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On motivating mathematics education

Here is a funny comic from the Fake Science blog.

Fake science - Use a ruler to find the third side of a triangle

 

The problem is, there is a kernel of truth in this satirical comic. Given most problems we will encounter in life, we would use a ruler to find the third side of a triangle. Obviously I think that there are good reasons to learn the Pythagorean theorem, but for most real life applications, one could draw a careful scale diagram (an incredibly useful skill in itself) and apply ratios to your measurements of your diagrams to find the missing length.

So why do we teach the Pythagorean theorem? Is it because of the power this abstract idea has? Are there other abstract ideas which have equal value? Could you imagine a mathematics curriculum which includes lots of rich abstract ideas, but happens to not include this theorem? How important is this theorem anyway?

 

I did professional development all wrong

Last year, I presented a lot on the need to improve mathematics instruction. I had pictures, I had questions, I had effective arguments, and my audience was engaged. I could present like the best of them on some of the ways that we can improve mathematics instruction. What I did not have was effective teaching.

The role of someone involved in professional development for teachers is to help the audience, teachers, improve their practice. It may be that they take part of what you do and use it, and it may be that they attempt to copy your method exactly. The problem is that the typical presentation does little to improve someone’s practice. It may inspire them, it may anger them (I’ve done both), and it may provide some helpful tips, but effective change in practice does not come from someone presenting on their practice. The best you can hope for from a presentation is small, temporary, surface level changes.

Improving one’s practice requires thinking. It requires time spent looking at the context of one’s school, on the way that one approaches one’s own teaching, and on what other practices one can incorporate into one’s own pedagogy. It requires discussion so that the learner can take the ideas they are assimilating and seek clarification and direction.

So instead of spending the entire time I present talking, I give participants much more opportunity to talk. Instead of participants sitting around listening, I give them opportunities to do. The last few workshops I’ve done have been more about conversations. They’ve involved rich, mathematical problem solving activities. They’ve involved teachers having insights, and sharing those insights, often things that never would have occurred to me. I’ve learned much more from my workshop participants than when I was a presenter.

I spent an afternoon talking with my colleagues about computational thinking, how computational thinking really is mathematical thinking, and how if our students get opportunities to program, then they are doing mathematics. My colleagues were working on a particularly challenging problem, and one of them stopped and said, "Okay, I get it. Solving problems is hard. I can see why the kids struggle with this stuff." This kind of insight, not directly related to my objectives, was probably the most valuable insight to come out of that workshop. It never would have happened had I not given participants a chance to think and to do.

The Math Emporium – The Walmart of Higher Education

Does it work? Who cares, as long as it is cheaper!
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.

Open-ended problems in elementary school 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).

10 tables with 10 chairs

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:

Separate science history from science inquiry

Zombie Feynman on Science
(Image credit: XKCD)

It occurrs to me that we have two goals for science education. One is to teach students what existing science is known, and how it can be applied to our lives, or how it is interesting to us. I call this first purpose, "Science History." The other goal is to teach the process of doing science, of thinking scientifically. This purpose, I call "Science Inquiry."

I think we should separate these two purposes into separate courses or domains, because the purpose of the first is diluting the effect of the second. Many children finish school thinking that science is a collection of facts known about the world, and do not spend enough time learning how those "facts" were derived.

Labs are a good start to learning science inquiry, but many experiments done in labs have issues.

  • The labs are rarely designed by students. This leads to underestimate the difficulty in designing a good experiment, and to over-emphasize the paperwork portion of science.
     
  • The labs rarely take more than 30 or 40 minutes to complete. Students rarely have to repeat a lab because of experimental error. They learn from this experience that laboratory science can be easily parcelled into sitcom-like episodes.
     
  • Students do not learn enough about the reasons why we have designed lab reports, and think of the portions of a lab report as blanks to be filled in. Quite often they will fake data so as to complete the boxes faster.

A Science Inquiry course would focus on the process of doing science, and less on the students learning existing scientific knowledge. Obviously students would be likely to find connections between the Science History course and the labs that they design, and hopefully they will also see connections between their Science Inquiry and other domains of knowledge.

