Archive for the ‘Classroom Life’ Category

Community Site for Algebra Activities


I rarely talk about the books I’ve written on this blog, but the Community Site for my new book, Algebra Activities,  just launched and I think it’s pretty cool.  Also, I now have an author page.  Yay!

The basic philosophy for the book is to provide easy-to-use classroom activities to instructors so that they can easily replace lecturing time with more active learning.  The book also provides instruction tips and lesson plans so that any algebra instructor, especially new ones, can have a “mentor” to guide them and help them reflect on how students learn.

If you go to the Samples section of the Community Site, you can print and use some of the activities from the book in your classes.  You can also see some of the fantastic new algebra cartoons that were commissioned as part of this project.

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Math ELITEs (Classrooms for Active Mathematics)

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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Wolfram Alpha for Inquiry Based Learning in Calculus


Now that all my Calculus II students know about Wolfram Alpha (I showed them), I have to make sure that the assignments I ask them to turn in can’t just be “walphaed” with no thought.  In Calc II, our topics list includes a lot of “techniques-oriented” topics (integration by partial fractions, integration by parts, etc.) and because of the need to keep this course transferable to 4-year schools, I can’t really get around this.  So now I’m in the position of having to reconcile the use of technology that easily evaluates the integrals with making sure that students actually understand the techniques of integration.  There are two ways I’m tackling this:

1. CCC (Concept Compare Contrast) Problems: I’m writing problems that focus on understanding the mathematical process and the compare/contrast nature of math problems.  While Wolfram Alpha can evaluate the integrals for them, the questions I’ve asked require (I hope) a deeper level of understanding about what happens when the techniques are used.  Here’s an example from my recent problem set:

There are two pairs of problems below that are exactly the same. You won’t see why until you do the integration, showing all the steps. Find the pairs and then explain how the matched integrals are fundamentally the same.


2. Inquiry Based Learning: One appropriate use for any CAS (computer algebra system) is to use it as a way for students to explore problem types that they have not learned about yet.  Here’s a definition of IBL, in case you’re not familiar with the terminology:

Designing and using activities where students learn new concepts by actively doing and reflecting on what they have done. The guiding principle is that instructors try not to talk in depth about a concept until students have had an opportunity to think about it first (Hastings, 2006).

It is relatively easy to use IBL in the really low levels of math (K-6) where there is not as much abstraction of concepts.  However, with the introduction of variables, rules, theorems, and definitions that come later in math, the use of IBL requires either that the instructor act as the inquiry tool or the use of CAS.

Back to the point (how to use Wolfram Alpha to do this):  I could have just taught the integral techniques straight up … here’s the technique, now apply it … repeat.  But learning the technique is not anywhere near as important (at least, in my mind) as learning to decide when to use a technique, i.e. what makes one integral different from another?

This semester, I’m doing it backwards.  In the problem set before we look at specific techniques of integration, the students will use Wolfram Alpha to evaluate twenty integrals.  Then they will look for patterns in the answers and the problems, and try, on their own, to make sense of what kinds of problems solicit different answers.  After they understand what characteristics make one integral fundamentally different (in technique of integration) from another, then we’ll look at how each technique works.  Below, you see a few examples of the integrals the students will explore.  You can view the whole assignment here.


For years, we’ve had CAS tools like Derive, Mathematica, Sage, Maple, etc. However, the use of these programs traditionally required so much coding minutia that the IBL often got lost in the coding.  How do I know? Because this was my experience as a student.  I had instructors that tried to teach me this way.  All I remember is how painful the coding was.  I followed the directions in the labs, I typed what I was supposed to type, and I answered the questions that were put forth to me.  But in the end, I never sat down at a computer and generated my own inquiries.  The details of using the programs were so painful that I just didn’t have any desire.

Here’s the sum total of the directions that were necessary for me to teach students how to evaluate integrals in Wolfram Alpha:

For example, here’s how to do the first one:^2-16)


I think Wolfram Alpha is a game-changing CAS (and no, I’m not being paid by someone to say this).  For better or for worse, my students are now using W|A on their own, without any prompting from me.  Their evidence of usage is showing up in emails, in discussion board questions, and in questions they ask in the classroom.  Maybe my class is unusual because I’ve given them the first push… but it’s just a matter of time before W|A is discovered by your students too.

