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.
Month: April 2012 (page 1 of 2)
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.
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:
- 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?
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
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?
I love this quote shared by Gary Stager via the Daily Papert.
“They first learn engineering, then from there they progress to learning the ideas behind it, and then they learn the mathematics. This would be inventing, it’s a little probe toward inventing a different kind of content. It’s not a different way of teaching; it’s not pedagogy. It’s different knowledge. It’s a good example of turning knowledge – turning learning – upside-down. Instead of starting with this abstract stuff we had from the nineteenth century, let’s start with stuff that’s really engaging for the children, out of which the deeper ideas can develop.” Seymour Papert, 2004
Instead of students learning a bunch of theory, and then being able to apply it to practice, they would engage in building and in creating, which would provide a need and a motivation for the knowledge behind the thing that they are building. One thing that is missing in great degrees from our school is motivation. Why do I need to learn this math when I can see no use for it?
It’s not to say that there aren’t fascinating ideas which are worth learning without something practical to support them, but this is not the general tendency of school. We do not usually teach (with obvious exceptions being good teachers) that which is interesting. We teach practical knowledge. We tend to say, "Oh, we’d like kids to be able to be doctors. Well, what do they need to know? Let’s teach them that." with the result that students do not see the connection between what we teach, and what they can do with it.
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.
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.
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.
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 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.
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).
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?
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.
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.
So I read some interesting research on student incentive progams which has a couple of very important paragraphs. Here’s the abstract:
This paper describes a series of school-based randomized trials in over 250 urban schools designed to test the impact of ﬁnancial incentives on student achievement. In stark contrast to simple economic models, our results suggest that student incentives increase achievement when the rewards are given for inputs to the educational production function, but incentives tied to output are not eﬀective. Relative to popular education reforms of the past few decades, student incentives based on inputs produce similar gains in achievement at lower costs. Qualitative data suggest that incentives for inputs may be more eﬀective because students do not know the educational production function, and thus have little clue how to turn their excitement about rewards into achievement. Several other models, including lack of self-control, complementary inputs in production, or the unpredictability of outputs, are also consistent with the experimental data.
Notice the sentence in bold. Incentives for outputs (like let’s say student test scores) does not improve performance. Incentive for inputs (like increased collaboration, more training, working in high needs schools) does. Of course, the benefits gained by the incentives are not terribly strong (with the exception in the study of students being paid to read more) and so any benefit from paying teachers for changing the inputs to education may be minimal.
The merit pay for teachers movement has it all wrong, and the very people in promoting merit pay for teachers have access to this research. In fact, the list of acknowledgements at the beginning of the paper is a veritable who’s who of the leading education reformers in the US.
One of my students came up (with some help) this procedure for converting between degrees and radians.
- Memorize the fact that 60° is π/3 and that 30° is π/6.
- Note that 10° is therefore π/18 and that similarly 1° is π/180.
- You can then take any degree measure and convert it by converting the number of degrees into sums of degrees where you know the conversions. For example, 70° is equal to 60° + 10° = π/3 + π/18 = 6π/18 + π/18 = 7π/18.
Obviously this procedure is not by any means the most efficient way to convert between radians and degrees. Although I showed a much more efficient algorithm for converting between degrees and radians, it didn’t make sense for this student, and so he and I came up with this procedure (which I drew out of him by asking him questions about the angles), which he does understand.
In general, I’d prefer students use inefficient techniques that they understand completely than highly efficient techniques that they do not understand. Hopefully this student will continue to work on his procedure to make it more efficient as he has to use it over and over again, but if not, at least he will be thinking with something that makes more sense in his head.
I’ve been learning how to program for a long time, a task that has much in common with mathematics. Both programming and mathematics involve being able to solve problems. Some of the problems in programming and mathematics have well established solutions and other problems do not. On a micro-level, programming involves manipulating code, a task much like the symbolic manipulation often used in mathematics. On a macro-level, programmers and mathematicians both need to be able to trouble-shoot, organize, and communicate their solutions.
When I learned how to program, I taught myself, and I know that as a result, the code I create does not always follow the most appropriate industry standards. I have some unconventional solutions to some of the standard problems in programming, and I have less than optimal solutions for some basic problems in programming. I’ve yet to develop my own library of solutions, a standard practice in the industry.
On the other hand, I’m not a professional programmer. I program to solve problems I run into in life, and I program for fun. I have many programming projects that I’ve started and not completed. I’m an amateur programmer. I don’t need my work to look exactly what professional programmers’ work looks like because I rarely, if ever, share my programming with other people. I often share the results of my programming though, and this has helped build some useful tools for my students.
There are many low-level tasks that I no longer need to reference. I don’t need to look up how to define variables or functions, and I don’t need to look up loops, conditionals, and other basic parts of the structure of the programming languages that I know. I still need to look up the methods and properties of some higher level objects in the programming languages I know though, and when I program in PHP, I have a reference manual for the hundreds of functions available in PHP always open. I could be said to have developed a certain amount of automaticity in learning how to program, especially for the more basic tasks.
This automaticity was not learned by memorizing programming structures. I didn’t develop automaticity by doing practice exercises. I didn’t develop automaticity by reading books on programming. I developed automaticity in the low-level programming tasks by programming, by giving myself projects to work on that required me to build my skill, and by repeatedly looking up reference material when I got stuck. I developed automaticity because it is frustrating to write code that doesn’t work. It’s frustrating to get error messages that are nearly incomprehensible back from the computer when you make a mistake in the structure of your code.
If we look at mathematics education, we see that many, many of the problems given to students which have standard solutions. We expect students to develop fluency in these problems, often before they ever get to see any of the non-standard problems. In fact, in k to 12 education, students can potentially never be given an open-ended non-standard problem. Unfortunately, I believe this approach has failed our students in the past, and I’m not alone.
I’d like to see a system without a focus on fluency and automaticity in mathematics. These are the wrong drivers of mathematics education. Instead of focusing on the lowest level tasks mathematicians do, and assuming that fluency in these tasks leads to mathematical reasoning, we should focus on the most interesting and challenging tasks, and expect that a certain degree of fluency and automaticity will be developed as a result of these tasks. Instead of expecting students to memorize recipes and algorithms, we should allow them to develop toolkits and libraries to use of their own that they can reference as needed. Instead of feeling that every problem students need to do has to be solved quickly or efficiently, we should allow for alternate solutions and methods to be used.