#MathEd – Math for My Nephew

A while ago I was asked to provide some resources I use to teach math for my nephew (well, nephew-in-law). I decided to gather them together into a blog post and thus get a two-fer.

Associations

Mathematical Association of America (http://www.maa.org/) – Their stated mission is “to advance the mathematical sciences, especially at the collegiate level.” Membership is open to students and teachers (K-12 and college), starting at $35/year (student with proof of status) and going up to $249/year (Member Plus). I’m a member and for me the real value of membership is access to a wide range of publications plus discounts on books, both e-book and printed. (Disclaimer: I currently write book reviews for the MAA web site but I am not compensated.) You can follow them on Twitter at @maanow and on Facebook as maanews.

National Council of Teachers of Mathematics (http://www.nctm.org/) – Like the MAA, the NCTM offers memberships to students, teachers (primary through college) and to organizations. They also offer the option of an ‘e-membership’ at each level for a slight discount. Membership annual dues range from $44 (Student and Emeritus) to $144 (Full Individual Membership). Membership gives you access to a host of instructional materials, NCTM’s ‘e-standards’ and NCTM’s E-Seminars, 60 minute on-demand video presentations on a variety of math education topics. You can follow them on Twitter at @NCTM or on Facebook as NCTM Illuminations.

Blogs

Math with Bad Drawings – This is one of my favorite sites. Ben Orlin provides an entertaining and educational view of math from a teachers perspective. The title comes from each post being illustrated by a series of stick drawings on a whiteboard. I wrote a review of his site here. I’ve used some of his posts as jumping-off points for discussions in my math class. You can follow Ben on Twitter as @benorlin. He’s also on Facebook but doesn’t appear to be too active.

MathBabe – Cathy O’Neil is a former Wall Street quantitative analyst who left for the wilds of higher education. She mainly blogs about Big Data and math education in higher ed. I wrote a review of her site here. A smart, interesting, fun site. You can also follow Cathy on Twitter.

Finding Ways to Nguyen Students Over – Fawn Nguyen is a California math teacher who has one of the best math blogs I’ve seen for primary and secondary educators. She clearly works hard to present math to her students in innovative and entertaining ways and she shares these techniques on her blog. (I wrote a review of her site here.) However, the real hidden gems of her site for educators are the affiliated sites like Visual Patterns, Math Munch and Would You Rather. These sites are a treasure trove of fun math problems and exercises you can share with your students or work on your own.

The NRICH Project – Not a blog per se, but The NRICH Project was started by the University of Cambridge, according to their “About” page:

“to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice.”

(More here.) The content is divided into material for teachers and students. Each category is further divided into primary and secondary education. The idea is to present tasks that target multiple learning styles. These are known as “rich tasks” and you can get more information about them from this article.  Teaching materials are printable and downloadable for ease of use. You can also follow the NRICH project on Twitter  and on Facebook. You can sign up on their mailing list to get updates. They also provide a guide for parents and caregivers.

Project Euler – Also not a blog but a cool site for more advanced math fans. (Yes, there are some of us out there.) It’s a collection of almost 500 math problems, some of which may require basic programming skills. Each problem builds on insights gained solving previous problems so I advise doing them in order. I’ve posted a few of the problems on my site. Working on the problems presents interesting insights and is definitely mind-expanding.

Videos

NumberPhile – This is a nice resource of short, entertaining videos about specific math concepts. I do a short segment at the beginning of my classes called “Math Minute” and I’ve used several of these as inspiration for material. You can also follow them on Twitter and Facebook.

ComputerPhile – Not strictly related to math, but computer science is based on math. Similar to NumberPhile, ComputerPhile provides short videos about topics like undecidability, cryptography and how computers use math to do animation. You can follow them on Twitter or on Facebook.

I’m always looking for good math resources, both for my classroom and this blog. If you have any interesting leads, post them in comments!

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#WeHateMath Apples v. Oranges

Recently I was asked to briefly explain the difference between math and arithmetic.  Here’s what I came up with:

 

Arithmetic can answer the question:

“What is 3 plus 2?”

Arithmetic cannot answer the question:

“What are 3 apples plus 2 oranges?”