A Science History course would focus on what existing scientific discoveries we have made, who made these discoveries, and what are the stories around these discoveries, and how these discoveries impact our lives. It might cover some principles behind the philosophy of science, as well as the connections between science and other domains of knowledge (like math for example).

Right now, many schools see Science Inquiry as optional or even an inconvenience. This suggests to me that some people think that thinking scientifically is an inconvenience or too troublesome to teach, and this scares me. While "thinking scientifically" isn’t the only way to think, it’s an important one, and certainly, we should all learn how to do it. I also think that Science Inquiry is indistinguishable from thinking scientifically. If you remove the inquiry, then it isn’t really science.

Another alternative to the traditional conference


(A typical conference presentation – Image credit: Emmanuelvivier)

 

I’d like to propose an alternative to the typical conference model. Chris Wejr got me thinking after he sent me a message suggesting that we host a conference sometime in 2013 that he called a ‘hybrid conference’ and this post by John Burk also influenced my thinking as well.

A typical conference

Some of the problems with a typical conference for many people are:

  • They don’t know anyone at the conference before they attend it, and so connections they could potentially make at the conference are not made,
  • At the conference itself, too much time is spent by presenters talking, and not enough time is spent by participants assimiliating what they learn,
  • Most conferences have no follow-up after the conference.

The best parts of a conference (in my experience) are:

  • The ability to meet and discuss ideas with other people in the same field as myself,
  • Being inspired by people doing amazing projects, and who give awesome presentations/keynotes,
  • Learning about ideas outside of our own personal areas of expertise, in other words, being pushed by others to improve ourselves.

A different conference model

First, we would assign people to cohorts (based on their interests, or on questions they answer during registration) after they register, and setting up email lists (since most people are more comfortable with email than with other social media, and it would automatically provide records of the partiicpant conversations) for those cohorts, along with a facilitator for each cohort. The job of the facilitator is to provide information to the cohort about the conference coming up, and to encourage conversation and introductions between participants before the conference. These cohorts would also be sent links to video presentations (which should be broken into small chunks and include searchable transcripts of the video) that they can watch in advance of attending the conference. Ideally, presenters would be part of these cohorts.

The people would then attend the conference, and potentially move around through their sessions (which would have to be scheduled in advance, like a typical conference, but with input from the registrations) as a cohort, with sufficient opportunities during the sessions to connect and discuss the ideas, or at least between sessions. Ideally each session would be run more like a workshop, rather than a lecture, since most (if not all) of the people in the cohort would have already listened to the presentation. In some of the more advanced sessions, participants would produce a product as a result of their time together.

Social media could be used during the presentation as a back-channel, so that people from outside of the conference could learn from the participants, and share their ideas back to the conference.

Naturally, most participants attending the conference would know some other people there. They would have conversations, and they could choose to eat together. Obviously, if one wanted to continue through the conference as a solo participant, this would still be supported by this model, one would just choose not to interact with their given cohort.

After the conference, the cohort email lists could be used for follow-up, as well as other social media. People would be expected to continue to ask questions and discuss ideas, as well as share their successes (and failures) back to the group after attempting to implement whatever strategies, techniques, or resources they learned about through the conference. Provided the participants made the effort to seek follow-up, they would have an avenue to receive it.

This conference format would help mitigate some of the problems with the typical conference format, while not taking away any of the benefits. It would further have the benefit of allowing people who could not afford to attend the conference in person to still participate in many meaningful activities related to the conference itself.

It would definitely require more work from participants than is typically expected for a conference, and I’m sure this would turn some people away from attending this conference. That being said, those people I think rarely get very much out of typical conferences anyway, and I’d rather not build a new conference model based on the lowest common denominator.

Do you see any flaws with this model? Can you think of any ways of improving it?

Intuition and research

There are a number of things which have been discovered over the years through research which are not entirely intuitive. In fact, many of the results that have been discovered are down-right odd.

 

  • If you pay people to perform simple, routine tasks, in general the more you pay the person, the better they perform. Oddly enough, if those tasks require even a bit of cognitive effort, extra pay reduces performance. What!? How does this apply to education? Well, first it seems that it would drive a nail into the coffin that we should give teachers merit pay (as opposed to just paying all teachers more) for improved student performance. It also suggests that other rewards, which are commonly used in education, may have the opposite of the intended effect; they may reduce performance.
     
