Category Archives: Reviews

Review: Desmos

Desmos is a powerful online graphing calculator available on all current platforms. There is a good Quick Start guide:

Desmos is integrated somewhat with Google Drive. “Integrated somewhat”? The third-party Chrome apps are generally not integrated as invisibly as the built-in apps (Docs, Spreadsheet, Presentation, etc.). In the case of Desmos, the integration looks like this:

  1. Desmos appears as an option when you use the Create option in Google Drive; however, a blank Desmos graph then appears, with nothing created in Drive.
  2. If you want to save anything from Desmos, you need to log in. Desmos lets you log in with your Google account.
  3. You can save a graph you’re working on, but it isn’t saved to Drive by default. (It’s saved in a special set of Desmos files, somewhere in the cloud, associated with your Desmos account.)
  4. If you choose to save a Desmos graph to Google Drive, it is saved there as a PDF. You can print this PDF, but you can’t open it and make changes to it, or use it interactively.

The assignment

I wanted to immerse my Algebra 2 students in graphing parabolas. (This topic is directly addressed in the CCSS standard CCSS.Math.Content.HSF-IF.C.7a.) I had them graph some simple examples on paper using a few sample points. They were required to label the vertex and the axis of symmetry. They then took pictures of their work using the built-in camera on the Chromebooks; they pasted these photos into a Google Document.

I reviewed their work, leaving comments in the document itself. The next day, they used Desmos to graph the same parabola. They were required to paste the Desmos image into the document, after labeling the vertex and axis of symmetry. I moved their final documents into a read-only folder and added a final comment with their overall rubric-based grade.

Working with Desmos in Google Drive

Each application (non-Google and Google) has a slightly different way of handling exports and imports. Once you’re logged in to Desmos, a Share Graph button appears in the toolbar at the top. There are four different export options, three for sharing the interactive graph itself (using a short URL) and one for sharing an image.

I asked my students to export the image. It then appears in the Downloads folder in the Files app. In their original document, they then used Insert Image | Upload to select the image from Downloads.

Desmos does not provide a way to add text to a graph, so the students were left to figure out for themselves how to do this. I envisioned them using Google Drawing, and most students did this, “crowd-sourcing” the solution. They pulled the PNG file created by Desmos in their Documents folder into Drawing, added the text, saved the Drawing, and imported that into their Google Document (using the web clipboard). A few used the third-party app Pixl Editor for its greater capabilities.

The screenshot below demonstrates my commentary, the graph itself (annotated by the student), and a file-naming convention.

Screenshot 2014-01-02 at 8



Cell Phones for Learning


The cell phone debate rages in classrooms, staff lounges and kitchens around the world: How can we keep these devices of the devil from derailing learning? Cell phones have the power to disrupt a classroom in many ways: students forget to turn them off and they ring in class, students use them for sending and receiving text messages (whose content may be innocuous and social or may sometimes be more serious, such as personal threats or drug deals), they get stolen or lost and instruction time is lost to their retrieval.

Literature Review

A review of the literature reveals two sides to the discussion. On the first side are the people who reluctantly admit that campus security requires that students have cell phones with them turned off and inactive except in emergencies. Opposing this view are the people who believe cell phones should be embraced as classroom tools that can engage and motivate students better than traditional methods and tools.

Historically, as cell phones became smaller, cheaper and more commonplace over the last 15 years, phones were clearly disruptive in the classroom and had no redeeming value; schools throughout the country banned them outright. After the events of Columbine and 9/11, Beth Lynne points out in her essay “Cell Phones and the Classroom: Schools and Parents Should Adopt Clear Policies to Benefit All,” parents and students began to push for more moderate policies, recognizing the safety benefits of having personal devices for instant communication (Lynne, 2007). However, the school districts’ limited acceptance of cell phones in schools raised as many questions as it resolved. Lynne’s article enumerates a few of these:

  • Should students be able to call their parents during class and have them listen in on the teacher’s reprimands to the class?
  • Should camera phones be allowed?
  • Should students be required to leave phones at home during state-mandated testing (makes cheating easier)?
  • Have there really been emergencies in which a student’s cell phone has been the savior?