NOTE: If you’ve missed all the other posts I’ve made about Wolfram|Alpha, you can see all of them here.

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Student Conceptions of Mathematics


Do you ever get the feeling that you’re not making any real progress with your students?  Sure, they pass tests and progress through the courses (well, most of them), but have you ever just had the uneasy feeling that they really don’t get what math is all about?  Suppose you were to ask your students the following question:

Think about the math that you’ve done so far.  What do you think mathematics is?

What do you think they would tell you?

Well, in 1994, a research group from Australia did ask 300 first-year university students this question (Crawford, Gordon, Nicholas, and Prosser).  They identified patterns in the responses, and classified all 300 responses into categories of conceptions in order to explore the relationships between (a) conceptions of mathematics, (b) approaches to learning mathematics, and (c) achievement.

If you want all the gory details about the study, you’ll have to track down the journal article, but let me attempt to summarize their findings.

Students’ conceptions of mathematics fell into one of five categories (Crawford et al., 1994, p. 335):

  1. Math is numbers, rules, and formulas.
  2. Math is numbers, rules, and formulas which can be applied to solve problems.
  3. Math is a complex logical system; a way of thinking.
  4. Math is a complex logical system which can be used to solve complex problems.
  5. Math is a complex logical system which can be used to solve complex problems and provides new insights used for understanding the world.

The first two categories represent a student view of mathematics that is termed “fragmented” while the last three categories present a more “cohesive” view of mathematics.  Note that the terms fragmented and cohesive are well-used throughout the international body research.  The categories above, as you may have noticed, form a hierarchical list, with each one building on the one above it.

Here’s where this study gets (even more) interesting.  The researchers also asked students about how they studied (changing the language slightly here to “americanize it” a bit):

Think about some math you understood really well. How did you go about studying that?  (It may help you to compare how you studied this with something you feel you didn’t fully understand.) How do you usually go about learning math?

Again, the researchers went through a meticulous process of categorization and came up with five categories (Crawford et al., 1994, p. 337):

  1. Learning by rote memorization, with an intention to reproduce knowledge and procedures.
  2. Learning by doing lots of examples, with an intention to reproduce knowledge and procedures.
  3. Learning by doing lots of examples with an intention of gaining a relational understanding of the theory and concepts.
  4. Learning by doing difficult problems, with an intention of gaining a relational understanding of the entire theory, and seeing its relationship with existing knowledge.
  5. Learning with the intention of gaining a relational understanding of the theory and looking for situations where the theory will apply.

Again, these five categories were grouped, this time according to intention, into two general categories: reproduction and understanding.  In the first two approaches to learning math, students simply try to reproduce the math using rote memorization and by doing lots of examples.  In the last three categories, students do try to understand the math, by doing examples, by doing difficult problems, and by applying theory.  Other researchers in this community have seen similar results on both general surveys of student learning and on subject-specific surveys and have termed this to be surface approach and deep approach to learning (see Marton, 1988).

Still reading?  Good.  Remember my first question? Do you ever get the feeling that you’re not making any real progress with your students? Let’s answer that now.

Here’s how the conceptions of math and the approaches to learning math correlated in this study (Crawford et al., p. 341):


Did you catch that?  Look at how strongly conception and approach correlates.  It’s probably what you’ve always suspected, deep down inside, … but there’s the cold, hard, proof.

Of course, all this is not so meaningful unless there’s a correlation with achievement.  At the end of their first year, the students’ final exam scores were compared to the conceptions and approaches to mathematics (again, for technical details, get the article). The researchers made two statistically significant findings:

  1. Students with a cohesive conception of math tended to achieve at a higher level (p < .05).
  2. Students with a deep approach to learning math tended to achieve at a higher level (p < .01).

Okay, so where does this leave us?  Well, we don’t have causation, only correlation (at least, that’s all we have from the 1994 study).  However, Crawford, Gordon, Nicholas, and Prosser were nice enough to use their research to develop a survey inventory that we can use to measure students’ conceptions of mathematics (1998).  The 19-item inventory (5-point Likert scales) has been thoroughly tested for validity and reliability, and can be found in their 1998 paper, University mathematics students’ conceptions of Mathematics (p. 91).

