What’s the Difference?

Two high school math teachers, both adept users of technology, work to engage their students in Algebra and Calculus. They use a variety of representations, including graphs, symbols, physical models, tables, written descriptions, and animations to explore math topics.

Engaging Students in math.


Melanie Coughman braces for her most difficult class.

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Melanie works diligently to help students master algebra concepts.

Melanie had been at school since seven-fifteen that morning, as usual. Her routine included a 5:45 AM kick-boxing class at the gym, a quick shower, and then at least an hour of uninterrupted work in her classroom before students arrived.

Since this was her hard teaching day—a morning section of Algebra II, followed by Honors Geometry, planning, another section of Algebra II, and finally an eighth period tutoring session—her planning time was especially important. She valued the standards set up by NCTM and worked hard to integrate many different presentations of material, especially in her low level classes. These students were S-L-O-W. It seemed to take forever for them to master the simplest of concepts, concepts her honors kids could grasp in a fraction of the time.

Overview: Principles for School Mathematics

As the morning announcements began, Melanie gritted her teeth and tried to hold on to the already fleeting sense of euphoria she’d felt after a great work-out and a productive hour of prep. She hoped her students’ lackluster reaction to the announcements was not an indication of how the rest of the block would go, but her expectations were low.



She’d already spent weeks on quadratic equations and today she was reviewing for the final quiz scheduled for next class. She’d worked hard throughout this unit to help students develop procedural facility and a conceptual understanding of the topic, using symbols, graphs, tables, and written descriptions of quadratic equations. Taking advantage of all the tools at her disposal—a permanently mounted SmartBoard, a graphing calculator hooked up to the TV, and easy access to a mobile laptop lab—she had provided many different ways for students to access the material. Today’s review should just be a quick exercise that would allow students to show their mastery of the different representations of quadratic equations.

Melanie’s pacing guide.



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Sei-Sei struggles through another math lesson.

Stacey, or Sei-Sei, as she liked to be called, was a senior—in Algebra II, a course most students finished by the end of their sophomore year. This was Sei-Sei’s second time taking the class—she failed on her first try—but she’d been doing the work this time around and passing. She still hated the class, though—that came through clearly. Melanie had decided early on that it was better to ignore Sei-Sei’s attitude. In her experience, there wasn’t much a teacher could do to help a student that closed off.

A journal entry Sei-Sei wrote about her most difficult class.

Closing the Achievement Gap.



There were bright spots, of course. Her trio of girls reminded Melanie of more advanced students—maybe they weren’t mathematically gifted, but they cared and put some energy into the class. It made them so much easier to work with.



Melanie dutifully made sure to check in with each group at least twice. She couldn’t wait to finish this unit. She wouldn’t be able to tell if it was a success until the quiz results were in, but she was keeping her fingers crossed that enough of them would pass, and they could finally move on.



Out of habit, Melanie saved the SmartBoard notes from today’s lesson. Sometimes she pulled these up during her eighth period tutoring session to review a concept covered in class that day. She hoped they would be ready to tackle the next unit after the quiz.

Melanie’s SmartBoard notes for the lesson.

Diana Hunt reflects on her role as an experienced mathematics teacher.

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Diana’s many responsibilities as math department chair occupy her time both in and out of school.

Diana Hunt liked to be involved. She’d taught ever since she graduated from college, more years ago than she liked to say. Back then, many women quit the profession after they got married, but she’d loved teaching too much to give it up.

So much had changed since those early days. Now Diana was chair of the department, taught a math methods course at the college, mentored student teachers, and attended every NCTM conference she could. In a typical day, she could go from teaching AP Calculus to general level geometry to coaching a struggling math teacher. She loved this variety and participating in a dynamic profession.

As the students began to file into the room for Calculus, Diana had the quickening sense of excitement that she still got before every class. Diana was especially pleased with this lesson that had evolved over the years to include many forms of symbolic, graphic and physical representations of concepts that could be so difficult for students to understand. She didn’t waste a lot of time on chit-chat but immediately began reviewing homework and showing an animation of topics from the previous class.



Diana held an optional weekend math session for her AP students at a coffee shop one afternoon each month. Sure, not all students came, but about a dozen showed up each time. Twelve out of fifty AP Calculus students wasn’t too bad! They would answer questions and challenge each other to tackle problems well beyond the scope of the course. They had fun, too! The cozy environment, warm lattes, and fresh muffins, gave them a chance to talk more than they could in the classroom.

That’s how Diana finally got to know one of the most promising math students she’d ever taught. Valentina’s family had emigrated to the United States while Valentina was in middle school. She quickly learned English and excelled in all of her classes, but her teachers often commented on her shy demeanor and reluctance to speak up in class. Valentina did not attend the first couple of Diana’s coffee shop sessions, but once she came to one she didn’t miss another. She quickly warmed to Diana and eagerly discussed various approaches to problem solving—both in and out of class. Diana could count on her to volunteer solutions to the most difficult problems and just a month ago, Valentina had won a state-wide mathematics competition.



Valentina seemed surprised when Diana first asked if she had considered studying mathematics in college. Diana gave her a copy of an address to Congress written by a young actress who had starred on a popular TV show called The Wonder Years and later went on to major in mathematics in college. Valentina wasn’t familiar with the actress—she hadn’t even lived in the U.S. when the show aired!—but she said she felt an immediate connection with the woman’s words. Valentina was now waiting to hear from several colleges with excellent mathematics programs and was considering pursuing a career in mathematics.

Speaking at Congress!

Diana could lose track of time sitting at the computer, looking for interesting demonstrations to show her students. She often visited the Math Forum Internet Mathematics Library site and sometimes made postings herself. When she heard the excitement in her students’ voices after seeing one of the examples she found online, it made the hours of time she still spent preparing for classes, even after all of these years, worth it.

The Math Forum@Drexel: Calculus.

When Diana thought back to how she’d presented calculus even ten years ago, it all seemed so antiquated. She had been so excited when she got wooden models of solid shapes, and she had made her own models out of carpet strips when she couldn’t find anything else that worked. Even though some of the physical models were heavy and difficult to wield, she still used them in her instruction. They were a solid complement to the displays and animations she projected using the SmartBoard. The combination of physical models and animations she was using in today’s lesson would be particularly helpful because the topic could be so difficult for students to visualize.



Once Diana gave the students a chance to work on practice problems, she glanced around the room. Most of her students seemed engaged, and she was able to help a few students who needed additional assistance. She was sure her students would be able to understand, speak and write about finding the volume of a solid of known cross section after her presentation today.



After school Diana had a curriculum committee meeting to begin collaborating on a review of the district’s mathematics scope-and-sequence. They had the difficult assignment of determining a structure for and sequence of mathematical topics that would be rigorous and coherent. Earlier, Diana had found an interview to share with her colleagues that explained coherence in terms that mirrored her own classroom experiences.

A Conversation with: William H. Schmidt on Mathematics.

Diana also had a meeting with a new math teacher during their shared planning period. Faculty turn-over had been so high, especially among math teachers, that the administration had implemented a mentoring program this year, and so Diana had added “mentoring new teachers” to her list of responsibilities as department chair. And, boy, did they need mentoring! She didn’t remember her first year of teaching as overwhelmingly difficult as so many new teachers seemed to find it. Sure, she sometimes had doubts about whether she was making a difference, but overall, she was pretty sure her impact on the profession and her students was positive.