Collette Georgii (2007) recommends the simple step of checking in phones with the teacher on entrance to the classroom, the purpose of which is to minimize disruptions due to confiscation or students’ sneaking phone activity. Apparently, this is practical for her, with cell phones previously labeled with name, address and homeroom.

As some teachers have noticed, although we block YouTube and confiscate MP3 players, there is a lot of truly dangerous technology that we allow students to work with, including rulers, solvent-loaded markers, and pencils. In an essay in, Wesley Fryer (2007) lists the rules we have developed over the years to accommodate having scissors in the classroom:

“No one (regardless of age) is permitted to run with scissors. Use of scissors to threaten or injure others is not tolerated. We keep scissors available in our classrooms to use at need, but we recognize the menu of uses for that tool must be limited by the attitudes, language, and actions of multiple stakeholders in the educational learning culture, not merely the teacher of record in the classroom.”

Outside the U.S., restrictions against classroom cell-phone use have fallen; in fact, many countries have embraced the new technology, especially in the university setting (Chinnery, 2006). Pilot programs in Korea, New Zealand, Australia, China and Japan have used SMS (text messaging) for polling and, most commonly, for second-language learners.

In contrast to the American teachers and administrators who reflexively restrict cell-phone possession and use, many teachers have made themselves into cell-phone proselytes. One of these is Marc Prensky, who has built an entire industry (web site, online persona, consultancy, dozens of articles throughout the web) around his own personality of Web 2.0 expertise (Prensky, 2006). Among other cell-phone ideas, Prensky suggests poetry games, phone-in caption contests, and classes conducted only via text messaging.

Probably the queen of cell-phone proponents is Liz Kolb, a former classroom teacher, district technology coordinator and current university professor. In her one-hour presentation, “Cell Phones as Classroom Learning Tools” (2007), she makes the case that cell phones have the capability to become the “Swiss Army Knife” of technological tools for the 21st-century classroom. Among the many creative cell-phone applications Kolb discusses on her blog, are the following:

  • “buzzers” (as used in classroom performance systems such as Qwizdom) where individual students or teams text messages to a special email account set up by the teacher;
  • cameras to be used in scavenger hunts (especially over winter and spring breaks);
  • clients for Powerpoint presentations;
  • mini-recording studios for oral history and other audio projects;
  • instant recording and upload devices for video projects (in conjunction with free video-storage sites such as YouTube and Eyespot.

Position Statement

Kolb’s enthusiasm and creativity is infectious. Her presentation takes viewers through many examples of integrated cell-phone use, in full detail. I was able to set up and use a free account at Gabcast in less than five minutes; I then called an 800 number from my cell phone and recorded an “episode” on Gabcast that was automatically published with a simple URL. She even addresses what you would do when you have a classroom with less than 100% cell phone ownership (give group assignments) and makes suggestions about minimizing families’ costs for cell phone use.

Unfortunately, Kolb doesn’t address two huge issues:

  • How will you get administration buy-in for in-classroom cell phone use?
  • With students using cell phones in the classroom, how can you keep them on the assignment, vs. playing around?

I suspect, the first issue would be simpler to address. You could begin by integrating cell phone assignments into your plans where phone use is only outside the classroom and work your way toward in-class use. When you’re ready to use phones in the classroom, you could either not mention the cell-phone use to the administration or fully document past use and your upcoming plans, overwhelming the administration with your massive organizational skills.

Regarding the second issue, how do we keep students on task currently? Whatever the system is, we know it’s imperfect: students continue to pass notes, throw paper airplanes, chat with each other, and … yes, even discreetly use cell phones for texting.


Chinnery, G. (2007). EMERGING TECHNOLOGIES. Going to the MALL: Mobile Assisted Language Learning. Language Learning & Technology. 10(1). 9-16.