Suppose you want to try something innovative in your math class, but you don’t know how to tell if it works.  You could, at the very least, try to measure a positive change on your students’ conceptions of math (there are other ways to measure the approach to learning, but this is already a long blog post and you’ll have to either wait for another week, or view my presentation How can we measure teaching and learning in math?).


To give the Conceptions of Mathematics Questionnaire (CMQ) would take approximately 10 minutes of class time (you should ask for permission from Michael Prosser before you launch into any potentially publishable research).  This would give a baseline of whether students’ conceptions are fragmented or cohesive.   If you were to give the survey again, at the end of the semester, you would be able to see if there is any significant gain in cohesive conceptions (or loss of fragmented conceptions).

So, I have permission (I met Michael Prosser this summer when he was at a conference in the U.S.), and I’m going to use this in all my classes starting next week.  I figure that when I start to see major differences on those pre and post-semester CMQ inventories, that I’m doing something right.  If I’m not seeing any gain in cohesive understanding of mathematics, then I’m going to keep changing my instructional practices until I do.

Papers that you will want to find (and read!):

Crawford, K., Gordon, S., Nicholas, J., & Prosser, M. (1994). Conceptions of Mathematics and how it is learned: The perspectives of students entering University. Learning and Instruction, 4, 331-345.

Crawford, K., Gordon, S., Nicholas, J., & Prosser, M. (1998, March). University mathematics students’ conceptions of Mathematics. Studies in Higher Education, 23, 87-94.

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Can we Teach Students to Understand Math Tests?

A few weeks ago, I gave a test where the grades were less than stellar. Whenever this happens, I try to sit down and reflect on whether the poor test grades were a result of something I did differently in class, a poorly written test, or a result of poor studying habits. After careful reflection and analysis of my own, I was pretty sure that this was the result of lack of studying (a theory which was verified … later in this blog post).

I was dreading the task of passing back these tests and prepared myself for an onslaught of questions aimed at trying to discredit the test (or the teaching).  Then I got one of those great last-minute ideas that come to you right before you walk in to face the students.  Maybe I should let them “pick apart” the test BEFORE they see their own tests.  The class in question is Math for Elementary Teachers (MathET) and I figured that a detailed test analysis would not be an inappropriate topic for us to spend class time on.

First, I made some blank copies of the test (enough for each group to have one).  I also created a handout with every single learning objective and assignment that I had given the students for each of the sections on the test (these are all available in their Blackboard shell, but  I compiled these in a paper-based handout that was 3 pages single-spaced). Here is what one section looked like:

5.1 Integers
• Describe operations on signed numbers using number line models.
• Demonstrate operations on signed numbers using colored counter models.
• Explain why a negative times a negative is a positive.
• Add, subtract, multiply, and divide signed numbers (integers).
• Know the mathematical properties of integers (closure, identity, inverse, etc.)
• Complete #1, 3, 7, 9, 15, 17, 19, 21, 23, 25, 37
• Be able to model addition, subtraction, and multiplication of signed numbers using a number-line model
• Be able to model addition, multiplication, and division of signed numbers using a colored counter model (why not subtraction? because subtraction is really the addition of a negative – treat it so)
• Read the blog posts about why a negative times a negative is a positive and be able to paraphrase at least two arguments in your own words.
• Study for your Gateway on Signed Numbers (be able to add, subtract, multiply, or divide signed numbers)
• Play with Manipulative: NLVM Color Chips Addition
• Play with Manipulative: NLVM Circle Game

When we met in class, I counted the class into groups of 3 students each. Each group received a copy of the learning objectives & assignments (by section) and a blank copy of the test. All of these were un-stapled so that the group could share and divide up the pages as they wanted.

Task #1: Look at all the objectives and assigned tasks/problems. Determine where these objectives, tasks, and problems showed up on the exam. (20-30 min)

Objective: Make the connection between what I tell them they need to learn/do and what shows up on the test.

Task #2: What didn’t show up on the exam? We discussed why these objectives might have been left off (for example, maybe it was not something that I emphasized in class) (5 min)

Objective: Make the connection between what is likely to show up on an exam and what is not likely to show up (with the caveat that any of the learning objectives are really fair game).

