AMATYC Keynote Notes: Interaction and Impasse

Nov 19, 2016 by

Thursday I had the honor of providing the opening keynote for the AMATYC Conference in Denver, “Learning Math is Not a Spectator Sport.” I expect the video of the talk will be available to share next week, and rather than provide the slides (124 mostly stick-figure drawings), I’ll point you to some resources that will likely give you the information you’re looking for between now and when the full presentation becomes available.

Selfie with room full of participants in the background

Keynote Selfie

We covered three main themes:

  1. Interaction & Impasse (this post)
  2. Challenge & Curiosity
  3. Durable Learning

I’ll provide resources for each of these categories, starting with Interaction and Impasse, in this post.

Interaction and Impasse


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Learning Notebooks for Online Math Homework

May 28, 2012 by

After teaching math at a community college for 10 years (and using online homework for at least 7 of those), I have noticed that my online math students don’t seem to have the same grasp on notation and the steps to “prove” the solution to a problem as when they did old-fashioned paper & pencil homework.  I have also found that the students who use online homework have become much more unorganized, and are unable to find the work for the problems they have questions on.


Example of student work in a Learning Notebook

This last year, I’ve been experimenting with what I call a “Learning Notebook” – where students keep an organized notebook of the handwritten work for selected problems from the online homework system. In these Learning Notebooks, the students have to show the steps required to complete the required problems (including all necessary graphs and proper notation).  They don’t have to keep a record of every problem, since some questions can be answered by inspection alone. For the Learning Notebook, I typically choose problems that would require me to show steps in order to complete it with a reasonable confidence in my answer.   The online homework, graded on accuracy, is worth 20 points per unit.  The Learning Notebook, showing sound mathematical thinking and notation for required problems, is worth an equal weight of 20 points.

The student is responsible for keeping the notebook organized, including a Table of Contents and page numbers (these help me to find assignments when I go to grade the notebooks).  While this may seem like busywork, keeping a notebook has several benefits to the student:

  • When studying for an exam, the student can find the work associated with each problem quickly.
  • When there’s a question on a specific problem, the student can quickly find their version of the problem and what they tried.
  • Repetition of the use of proper notation leads to better outcomes on the exams (since they don’t “forget” to include the notation there when they are required to have it in their notebooks).
  • Thoughtful reflection on the problem steps may be more likely when they slow down to write the steps down instead of trying to do too much in their head.
  • Students get points for showing their work, which can act as a slight padding of their grade when the tests are hard (which they inevitably are).
One of the additional benefits of the Learning Notebooks is that it gives me a “place” to collect additional assignments that can’t easily be covered by online homework.  For example:
  • Sketching the graph of a function given a list of properties
  • Explaining the transformations of a graph in multiple steps
  • Proving that a series converges or diverges
  • Explaining all the properties of a rational function

A collection of Learning Notebooks on exam day.

For my traditional classes (that include an in-person meeting) I grade the Learning Notebooks while the students give the exam. I select ten problems at random to check for completion, notation, and supporting steps.  I typically give a 2-hour exam, and I can grade the notebooks for 15-25 students by the end of that time.  This is when it becomes vitally important to me that the students include a Table of Contents and numbered pages.  Without those, I would spend a lot of extra time searching for assignments.  I use a 0-1-2 point scale for each of the ten problems.

  • 0 points = the problem cannot be found, there was only a problem and answer,  or there was no reasonable attempt to solve the problem
  • 1 point = some reasonable attempt to solve the problem, but details missing or problem is incomplete
  • 2 points = problem is completely solved, with all appropriate details included
After I have worked through all 10 problems, I give the student a score out of 20.
To help you understand the process a little better, I asked a few students to let me share their notebooks and the grading process.  They agreed, so here’s a little video explanation of how the process works.

Video: Learning Notebooks for Online Math Homework

Here is a Sample Table of Contents and Sample Notebook Check for the Learning Notebooks.

Because they have to keep a Learning Notebook, students know that they shouldn’t cut corners when they work through problems.  At first, many will try, rushing through the online homework (probably with the aid of calculators and WolframAlpha) with the belief that they will just “take a few minutes to go back and write up the steps.”  For this reason, you shouldn’t be surprised if the grades for the first set of Notebooks are pretty bipolar (half will be great, half will be awful).  It turns out that to actually think through and write the math takes time, time that some of these students have been cutting corners on ever since online homework was first introduced.

I’ve been using these notebooks in Math for Elementary Teachers, College Algebra, Calculus I, and Calculus II over the last year, and have seen an improvement in mathematical thinking, use of notation, and study habits for those students that keep good notebooks.  I don’t have any scientific evidence, but overall, I feel like these Learning Notebooks are helping improve my students’ success.

NOTE: In about a week, I will share how I’m using the same strategy in my online classes.  I want to get all the way through the process of collection once before I write about it.  Hint: It involves webcams and cell phone cameras.

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Math ELITEs (Classrooms for Active Mathematics)

Feb 24, 2010 by

Thanks to Diane, Gary, and Tom … who also contributed ideas to this classroom redesign project idea.

Objective: Create classroom spaces specifically for a) actively learning mathematics and b) using technology to demonstrate, teach, and learn mathematics.