Freyer, W. (2007). Scissors and Cell Phones. Retrieved December 2, 2007, from

Georgii, C. (2007). Teacher tips: How to deal with student cell phones in the classroom. Retrieved December 2, 2007, from

Lynne, B. (2007). Cell Phones and the Classroom. Retrieved December 2, 2007, from

Kolb, L. (2007). Teen Content Creators and Consumers. Retrieved December 2, 2007, from

Kolb, L. (2007). From Toy to Tool: Cell Phones in Schools. Retrieved December 2, 2007, from

Prensky, M. (2006). Using Cell Phones in School for Learning. Retrieved December 2, 2007, from


Review: “Computer support for learning mathematics: A learning environment based on recreational learning objects”

In this era of ubiquitous Internet access, cell phones, PDAs and other digital technologies, today’s teachers face a time-honored dilemma: students who won’t think in school and who avoid homework will spend all their free time on something recreational yet mentally challenging, especially Massively Multi-Player Online Role-Playing Games (MMORPGs). If we could only find an activity which motivated the kids to work as hard on learning as they do on playing!

The authors of the paper “Computer support for learning mathematics: A learning environment based on recreational learning objects” may be on to something. They describe an “electronic collaborative learning environment” and report on its power to motivate high-school math students.

The environment Lopez-Morteo and Gilberto López have built is essentially a portal populated by “portlets” (portal elements), their word for open-source objects such as Jabber (chat rooms and instant messaging) and email clients. In addition to these familiar objects, there are math objects called “Interactive Instructors of Recreational Mathematics” (IIRM). These include games, simulations and other applications, designed to encourage student involvement through problem solving.

Students log into the system and can customize the appearance and contents of their environments – exactly as users of MySpace or other social networking sites might. The authors describe a specific math object, a memory game built in Java called “ArithMem.”

Having established a full-featured environment, the authors tested its ability to motivate math students. Groups of students logged onto the system, watched a lesson presented by the teacher, and then used the interactive elements of the system (programs, spreadsheets, animations) freely. Students then filled out a survey about their attitudes toward mathematics.

Although the authors seemed satisfied with the results, I did not see any convincing statistical evidence that the environment served its primary goal: to motivate students to learn math. Rather than overwhelming evidence collected in the survey, the authors provided opinion and a few weak statistics to support this claim, along with anecdotes to back it up. Nevertheless, I would probably use such a system if I had access to a computer lab for a year and a set of classes with which to try it.


1. Lopez-Morteo, G., and López, G. (2004). Computer support for learning mathematics: A learning environment based on recreational learning objects. Computers & Education, 48(4), 618-641.

Review: “Building Equations Using M&M’s”

The article describes an activity to support using algebraic techniques to solve linear equations in one variable, intended to directly follow instruction in equation-solving techniques. The author claims that the activity “actively involves students in identifying the variable, formulating an equation, and then solving the equation.” (Borlaug, 1997)

To begin, the teacher distributes small bags of M&M’s to each student and keeps a bag to use for him or herself. The teacher counts each color in his or her own bag and draws and completes a table on the board listing each color and the number of that color found in the bag. The students count their own M&M’s and make their own tables, but they keep their results secret.

The students then receive a list of questions, designed to be set up as equations and solved. The list begins with easy questions that the student will be inclined to “just answer” without algebra; the questions get progressively more difficult, encouraging the students to use algebra to solve them. The teacher selects a student and asks that student to complete one of the questions and then pose the question to the class for their consideration. For example, “I would have to add (or eat) ____ red candies to have the same number of red candies as the teacher. How many red candies do I have?”

The rest of the class works on a solution to this problem. As the questions become more complicated – “If I tripled the number of yellow candies I have, I would have ____ more candies than the teacher. How many yellow candies do I have?” – many students will decide that the algebra techniques they’ve been studying are – surprise, surprise! – useful for solving these problems. For those who don’t gravitate to algebra but who are stuck, the teacher provides friendly nudging. So, the next phase is to build and solve the appropriate equations. A side effect of this effort is that students sometimes discover that there is more than one appropriate equation, depending primarily on what color they choose for the variable x.

The author finds that after the main activity, some classes will be comfortable moving on and inventing their own problems. Here’s one actual example from the article:

I have a total of 61 candies in my bag. I have 9 more brown candies than orange candies. If I eat all my brown and orange candies, I will have 32 candies left. How many brown candies did I eat? How many orange candies did I eat?