Task #3: Make a chart that shows how the points for each test question were distributed between the sections that were covered on the exam. We then compared the results from each group and compared these results to my analysis of the point-distribution for the test. (5-10 min)

Objective: Clearly see that it is necessary to study ALL sections, not just a couple of them.

Task #4: I passed back the individual tests. Each student was given three questions to answer as they looked through their tests. 1. Where were the gaps in your knowledge? 2. What mistakes should you have caught before turning in your test? (read directions more carefully, do all the problems, etc.) 3. What can you do to better prepare for the NEXT test?  I collected these, made copies, and passed them back to the students the next class. (10 minutes)

Objective: Take responsibility for your own studying.

It was here that students surprised me by being honest on Question #3. Most of them confessed that they had not studied at all, but now realized that they needed to start studying. I cannot help but wonder whether I would have gotten the same result if I had simply passed back the tests with no analysis.

Task #5: At the same time the students were looking over their own exam, I passed around one more blank copy of the test and asked students to write their score  for each problem (no names) on that problem so that we could see what the score distributions looked like.  When this was done, I placed the pages on the document camera one-by-one so that they could see the scores problem by problem.

Objective: To show the students that for many questions, students either get the question almost completely right, or completely wrong (you know it or you don’t).

Task #6: We had already covered the first section of the next unit, so I had the students begin a set of “How to start the problem” flashcards. On the front of the flash card, they wrote a “test question” for the new unit. On the back of the card, they wrote some tips for starting the problem and details they might otherwise forget.

Objective: To begin to see tests from the perspective of a test-writer instead of a test-taker.

Take-home assignment: I told each student that they must come to the next test with at least 5 flash cards per section (35+ flash cards). I suggested that a great way to study would be to swap cards with each other and practice with someone else’s questions. I checked to see if they carried through (although I assigned no consequences if they didn’t). All but two carried through. Anyone want to guess how those two fared?

Results: I just finished grading the latest test (signed numbers, fractions, and decimals – not easy topics), and the test results were almost all A’s and B’s (instead of C’s and D’s on the previous test). Most students left the test with a smile on their face, and several finally got the “A” they had been trying for all semester.

Three questions for you, my busy and wise readers:
1. Will the students persist in better study habits now? For example, will they study like this for the next exam?
2. Would this detailed test analysis have had the same effect if the class hadn’t just crashed & burned on a test?  (i.e. is it necessary to “fail” first)
3. Was this a good use of class time? Would it be good use of class time in a different math course, like Calculus?

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What we’re doing with Wolfram|Alpha

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.


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Transforming Math for Elementary Ed

After several months alone to think about why education has become so transactional, I decided that I’d have to “walk the walk” and not just “talk the talk” and so I set about revamping my own classes.  For several weeks, my brain processors whirled while I tried to figure out how to make courses that have a highly structured and full curricula into courses that are transformational and revolve around learning.  Eventually, I hit upon the solution: Learning Projects.  Each student in Math for Elementary Teachers (MathET, as I like to call it) has to do five learning projects during the semester:

  1. Writing a Learning Blog
  2. Building a Mindmap
  3. Giving an Inquiry-Based Learning Presentation in class
  4. Creating a Video for the Internet
  5. Creating a Digital Portfolio to house their projects (this will be done by everyone last)

We cover four “units” in MathET, and each student completes the first four learning projects in a random pre-assigned order (I made a chart of all project assignments at the beginning of the semester).  This means that at any time, 25% of the students are blogging, 25% are building mindmaps, 25% are working on a 10-minute presentation for class, and 25% are building a video on a specific topic.  Projects are due two days before the unit exam so that everyone can learn from reading and clicking through each others’ projects.

No lies.  This required a large amount of time to get a new syllabus in place, verbage about privacy and appropriate computer use, tutorials on the LMS, and grading rubrics (and I already knew how to use all the technology).  I had to move one hour of class (4 hours each week) into a computer lab (and lab time is as precious as gold on our campus).   I set up an RSS feed (via a class netvibes page) to put news about math and teaching at the fingertips of the students.   I have to create a page to hold all the RSS feeds from student blogs, videos, and mindmaps (see the Unit 1 Tab of the class netvibes page).  This project also required a pep talk on the first day of class to explain why I was requiring that students use technology as they learned (because it will help them find jobs and provide them with valuable ways to teach and learn).  It was a bit of a shock, especially to those students who had barely touched a computer before.


