Education ∪ Math ∪ Technology

# Tag: Computer based math(page 1 of 2)

This image is an attempt to capture the important stages of doing mathematics. As pointed by other people, mathematics is not a linear process, which I am attempting to share via this image. I see analytical reasoning, flashes of insight, and exploratory calculations as the glue that holds these stages of mathematical thinking together.

How do you see the process of "doing math"? Is it possible that what sets mathematics apart from other disciplines is the formalism, and the calculations involved? How does this process compare to other things that we do in life?

Here are the slides from my presentation for the Global ed conference on Computer Based Math. I will share the presentation recording when I have it, as I had some great questions from the audience. If you are interested in discussing these ideas further, join the Linked In group. If you want to challenge my thinking on this, please feel free to do so. I don’t want to end up moving mathematics education in the wrong direction, but I am becoming more and more convinced that the full use of computers in education is the right way to go. See this page for some responses to common objections to the use of computers in mathematics.

Update: You can view a recording of the presentation here (requires Java).

At the conference I was at in London, we were discussing, what would a mathematics curriculum look like if the computational step of doing mathematics was something students did using a computer?

Update: The video from this session has been posted by the Computer Based Math organization. See below.

Here are some objections shared by Conrad Wolfram and Jon McLoone at the Computer Based Math summit that happened in London, England. I just thought I’d add my two cents, and offer some more possible objections not in this list.

1. You’ve got to know the basics first.

First, what are the basics, and why did we define them that way? Are they basic because the concepts are basic, or because they were discovered in a particular order? Are they considered basic because of computational complexity, or because of conceptual complexity?

Seymour Papert, in his book Mindstorms, suggested that much of what we teach in mathematics classes is the result of historical accidents. He also suggested that we teach some concepts, not because they are the most valuable to teach, but because they can be solved with paper and pencil.

2. Computers dumb math down.

Jon McLoone has a terrific rebuttal to this argument. His basic premise: we’ve dumbed down mathematics education to limit us to what we are capable of doing with pencil and paper. I’d like to add that we have already turned mathematics education into sitcom-like instruction, where each topic can be taught in a single lesson (or a series of topics can be taught in a single unit), and where older topics are rarely, if ever, revisited. Having taught students how to use a particular topic, we then abandon it to learn new techniques.

3. Hand calculating procedures teach understanding.

While I think that it possible that hand-calculating can teach something, too often I see people learn recipes for doing math, rather than actually learning mathematical reasoning. I don’t see this procedure necessarily helped by computer based math, but I don’t see that it is hurt either. Whether students do a procedure by hand, or by their computer, if they don’t understand the underlining concepts, they will struggle to use the mathematics in any meaningful context.

Really? There are some small pockets which are using computers as the tool for computation in mathematics, but not on any reasonable scale at the k to 12 level. Students do use calculators, but not consistently across the curriculum, and many potential applications of computers are not well represented by calculators.

5. It isn’t math.

Here’s a diagram I’ve created to help capture the process of doing math.

The big place in this process where computations happen is in the formulation (and sharing) step shown in the bottom right-hand corner. Note that actually doing the computation, according to this diagram, is only a tiny piece of doing mathematics. If you agree with my premise, that doing mathematics is more than the computations, you might be willing to accept that actually doing the computation step is a tiny piece of the mathematical process. Can we really say that students aren’t doing math if they hand-off that step to their computer?

Do you think that children often get to do the entire process of math in schools, or are they often stuck at the computation step?

6. You are making people over-reliant on computers.

I’d like to have students doing more of the mathematical process. Not everything lends itself well to using a computer, and these types of things will still happen in classrooms. Some concepts and ideas are actually not often taught in schools (such as the applications of origami to mathematics) and should be. I want to see students doing more thinking in classes, not less. Mathematics is not entirely in the tool one uses to do computations; most of it happens in the head.

So rather than seeing people be reliant on computers, I’d like to see some resources available (in the public domain) so that every computation students do on their computers has the "by hand" method carefully catalogged and available for students to use. I’d like to see the computations become part of a toolset, rather than what our students focus on learning.