This example represents an impressive level of involvement with the material; however, a number of issues come to mind as I consider the realities of using this activity in my own classroom. I would be concerned about my students just eating the M&M’s. I’m thinking of one student in particular, who has no self-control and who is frequently out on suspension for various discipline incidents. He has learned almost none of the algebra techniques that the rest of the class has picked up and so would not naturally be able to participate in this activity without hand-holding. An obvious approach would be to have students work in small groups; however, in this case I can’t imagine him contributing to a group larger than two. Even in pairs, it’s hard to imagine this student not just eating the materials. (I’ve seen him eat paper. Could he resist M&M’s?)

Another drawback I envision in a high-school classroom would be the non-traditional math student who prides himself on creative ways not to do the assigned work. If this student has access to the Internet, he could easily look up the distribution of colors in an average bag of M&M’s and use that to “cheat.” Of course, this would be more complicated than just doing the work and would only lead to approximate answers (due to the actual randomness of filling an M&M’s bag at the factory), but the student would have found another distracting way to avoid learning the material. This scenario is unlikely in most classrooms, especially a middle-school classroom such as mine.

Also, I’m not comfortable with giving candy to kids. Is that what the parents want? What about the teachers who have these kids later in the day? And do I want to fight the trash battle? But are there alternative manipulatives? They need to be similar in size and shape to each other and randomly distributed with a limited number of colors. I’ve seen little plastic colored circular pieces, like transparent tiddly-winks, that are made for overheads. Maybe the simplest solution is colored construction paper cut into one-inch squares and mixed together.


1. Borlaug, Victoria. (1997, February). Building Equations Using M&M’s. Mathematics Teaching in the Middle School, 2(4), 290-292.

Review: “If I Only Had One Question: Partner Quizzes in Middle School Mathematics”

The authors of “Partner Quizzes” have followed the advice of the Assessment Principle in the NCTM Principles and Standards by developing and administering assessments which are “not limited to individual, graded, end-of-unit examinations” (Danielson and Luke, 2006). In the middle of a unit, students work in teams of two and produce one quiz for each individual over two days, obeying strict rules of collaboration. According to the authors, “Partner quizzes are useful as formative assessments that help us to monitor and adjust our instruction during the remainder of the unit.”

The authors state explicitly two crucial advantages of these assessments: they demonstrate that working together is important, even (especially) during assessment; and they provide an environment in which “discussion is essential to doing good work.” They claim that because of the structure of the activity, the questions are “deeper and more complex than on an individual assessment.” These characteristics would make the assessments attractive to any teacher.

The process has three strict rules:

  1. Partnerships are completely private and exclusive. Teams are only to work with other team members. The teams can choose their own work style, but there are strict consequences for collaboration outside the team.
  2. Each team is allowed to ask the teacher exactly one question during the quiz.
  3. Teams work on their quizzes on Day 1, and then submit both copies for teacher input. The teacher selects one quiz from each team and provides cryptic feedback, meant to provoke further discussion during Day 2 without guiding the students too directly.

The authors find the “one question” rule valuable because it forces the teams to ration their requests, eliminating the superficial questions such as “What is an outlier?” in favor of the deeper questions such as “Can there be two outliers?”

The article includes two sets of actual quizzes, revealing within one class a range of understanding and sophistication that seems typical of any classroom. The authors analyze the student work as well as the feedback and revision process, making a relatively complex system almost easy to understand.

I’m eager to try this out in my classroom for all the reasons cited above. Certainly this isn’t the most important reason, but I’d like to see my students step up and not ask the same procedural question five times in two minutes. On a slightly deeper level, I’ve been looking for a way to pull the introverts and outsiders into the group so they can participate – and learn.


1. Danielson, Christopher, and Michele Luke. (2006, November). If I Only Had One Question: Partner Quizzes in Middle School Mathematics. Mathematics Teaching in the Middle School, 12(4), 206-213.

Review: “How to Buy a Car 101”

“The style of PBL units is designed for teachers who are willing and able to hand over control of the classroom to the students.” This chilling statement is found in the core of the article “How to Buy a Car 101,” an overview of Problem-Based Learning.

