Tag: math (page 4 of 10)

I noticed through this blog, that the CBC had published the PISA results for Manitoba (released as charts) for 2000, 2003, 2006, and 2009. I wanted to verify the results they had posted, especially the mathematics data, so I went and looked up the data for myself on the Stats Canada website (which you can access yourself here, here, here, and here). Using this data, I created th graph below, which shows the scores in math for Canada and for each province (get the raw data here).

I’m not sure what this data shows, although I can see some trends. Of course, if I change the scale, the overall trend seems more clear.

It looks to me like overall the results have been somewhat stable, at least at this scale. While the trend in Manitoba definitely looks like a downward turn for the last few years, and this trend is probably statistically significant, overall for Canada, it looks like the results have moved somewhat randomly, as one would expect from year to year.

Not convinced that there are cultural nuances in how we understand and define math? Watch the following short video (see http://www.culturecognition.com/ for the source) in which a child explains the number system his culture uses to another child.

There are other areas in which we understand mathematical concepts differently depending on our culture. For example, this recent study suggests that something like ‘numbers come in a certain order’ may be a cultural representation, and not one of which most of us are aware.

One wonders, if we can see such dramatic differences between different cultures in terms of understanding something fundamental like number, how likely is it that there are other differences within our own culture?

My wife, for example, tends to rely on landmarks for navigation, but I tend to rely on an internal map based on the names and numbers of the roads. She and I therefore have a different understanding of how one should navigate. I can remember meeting people who could not read a map (but who were otherwise able to navigate with ease) suggesting that our representations of geographical information may differ greatly between different people.

How does this influence how we should teach?

Yesterday, I was covering a colleague’s math class at the last minute, and he had made photocopies of a chapter 1 to 7 review. I looked at the review sheets, and the grade 10 students in front of me, and decided that it was unlikely that the review sheets were going to be useful. I handed them out, and then started putting puzzles up on the board.

The first puzzle I put up was the Seven Bridges of Königsberg problem. Within  a couple of minutes, every student was trying to figure out the path across the 7 bridges that doesn’t cross any of the bridges more than once. Before the students got completely frustrated with this problem (since it is deceptively simple to state, but "difficult" to solve), I put up a couple more problems, including a gem from Dr. Gordon Hamilton. I added the frog hopping problem to the board, and taught two students the game of Nim.

Each problem had some students who were working on it intensely. Every student found some problem which was interesting to them, and almost all students were working in small groups on the problems and puzzles. Eventually, a small group of students gave up on all of the puzzles and worked on the review sheets while the rest of the students continued to work on the puzzles until the end of class.

Some students asked for a hint on the bridge problem, and I led them (through questioning) to Euler’s formulation of graph theory. From this, we discussed that there could be at most one starting spot, and one ending spot, and that only a starting and ending spot could have an odd number of paths leading in and out of it. I then put up the 5 rooms puzzle, which one of the girls said within seconds was unsolveable by applying Euler’s analysis to the graph.

A group of boys worked on the frog problem, and went from struggling to even find a single solution to the 3 frog problem to being able to generalize a solution for n-frogs on either side (and a formula for determining the number of moves for each frog puzzle).

The next day, I spoke to my colleague, and asked him if he was okay that I had not done the worksheet with the students. As expected, he was fine with it. I asked him what the students said. He said that students said that they enjoyed the day before, but one student had said, "We didn’t even do any math yesterday."

I’m not sure I agree with that student, and I’m slightly distressed that he didn’t see the problem solving activities we did as being part of math. What do you think? Are problems like these important in mathematics? If so, why aren’t more of them in our curriculum?

I recently learned of a massive project at Virginia Tech called the Math Emporium. Here’s a quote from the original article.