However, the work was 100% worth it (maybe even 200% worth it).  We have never (and I mean never) had so much fun with a class before.  Every day of class I automatically get fresh learning assessments from the students who are blogging or mapping out the concepts we’ve learned.  The students really enjoy participating in each others’ active presentations and gain lots of fresh ideas about how to incorporate different teaching strategies into their own classes.  It’s also fun to watch the students get more brave (technology-wise) as the semester progresses – I really can’t wait to see what these projects look like by the end of the semester!  As I walk through the lab or peek at laptop screens before class,  I see students getting sucked in to reading blog posts and news articles that they might not otherwise even see (e.g. Math in the News).  I see them playing with interactive manipulatives from NLVM, and getting hooked on logic puzzles.

Because every single project is organized around learning, they all enhance the students’ understanding of the material.   How do I know?   There were no failing grades on the first test.  Students write and talk about how learning Venn Diagrams is “awesome” and how learning base-5 arithmetic is “tricky but cool” … it’s like math has gotten turned upside-down. What was once scary and difficult is now fun and interesting (maybe still difficult, but more tolerable now).  I think it may even be possible that students are now more likely to study for the exams because they actually enjoy learning the material (this is just conjecture on my part).

There are lots more details to share about how, exactly, I’ve pulled this off (release forms, privacy issues, etc), but for now I’d like to share a few of the best projects from Round 1 of the Student Learning Projects.  I hope that by the end of the semester, every one of my students will have found a project where they had a chance to shine the best and brightest!

Best Student Web-based Projects: Round 1

Honestly, I wish I had recorded more of the student IBL presentations, because many of them have been clever and well-designed.

In addition to the projects, we’ve found ourselves doing some other fun things:


One more thing I’ve changed in all my classes this semester, I try to begin every class by asking students what they’ve learned in their other classes (an acknowledgment that these things are important too).  The only way to refocus education on learning is to make sure it actually is the focus.

Learning Projects Round 2 are already well underway!  Students can see each others’ blogs and mindmaps in progress from day one of the unit.  This (hopefully) encourages them to explore and read more about each topic as they follow links to resources and read about how math has been applied.  Stay tuned for more in our little learning experiment.


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Notesharing in the Digital Age

What do you see when you look out at your students?  Do you see them writing down everything you write and everything you say?  How does it make you feel?  Honored, proud, powerful?

What if those same students then put those notes online and share them with the rest of the class, or the world?  What if they sold those notes to a note-selling business like Einstein’s Notes, for profit?  Would you be okay with that?

Michael Moulton, a University of Florida professor felt violated when it happened to him.  So much so, that he filed a lawsuit in 2008.  An article in The Chronicle of Higher Ed shows Moulton’s frustration with students who participate in these activities.

A more recent altercation took place at San Jose State University.  Here, it was determined that the student does have the right to display homework results online.

Many professors invite the use of shared notes amongst classmates.  They see it as an opportunity for collaborative study.  A research paper by DeZure, Kaplan & Deerman indicates that  students (in general) fail to record 40% of the important points in a typical lecture.  First-year students, on average, do considerably worse.

Whatever your take on this, there are several note taking and sharing sites available today.

Here are just a few sites available for free:

  • NoteMesh — this site seems like the most honest of the bunch in that students collaborate to build a set of good notes and there is no profit to be made.  Students have to  indicate their college/university and add their classes to their profile upon registering.  Students in the same class can then post and edit their own notes.  Since each class uses a wiki, students are able to view and edit their peers notes as well.  Like most wikis, there is a “history feature” which allows you to remove any changes if necessary.  The real question, in my mind, is whether the site has build in LaTex compatibility so that math notes can be shared (I doubt it, as this is not addressed anywhere in the FAQs).