The article describes a project for 7th graders where they are given a few specific criteria for a fictional car purchase and then must research, assemble, present and defend their choices. The final product covers four state math standards, and is graded according to a simple rubric. According to the author, the teachers “no longer act as the experts; they serve as a resource.”

The authors’ students research the requirements and available options (car and financing) on the web, and then present their findings using Powerpoint. This is not feasible in all classes, partly due to the classroom setup (few computers, little experience with Powerpoint) and partly due to limited experience by the students. (It’s been my experience that at the middle-school level, kids are still as a whole relatively unsophisticated regarding technology. In my own class recently, I had to explain what Excel was.)

The project is fascinating, and my impression of it is that if you can pull it off, it’s wonderful for stimulating discussion and high-order thinking. Some drawbacks, as I see them:

  • For 7th graders, the project is relatively complicated, including the commuting distance of the fictional car purchaser, is budget, his target down payment, and the current interest rate.
  • How much time do you want to spend teaching and coaching Powerpoint? And what about showing students how to use the web?
  • In some areas, students have no Internet access at home. Of course, this means they will have to use the library, but does this give an unfair advantage to the kids with broadband Internet in their own bedrooms?
  • Powerpoint is not available on all computers, and actually costs money to purchase. If a student has a computer at home, will he be required to buy a Powerpoint license?

I was happy to see the author list a set of warnings of her own. She says you can never be overprepared; you should get the help of the students for planning future units; and be sure not to do the students’ work for them.

One very interesting note: during the project, the students developed a relationship with local car dealers, who bring new cars to the school for the students to see. They also “kept in contact with the students about any incentives and promotions they might find interesting.” This sounds a lot like indoctrination and commercial promotion to children, and I would want to be careful with this.


1. Flores, C. (2006, October). Using Friday Puzzlers to Discover Arithmetic Sequences. Mathematics Teaching in the Middle School, 12(3), 161-164.

Review: “Using Friday Puzzlers to Discover Arithmetic Sequences”

The author teaches eighth-grade mathematics and writes about using related puzzles to lead students to discover the formula for the sum of an arithmetic sequence. The puzzles are presented each Friday, and students look forward to the experience each week.

The guidelines are presented before the first puzzler:

  1. Each puzzler can be completed within a class period.
  2. Your current mathematics knowledge is all you need to solve the puzzle.
  3. You will never be given a puzzle that is unsolvable.
  4. Often by first thinking about it, a technique will emerge that allow you to complete the puzzle in less time than if you plod ahead without thought.

Yolles uses the following five core Puzzlers:

  1. Gauss: A Child Prodigy. This is the story of Gauss’s 3rd-grade teacher assigning the class “find the sum of the numbers from 1 to 100” and having Gauss almost instantly produce the answer.
  2. Handshakes All Around. Ten friends attend a party where each person shakes everyone else’s hand, exactly once. How many handshakes occur?
  3. Lighting the Chanukah Menorah. Over the course of eight days, how many candles are lit in all on a menorah?
  4. The Twelve Days of Christmas. How many gifts were sent on Day 12?
  5. Clock Face Puzzle. Can you locate one straight line to split a clock face in two so that the sums of the numbers on the two parts are equal?

Students work on each problem for an entire class period, earning stickers if they solve the problem. The author says that few succeed at first, but the proportion of students solving each problem grows through the year. Puzzlers are not provided each Friday; the students take a few weeks between sessions.

The author provides samples of student work; her students show surprising creativity in their approaches to these problems of counting. She also describes Socratic conversations which attempt to bring together the results of multiple weeks’ work.

By the end of the Friday Puzzler sequence, she says, “all students have earned an ‘I got the Friday Puzzler!’ sticker,” and the class has discovered the formula S = (n/2)(a + l), where n is the number of terms in the sequence, a is the first term, and l is the last term.

I found the article fascinating and inspiring, and I look forward to using it in my own classroom.


1. Yolles, A. (2003, November). Using Friday Puzzlers to Discover Arithmetic Sequences. Mathematics Teaching in the Middle School, 9(3), 180-185.