The Emporium is the Wal-Mart of higher education, a triumph in economy of scale and a glimpse at a possible future of computer-led learning. Eight thousand students a year take introductory math in a space that once housed a discount department store. Four math instructors, none of them professors, lead seven courses with enrollments of 200 to 2,000. Students walk to class through a shopping mall, past a health club and a tanning salon, as ambient Muzak plays. – Daniel de Vise

Students sit down at computer terminals and read mathematics lessons, and then take quizzes based on those lessons. The idea is compelling for those wishing to reduce the cost of higher education, because if you can successful replace people with computers to teach the classes, you don’t have to worry about benefits, salaries, and other major expenses of a university. According to the article graduation rates for the introductory courses are up, and costs are way down, as the Emporium is almost 1/3 cheaper than the previous model used at Virginia Tech.

So what do the students think? I was recently given a link to a public Facebook page where Virginia Tech staff had linked to the story.I took some screenshots of what a (probably biased) sample of the students think of the Math Emporium, just in case Virginia Tech ever decides to remove the public feedback they got on their Emporium. Here are some quotes from that page.

“How about being taught in actual classrooms… The concept that the Empo improves anything is an outright joke. It’s horrendous that I have to pay exorbitant amounts of money so I can take 30 minute bus rides to this soul-killing place and stare at a computer screen under the guise of “education.” What a load.” ~ Andrew Michael Burns

“[P]aying a lot of money to get no teacher for math. that is what i remember” ~ John Hawley

“None … it was a nightmare & I ended up having to enroll in pre calc & calc at the community college over summer because I couldn’t learn a thing online in math” ~ Amy Domianus

“I remember vividly the obnoxious, intrusive hum of the fluorescent light fixtures; the ‘tutors’ that clearly understood the problem you were asking about, but couldn’t answer your question because they barely spoke English; the feeling of overwhelming despair that seeped into my bones with every second spent glued in front of a screen; the nagging thought that my education was being reduced to an assembly-line process; the vertigo that overtook me as I glanced down the isles and beheld row upon row of workstations stretching into infinity. In my time as a college student, I never experienced anything so degrading, time-wasting, blatantly bureaucratic, and soul-less as the wretched Hell-spawned Math Emporium.” ~ Andrew Lord Wolf

There was one somewhat positive comment on the thread.

“I’m going to go against the crowd and say that I actually really like the math emporium as a place to study. I never took the classes that were solely empo based, but I did take a few that involved having to go and take quizzes. In helping people that have taken empo based classes though, I have realized that the classes aren’t so much about learning calculus as much as it is learning the tricks to the quizzes. There are only a certain number of different types of questions, and most of the questiosn have answer patterns. So basically if you do enough of them, you don’t really even need to know much calculus to be able to do well.

Study wise, I think it’s a great place to get work done. It’s bland enough that you can sit down and do work without too many distractions, and if you take your computer as well as using one of the work stations you have tons of monitor space to use, so you can look through powerpoints and take notes at the same time and such. At the same time though, if you get bored there’s always people there to talk to/take a break with.” ~ Malou Flintsch

I’ve bolded a couple of statements in this quote because they are pretty important. First, Malou never actually took any classes in the Emporium, and she is one of only two positive comments about the experience in the thread. Second, as a tutor for the Emporium, she realized that the classes weren’t about learning calculus as passing quizzes.

I interviewed someone directly who took a number of courses in the Emporium when she was an undergrad at Virginia Tech. Her name is Jessy Irwin, and she works for a technology company that offers online lessons and instructional support for mathematics. She commented that:

• There was no video explanation, just text on the screen. Often the text on the screen, and the text from her textbook used different terminology, and she would work out the solution to a problem, and then spend 20 minutes figuring which of the multiple choice responses matched her solution.
• She didn’t feel like part of a community because there was no course community. It was possible, even likely, that the people next to you in the Emporium were working on different courses, or were in a different stage in the same course.
• Everyone had to be a self-sufficient island. You could put a red cup on top of your monitor, which would tell the roaming assistants that you needed help, but she often had to wait up to 45 minutes for someone to help her, stuck on a single question that she couldn’t skip because of limitations in the software design.
• She almost hired a tutor to help her through the first year calculus course, which she ended up failing 4 times. She eventually found a math-for-liberal-arts-students course and took and passed it. Notably, no one helped her find this option after her first failure, which suggests a lack of counselling support for this program.
• She found the Emporium to be the “worst educational experience of her life.”