  • Notecentric — this site is similar to NoteMesh but also gives the user the ability to “spy” on other classes.
  • Knetwit – students can (try to) make money off their class notes (one muses to oneself why the student without notes does not just pick up and read their textbook instead)
  • Sharenotes –  students (or presumably the professor) can post notes and charge by the download if you’d like.  You can also browse institutions  for specific notes on specific classes.  Some notes are shared free of charge.  Should professors really be charging for notes?  I think not.  Nor should students, though.
  • University Notes — in addition to sharing notes and/or tests nationwide, students can also rate their professors here and use the on-site blog.

Here are some links to other blog posts / articles on this topic in case you are, like us, morbidly fascinated with this industry that is emerging around the economy of notes:

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Crowdsource Your Syllabus Presentation

You know what I hate about the first day of class?  Going over the syllabus.  You know that the students do nothing but listen to instructors read them the syllabii the first two days of class as they meet all their instructors for the first time.  Not only is this tedious for them (and I wonder if they even remotely pay attention), but it’s tedious for me too.

This year I vowed to turn all my classes into student-oriented learning as much as possible, starting with DAY ONE!

The syllabus was five pages long.  I had students count off by 5 and put them in groups.  Each group received copies of one (and only one) page of the syllabus.  They had about 5-8 minutes to read that page and then decide what to present and how to present it.

During this time, I circulated to answer questions that the groups might had (clarifying points mostly).

The five groups then presented the five pages of the syllabus, highlighting what was important to THEM and phrasing the main points in their own words.  The class was very attentive (especially since they had not seen any page but the one they had).

Then I passed out the syllabus to everyone.

Completely painless.  I wish I had thought of this one years ago.

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Don’t get W|A Implications? Try these examples.

Wolfram|Alpha is a “computational search engine” built by Wolfram Research (the developers of Mathematica). W|A (pronounce this as “walpha” if you’d like) is similar in appearance to the search engines that we are used to and easy to use. It’s not that W|A will replace other search engines, because it won’t. It’s more of a missing piece in the search engine puzzle. W|A provides a collection of data, formulas, computations, and interpretations that are different from other search engines.

Although the media has stressed data-driven examples (for example, type your first name to see a graph of the frequency of that name over time), the ability of W|A to function as a combination of CAS and natural language computational system is stunning. Let me illustrate with a couple of examples for you to try yourself.  Simply follow the links below to see how W|A handles these search requests:

126 (make sure to click on “other historical numerals”)
convert 125 m^3 to gallons
sphere r=7 cm
Line (2,7) and (3,1/2)
Solve x^2-6x=16 (make sure to click on “show steps”)
4 – x^2
Triangle 7,8,9
limit x->3 (x-3)/(x^2-9) (again, make sure to click on “show steps”)
integral (x^2)sin(x^3) (“show steps”)
sum 1/n^2
New York City, Chicago
convert 78 to base 5

There are several differences between W|A and traditional CAS systems. The first, which you should have noticed after those examples, is that the less you ask for, the more you get. W|A just assumes you want all relevant computations and information that it can generate: graphs, solutions, alternate forms, derivatives, integrals, area under the curve (if bounded), and steps (if they are available). W|A provides quick and painless access to all sorts of data that has been organized so that it can be cross-referenced. In this sense, W|A could be a valuable tool for us in helping students to see the connections between concepts within mathematics and in relating mathematics to the real world.

On the other hand, you’re probably also seeing that there could be implications with academic dishonesty, especially in online and hybrid courses. We will all have to individually decide whether W|A is off limits, and if so, how we can possibly enforce it. Ready or not, W|A is now available on any computer with Internet access and on most SmartPhones.

It is up to us to think about (with as much advanced notice as possible) how we want to embrace, adopt, accommodate, or regulate the use of W|A in our courses. This is a conversation we should have in every department at every level of mathematics, including both full-time and part-time instructors. It is a conversation that we should have with our colleagues in the partner disciplines and with our colleagues at our transfer institutions.

Note: There are additional resources you may wish to view. A longer analysis regarding the rate of adoption and the impact of large-scale change in mathematics on the higher education system can be found here.  A wiki (WalphaWiki) documenting some of the capabilities of W|A for math courses and the implications for teaching has also been started  by Derek Bruff.

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