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Teaching probability

My colleague found an activity to do with his 5th grade class, similar to this one. Basically, he gave the students 10 coins each, and asked them to put the 10 coins on a number line (with numbers from 1 to 12) with a partner. Each round they roll 2 six-sided dice, find the total, and remove a coin from their number line if it matches the roll. They keep going until one of the two students has no coins on the number line.

 

At first, most student’s starting positions looked like this:

Student 1 - flat distribution

or this:

Student 2 - another flat distribution

 

At the end of the lesson, we played a 5 coin version of the game, and one student’s paper (after 1 round) looked like this:

Student 3 - All 4 coins on number 7

 

Unfortunately, although most of the students did notice that some numbers came up more frequently than others, as evident by their distributions looking a bit less flat, and bit more centred on 7 on the number line, many of the students still had obvious misconceptions of probabilty. Students made comments like:

"If I spin around twice before rolling, I get a more lucky roll."

"I got a few 11s last game, so I’m going to put a few more coins on 11."

"8 is my lucky number! I’m going to put 3 coins on 8."

"I need to spread out my numbers so I have more chance of getting a coin taken on each roll."

Our plan for next class is to have students switch up groups, discuss insights they’ve had on the game, play a couple of rounds, switch them up again, while we walk around and see what strategies they use. Hopefully we’ll see less students spinning around in order to improve their dice luck…

I also asked for resources on Twitter for using a probability game as part of a lesson on probability, and had the following 5 games recommended (all of these look good because they are relatively easily produced and used in an elementary school classroom).

 

Update:

I wrote a simulation to test to see what distribution of coins is the best. I am somewhat surprised by the results. Check it out here.

Why am I surprised by the results? My intuition about how this game works failed me. I thought that the likelihood of each number coming up was the most important factor in deciding a good strategy for the game. It turns out that one has to balance out the knowledge of the likelihood of each number being rolled with having a selection of numbers available. It may be that the probability of a 7 being rolled is 6/36 but the probability of a 6 or a 7 being rolled is 11/36.

In fact, it turns out that having a selection of different rolls available is fairly important. I updated the simulation so I could choose the number of coins to place, and with 3 coins placed, choosing 4, 5, 6 is better than having all three coins on the number 7.  With 2 coins, it is better to choose 6 and 8 than to put both of the coins on 7 (although 6, 7 is better than 6, 8 – but only slightly).

There are three messages I get from running this simulation.

  1. One should, in general, not make too many intuitive assumptions about how probability works, particularly with somewhat complicated examples.
  2. One should be careful how one uses even simple games to teach probability. All of the students saw that 7 was the most common number, but their intuition of "choosing a wide spread" is a valuable one for this game, but it doesn’t help us get at the idea of some numbers being more likely than others. I think we’d need a different game for that.
  3. It is probably a good idea to build the simulation before you play the game with students, if at all possible.

PISA results from 2000 to 2009 for Canada

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.

Culture and counting

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?

We didn’t do any math yesterday

Practice makes perfect comic

 

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.

 

Seven Bridges problem

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?

Interesting ways to use Google Apps in the math classroom

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.

 

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?

 

Not a math person

Original blueprint

 

Someone I know produced the diagram above in her planning steps to produce the shelves seen below.

 

Finished shelves

 

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. 
 

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:

Learning mathematical ideas through literature

I looked through our school library today to see if we had any books which would tell mathematical narratives, and I found the following collection. Some of these stories are more "mathy" than others, but each of them has a narrative written around a mathematical concept. Some of these stories could be used to develop context for your students.

 

Anno’s Mysterious Multiplying Jar

This short book tells a very interesting story about a mysterious land with 2 countries, 3 mountains inside each country, and 4 kingdoms inside those countries, and so on, ending with 10 jars. The first pages are essentially describing what a factorial looks like, and if the story ended with question "how many jars are on the island?" I think it would have been an excellent lesson hook into factorials. Unfortunately, the book continues after describing this very interesting narrative with a fairly complete description of factorials. If you are a home-schooling parent, and you want to understand factorials better, and have a story to share with your students, this could be a fabulous resource. If you are a teacher, I recommend ending the story on the page where it first asks how many jars are in the boxes (read the story), and using this as an introductory activity with your students into factorials. Alternatively, this could be an interesting lead up to a less interesting arithmetic problem, wherein the students actually calculate how many jars there are.