  • If you tell children how to play with a toy, they are less likely to perform irrelevant actions with that toy; but they are also less likely to do anything novel with it, or discover anything beyond what you told them about the toy. One would think that if one knew how to use a toy effectively, you’d have a base of knowledge necessary to expand upon and to make new discoveries. It turns out; sometimes even a little bit of knowledge is too much.
     
  • In a pivotal study done in the 1980s, researcher Jean Lave sought to find out how successfully people applied math in their everyday lives. Her surprising answer is that people actually use mathematics reasonably reliably, at nearly 98% accuracy in the supermarket, for example. What is somewhat shocking is that when the very same people were given a pencil and paper test on the very same skills they had successfully solved in the supermarket, the percentage they got right dropped to 59%. The conclusion Jean Lave had was that the subjects were using strategies in the supermarket that they had developed themselves, but fell back into the strategies they had learned in school for the test.
     
  • What do you think would happen if you didn’t teach arithmetic at all to students? In a highly unethical study done in the 1930s, a group of students was given no arithmetic instruction at all until 6th grade. Instead, the students spent this time discussing things that came up in their lives, and some practice in measuring and counting. In 6th grade, the students were taught arithmetic. At the end of the 6th grade, this group of students (who came from the poorest parts of the district) exceeded their peers from the other schools in solving story problems, and had caught up in arithmetic. In other words, not teaching math for 5 years (and spending this time reasoning through discussion instead) improved their mathematical reasoning skills.
     
  • A longer work week does not necessarily lead to more productive employees. In fact, most often it reduces overall employee productivity. 40 hours a week seems about optimal (for maximizing productivity, if not morale). What are the implications of this research on education? Should we be looking at less time in school (or at least doing "work" like activities for students) rather than more?

 

What these studies show is that our intuitive sense of what may be true is often not true, or at least can be shown to be not true under certain circumstances. We must then shy away from relying entirely on our intution, especially when examining large-scale educational practices. We must do a better job in education in funding and supporting effective research in our schools. We also need to be less reactionary when it comes to approaches that don’t fit into our personal perspective on how certain things should be taught, and focus more on dialogue and research to satisfy our reactions.

Copyright for Canadian Educators

I’ve created a brief presentation on copyright which simplifies (perhaps too much?) copyright for teachers. Please give me some feedback on this presentation before I use it with my colleagues. Note: These tips on copyright only apply to Canadian educators as copyright rules are specific to each country. For example, Canada has no "fair use" provision.

Update: This presentation needs to be updated with the recent changes to Canadian Copyright law. See Copyright Matters! for more accurate information.

 

Articles I’ve written on Math Education

Here is a list of some of the articles I’ve written on Math Education

On Mathematics education reform:

 

On the use of technology in mathematics education:

 

Other articles on math education

 

Some of my favourite articles/videos on mathematics education by other people (incomplete):

 

Let them solve the problem

Arts week poster

When I came in this morning to school, the poster in the photo above had fallen down. I noticed it, and thought to myself, "I should fix that." I went and dropped off my stuff, and when I came back to the poster, one of our students was standing there staring at the poster. He said to me, "It looks like the poster came undone here," and he pointed to a spot on the back of the poster. "I’m going to fix it," he said.

"Do you need any help?" I responded.

"Nah, I can do it. Thanks anyway," he said.

Later that morning, the student said "It’s a bit crooked, but it’s up." "It doesn’t matter if it’s a bit crooked," I replied, "The most important thing is that it’s up."

I could have stepped in and solved this problem for my student, but then he wouldn’t have learned how to solve it himself. "Be less helpful," says Dan Meyer of the math class, but of course the same is true all over the place in education. The objective isn’t to show students how to solve problems, our objective is for students to learn how to solve problems. Sometimes our role is to stand back and let students solve problems for themselves. Will they always come back with the "perfect" solution? Probably not, but they will have learned more in the process.

Update: 

Sign fallen down again

So it turns out the solution the student tried didn’t work. When I talked to him, he pointed out the flaw in his solution, and suggested a solution. In other words, the problem he solved gave him feedback directly as to whether or not his solution worked.