Learning Notebooks for Online Math Homework

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.

Possibly Related Posts:

Math Technology to Engage, Delight, and Excite

This is the Plenary Address from MAA Michigan last week.

Possibly Related Posts:

Teaching Math with Technology (Discussion Panel)

While I was at Wolfram Alpha Homework Day, I participated in a Panel Discussion about the Myths about Teaching with Technology. The panel ran 30 minutes and was mediated by Elizabeth Corcoran. There were three of us (all women, weirdly enough), Debra Woods, a mathematics professor at the University of Illinois at Urbana-Champaign; Abby Brown, a math teacher at Torrey Pines High School; and myself.

I no longer remembered anything that I said in this panel, so it was fun to watch the discussion from an outside point-of-view. I am glad to see that I talked about the value of play during the discussion, because I am finding more and more that introducing play (and exploration) back into learning makes a big difference in engagement and in retention of the subject.

Possibly Related Posts:

WolframAlpha: Recalculating Teaching and Learning

For at least a decade, we have had the ability to let CAS software perform computational mathematics, yet computational skills are still a large portion of the mathematics curriculum. Enter Wolfram|Alpha. Unlike traditional CAS systems, Wolfram|Alpha has trialability: Anyone with Internet access can try it and there is no cost. It has high observability: Share anything you find with your peers using a hyperlink. It has low complexity: You can use natural language input and, in general, the less you ask for in the search, the more information Wolfram|Alpha tends to give you. Diffusion of innovation theories predict that these features of Wolfram|Alpha make it likely that there will be wide-spread adoption by students. What does this mean for math instructors?

This could be the time for us to reach out and embrace a tool that might allow us to jettison some of the computational knowledge from the curriculum, and give math instructors greater flexibility in supplemental topics in the classroom. Wolfram|Alpha could help our students to make connections between a variety of mathematical concepts. The curated data sets can be easily incorporated into classroom examples to bring in real-world data. On the other hand, instructors have valid concerns about appropriate use of Wolfram|Alpha. Higher-level mathematics is laid on a foundation of symbology, logic, and algebraic manipulation. How much of this “foundation” is necessary to retain quantitative savvy at the higher levels? Answering this question will require us to recalculate how we teach and learn mathematics.

There are two videos embedded in the slideshow. You should be able to click on the slide to open the videos in a anew web browser. However, if you’d just like to watch the video demos, here are direct links:

Note that I’ve turned ON commenting for these two video demonstrations and I will try to load them into YouTube later this weekend.

There are several other posts about Wolfram|Alpha that you may want to check out:


Possibly Related Posts:

What we’re doing with WolframAlpha

Originally, I started this post with the title “What I’m doing with Wolfram|Alpha” and then I revised it, because it’s not just me using Wolfram|Alpha. My students are using it too. Here are some of the things we’re doing:

Discussion Boards: Wolfram|Alpha + Jing = Awesome

Before Wolfram|Alpha, it could take several steps to get a graph or the solution to solving an equation to the discussion board in an online class. You had to use some program to generate the graph or the equations, then make a screenshot of the work, then get that hyperlink, image, or embed code to the discussion board.

With Wolfram|Alpha, sometimes a simple link suffices. Suppose, for example, I needed to explain the last step in a calculus problem where the students have to find where there is a horizontal tangent line. After finding the derivative, they have to set it equal to zero and solve the equation (and calculus students notoriously struggle with their algebra skills). Rather than writing out all the steps to help a student on the discussion board, I could just provide the link to the solution and tell them to click on “Show Steps.”


Sometimes, a bit more explanation may be required, and in these circumstances, Jing + Wolfram|Alpha really comes in handy. For instance, I needed to show how to reflect a function over the line y=1.

Here’s what the reflection over y=1 looks like. If you graph y=sqrt(x) and y=-sqrt(x)+1 you will see that they are not reflected over y=1.

Here’s another example of Wolfram|Alpha + Jing:


Classroom Demonstrations

We’re also finding that Wolfram|Alpha can be a good program to use for exploratory learning. One of the subjects we cover in Math for Elementary Teachers (MathET) is ancient numeration systems. Rather than just tell students how the Babylonian number system worked, students can use Wolfram|Alpha to explore the number systems until they’ve worked out the pattern.


  1. Start by exploring numbers under 50 (42, 37, 15, 29).
  2. Now ask students to figure out where the pattern changes (hint: it’s between 50 and 100).
  3. Explore numbers in the next tier and see if they can figure out at what number the next place digit gets added.
  4. Discuss how a zero is written (and why this is problematic).

Supplement to Online Course Shell

Another topic in Math for Elementary Teachers is learning to perform operations in alternate-base systems (like Base 5 and Base 12). You can easily supplement your online course shell by providing additional practice problems and then linking to the answers with Wolfram|Alpha.

  1. Find the sum of 234 and 313 in base 5. (answer)
  2. Subtract 234 from 412 in base 5. (answer)
  3. Multiply 234 by 3 in base 5. (answer)


Student Projects

Wolfram|Alpha has also started making its way into student projects because of the ease of just linking to the mathematics instead of writing out or drawing the math. Here are a few examples.

For one of the calculus learning projects, the group built a mindmap that demonstrates the graphs and translations of exponential and logarithmic functions.


Another group recorded some help tutorials on using Wolfram|Alpha for evaluating limits. Here are two of their videos (one with sound and one without).

Several of the MathET students have used Wolfram|Alpha and Wolfram Demonstration links as they mapped out the concepts in our units.


Checking Solutions and Writing Tests

Personally, I’m finding that I use Wolfram|Alpha from a simple calculator to a CAS for checking answers as I write a test. I’ve also been snagging images of graphs from Wolfram|Alpha to use on tests (use Jing for simple screenshots). Here’s a short 1-minute tutorial on how to change the plot windows to get the image you desire.

Homework Day

Oh, I almost forgot to tell you. I’ll be down in Champaign, IL for the rest of the week at Wolfram Research. Tomorrow I’ll be one of the “experts” participating in Wolfram|Alpha Homework Day (a live, interactive web event). The events begin at noon (CST) and end around 2am. I’ll be interviewed somewhere around 3 pm and participate in a panel discussion about technology and math education at 8pm.


Possibly Related Posts: