(im)possible

I was talking to one of my daughters today and remembered two experiences from my life as a young student:

1) In sixth grade the math teacher asked the class  for a show of hands indicating who was interested in math field day.  I raised my hand.  He looked square at me and said something like “You’ll never make it very far in the competition.  Don’t try.”  I didn’t, until much later.  During my junior or senior year in high school I finally participated and advanced to the regional competition.  I bombed there in spectacular fashion, but I went on to earn two math degrees and work for 17 years as a college math instructor.

2) In high school I wanted to skip a particular introductory class.  I had owned a computer for  few years and could already do everything we were doing in the introductory class.  The teacher would not allow it so I decided not to take her classes.  She commented in a not-so-nice tone that unless I had her classes, I would never pass college.  I did pass college (3 times) and earned a minor in Computer Science with one degree.  Now I am an Associate Professor.

I do not recall either incident inspiring some deep rooted resolve to prove these teachers wrong, but I am glad I did in the end. I am sure they have no idea and probably don’t even remember these incidents, but I get some self-satisfaction out of the path I have taken.  Something curious to me is that not only did I go on to have life experiences that proved these two instances wrong, my main line of research has focused on learner motivation issues and some on feedback to students.  Is that a total coincidence, or was something planted in me as early as the 6th grade?

What about other students?  I was lucky to take a path that lead me past these comments. I’ll bet many other students didn’t have my luck.  Teachers, choose your words carefully.

I told my daughter that in life you need to do this sometimes, and that as a girl/woman, she’ll need to do it even more often — prove “them”, whoever “them” might be, wrong.

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Is technology pushing college-level Math Courses up Bloom’s Taxonomy?

prof lecturing about math

public domain image from: http://bit.ly/W3xJsE

Many college or university-level math courses are delivered with the following general structure. A course meets 3 days per week for the length of the semester.  In each class meeting there is a one-to-many lecture delivered by the instructor, and possibly some questions answered about the lecture or the homework assigned during the previous class period.  There are variations on this structure; a quiz might be involved for example, or graded homework might be collected or returned.  Periodically there will be an assessment of some type, often a closed-book, written exam.  Calculators may or may not be allowed on the exam.  Of course there are exceptions to this type of structure, but my experience as a college math teacher and my work at universities gives me this gut feeling that this is the most common course structure.

Homework is a staple of most college-level math courses.  Instructors often will assign students the even-numbered exercises in the text book to be turned in as part of a homework grade.  The beauty of the even-numbered exercises are that their answers are not provided in the back of the book.  If you work the odd-numbered problems, many books include at least the correct final answer so that you can check your work.  In some cases there are hints for how the solution can be found there too.  Expanded solutions manuals are sometimes available, at an extra cost to the student that show you in a step-by-step fashion how to work those odd-numbered exercises.  You see the importance of assigning the even-numbered exercises; the ones with no solution available.  The students can use the resources available to them with the odd-numbered exercises to learn and self-evaluate their learning, but when it comes time to do some grading, you need to see how they do with no such resources.  Enter some modern technologies…

Technology in math courses has long been a source of debate.  Many of us have had math courses where the instructor commanded “no calculators allowed”.  Others may have been able to use them sometimes, and others yet perhaps all the time.  A strategy I have often seen enacted is for the math instructor to construct exercises and/or quiz and test questions where the use of a calculator was really no help.  Two freely available online resources are changing a lot of how this might be working: Wolfram|Alpha and YouTube.

wolfram|alpha logo           

The combination of these two technologies creates the largest solution manual ever created and renders the even-numbered math exercise strategy pretty much useless. Wolfram|Alpha has tagged itself a “computational knowledge engine for computing answers and providing knowledge“.  The stated long-term goal for Wolfram|Alpha is to “make all systematic knowledge computable and broadly accessible“. Since a lot of the exercises assigned for math homework fall into the category of being solved by the use of various algorithms, it seems like Wolfram|Alpha will have it done.  Actually, it already does quite a lot. You can enter standard calculus problems of calculating derivatives and integrals and Wolfram|Alpha will produce the final answer, produce graphs, and here’s the kicker — show you the step-by-step work from the statement of the problem through to the final answer.  You can get up to three step-by-step solutions per day for free, but a very modest fee eliminates that limit.  If you have not tried Wolfram|Alpha, do it.  I taught university-level calculus and differential equations for several years and I am impressed with the information Wolfram|Alpha provides, and it does it all quickly on just about any Internet connected device.

Some of you reading may be thinking that Wolfram|Alpha is impressive, but it cannot do “real” math.  It cannot do a mathematical proof.  Enter YouTube.  Do you need to prove that a subgroup of a cyclic group is cyclic? Click here.  Do you need to prove that a scalar multiple of a continuous function is continuous?  Click here. Will there be incorrect proofs posted online?  Sure.  However, the comments and “likes” available will allow users to find the best versions available.  What does all of this mean for college and university-level math teaching?

Some math students who choose to rely on these resources for homework solutions will be discovered when their test scores show that they cannot solve the problems.  Some math students will regulate their learning using these resources and perform well on tests.  But should an instructor wait until the tests to see which students are really engaged with the course material in a meaningful way?

What about flipping your classroom? Make use of the online resources like lectures and examples of proofs and then utilize class time for collaboration and problem solving.

Many of you have heard of Bloom’s Taxonomy for learning objectives.  Where do you think “traditional” math homework falls in this taxonomy? Do tools like Wolfram|Alpha and YouTube push assignments up the taxonomy to make them worthwhile? The fact is that Wolfram|Alpha and YouTube are becoming ubiquitous thanks to mobile devices and increasing access to wi-fi and cellular networks.

bloom taxonomy pyramid

Bloom’s Taxonomy from: http://bit.ly/13QkBKE

Acknowledge that these tools exist.  Design your courses to make use of them.  Leverage their power to help you create courses that require students to function higher on the Bloom pyramid.   Will you do it?

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