Education ∪ Math ∪ Technology

Tag: mathematics (page 2 of 4)

What should be on a high school exit exam in mathematics?

Personally, I think an exit exam for school (an exam a student needs to graduate from secondary school) is not necessarily the best way to determine if a student has been prepared by their school. That aside, some of sort of assessment of what a student has learned from their school, whatever form that would take, should satisfy an important criterion; that the student is somewhat prepared for the challenges that life will throw at them.

A typical high school exit exam is testing a student’s preparation for one component of life, specifically college academics. It seems obvious to me that this narrow definition of "preparation" doesn’t actually prepare students for the challenges of life. A student could quite easily pass the NY Regent’s exam in mathematics, any of the IB mathematics exams, their SAT, and any number of other standardized exams, and not know a lick about how to apply the mathematics they are learning in school to solving problems they will encounter in life.

While this shouldn’t be the only goal for mathematics education from K to 12, it seems to me to be a minimal goal, and one which at which we are failing quite dramaticly. Some evidence of this failure is seen by our mostly innumerate public who; lack basic literacy of graphs & statistics, are largely mathphobic, do not understand probability (casinos are good evidence for this), and generally only use relatively simplistic mathematics in their day to day life for problem solving. 

There is nothing inherently wrong with teaching how to do a calculation for it’s own sake, or for sharing some of the beauty and power of mathematics, but it should be framed by the notion that our education of mathematics is intended for a greater purpose. If we only focus on the 4 years people spend in college, we do a disservice to the decades of life they have after college.

Mumbo Jumbo

Algebra is just mumbo jumbo to most people. Seriously.

If you asked 100 high school graduates to explain how algebra works, and why it works, I’d guess that 99% of them couldn’t, not in sufficient detail to show that they really deeply understand it. Remember that I am talking about high school graduates, so these people have almost certainly had many years of algebra and algebraic concepts taught to them. Most of these people will only be able to give you some of the rules of algebra at best, and some of them don’t even remember that much.

Algebra is an amazing tool for solving problems though! Formulate a problem as an equation, and unless the equation is too complex, there is an algebraic algorithm to solve that equation, and hence the problem you formulated.

Maybe it is such a useful tool that people don’t really need to understand how it works, maybe they can get by without a deep understanding, but still be able to follow the rules of algebra and use it to solve problems. I don’t really buy that argument though, simply because people who don’t understand something are prone to make mistakes, and not be able to check their work with a reasonable level of accuracy.

Computers are also mumbo jumbo to most people. If you asked people to explain how computers work, most of them cannot. There are actually very few people in the world who can explain from start to finish how a computer works, and there is no one that can explain every single piece of a computer. Computers are still amazing tools though, and give people the ability to solve problems that would otherwise be intractable.

I think computers are a useful tool despite our lack of understanding of how they work. Like algebra, computers are a block box in which we put our inputs and get outputs and don’t understand how the inputs are related to the outputs. Given this similarity, we should look at other reasons why using a computer might be superior to algebra.

There are some significant differences between using computers to do computation, and using algebra to do computation. The first is that using a computer, the error rate is much lower. Obviously you can still press the wrong buttons, enter the wrong information, read the information the computer gives back to you improperly, so there is error, but I’d argue that this error is much less than the standard error rate for algebra. The second benefit of using computers is that they are much faster than doing even moderately complicated algebra by hand, including entering the computation into the computer. In the case that doing it by hand is faster, then I’d say you should do the calculation by hand. 

The largest difference between using a computer to do the calculation and using algebra is that algebra is a single use tool. It can only be used to turn an equation into a solution. A computer can be used for so much more.

Granted we should consider computational mathematics to be a broader tool than just plain algebra, if we want a more fair comparison with a computer, but I’d argue that all of the same problems exist with other areas of computational mathematics. As we increase the scope of computations we can learn how to use, the power of the computer becomes even more evident. It takes much less effort to learn how to compute a broader scope of problems using a computer than learning all of the individual computational methods. Witness the power of Wolfram Alpha, for example. Enter in a search phrase and all sorts of useful information comes up.

