?**

Students often view mathematical ideas as disjointed topics, where we see them as various manifestations of one concept. For instance, calculus students often choose to memorize summation formulas for different numerical integration techniques. I?d prefer that they consider all the techniques as consisting simply of dividing the planar region determined by the integral into various shapes ? rectangles, trapezoids, and figures formed by replacing the top of a rectangle with a parabolic segment. This unifies all the Rules typically taught in calculus into variations on a theme: subdivide the interval of integration, form the appropriate shapes, find the areas of these shapes, and sum. We work our way through a numerical integration sheet together, with them supplying the first two (review) rows of the table verbally and me leading them through developing the last three rows. (The pictures serve as reinforcement of which shape corresponds to which rule.) The worksheet provides a handy place for them to summarize the critical information. I make use of it during parts of two classes: first reviewing and introducing the different rules, then developing the material on error bounds.

Similarly, I use a worksheet to introduce the concept of a probability distribution function when teaching introductory Probability and Statistics. At this point the students have worked with probability mass functions for discrete distributions. Before handing out the worksheet I use a computer simulation to motivate a discussion about what is likely to happen when larger samples are collected from the same population. The pictures on the worksheet (with their notes) then help them retain our conclusions. I ask them about the properties of a probability mass function and write these on the board. We then figure out how to ?translate? these properties into analogous properties pertaining to continuous functions, which they then write into the spaces provided on the front side. Note that there are two copies of the back page: theirs is mostly blank, with just enough information for them to figure out what I?m asking them to provide. We then use what they know about probability mass functions to determine how probability distributions functions ?should? work. This process may take up to one and one-half class periods.

**Bridging the gap between watching and doing**

Worksheets can provide an excellent means of teaching mathematical conceptsthat are somewhat algorithmic. For instance, mathematicians generally consider proofs by induction to be the easiest kind of proofs to teach students: we can (almost) tell them exactly what to do! Start with the basis step, write out the inductive hypothesis, and use the hypothesis (as necessary) to complete the inductive step. What could be easier? Yet students frequently have trouble making the transition from watching us write an inductive proof to writing one themselves. My colleague walks her students through an induction worksheet that helps them understand what they should do at each point. A significant side benefit is that they then have a template for out-of-class work. As they become more comfortable with the ideas they use the template less.Of course, by exam time they're expected to have mastered the technique.

Similar "bridging the gap" worksheets can be handy when teaching epsilon-delta proofs, creating general solutions to systems of equations, and in fact, for teaching any multi-step solution procedures. Examples:breaking down the pieces needed to implement the Chain Rule, interpreting and applying the definition of a vector space, and working through the steps to carry out a reduction in order of a Cauchy-Euler equation.

**Focusing students? attention in class**

Examples introducing new material: Stokes? Theorem (first part), vector space definition, characterizing differential equations

Examples applying new concepts: Stokes? Theorem (second part), polar areas

** Delivering and/or summarizing content efficiently**

For example, Section 4.1 of Zill?s *A First Course in Differential Equations, with Modeling Application, 7 ^{th} Ed.* (Brooks Cole, 2000) introduces several new terms including the key ideas of linear independence and forming general solutions. I?ve used this Chap 4 terms worksheet to guide students through the main ideas of the section, pausing to have them generate responses and write out explanations. Other examples include the induction worksheet referred to earlier, and the pdfs_pmfs worksheet leading students familiar with probability mass functions to the analogous probability density functions.

**Encouraging students to communicate their mathematical ideas**

I use the integration strategies worksheet once the class has worked through sections on various techniques of integration one by one. Students go to the boards (or work at their desks if the classroom has insufficiently many boards) in groups of three. They are specifically and repeatedly instructed to complete the ?Novice? activity before attempting to evaluate any integrals: the focus is on analyzing integrands to determine which integration strategy is most likely to be effective. Groups must discuss their choices thoroughly and provide strong reasoning for them. They interact with the instructor and with other groups, defending their choices. I may send emissaries from one group to another. Most groups have time to begin evaluating several of the integrals and thereby to see if their chosen technique will work. Note that structuring the activities by different levels diffuses the tendency to race through to completion of individual integrals.

Students begin work on the polar integration worksheet individually. They?re asked to get as far as time (4 ? 5 min.) allows on the first four problems, then to compare their responses with their neighbors. Once pairs of students agree (or are stuck), they compare their work to that of another pair. The whole class then discusses the first four problems ? often by having pairs present their solutions to the class ? before moving on to the last two. The whole process takes about 20 minutes. Separating out the process of setting up the region of integration breaks down polar integration problems and helps the students feel comfortable with later multi-step problems.

** Teaching students how to learn from their textbooks**

Examples: a worksheet introducing Stokes? Theorem, a worksheet designed to be used when working through Section 4.1 of Zill?s *A First Course in Differential Equations, with Modeling Application, 7 ^{th} Ed.* (Brooks Cole, 2000).

**Connecting new material to previously-covered material**

Learning theory shows that students absorb new material best when they have mental ?hooks? upon which to hang new ideas. These hooks, of course, consist of previously-grasped concepts; making connections between concepts equates to hanging the new idea on an appropriate hook. Making connections can be achieved by asking students to review, and possibly rephrase, previously-covered material at the beginning of a worksheet, and then to use this material to investigate new ideas. For example, a worksheet introducing Stokes? Theorem directs students to review Green?s Theorem in the context of a 3D vector field with zero *z*-component. Through completion of the worksheet and simultaneous discussion in class, students then approach Stokes? Theorem as a generalization of Green?s Theorem. Similarly, students in an introductory Probability and Statistics class begin study of probability density functions by reviewing the key characteristics of probability mass functions, then tying these ideas together. See p. 2 of this example.

Cindy Wyels (cindy.wyels@csuci.edu) is an Associate Professor at CSU Channel Islands, California's youngest state university, where she directs the MS in Mathematics program. Her BA is from Pomona College and her Ph.D. is from U.C., Santa Barbara. Her recent research interests have focused on graph pebbling and graph labeling, and she strongly advocates student participation in research. Her pedagogical interests include incorporating technology to aid student learning, improving students' communication skills, and increasing the participation of underrepresented minorities in mathematics.

The Innovative Teaching Exchange is edited by Bonnie Gold.