7. Traditional math is part of our culture.

I’d love to see mathematical history taught as an option in schools, so as to preserve the culture of mathematical tradition. That being said, culture changes, and we grow and adopt new traditions. For example, almost no one uses quill pens anymore, and it’s certainly not a skill we teach anymore in schools.

8. Good idea – but it can’t be done.

People have already been teaching mathematics with computers as a tool for computation for a few decades now. I took a course myself in mathematical computation at UBC, and loved it. Rather than saying it cannot happen, since it has happened, we should look to see how we can expand and learn from the current iniatives.

Some further objections I can imagine people having:

1. It isn’t fair; some students have access to technology, some do not.

This is one objection that I think has some merit behind it. We need to ensure that if we do move toward a model where computers replace the by-hand methods, we need to ensure that everyone has equitable access. As Seymour Papert (and others) have noted, a computer is only a tiny fraction of the total amount of money we spend on a student’s education, and so objections based on money seem to assume that we need to keep all of our existing structures, and that we can’t shrink some of them to pay for computers. How much money do we spend encouraging disengaged learners to remain in schools?

2. It isn’t healthy.

It’s also not healthy to lack mathematical reasoning, literacy and analytical reasoning skills, but we let plenty of students graduate without these vital skills for life. We do need to balance screen-time versus other forms of more interactive and kinethestic learning, and this will be one of our challenges going forward in education.

What are some other objections you can imagine people having to this kind of change in mathematics? Can you extend my rebuttals to these objections?

I’m to be on a panel for the Computer Based Math summit happening in a couple of weeks, and I have to construct a 5 to 10 minute presentation on the following question:

Where do we draw the line between what should be done "by hand" and what calculations can be done on a computer in mathematics education?

If you could help me with some feedback (and potential challenges) on my position (see below) that would be helpful. I’d like to thank the various people who have influenced what I’ve written so far as well.

My existing posts on this topic:

Summary:

Conceptual knowledge is necessary to be successful at mathematics, but I believe that for many of the algorithms we teach students, there is little difference between using a computer to do the algorithm and using pencil and paper. Some of the algorithms themselves have embedded conceptual knowledge, and are of course important to learn, but should be learned for understanding how the algorithm itself works, rather than necessarily memorizing the algorithm.

Bio:

David is a mathematics teacher and learning specialist for technology at Stratford Hall, a small independent school in Vancouver, BC. He is an experienced international educator, having worked in the USA, England, Thailand, and Canada. He has his Masters of Educational Technology from UBC, and Bachelor degrees in Mathematics, and Secondary School Education. He has written numerous articles for magazines, and blogs regularly at https://davidwees.com

Position:

I want to challenge the broad assumption that seems to exist, at least in k to 12 education, that there is a best set of content for learning mathematics. Aside from some numeracy skills, and arithmetic, the vast majority of the mathematics we learn tends to focus on algebraic (and eventually calculus) thinking. I suggest that what would be better would be to focus on mathematical thinking, and to allow much more room for many different kinds of math to creep into our schools. Learning algebra, for a dedicated individual interested in using it in a science, math, or engineering career, is not that difficult and would only take a year. Instead of the issue being hand versus computer, we could focus on ensuring that students learn how to think mathematically, in a variety of different ways.

Specifically related to calculating using a by-hand method or a computer, both are mechanical operations; without understand the algorithm, one cannot really be considered to be doing math.

Paper, pencil, and language itself, are all forms of technology. If the technology changes, the way the algorithm is done changes. When we use a computer to do a calculation rather than doing it by hand, we are merely trading one algorithm which students could potentially understand or not understand for a different one.

Critically, pushing around symbols on paper is just a symbolic representation of the real math taking place within one’s head. When one does a calculation, whether it is by hand, or by machine, an important feature of whether or not one can be said to be doing the calculation is whether or not one can predict the potential output from the algorithm, or if one understands the process they are using. By prediction, I mean, have the ability to recognize nonsensical answers, and to have a feel as to the approximate size of your answer at least, if not always the exact value.