 

The Number Devil

This is a novel about a boy who has a fantastic series of dreams full of interesting mathematical ideas, described in language he understands by a character called the Number Devil. The ideas in this story are very interesting to me as a mathematics teacher, and although I’m not sure every kid would enjoy this story, certainly those kids (and adults!) who are interested in mathematical ideas would find this story very interesting. Teachers may also find this book a useful resource for analogies and narratives to help students understand some complicated mathematical concepts. Disclaimer: I have not yet read this entire book, but have enjoyed the 1/3 of it or so that I have read.

 

The Fly on the Ceiling

This story is a historical account of how Rene Descartes may have come up with the idea of the Cartesian plane. According to some other sources I consulted, unfortunately the story is either not true, or incomplete, as it somewhat ignores the series of other inventions made by other mathematicians in advance of Descartes. That being said, it does have an excellent description of how one might make their own Cartesian plane, and teachers may find some inspiration for activities related to introducing the Cartesian plane from this book.

 

A remainder of one

This short story is an excellent description of remainders when doing division, and presents an interesting puzzle. How can 25 soldiers be divided into rows evenly with no remainders left over? One could easily find other such puzzles that are related, and make the concept of remainders much more tangible for students. I think that students may find this book interesting (even younger readers who have not yet learned about division) as well as parents and teachers looking for a lesson idea.

 

Sir Cumference and the Great Knight of Angleland

This story is an attempt to justify the use of angles to solve a problem of navigating through a maze. In terms of giving students some context for understanding angles, I think it does an okay job. One of the benefits of this book is that it does not show the same angles using exactly the same diagrams, which may help students understand that just because two angles have different size rays (or line segments) attached to them, they may still be the same size. I think some students will enjoy this, and teachers may find some useful activities for students to do related to the concept of angles.

 

One Hundred Angry Ants

One hundred angry ants is a short story about 100 ants trying to get to a picnic quickly, and trying different arrangements of rows of ants and the number of ants in each row. The story is appropriate for talking about factors of 100, and could be easily turned into a problem about factors of other numbers. Students will probably find this story interesting up until about 7 or 8, but teachers may find it a source of an idea about teaching that some numbers have multiple factors.

 

Among the Odds and Evens

This short story describes what happens when X and Y visit the land of the numbers. They find to their surprise that there is some strange relationship between the oddness or evenness of the parents, and their offspring. I think this book is interesting to help children remember the fact that odd numbers when added together always add up to an even number, and that even numbers always add up to be even, and that an odd and an even number add up to be odd. However, I suspect that this will be more interesting to student to discover this relationship between numbers (among the many other relationships out there).

 

One Grain of Rice

This is a retelling of the classic story where a peasant outwits the ruler of the land by asking for a doubling reward each day, and ends up with a much larger reward than the ruler expected. It would be good for introducing exponential growth. I would recommend stopping through the story occasionally and asking for predictions from the students about how good a deal the Raja gets. There is an interesting follow-up question for the students as well which is somewhat open-ended: how many years has the Raja been collecting rice? Is it possible for him to have collected 1 billion grains from the lands in his kingdom?

 

Two of Everything

In this story, an old woman and her husband discover a magical pot that allows them to double everything. It could lead to some interesting questions, like "how long will it take the couple to gain enough money from their pot to be comfortable for the rest of their lives?" I’d recommend this for parents who would like to develop more number sense in their children, and for teachers who would like a hook for a lesson around symmetry, doubling, or multiplication.


The Phantom Tollbooth

This book is a treasure trove of logical puzzles, mathematical ideas, and will get kids thinking about different ways of viewing the world. I remember reading it when I was a kid, and I thought it was excellent. Years later, I realized just how many mathematical ideas were in the book. I would recommend this as reading material for students, parents, and teachers.

 

If you know of more books like this, which have a mathematical concept (more interesting than counting books please, there are SO many of those) embedded within the storyline in some way, please share them.

Update: I saw this huge list of books with mathematical ideas shared via Twitter. No reviews, but each book has a very short word description of the math idea to which it links.