Feb 11

Assessing the Good Stuff

A student sent this vid over. I’ve actually seen this before, but I watched it again because it’s so good. Dan Meyer is saying all the right things, and it’s exactly how I’d like my classes to work. Where I always get stuck is how to assess this kind of assignment. I checked out Dan’s blog Dy/Dan and searched his assessment tag. There was plenty of good info, and he has done a lot of thinking about how to assess skills, the boring stuff. But, how do you assess deep/critical thinking, collaborative, and creative processes? I’ll email Dan and ask, but let me know if you all have any ideas, or have tried anything.

Thanks, Jay, for sending this along!

Update: Dan was nice enough to get back to me, and permit me to post his reply:
Hi Hendree, I think you have my game pretty well figured out. I do a lot of work with project- and inquiry-based learning in class but my assessments (which are used to determine whether or not a student will repeat my class) focus more on the basic, must-have skills. In other words, I don’t assess the soft skills in any meaningful way, certainly in the largest part because I never found an easy way to do it but also because I don’t think I should force a student to repeat a year who isn’t good at creative problem solving.

Dec 10

Mathematical Nonsense

nonsense mathMy daughter is a fan of the “If you give a x a y” childrens books. We checked out Time for School, Mouse! from the library, and I got to this page and was immediately curious. Some of the expressions seem well formed, like , others seem like nonsense, . Anyone recognize anything familiar in there? Could this be an instance of math-washing, using the veneer of mathmatical coolness for one’s own non-mathematical ends?

Oct 10

Triangular Multiplication

My mind recently drifted onto the rectangular array model of multiplication. And I began to wonder what an operation based on a triangular array might look like. This is what I came up with for an array base*height. I’ve created a diagram below to illustrate. You end up with a lot more degenerate triangles than you do rectangles.
tri num 2 crop

Here would be the Caley table:
trin num table crop

You’ve got the triangle numbers on the diagonal and a definitely non-commutative operation. It would be easy to iterate this and form an exponentiation. (eg. 2^3=5) Would it be possible to discover an addition from this multiplication?

Apr 10

Teaching Problem

I use MyMathLab to run my classes and I have all this great info on each assignment called an “Item Analysis”. Below is an item analysis for a sample exam in one of my algebra classes. Here’s my question, which are the most urgent problems to cover when I go to give my class the review? I’m not looking for an intuitive listing of the top problems, but a spreadsheet function that I can use to sort the problems. Note, something that makes this problem harder is that the questions aren’t given with the same frequency. (Hint: the most obvious is calculating the percent of time a question is gotten correct, but there are certain problems with that.)

Dec 09

What would a teachers report card look like? (Part 2)

This follows up on yesterday’s post, which had gotten a bit lengthy. I described in the last email the two measurements I’m currently taking: cumulative exam average and cumulative final exam passing percentage. I’d like to get a little more sophisticated. I know one huge thing I’m missing is retention. At the moment, the only students I’m judging my performance on are those taking the final exam. I’m not at the moment looking at how many I get to the finnish line.

Another metric might be student satisfaction, most institutions do teaching evaluations. These scores might be incorporated. All performance benchmarks being equal, it is better to have satisfied students.

What I’d also like to know is how my students are doing in the next class. This data can prove a little harder to track down, but is an excellent way to measure the attainment of basic skills. Considerations do have to be made though. Sometimes students don’t take a class directly after mine. To draw up an extreme case, what if they waited 4 semesters to take the next course? How much of what they do in that course reflects on what they learned in my class?

Let me know if you have any more ideas on how to measure teacher performance. What I would really love is to have a dashboard like one has in Google Analytics. One that would present at the end of each semester several key numbers on the mainpage, and then let you drill down from there. That would be great!
google analytics

pic by vrypan

Dec 09

What would a teachers report card look like?

report card cookieI have been getting more and more interested in this question. It is curious that in a profession in which we are constantly evaluating other’s work we get very little feedback on our own. Up to this point, I’ve been doing a little rudimentary record keeping. I teach three courses that have departmental finals. I figure the final is about as good a benchmark as I have on performance, so I basically keep track of two statistics each semester: my student’s final exam average and what percent passed the final (70 being passing). Here’s a link to the data. The semester by semester scores vary a lot. This is true for almost all my classes. Here’s the picture of my Intermediate Algebra course over the last 7 semesters.
sem by sem 137
Semester by semester performance is volatile for a number of reasons (class size, time of day, class time…). It wasn’t helping me much to set goals or understand my own performance. I switched over to a cumulative exam average and cumulative passing percent. What that means is that is if my cumulative exam average for Fall 07 is 65% that all students that I’ve taught prior to and during Fall 07 taken together averaged a 65% on the departmental final. Here’s a comparison of average score vs. cumulative average score.
You can see how much smoother the cumulative averages are. They aren’t as susceptible to all the noise that single semester stats are. They also give me a benchmark for each semester. I don’t try to beat last semester’s averages, but simply my cumulative average up to that point. So I hope I can get my intermediate algebra class next semester average more than a 72.9.