Math, on the other hand, can take a higher order look at the problem and answer:

“3 apples plus 2 oranges are 5 pieces of fruit.”

Math is about relationships and patterns.

Math is arithmetic plus context.

#WeHateMath Book Review – How to Think Like a Mathematician

It’s not often that you find a math book that opens with a joke. I mean, literally in the opening, just before the preface:

Question: How many months have 28 days?
Mathematician’s answer: All of them.

(Okay, it’s no ‘man from Nantucket’ but I got a chuckle out of it.) I also found one of my favorite footnotes:

….use or like mathematics are considered geeks or nerds*

* Add your own favorite term of abuse for the intelligent but unstylish.

This was just in the preface (ProTip: always read the preface, kids!).

The author, Kevin Houston, teaches at the School of Mathematics at the University of Leeds. His dry British wit is evident throughout this very readable text. But this isn’t just a math-oriented joke book. It’s a solid collection of mathematical ideas and skills starting with sets and functions through proof techniques and equivalence relations. Houston wraps up with a discussion of how to recognize true mathematical understanding.

References

Houston, K. (2009). How to think like a mathematician: A companion to undergraduate mathematics. New York: Cambridge University Press.

#WeHateMath A Better Way to Teach Math?

Ever since I started teaching math, I’ve been trying to figure out why so many students (and non-students) seem to have problems with it and what I could do about that. With that in mind, I’ve been trying different techniques to see which ones are effective.

The first one was what I call the “Age of Aquarius” technique. It usually has me saying things like “Math is the secret language of the Universe” and “We’re all made of math” in an attempt to engage student imaginations. Unfortunately these work best with an audience that is already receptive and math classes are generally full of people who want to be anywhere else. As a result, saying these things makes them think that either you are high or you’re trying to make your job sound more interesting than it is.

I call the second method “Eat Your Vegetables”. This consists of explaining how learning math is good for you, usually by citing the Stanford Medical School study that showed improved brain function from the very act of learning math. In other words, even if you never use this stuff in your real life, it’s the mental equivalent of CrossFit. This also fell flat, being a bit too abstract for a group that just wanted to get through the class with a minimum of effort.

I decided to tackle this from the other end. I wanted to find the source of this distaste for math. I’ve always felt that hate is a fear-based emotion so if I can lower the general anxiety level in the room, I should get better responses.

One technique I used was to use tools to handle the mechanics of problem-solving. For example, whenever convenient I encouraged the use of calculators once the math portion of the problem had been processed. (If you understand the math well enough to explain it to a machine, this is a good thing.) For statistics, I showed how to use spreadsheets to quickly and easily observe the effects of different sample data on results as well as how to create different types of charts to visualize the numbers. For probability, I brought in decks of cards. I used a drawing program to sketch out problems and scenarios.

I also tried to minimize the use of jargon during class. While it’s useful to have a common vocabulary, I wanted students to pay more attention to the relationships and patterns than to worry about coming up with the correct terminology. We introduced terms as needed but it was okay if they had to use a few more words to explain what they were talking about or if they had to draw a picture or diagram.

This was the first term I used these techniques so it’s early days. But my last class session was today and overall I feel quite positive. Several students have remarked that this was the best math class they’ve ever had and my retention rate was pretty good. For math classes, it’s not unusual for half or more of registered students to withdraw before end of term. Out of an initial class of twenty one, I lost just seven and of those, four withdrew before the start of the term.

All in all, a good start. I’m teaching two sections of the same class next term so I’ll continue to refine my techniques and report my data here.

#WeHateMath Calculators Considered Harmful

“For this invention will produce forgetfulness in the minds of those who learn to use it, because they will not practice their memory. Their trust in writing, produced by external characters which are no part of themselves, will discourage the use of their own memory within them. You have invented an elixir not of memory, but of reminding; and you offer your pupils the appearance of wisdom, not true wisdom, for they will read many things without instruction and will therefore seem to know many things, when they are for the most part ignorant and hard to get along with, since they are not wise, but only appear wise.”(Phaedrus 274c-275b)

 

The above quote is Socrates complaining about the invention of the written word. He claimed that allowing students to write down facts instead of memorizing them would weaken their minds.