It is important to recognize that this is not a new perspective. Consider this statement from the Agenda for Action produced by the NCTM in the 1980s.

"It is recognized that a significant portion of instruction in the early grades must be devoted to the direct acquisition of number concepts and skills without the use of calculators. However, when the burden of lengthy computations outweighs the educational contribution of the process, the calculator should become readily available."

Obviously we can easily substitute calculator for computer. So the NCTM draws the line between that which is educationally useful versus a “burdensome” calculation. Clearly this is a fuzzy line and needs clarification, which is part of the purpose of this discussion.

Control over what one does is a key aspect of “doing something” and is often the chief complaint against using a computer to do mathematics. “If you just enter it into the machine, you aren’t doing mathematics, the machine is doing it for you.” A story might be useful here, so you can understand my perspective on this.

One of my friends is an oceanographer, and at the end of the summer, he and I had a conversation at a party about what he does for a living. I asked him if he does any math as part of his job, since I am, of course, naturally interested in where mathematics is used outside of school. He replied, “No. My computer does all of the math for me.”

He explained to me that he spends about half of his time creating mathematical models to describe ocean currents and climate on a small scale, and then uses the computer to crunch data and compare it to his model. For example, he recently proved that of three data collecting stations a company he is working for deploys, one of them is unnecessary since the other two can predict the conditions at the 3rd station with 88% accuracy.

So here is this person who is creating complex models involving differential equations, writing Matlab scripts to crunch data, and comparing the output of the scripts to his models, and then communicating his analysis to his employer, and he doesn’t consider himself to be doing mathematics because the calculation step is done by his computer.

I think we probably agree that my friend has done a great deal of mathematics, and that what he does for a living models some of the mathematics we’d like our students to be able to do. His creation of a model, programming of that model into his computer, analysis and organization of the resulting data afterward is all highly mathematical, and is the kind of stuff that we could consider to be done “by hand.”

What I also see from this story is that my friend is most definitely “in control” of what he is doing. He has both control over the process he is following, and over the machine which is helping with calculations he could not possibly do “by hand.”

Further, when you program the machine, you are in control of what it does. If you make a mistake in your program, the computer complains.

So we require then an ability to predict and understand an algorithm, an ability to use it to model contextual situations, and an ability to use the output of an algorithm to reason and communicate mathematics. We also require, as a system, much more flexibility in the mathematics taught at the k to 12 level.

Here are some tools which I’ve either used (or explored) for mathematics education. They aren’t all open source, but they are all extremely useful, and they are all free to use (free as in free beer, some of them are also free as in free speech).

### Geogebra

This program lets you explore algebra and geometry, much like it’s proprietary cousin, Geometer’s Sketchpad. Having used both, I actually prefer Geogebra because I find it to be more flexible and easier to use. It will run on many different platforms including Windows, Mac, Linux, Android, and iOS.

### Mathematics Visualization Toolkit

The Mathematics Visualization Toolkit is exactly that, a program which lets you visualize mathematics. You can use it to build complex visualizations, or you can use the visualizations which are already included (which are awesome by themselves). You can either use the web start version of the toolkit, or download an offline installer.

### Scratch

Scratch is an excellent program for learning programming but also mathematics like variables, sequences, Cartesian coordinates, and other useful mathematical concepts. Developed at MIT, it is a free download and includes a strong user community to seek help, and see what else can be done with the program.

### Netlogo

Netlogo is “a multi-agent programming modelling environment” (According to the Netlogo website). It comes with hundreds of models for all areas of science and mathematics preprogrammed. It is a free download and will work on any computer which has Java 5 or later installed.

### Audacity

Audacity is an open source audio editor and recorder. One example use in mathematics is to record a bouncing ball, and use the visual data from audacities recording to turn this into a graph of bounce versus time between bounces. You can also use it so students can record 60 second podcasts explaining some aspect of mathematics.

### Calculize

Calculize is a free (currently) web app which lets students perform mathematical computations using a reasonably simple programming language.

