I'm really keen to explore ways to make the teaching of maths concepts as authentic as possible for students. Mathematical concepts can seem quite abstract but, with practical application in an authentic problem-solving concept, abstract can become concrete.
I listened to this TED Talk...
I listened to this TED Talk...
...and it reinforced what had been burbling away in the back of my mind.
For those who don't have time to watch the video here are the excerpts of the transcript that I think capture its essence:
What is math? What do we mean when we say we're doing math, or educating people to do math? Well, I think it's about four steps, roughly speaking, starting with posing the right question. What is it that we want to ask? What is it we're trying to find out here? And this is the thing most screwed up in the outside world, beyond virtually any other part of doing math. People ask the wrong question, and surprisingly enough, they get the wrong answer, for that reason, if not for others. So the next thing is take that problem and turn it from a real world problem into a math problem. That's stage two. Once you've done that, then there's the computation step. Turn it from that into some answerin a mathematical form. And of course, math is very powerful at doing that. And then finally, turn it back to the real world. Did it answer the question? And also verify it -- crucial step. Now here's the crazy thing right now. In math education, we're spending about perhaps 80 percent of the time teaching people to do step three by hand. Yet, that's the one step computers can do better than any human after years of practice. Instead, we ought to be using computers to do step three and using the students to spend much more effort on learning how to do steps one, two and four -- conceptualizing problems, applying them, getting the teacher to run them through how to do that...
...Do we really believe that the math that most people are doing in school practically today is more than applying procedures to problems they don't really understand, for reasons they don't get? I don't think so. And what's worse, what they're learning there isn't even practically useful anymore. Might have been 50 years ago, but it isn't anymore. When they're out of education, they do it on a computer.Just to be clear, I think computers can really help with this problem, actually make it more conceptual.Now, of course, like any great tool, they can be used completely mindlessly, like turning everything into a multimedia show, like the example I was shown of solving an equation by hand, where the computer was the teacher -- show the student how to manipulate and solve it by hand. This is just nuts. Why are we using computers to show a student how to solve a problem by hand that the computer should be doing anyway? All backwards....
...Programming is how most procedures and processes get written down these days, and it's also a great way to engage students much more and to check they really understand. If you really want to check you understand math then write a program to do it. So programming is the way I think we should be doing that. So to be clear, what I really am suggesting here is we have a unique opportunity to make maths both more practical and more conceptual, simultaneously. I can't think of any other subject where that's recently been possible. It's usually some kind of choice between the vocational and the intellectual. But I think we can do both at the same time here. And we open up so many more possibilities. You can do so many more problems. What I really think we gain from this is students getting intuition and experience in far greater quantities than they've ever got before. And experience of harder problems -- being able to play with the math, interact with it, feel it. We want people who can feel the math instinctively. That's what computers allow us to do....
...So I want to see a completely renewed, changed math curriculum built from the ground up, based on computers being there, computers that are now ubiquitous almost. Calculating machines are everywhereand will be completely everywhere in a small number of years. Now I'm not even sure if we should brand the subject as math, but what I am sure is it's the mainstream subject of the future...
For those who don't have time to watch the video here are the excerpts of the transcript that I think capture its essence:
What is math? What do we mean when we say we're doing math, or educating people to do math? Well, I think it's about four steps, roughly speaking, starting with posing the right question. What is it that we want to ask? What is it we're trying to find out here? And this is the thing most screwed up in the outside world, beyond virtually any other part of doing math. People ask the wrong question, and surprisingly enough, they get the wrong answer, for that reason, if not for others. So the next thing is take that problem and turn it from a real world problem into a math problem. That's stage two. Once you've done that, then there's the computation step. Turn it from that into some answerin a mathematical form. And of course, math is very powerful at doing that. And then finally, turn it back to the real world. Did it answer the question? And also verify it -- crucial step. Now here's the crazy thing right now. In math education, we're spending about perhaps 80 percent of the time teaching people to do step three by hand. Yet, that's the one step computers can do better than any human after years of practice. Instead, we ought to be using computers to do step three and using the students to spend much more effort on learning how to do steps one, two and four -- conceptualizing problems, applying them, getting the teacher to run them through how to do that...
...Do we really believe that the math that most people are doing in school practically today is more than applying procedures to problems they don't really understand, for reasons they don't get? I don't think so. And what's worse, what they're learning there isn't even practically useful anymore. Might have been 50 years ago, but it isn't anymore. When they're out of education, they do it on a computer.Just to be clear, I think computers can really help with this problem, actually make it more conceptual.Now, of course, like any great tool, they can be used completely mindlessly, like turning everything into a multimedia show, like the example I was shown of solving an equation by hand, where the computer was the teacher -- show the student how to manipulate and solve it by hand. This is just nuts. Why are we using computers to show a student how to solve a problem by hand that the computer should be doing anyway? All backwards....