I’m reminded of this when I see some of the reactions to suggestions that we embrace the use of computer technology in the way we teach math. The most noted advocate for this is Conrad Wolfram, who described his philosophy in a 2010 TED Talk.

My first introduction to technology-assisted math was in my high school chemistry class, where we learned how to use a slide rule. (Yes, I’m thatold.) We didn’t have them in math class (or calculators for that matter) so all calculations was done by hand. I didn’t get my first calculator until I went off to college. As this technology got cheaper and more readily available, it began to filter down towards K-12, where I recall a lot of controversy over allowing their use in class. It was felt at the time that it would make students unable to do arithmetic by hand and therefore become dependent entirely on machines to do it for them.

Back in 1957, Isaac Asimov wrote a short story “The Feeling of Power”, about just that kind of future and how society had to rediscover the technique of doing arithmetic by hand. The comments at this page about this story are…..okay, I’m trying to think of how to say this….well, let’s just say there’s a certain ‘you kids get off of my lawn’ quality to them.

Please don’t misunderstand. I absolutely believe that grade school kids should learn how to do arithmetic by hand, to start. But I don’t believe that it’s as simple as we either do all of our math by hand or become completely dependent on machines.

First of all, I have no problem doing arithmetic using a machine. Arithmetic is a very mechanical activity, which is one of the reasons why teaching it involves so much memorization. It isn’t something that comes naturally to us humans. However, we should be comfortable with arithmetic so that we can visualize what our answer should look like and to make sure the problem was entered correctly.  (I prefer using a text editor to writing by hand. This doesn’t make me illiterate.)

Math, on the other hand, involves intuition, creativity, imagination and logical thinking. Machines can make the arithmetic part of it easier but you still need to understand the problem well enough to explain it to the machine. We teachers don’t have to fear the use of calculators or computers in a math class if we use them intelligently. Where these machines can be used to our advantage is to reduce student anxiety about the mechanical parts of the problem so they can focus on the part requiring human-based thinking.

In my experience, students hate math because they fear arithmetic. They are so scarred by their grade school arithmetic classes where the slightest error in a long chain of arithmetic would ripple down and cause them to get the problem wrong (and fail the test) that they don’t want to be anywhere near any class that reminds them of that.

I encourage the use of calculators (and spreadsheets and Wolfram Alpha) in my college math class. They take away fear and give us more time to actually talk about math.

#WeHateMath Two Teachers

We recently celebrated Teacher Appreciation Week so I’ve been thinking about the teachers in my life who have influenced me. Since this is a blog about math, I decided to mention two of my math teachers who have made a lasting impression on me.

(Unfortunately that impression didn’t extend to their names, since I’m terrible with names and it was about forty years ago. However, they know who they are.)

Middle School Algebra – My teacher was a tall, lean, severe-looking man whose horn-rim glasses and dark suits gave him the air of a church deacon. He had a deep, mildly monotonic voice, with an odd rhythm that forced you to pay closer attention. He addressed all of us as “Mr. X” or “Miss Y” and had a way of looking at you that felt like he could see inside of your skull. He gave off an air of mild disdain that for some reason pushed us to work harder to prove ourselves.

High School Geometry – My geometry teacher was a former Marine D.I., with a gymnast’s build and a wicked Van Dyke. He seemed to radiate energy and his eyes took in everything and missed nothing. His enthusiasm for his subject was contagious and he spent the majority of the class going through proofs. This might sound dull, but he presented them as if he was letting us in on this Great Secret and before long we were excited to get to the proofs and solving the puzzle. The most important thing I took away from this class was that I could assert that something was true but that was no better than simple faith. If the only evidence of a truth is belief, what happens when the believer goes away?

But if I can assert a truth through a rigorously logical proof, then this is not just true because I say so. The Universe says so. Not only that, but my proven truth will always be true and will have always been true. It doesn’t matter what happens to me.

Despite their different styles, both teachers had something in common. They treated their students as individuals and were able to express their passion for the subject matter in a way that energized us to learn.

With each new class I teach, I try to live up to the standards set for me by these two men.