So in the consideration of using computers for solving computations, over a by hand approach, we can see postulate that the computer will produce less errors, be generally faster, and is more multipurpose than the pencil and paper model is. Furthermore, the computers can do a lot more as a tool than what you can do with algebra.

Another issue I see is that our current mathematics curriculums leave very little time to learn more important skills than computation. As Dan Meyer (@ddmeyer) points out, the formulation of a problem is more important than the actual solution. Learn how to formulate problems and understand how to verify that what you are doing makes sense, then spotting errors in computation becomes that much easier. Furthermore, I’d like to see mathematics education be much more grounded in what is relevant, than be a collection of different types of math which are taught for historical purposes or because they are the ground-work for calculus.

The question for me is, why aren’t we using computers more to do mathematics in elementary and secondary education? It can’t just be because people are scared of change, can it?

You need to give them the tools

Container of coins

Every elementary school classroom should have about $20 in change. Not fake money printed on a piece of paper, but real money. Yes, some of it will go missing over time, and you might need to lock it up depending on your community, but honestly it’s worth the risk. It’s only $20.

Like it or hate it, money is an integral part of our lives. If you want your students to understand the world, you have to provide them with the tools they need. Your students aren’t going to learn about how to add up change without holding it in their hands.

Computers should transform mathematics education

Stephen Shankland posted an interesting article on CNET today. Here is an exerpt from his article, which you should read in full. He says:

Clearly, children need some understanding on their own of math, and reliance on a computer has a lot of drawbacks. But computers can also aid those who otherwise would fall by the mathematical wayside, or let people with more advanced abilities bypass drudgery and move on to the challenging material. Graphing calculators can let many students explore curves and functions that realistically they’d more likely ignore if they had to plot them by hand.

My response to some of the negative comments about his article is:

Some of you have decided that using technology to handle calculations in mathematics is going to weaken student’s understanding of mathematics. I have to tell you, our student’s understanding of mathematics, and even the vast majority of people’s understanding of what mathematics is pretty bad. Awful. Horrible. I mean, really, really bad.

Mathematics is not about calculations. Mathematics is about understanding how our world works through the lens of logical reasoning and pattern forming, and then communicating our understanding of that process to other people.

Calculations are a tool in mathematics to understand a process. In my opinion, I want students to understand the processes and ideas that mathematics represents, not the calculations which short-cut that understanding.

Here’s an example that Gary Stager suggested to highlight this problem. Ask a typical math teacher to explain to you why "you invert the 2nd fraction and multiply instead" when dividing two fractions works. Ask them to explain the concept behind "inverting and multiplying" two fractions, and you know what, they can’t. They’ve learned a recipe for doing a calculation but have no conceptual understanding of why that rule works, and these are people who are teaching our children about mathematics!

We need to move away from the mindset that the most important part of the mathematics curriculum we teach is the rote calculations which can generally be done much faster on a computer, and towards the mindset that students need to be able to formulate problems, decide on appropriate mathematics to use to solve these problems, and then do the calculations on an appropriate device, and finally check that these solutions make sense. These are the steps that Conrad Wolfram and Dan Meyer (in their TED talks) outline as crucial to mathematical understanding, and I completely agree.

Mathematics education needs to change. Those people who want a "back to basics" approach and get rid of the calculators seem to think that this will improve the mathematics education in our schools. This is flatly not true.

If you ask a random sample of people, they either "weren’t very good at mathematics" and generally hated it, or a very small minority loved it. This opinion spans all age groups and goes back many years, far before the introduction of calculators in schools. If we judge the success of an educational approach by the number of people who enjoy working in a subject, why are so many people who were exposed to that approach before the introduction of calculators hate mathematics so much?

Maybe we need to rethink our approach?

Mathematical problem solving

Today I decided to record the process of solving a mathematical puzzle I found at the Project Euler website, in an effort to try and begin to analyze the problem solving techniques I use. My interest here is mostly in how the process unfolds, and the skills I use to solve these problems, rather than the actual problems themselves, although those are interesting. Below is the video I recorded when solving this problem.