There are obvious problems with such a program. First, too many students hated the experience, and this is unlikely to have encouraged these students to continue learning mathematics, which is a primary purpose of mathematics courses in university! A second objective of university level mathematics is to help students continue to develop analytical and mathematical reasoning, which it seems unlikely that the Emporium is successful in doing. One does not develop analytical reasoning from guessing which multiple choice answer matches your solution, or learning the tricks to passing the course quizzes. Another purpose of university in general is to help students foster connections with other students, and begin to develop a network of peers that they will carry with them throughout their life. This purpose is not possible when students are isolated from each other so completely.

The two benefits of the Emporium are themselves contestable. Costs may be down for the university, but according to Jessy, many students have paid for private tutoring to get through the Emporium courses, or taken equivalent courses at the local community college instead. This means that some of the students, who are already paying significant tuition fees, are being forced to pay additional fees as a result of this program, which is essentially transfering the cost of instruction from the university to the student. The other benefit – the increased graduation rates – is impossible to compare to the model Virginia Tech used before the Emporium for these courses, since the courses are so different. More important than graduation rates is the amount of mathematical knowledge and reasoning skills gained by the students, for which there appears to be no data.

Unfortunately, the Emporium has spread to about 100 other colleges since it was invented, which suggests that there are hundreds of thousands of students forced to experience it. This kind of reduction of education to what can be easily measured by a computer is dangerous since we could quite possibly end up with many people believing they understand mathematical principles, when in fact they do not.

The worst part of the Emporium? Four of the courses offered in the Math Emporium are required courses for future mathematics educators. Hopefully these educators will be able to see the Emporium for what it is – a poor way to teach mathematics.

I’m hoping to find (or potentially build, given how well my search is going) some open-ended problems appropriate for elementary school math classes. By open-ended problems, I mean problems which:

• do not have an obvious solution,
• require some time to figure out,
• have multiple solutions,
• may require some assumptions are made by the students,
• are extendable in some way,
• require that the solution be explained, rather than a single number given as the answer.

I’ve found that the definition of open-ended problem seems to vary quite a bit, with many sources that I’ve found using free-response or open-response as a synonym for open-ended.

Here’s a sample question (forgive the wording, it may need improvement).

Ellen is planning a party for her friends. She has invited 100 of them, but she doesn’t know exactly how many of her friends will attend. She wants to put out tables for her friends, and she wants to put enough chairs at each table so that none of her friends has to sit alone. Assume that her friends will fill up each table as they arrive. How many tables should she put out, with how many chairs at each table?

The curriculum link here is either counting (likely to be a slow technique so I’d recommend reducing the number of friends if this is the strategy your students are going to use), addition, multiplication, or division. Note that if you do questions like this, it is important for students to explain their reasoning, and you may need to help some students do this. You may also have to point out that since Ellen doesn’t know exactly how many of her friends will attend, this problem is harder than it looks. Also, I may or may not give the actual diagram as this likely gives away too much of the problem to students. Once students have drawn a diagram though, one could turn this into a bit of a probability question (given the diagram above, how likely is it that one of Ellen’s friends will have to sit alone?).

Does anyone else know a source of questions which are this open-ended, and are designed for elementary school students? 

Update:

Here are some resources I’ve been given or found so far:

• Numeracy tasks organized by Peter Liljedahl
• Good Questions: Great Ways to Differentiate Mathematics Instruction by Marian Small
• More Good Questions: Great Ways to Differentiate Secondary Mathematics Instruction by Marion Small
• Creative Problem Solving in School Mathematics by George Lenchner
• Galileo Math problems
• Nrich Maths
• Open ended questions in elementary school math by Mary Kay Dyer and Christine Moynihan
• Open ended assessment tasks