This post is growing a little longer than I like. I know in only recording these two stats I am missing quite a bit of the picture. There are so many others to talk about. You’ll have to wait till tomorrow night.

pic by Cookie Jan’s Creations

Dec 09

Generalizing the concept of prime

primesTeaching a lot of developmental classes, I spend a lot of time with basic concepts like the fundamental theorem of arithmetic. If you are as geeky as me, it charges your card to no end that every natural number can be written uniquely as a product of primes. It is so strange (and wonderful) that a certain subset of numbers, the primes, are the multiplicative backbone of all the natural numbers. I think of the fundamental theorem as a decompostion theorem in the sense that shows us that all the naturals can be broken down in a particular way. Another of these “decompostion” theorems is Lagrange’s four square theorem, that any number can be expressed as the sum of four or less perfect squares. Why this theorem is cool but not as cool as the fundamental theorem is that the decompostion of a number in squares is non-unique. Think 64 = 8^2 and 64=4^2+4^2+4^2+4^2. Also Lagrange’s decompostion is a decomposition in two operations addition and exponentiation. So the fundamental theorem is one of many, but also special.

Lately, I have been wondering what “prime” would be for operations other than multiplication? This is what I was thinking about: a natural number n is prime in operation @ if the only way to decompose n in @ for the naturals is n@e, where e is the identity for @. Applying this to addition there is only one prime number, 1. Every other number is composite. Take 4, it can be written as 3+1, 2+2, or 1+1+1+1. The latter is in fact the additive prime factorization of 4. Addition is kind of boring though. Not enough primes.

Note, I left the identity on the right-side in the definition, so that we could apply the definition to exponentiation. Here we have a lot more primes. In fact the only composites are the perfect squares, perfect cubes, perfect….. The first ten exponential primes are 2, 3, 5, 6, 7, 10, 11, 12, 13, 14. Compare that to the first ten multiplicative primes 1, 2, 3, 5, 7, 11, 13, 17, 19, 23. We can’t really say that there are more exponential primes than multiplicative primes. (Showing that there are an infinite number of exponential primes is, I think, just a matter of adapting the same old proof we had in multiplication.) What we can say is that the exponential primes definitely seem more dense. In fact, I think it stands to reason that (other than 1) every multiplicative prime is by necessity an exponential prime. Now imagine primes for super/hyper-exponentiation. The primes get more and more dense.

Now let’s play. What is the exponential prime factorization of 16?
What about the hyper-exponential prime factorization of 16?

pic by Adrian Leatherland

Nov 09

With out math, you can’t even…tie your shoes?


I found I neat post on a new combinatorics proof that shows which manner of lacing shoes is the strongest. I came upon it via a great site called The constructionists would definitely approve.

pic by and within…

Oct 09

Abilities Based Education

quantitative reasoning
Tunxis is going through an assessment change. We are keeping traditional course and program related assessments, but we are beginning to assess core skills across the curriculum. Intermediate Algebra is one of many courses that assesses for quantitative reasoning. The above chart is taken from our syllabus. It indicates three levels of quantitative reasoning: unsatisfactory, satisfactory, and distinguished. The question the math department is now facing is how to assess for this skill. Some feel a student’s ability to quantitatively reason correlates to their performance in the course. Others feel that it is their performance on word problems that most closely correlates to the student’s ability to quantitatively reason. How would you assess this? Are you already doing something like this?

Oct 09

Living algebra, living wage

living wage
I was sent an interesting link to an extended lesson on getting middle students to explore the minimum wage in a math class. It definitely deserves a read. I think these issues might be even more real for community college students. I would love an extended exploration like this, but am not sure how to work it in with as little class time as I have. Maybe this is a good place to use the online discussion tool in my on-ground classes? Anyone tried something like this on a three hour a week schedule? BTW, I have lost track of who sent this my way. Thank you, and please take credit in the comments.

pic by stemlimited