Find the problem at

I think it would be interesting if we could create more examples of these videos in action, where we solve mathematical problems. I’m kind of inspired by Vi Hart, whose creative genius in mathematical doodling I cannot match.

Check out my discussion of a solution to the problem I solved below. Could you do this for another problem?

Revamping Mathematics Curriculum

What if we revamped the mathematics curriculum to match the style of teaching Dan Meyer recommends? What would that look like? Watch the @ddmeyer video below from his TED talk, and then let’s look at how we can make specific changes to our own teaching practice, and talk about whether or not these are changes worth making.

I’m sure we are all guilty of creating problems for our students which are too well defined. I know I have. I’m trying to change how I do my own teaching practice, but it is always helpful to do this with a team of other people. Does anyone want to jump in and help take a set of math curriculum and turn it into something which is more useful for our students’ learning? Let’s create a problem forming curriculum instead of a strict problem solving curriculum?

I’m putting the call out to collaborators here (or for anyone to point me at a similar project with which I can join efforts). Please check out and ask for an invite to the wiki if you are interested in helping with the mathematics curriculum revamp.

Introducing Probability Using Settlers

This past week I was looking for a way to introduce probability to my 9th grade students.  One of the problems students have when they are first learning probability is developing some intuition about what to expect.

I decided that one of the best ways to develop intuition about probability is to have some strong emotions associated with the results of their initial probability experiments, so I decided to teach my 9th grade students how to play Settlers of Catan.  I didn’t give them any information about best strategies to play the game, I just taught them the basic rules and set them loose.  Here are some rules for your reference.

Settlers Map

The basic idea is, each hexagon produces resources, but only when the number shown on the hexagon is rolled as the total of 2 six-sided dice.  If you have a settlement located at one of the vertexes of a hexagon which has just produced resources, you gain 1 of those resources.  You can then save up these resources, trade them with other players, or then use them to buy more settlements, cities, etc… Essentially if you gain enough resources of the right type before your opponent, and you win.

The actual system we used to play is called JSettlers, and it is an open source Settlers of Catan server.  I hosted it on my laptop with no difficulty and shared the link to my students to play it.  This way I didn’t have to pay for a class set of expensive Settlers of Catan games.

It only took about 10 or 15 minutes of playing for the kids to realize when they had made poor choices, or when someone had an obvious advantage.  The question I had once we had played for enough time that they had gathered some data (I required them to keep track of what was rolled as they played), which starting settlements were poorly placed, and which were in the best locations.  Students looked at the following situation and decided that this intersection of hexagons was a good place to put a settlement.

Good choice of settlement

They looked at an intersection like the following and decided that this was a poor place to put a settlement.

Poor settlement location

I asked them why they liked the first spot and didn’t like the second?  One of them said it perfectly, "well, the numbers 8,9, and 10 are WAY more likely to come up than 2, 4, and 11."

We followed with a discussion of why each number was not equally likely to come up using a typically sample space table, and then we kept playing, having both put some context on the probability we were learning, and developing some intuition about which numbers were more likely to come up.  I was able to extend their thinking quite a bit, as there were several different games being played, none of which had exactly the same set of numbers rolled.  It really worked well, and I’ll continue to use an example like this in my practice.

Museum of Math

An organization called the Museum of Math is having a contest to promote how Mathematics can be fun and exciting.  As per their website:

"Mathematics illuminates the patterns and structures all around us. Our dynamic exhibits and programs will stimulate inquiry, spark curiosity, and reveal the wonders of mathematics."

The idea of an organization which is dedicated to spreading the word about how mathematics is evident in the real world is fascinating to me, because this is a question I always have to answer.  Every single one of my classes, every year that I teach, asks me why they are learning math.  The relevance of mathematics for the typical high school student to the real world is really hard to see, especially if one is is learning the material in the more traditional way mathematics can be taught.

They are having a contest to promote both their organization and the love of mathematics.  From their website again:

Enter our Twitter contest and tell the world why you love math! The best tweet with hashtag #MathIsFun will win a free iPad. Contest ends June 1, 2010.


Why don’t all students react the same to feedback from their teachers?