### Wolfram Alpha

Wolfram Alpha is a computational engine built on top of the Mathematica architecture. It is amazingly powerful, and turns some homework assignments into a breeze. Recommendation: change your homework assignments, or do away with them all together.

Desmos

This is a free online graphing calculator. It emulates a lot of the functionality of a typical graphing calculator but with a much easier to follow user interface and without much of the non-graphing functionality of a graphing calculator. It is easy to create graphs, and then share those graphs with other people. It is also currently in development, so it is still improving over time with new features being added every couple of months.

### Logo

This Logo emulator lets students play with the classic programming environment Logo, built for kids by Seymour Papert and his colleagues at MIT, all online. It requires Java, but should run on most computers (sorry, no iPads…).

Google Earth is free (but proprietary) software that allows students to explore the world in 3d. One could use it for GIS applications, or even to explore the relationship between our 2d mapping system (longitude/latitude) and 3d space.

Google Sketchup (another free, but proprietary program) that allows students to create highly complex (or very simple, if they prefer) models. I’ve used it to have students construct their “ideal” school, and then from this model, they calculate the cost to build their school.

### Screenr

Screenr is a free (for up to 5 minute recordings) screen-casting (think record your screen as a video) software. Some possible uses of it are for students to use it to create video tutorials, record their process of solving a problem, or create their own video word problems. Another alternative for screen-casting is Jing, but it publishes to a format which is harder to share in the free version.

### Endlos

Endlos is an open source fractal generator which I’ve found runs very fast. It runs in Java, so it should run on any computer capable of supporting Java. The ability to experiment with, and explore fractals is a very interesting thing for students to do, but very tedious to do by hand…

### The Number Race

The Number Race is an open source program intended to help students who have dyscalculia develop their number sense. It has many levels of difficulty, and runs in Java, which means it should run on a wide variety of computers.

### Code Cogs equation editor

This free to use online equation editor could be a nice way for students (and teachers potentially) to construct equation images for use in a website.

### Eigenmath

Eigenmath is an open source program for symbolic manipulation in math. It runs either in Windows or on a Mac. Some examples of what it can do are shown above.

### Peanut math programs

These 9 free programs cover a wide range of different types of mathematics. Above is the popular statistics calculation and visualization program included in the package.

### Yacas

Yacas (Yet Another Computer Algebra System) is a command line program which allows for the symbolic manipulation and calculation of mathematical expressions. One thing I like about it is that it calculated 600! in a fraction of a second, so it is very fast (an aside, ever wondered what 6000! factorial is?)

Free CAS programs

Update: Just found an open source implementation of LOGO (as described in Seymour Papert’s Mindstorms) here: http://www.softronix.com/logo.html

Other free programs which I have used either for constructing mathematical diagrams/simulations or with students in some way include:

The Gimp, Programmer’s Notepad, Flex Builder (free with an education license), Open Simulator, VLC PLayer,
Wolfram Demonstrations (requires a free browser plugin), and Project Euler.

You might find these programs as useful alternatives to the “free apps” which “help” students memorize formulas & algorithms. For an enormous list of other free programs see this helpful list.

What other free programs for mathematics education do you use with or for your students?

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?

One of the parents of a child I taught last year shared this with me. Here is what I think the math class should look like. Forget teaching kids computation, especially when a computer can do it faster, cheaper, and more reliably.

This classroom more closely resembles Dan Meyer’s math classroom where students are expected to formulate problems but taken to a further degree. Let’s do away with the repetitive tasks that a computer can easily do by hand, make sure all students have those devices that they need to do these repetitive tasks, and then focus on how to use the computations in the real world.

I was asked if I thought that including interactivity in a mathematics was important. The answer to me is most definitively YES! In fact, I believe that if your mathematics and science classrooms do not include at least some of the features that I will describe below then you are doing a great disservice to your students. It may not be possible to include all of these examples in every context, but at least some of them are crucial to a deep understanding of mathematics and a recognition of its importance in our lives.