...Programming is how most procedures and processes get written down these days, and it's also a great way to engage students much more and to check they really understand. If you really want to check you understand math then write a program to do it. So programming is the way I think we should be doing that. So to be clear, what I really am suggesting here is we have a unique opportunity to make maths both more practical and more conceptual, simultaneously. I can't think of any other subject where that's recently been possible. It's usually some kind of choice between the vocational and the intellectual. But I think we can do both at the same time here. And we open up so many more possibilities. You can do so many more problems. What I really think we gain from this is students getting intuition and experience in far greater quantities than they've ever got before. And experience of harder problems -- being able to play with the math, interact with it, feel it. We want people who can feel the math instinctively. That's what computers allow us to do....
...So I want to see a completely renewed, changed math curriculum built from the ground up, based on computers being there, computers that are now ubiquitous almost. Calculating machines are everywhereand will be completely everywhere in a small number of years. Now I'm not even sure if we should brand the subject as math, but what I am sure is it's the mainstream subject of the future...
It frustrates me that there are kids who still think using a calculator is cheating. Is this the same argument we're battling with computers? If we've got a tool that helps us to do higher order thinking more efficiently then let's use it!
As a teacher who was passionate about making maths as authentic as possible, I found it really challenging to find resources which supported this. This year, I plan to develop some. If you're reading this and want to get involved, please contact me.
Conrad Wolfman talks about computer programming as a way to combine conceptual and practical maths. I have been taking code workshops with classes. The questions that come up most frequently with the age group I teach are questions about angles and x,y co-ordinates. They are basic knowledge blocks needed for the drag-and-drop coding they are completing. Those who don't know, want to find out pretty quickly....just-in-time teaching. They need the knowledge to move forward and then consistently reinforce and expand on it.
The Numeracy Project in New Zealand has put a huge focus on multiple methods of mental calculation. Estimation and mental calculation are still important. However, I believe, that there are many students who are confused by learning a battery of methods which are often taught out of context. We need to be spending more time on authentic problem solving and creating higher levels of positive engagement in maths.
This article was recently published by the NZ Herald, the key quote from Lisa Rodgers (Ministry of Education) being:
"Students are not learning space and shape mathematical thinking. Our students are some of the strongest in the world in terms of data and statistics, but they are missing out a huge part of the curriculum in terms of mathematical thinking."
I believe a big part of developing this type of mathematical thinking is through computer programming. Why are we 15 years into the 21st Century, working with students who are digital natives, to find that they don't even know they can create their own digital content, let alone how.
Computer programming is not the only answer. I believe it also lies in making maths multi-disciplinary and hands-on. We need a mathematical revolution where money that was traditionally spent on text books is spent on devices and equipment which will facilitate authentic problem solving.
As a teacher who was passionate about making maths as authentic as possible, I found it really challenging to find resources which supported this. This year, I plan to develop some. If you're reading this and want to get involved, please contact me.
Conrad Wolfman talks about computer programming as a way to combine conceptual and practical maths. I have been taking code workshops with classes. The questions that come up most frequently with the age group I teach are questions about angles and x,y co-ordinates. They are basic knowledge blocks needed for the drag-and-drop coding they are completing. Those who don't know, want to find out pretty quickly....just-in-time teaching. They need the knowledge to move forward and then consistently reinforce and expand on it.
The Numeracy Project in New Zealand has put a huge focus on multiple methods of mental calculation. Estimation and mental calculation are still important. However, I believe, that there are many students who are confused by learning a battery of methods which are often taught out of context. We need to be spending more time on authentic problem solving and creating higher levels of positive engagement in maths.
This article was recently published by the NZ Herald, the key quote from Lisa Rodgers (Ministry of Education) being:
"Students are not learning space and shape mathematical thinking. Our students are some of the strongest in the world in terms of data and statistics, but they are missing out a huge part of the curriculum in terms of mathematical thinking."
I believe a big part of developing this type of mathematical thinking is through computer programming. Why are we 15 years into the 21st Century, working with students who are digital natives, to find that they don't even know they can create their own digital content, let alone how.
Computer programming is not the only answer. I believe it also lies in making maths multi-disciplinary and hands-on. We need a mathematical revolution where money that was traditionally spent on text books is spent on devices and equipment which will facilitate authentic problem solving.