A Mathematics ELITE is an Engaged Learning Interactive Technology Environment and consists of:

1. Multiple Whiteboards

There should be enough whiteboards in the room so that 24-30 students can work in pairs at the boards. One set of boards should be lowered so that shorter students or a student in a wheelchair could participate more easily (another modification could be to use a portable whiteboard for disabled students).


Students rarely learn mathematics from copying the instructor’s work. When students work on the whiteboards in class, it is relatively easy for the instructor to monitor the work of all student pairs at once, stepping in to answer questions, give hints, and correct notation. Students take turns being the writer and the helper, talking over the mathematics as they learn to solve new types of problems. With an interactive board in the room, one pair of students can record their work on the interactive board, creating a record (PDF file) of all the problems worked in class that day.


2. Document cameradoc_camera4

Can be used for displaying documents (i.e. worksheets, going over a test key, etc.). In a classroom with math manipulatives (i.e. fraction strips, base‐10 blocks, etc.) , these can also be displayed to the class via the document camera.  Scientific and graphing calculator demonstrations can also be shown using a document camera.  It can be helpful to have a mini whiteboard to use with the document camera.

3. Computer station, laptop connection, and ceiling projector

Many simulations and interactive demonstrations (e.g. NLVM, Wolfram Demonstrations) are now emerging on the Web as a way to demonstrate mathematics. A fixed computer station eliminates the problem of tripping over the cord and solves the problem of obstructed student views of the board. Through a laptop connection, instructors can connect their own laptop or tablet.

4. Ceiling speakers

Connecting math to real‐world applications can require extensive knowledge of other disciplines. However, the use of short video clips from the Internet can mitigate this (i.e. TED, Science Friday Videos, etc.).

5. Interactive Whiteboard

Math demonstrations can be shown by the instructor on a traditional computer. Using an interactive whiteboard (e.g. SMART board), students will be able to participate in the demonstrations up at the screen. Interactive whiteboards can be used to record the student board work (see #1). Lessons written on an interactive board can be recorded as video or as documents (PDF files). Many of our future elementary teachers will eventually be teaching in classroom spaces with interactive whiteboard technology, and it’s important that they begin to see how to use these tools effectively.


6. Math manipulatives and storage space

For many math classes (in particular, Developmental Math, Algebra, Math for Elementary Teachers, Excursions in Mathematics, College Algebra with Applications, and Statistics), the students’ understanding of mathematics can be enhanced by playing with math manipulatives. Manipulatives help students make connections between the physical world and abstract concepts. Some math manipulatives must be purchased and some can be assembled using everyday materials, but it is important to have some storage space for these close to the learning space.

7. Half‐round tableskidney-shaped-activity-table

Student seating in clusters instead of rows makes it easier to facilitate group work. Students must be able to view lessons on either the interactive whiteboard or the white board (placed on an adjacent wall), so half‐round tables are used. These tables are also nice because they provide the instructor a space to “drop in” on the group and check their progress by simply walking down the main aisles in the room.

8. Wireless Internet

Anticipating the likely possibility that most students will have a laptop, netbook, or smartphone capable of Internet access in the near future, wireless Internet is a good option for bringing computing power to the hands of students in the classroom.

9. Recording Equipment

The room should also contain some kind of easy way to record classroom activities (for later posting to the web or to help students with recording digital projects).  An easy and relatively inexpensive way to do this is with a Flip Video Camcorder and a tripod.

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Record with a Document Camera and a Flip

Feb 18, 2010 by


In my Math for Elementary Teachers (MathET) course, we do a lot of work with math manipulatives, puzzles, and games of various sorts.  Some of this work can be done with virtual manipulatives, but only if all the students have a computer too.  As a result, we do a lot of classroom work with old-fashioned hands-on math manipulatives, and I demonstrate using a document camera.

Since the beginning of Fall semester, I’ve been trying to figure out how to record these hands-on demonstrations to put in the online course shell, but the best I could figure out was to hold my little Flip video camcorder with my left hand while I write and rearrange the board with my right hand. (Note that there is not room on the document camera station for a tripod.)  Unfortunately, this results in a shaky video, it is tiring, and it’s hard to do everything with one hand.

After doing this for about six months, on Monday I had this flash of insight (one of those ideas where you wonder why it took that long to have the idea).  I was considering the idea of using masking tape to affix the Flip to the Doc Camera during class (which wouldn’t work because of the need to press the on/off button) … and I realized that I had a very simple solution in my pocket.


Here’s a closeup:


This works surprisingly well.  The top and the bottom of the viewing area are a bit cut off, but with a little experimenting, and knowledge of where the working area is, this is a surprisingly slick and cheap way to record.  I also recommend having a mini-whiteboard so that you can circle items, write notes, and generally “mark up” the viewing area without doing any damage to your document camera.  The glare off the whiteboard does create a slight glare spot on the image, but it’s much easier than using sheet after sheet of paper (picking up the manipulatives between each sheet of paper).

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Mathematica 7 is for EVERYONE

Nov 18, 2008 by

Finally, Mathematica 7 is released and I can blog about it! This blog post has been sitting in my queue for several weeks now.

Last month I was at the Mathematica User Conference in Champaign, IL and got a sneak peak at Mathematica 7. Sometimes it’s hard to tell what’s worth upgrading for, but this is NOT one of those times.

Eric Shultz (a fellow Community College professor) did a great job of helping to design, and then presenting the new Classroom Assistant palette that is in Mathematica 7 – an interactive feature designed for

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