Graphing

When I attended high school in the early 90s, every graph I had to produce I did by hand. As a result my graphs looked something the following.

image credit: http://xkcd.com/418/

Now the problem with this of course is that if I want to modify the graph above and compare my modification against my old graph, I need to redraw the entire graph from scratch every single time. While there is some merit in learning the skill of creating a crisp neat graph, it is difficult to progress to more advanced graphing concepts when it takes you 15 minutes a graph to produce something worth reading.

Imagine the situation today where I can produce a graph immediately and then modify it, add an extra graph to compare two graphs, save my graph to look at later, etc… This is what modern graphing software allows us to do. This kind of interactivity allows students to look at much deeper concepts involved with graphing and of functions. Look at this example graph.

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

Multimedia

Imagine you are working on learning how to find the equation of a line from the graph. Which would you rather do; a worksheet with 20 different graphs on it, or would you rather look at a picture like the one below and find 20 lines first, and then find their equations. At least one of these examples involves recognizing that the lines come from nature.

image credit: Toomas & Marit Hinnosaar

What if you are looking at the properties of quadratic functions? Do you want to stare at a bunch of graphs of quadratic functions, or do you want to look at video footage you created yourself? Here’s what that looks like. Now the students can collect data about the graph, learn about the relationship between thrown objects in our world and parabolas, and then finally they can analyze their data and come to a conclusion about the motion of thrown objects. Here’s what that kind of video looks like (this was video collected from a webcam,  transparent graph paper over-laid on top of the video and the entire clip slowed down; all in iMovie).

Want students to create their own trigonometry word problems for practice? Maybe you’ve recognized that the best way to understand the formatting of a word problem is to create one for yourself. Instead of having students write out a word problem on paper, look at what happens when you have them create a story-line and video tape themselves performing their word problem.

Other ideas

In probability class you have them running simulations with dice (or even better playing a game involving dice). In calculus, have them compare the instantaneous speed versus the average speed as a homework assignment when they take a drive with their parents. In statistics class have them gather data from their peers and do the statistical analysis of information gathered from class. In geometry, have them prove the Pythagorean theorem by measuring out giant right triangles on a soccer field and compare the known length of the hypotenuse (which they measure) to the expected hypotenuse (which they calculate using the Pythagorean theorem). Use the last idea and talk about experimental error.

Summary

There are lots of other types of interactivity in the mathematics classroom that I haven’t shown here. Interactivity doesn’t have to include the use of technology, but at the very least you should have your students doing something each class, rather than sitting there and being passive recipients of the information.

I read an article one time which questioned why we choose calculus to be the top of the math pyramid in school.  Basically, most of the mathematics students learn once they master the basics aims toward preparing the students to take calculus at the end of K-12 school.  The article I read suggested that statistics instead of calculus should be at the top because it is much more practical to real life than calculus is.

We deliberately choose calculus to be at the top because we want our society to produce more engineers and scientists.  This helped produce a generation of engineers and scientists.

However, although engineers and scientists are still needed, the US Department of Labor predicts that neither engineers nor scientists will be in the fastest growing jobs in the future.  They have predicted the 30 fasted growing jobs in the United States and there is something interesting about the list.  5 of the jobs involve the use of computers.  Jobs number 25, 24, 23, 4, and 1 all include the significant use of computers in a highly technical fashion.  In fact all 5 of these jobs require computer programming skills to some degree.

So I propose that we make computer programming skills should be at the top of the list.  This way we will be preparing our students for careers in the future rather than the careers of the past.

Now we will still end up producing engineers and scientists because there is a huge overlap between the mathematics required to master calculus and the skills required to master computer programming.  We will end up producing a lot people who are totally capable of programming a computer.  Students who do not end up completing the stream will still end up having a very good understanding of how a computer works, which is obviously going to be an advantage in the future anyway.

I suspect that the current stream of math would end up diverging just after algebra.  It would end up involving a lot more number theory and logical reasoning and a lot less graphing and physics based mathematics (except for the stream of students interested in game programming).  I don’t know that students would find this much more interesting, but at least it would pretty easy for them to use the math they were learning and use it in direct applications involving their favorite technological devices.

Maybe kids might enjoy math more?

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 Pixorial.com 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.