There is lots of research which shows that human beings are complex.  In fact, although we can be modeled as groups of people mathematically in many circumstances, individual humans are too complex for mathematical analysis to much use in exactly predicting our behaviour.  However humans do follow patterns of behaviour, and we can predict what a possible range of behaviours are we expect to see.  Obviously this is why our social structures work because this predictive ability is easy enough that one can do it without the aid of a computer.

However this behaviour does follow the mathematical rules of a chaotic system.  Chaotic systems are systems which typically follow fairly predictable patterns, but for which small differences in input can lead to widely different outputs.  If you want to learn more about this I recommend reading some of Keith Devlin’s work in this area, he explains it in an easy to follow way.  Humans are chaotic systems because they take input from the outside world, process it, and modify their own behaviour, which leads to changes in their environment to which they again react, etc… This leads to what we call a feedback loop.  Often these feedback loops stabilize, which leads to predictable behaviour, but occasionally they can destabilize and chaos erupts.

We have all seen this as educators.  Johnny is a perfect angel every day, and then one day he comes into school and gets into a fight in his first period class.  I think sometimes we blame ourselves when these things happen, and we wonder why they happened.  Assuming you treat all students very similarly, and they come to school with pretty much the same kinds of things happening in their lives, you might wonder why some students are bright and cheerful despite their possible misfortunes, and others are a bad mood.  I think if you look at it from the perspective that they are chaotic systems, you can assign a lot less of the blame onto yourself.

I think that this boils down to, not all students, even ones who seem very similar on the surface, are going to react to you in the same way.  Even a minor variation in what is going in their lives can lead to very different attitudes from students, and you have to accept that.

There is some recent research though that shows that a lot of chaotic systems have large areas of stability.  The implication of this fact to educators is that if we moderate our own behaviour and feedback to guide students into these areas of stability, we can encourage students to behave in a more predictable manner.  I think it is well established that a good teacher respects their students, treats them fairly, cares for their students in such a way as to make them feel comfortable in the classroom.  These types of behaviours from the teacher are reinforcing the stability that our students need so much in their sometimes chaotic lives.  

Turning Math word problems into Math video problems

Last year I tried an experiment after being exposed to research about the Jasper project.  The basic idea of this project is, turn difficult word problems into authentic video problems which include potential extensions.  The experiment was this, have my students create the video word problems, and start creating a library of these problems to use with my future classes.

The experience of creating the problem has some minor mathematics in it, after all the students need to formulate a difficult problem, verify that they are able to shoot the problem on video and then show a working solution to the problem (on paper or handed in separately in digital form).  These skills are quite difficult, and are higher order skills in Bloom’s taxonomy.

Here’s an example of one of these word problems on the right.

It’s important to note here that there are some very difficult mathematical concepts embedded in this video.  Students will need to be able to understand rate problems, solve for the distance of the falling object using kinematics, and use trigonometry to determine the distance that needs to be traveled, and then go back to rate problems to answer the question.

The whole process from start to finish took about 2 weeks (or 8 classes).  One class to brainstorm the idea, one class to decide on the script and come up with the text version of the problem, and a few classes to solve the problem and do some in-class video editing.  Yes, this is a lot of time, but in terms of building student self-esteem, working on very important collaboration and planning skills, it is worth it.

There’s no way that is actually enough time to produce such a high quality (for a student group) video, so I know for sure that lots of time was spent on this video outside of class, probably many hours of time.

So this process also inspires the students and gets them excited about your material.  They will work much harder when they are excited about coming to class.  

The video editing process itself was fairly straight forward.  Most groups shot the clips with standard digital cameras, and then recorded the audio tracks after their video was done on their computers using Audacity.  One group used iMovie for their editing and production, and the other 3 groups used Windows Movie Maker which was totally sufficient for their needs.  If you want a no-install option, you can look at using which I’ve tested out myself and works fairly well.  It only really lacks two important features, the ability to edit the audio track separate from the video, and the ability to modify the video itself (instead of just moving it around), such as slow-motion, etc…

Check out these other two videos.  Maybe use them with your class and try and solve the problems.  As far as I remember, all of them have solutions, although some will require students to estimate distances.