 

#WeHateMath One Man’s Meat is Another Man’s Poisson

Recently I got to thinking about my grad school course on system performance. The instructor had a day job at Bell Labs where he did the math that he tried to teach to us. It was all about things like queuing theory, think time and all sorts of statistical and stochastic analysis of information systems. (It was pretty overwhelming at the time but I’d like to take the class again.)

Anyway…one of the concepts that I took away from the class was about Poisson distributions. It got me wondering about how I would explain this concept to my College Mathematics students.

Let me explain.

College Mathematics at my school is a 100-level class for non-technical students. Most of my class are in programs like Medical Assisting, Graphic Design and Business Administration. This is the course description:

This course develops problem-solving and decision-making strategies using mathematical tools from arithmetic, algebra, geometry, and statistics. Topics include consumer mathematics, key concepts in statistics and probability, sets of numbers, and geometry. Upon successful completion of this course, students will be able to apply mathematical tools and methods to solve real-world problems.

Basically, it’s a functional numeracy class. Don’t get me wrong — it’s fun to teach and I enjoy the challenge of presenting math concepts in an interesting and accessible way. One of the ways I do this is start each class session with a ‘Math Minute’ where I run through a quick math concept to give my students a little something to ponder.

The Poisson distribution was developed by Siméon Denis Poisson in 1837. According to StatTrek:

A Poisson distribution is the probability distribution that results from a Poisson experiment.

Okay, a Poisson something-or-other comes from a Poisson something-else. Could you vague that up for me?:

A Poisson experiment is a statistical experiment that has the following properties:

  • The experiment results in outcomes that can be classified as successes or failures.
  • The average number of successes (μ) that occurs in a specified region is known.
  • The probability that a success will occur is proportional to the size of the region.
  • The probability that a success will occur in an extremely small region is virtually zero.

Note that the specified region could take many forms. For instance, it could be a length, an area, a volume, a period of time, etc.

That’s a little better. Let’s look at the classic example of a Poisson distribution: bus arrivals.

On your route, a bus comes by, on average, every ten minutes. On this particular day, you arrive at the bus stop and there is no bus to be seen. How long, on average, do you think you’ll have to wait for the next bus?

At this point, your audience is expected to say “Why, five minutes, of course.” The standard answer is actually ten minutes. The problem with this example is that it doesn’t give the right picture so the actual answer doesn’t fit our mathematical intuition. That’s why at this point the explainer hauls out the equation:

Via StatTrek:

Poisson Formula: Suppose we conduct a Poisson experiment, in which the average number of successes within a given region is μ. Then, the Poisson probability is:

P(x; μ) = (e) (μx) / x!

where x is the actual number of successes that result from the experiment, and e is approximately equal to 2.71828.

And everyone’s eyes glaze over.

I’m not complaining but I’m looking for a way to explain this to a non-technical crowd. So the bus stop example can be much more effective if we just tweak a few parameters. A Poisson distribution is generated out of experimental data so let’s adjust our mental picture.

Instead of being the person waiting for the bus, let’s instead picture ourselves sitting on a park bench across the street from the bus stop. In one hand we have a stopwatch and the other a spreadsheet. We can start our experiment any time but let’s say that we begin when a bus arrives. As we observe busses and passengers, we record two things:

  1. The time the bus arrives
  2. The time that each person arrives at the bus stop.

We need to gather enough data for a useful answer, so we’re going to be here for a while. It could be an hour, two hours, twelve, whatever. At some point we get bored and go home to crunch our data. Lo and behold, we discover that the bus interarrival time does in fact average ten minutes but the average wait time is greater than five minutes.

Why is that? Well, from our perspective as an outside observer, this phenomenon makes a lot more sense.  Let’s look at our bus arrivals over a timeline:

bus 1  –  5 mins. – bus 2  – 15 mins.- bus 3 – 10 mins.bus 420 mins.bus 5 and so on…

As an individual waiting at the bus stop we didn’t have this perspective. But when we look at it from outside, we have a better picture of what’s going on. More passengers arrive during the longer intervals than during the short ones. As a result, the wait times for these passengers are going to skew our average wait time, giving us our previously unintuitive result of greater than five minutes.

Uncertainty is part of our lives, so it makes sense that it would show up in math as well. Probability (and statistics) don’t eliminate uncertainty but rathergive us a useful way to acknowledge  and honor uncertainty.