Education ∪ Math ∪ Technology
I just found this presentation from more than a year ago on some interesting ways to use Google Apps in a mathematics classroom. I noticed that it had been edited slightly, so I did some more edits and thought I would share it here.
You can help edit and curate it here. I could imagine that Google+ would be useful, and that some of the file sharing options through Google Drive have improved, neither of which has made it into this presentation yet.
Here is a funny comic from the Fake Science blog.
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?
Derek Muller has done research on the effectiveness of science videos. To summarize his research in brief – when you present only the correct information in a science video without the possible misconceptions that students may have, students learn less (but feel better about the experience) than if you present information in a science video and include the misconceptions.
Of course, we should recognize that this effect probably does not depend on the medium of instruction, only on the nature of the medium. Videos are a unidirectional medium because they present information without the ability of the learner to ask questions. One might guess that any unidirectional medium may have the same effect. So textbooks, lectures, and other unidirectional mediums may suffer from this same deficit; without common misconceptions addressed in these mediums, the learners learn much less than if those misconceptions are addressed.
Dr. Eric Mazur shares essentially the same message – unidirectional instruction (in his case lecture) – has flaws. He relies on peer instruction and student response devices (clickers) to change the nature of the instruction so that it is more bidirectional (from each student’s perspective). The key here is that he has embedded more opportunities for feedback to reduce the chance that students incorporate the new information they are receiving into their existing misconceptions.
Textbooks (another unidirectional instructional tool) rarely present misconceptions and address them. Most students rarely use their textbooks as a learning resource (at least in k – 12), prefering to rely on the bidirectional instruction their teacher (or parent) provides. This means that the vast majority of information presented in a textbook goes unused. There are some changes to the textbook I’d like to see, which would allow for them to be a more bidirectional learning tool.
While it is clear that the medium of instruction influences the type of cognition that occurs, as Marshall McLuhan has pointed out, it should also be clear that different mediums have similarities in how they affect cognition or learning. If we find out that failing to address misconceptions in video instruction results in poor learning of the concepts, we may be able to transfer this finding to other modes of instruction. If that is the case, then we need to look at our instruction carefully, and ask ourselves, how much opportunity do we give students to address their existing models and resolve conflicts between their misconceptions, and the models we suggest?
Someone I know produced the diagram above in her planning steps to produce the shelves seen below.
This person describes herself as "not a math person." What do you think? Is she a math person or not? It worries me that we have all these people walking around thinking they aren’t "math people" when in fact, they quite obviously are. We need to do a better job of explaining the difference between every day mathematical reasoning, which quite a lot of people are good at, and the formal systems of mathematics that have taken generations to develop.
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.
I recommend watching this video on how to produce math anxiety. All 10 of his "tips" are good starters for conversations on what to avoid in mathematics classrooms.
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 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.
From the National Mathematics Advisory Council of 2008 final report:
The use of “real-world” contexts to introduce mathematical ideas has been advocated, with the term “real world” being used in varied ways. A synthesis of findings from a small number of high-quality studies indicates that if mathematical ideas are taught using “real-world” contexts, then students’ performance on assessments involving similar “real-world” problems is improved. However, performance on assessments more focused on other aspects of mathematics learning, such as computation, simple word problems, and equation solving, is not improved.
It seems to me that if a real world focus in mathematics makes students more able to use mathematics to solve problems from the real world, then this would make much of the mathematics instruction we do more useful. What exactly is the goal of those simple word problems anyway? Aren’t they an effort to add some context to the learning of computations, so that students are able to use mathematics to solve problems? And if students are going to graduate from school, aren’t the problems they are most likely to face going to come from the real world?
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:
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:
Well, okay, some people do burn themselves twice, but hardly anyone. The message is loud and clear, touch the stove and you get burnt, which hurts. There are lots of other things in life people learn the first time.
I was mouthing off in class one time about how much I hated the school newspaper, when Michelle turned to me and nearly with tears in her eyes said, "Do you really think that, David?" Of course, I was just making noise, and didn’t really mean it, but nothing I could do could fix that moment. I had jammed my foot far into my mouth, and I was not able to back-pedal smoothly. 20 years later, and I can still remember this exact moment. I didn’t need 15 slightly different examples with the answers in the back of the book to remember this moment, and the lesson that came with it.
How can we make more of the learning in school have the same kind of stickiness? Do I remember these lessons because they hurt, or because I had a strong emotion attached to that moment?