Revolutionizing Math Education: Learn.org Speaks with Conrad Wolfram
Conrad Wolfram is the strategic director of Wolfram Research, where he finds innovative new uses for the software Mathematica. Mr. Wolfram argues passionately for a transformation in the math curriculum, moving away from hand calculation. Teaching computer-based math, he claims, can open students up to a whole new world of mathematical analysis. Learn.org recently spoke with him to learn more about this concept and Mr. Wolfram's vision for the future of math education.
Learn.org: What's your educational and professional background, and what led you to join your brother at Wolfram Research?
Conrad Wolfram: I took natural sciences - choosing mostly physics - at Cambridge University in the U.K. and then switched to math (really more theoretical physics!). Wolfram Research was just getting going at the time, so I became involved and got the European company up and running, which nicely distracted me from my math course.
Learn.org Part of your job at Wolfram Research is to find and promote new uses of the Mathematica technology. Can you describe what Mathematica is for our readers?
CW: Mathematica is software for computing, developing and deploying solutions in STEM (science, engineering, technology and math) subjects and beyond - in fact, anywhere where mathematically computing results is important.
Companies and governments have adopted Mathematica for everything from modeling financial risk to space exploration to working on new medicines. (It's also what our own Wolfram|Alpha knowledge engine is built on and computes with.) Many schools and colleges around the world have Mathematica site licenses, often with student computer-use included. Check out your college! Students and teachers can also get fully-featured but heavily discounted versions.
Learn.org In your research into the uses of computational software, you've come to the conclusion that K-12 schools are taking the wrong approach to teaching math, focusing on hand computation that can be more efficiently done by computers. Can you explain the four steps of mathematical problem solving, and how computers might be used in the process without 'replacing' humans?
CW: I think of math as an amazingly powerful process for getting answers to questions and I define that process as:
- Posing the right questions: What question is it that you need to ask to work out what's important about a given situation?
- Real world -> math formulation: Take what you're asking and turn into math, whether that's formulas or equations or whatever. Math is a really powerful language; getting stuff appropriately into math is important.
- Computation: Transforming the math you formulated to math that represents your answer.
- Math formulation -> real world, verification: Turning 'math' back to 'real world' and checking that it really makes sense, that you didn't do something wrong in one of the steps.
Right now about 80 percent of a student's math course is hand-doing step 3, yet it's the one step computers can do better than any human, even a human with years of practice. It's why outside education, few computations (except very basic mental arithmetic) are done by hand. We're talking not just working with numbers but symbolic math, too.
Training for the other three steps is majorly lacking, yet it's central to people's jobs and everyday existence in the modern economy. It's crazy that we're only spending 20 percent of school math time on those steps. Instead, if we spent 80 percent on steps 1, 2 and 4 and used computers except in a few cases for step 3, students could learn conceptually harder math that is also more applicable in their lives and they could learn it much earlier. It really is a crucial change.
Learn.org Many people argue that students need to learn hand computation in order to better understand math, yet you have pointed out that computation is precisely the mathematical skill that students will have the least use for in the rest of their lives. Can you explain what other skills math can impart, and how using computers for calculation can help students bolster those abilities?
CW: So many mistakes happen in real-life because no one asked the pivotal question. What happens if all the banks fail at the same time? What will happen to the bridge if the winds hits this particular speed?
Experience is key. And one of the great benefits of using computers for calculating is the far larger number of problems you can run, each more complex and more realistic than the easy-enough-to-hand-calculate variety often seen now at school. And you can interact with the problems, adjusting parameters in real time, really 'feeling the math' - accelerating students' experience of what works, what doesn't.
Contrary to what's often claimed, I don't believe hand calculating experience is that related to skills for the other math steps I described; and worse, it's a time sink and a huge turn off for many students.
An analogy I often give is that to learn driving you don't need to learn the details of car mechanics, at least not now: Driving is a separate skill from mechanics or automotive engineering. Some students will later specialize (which is great) but driving is the mainstream subject. Likewise, computer-based math is the mainstream subject, not the history of hand-calculating or knowing how to program computers to do the fundamentals of calculating.
Learn.org You've also argued that studying programming can offer the same process-based learning of mathematical principles as hand calculation. Do you think that programming could actually replace calculation in the curriculum, and how would that work?
CW: My key point is that learning about processes - how to use, apply and make them - is important in life and some argue that's the reason to teach hand-calculating. But nowadays programming is the central medium for processes outside education; also it's often much more inspiring because you can see your process work and its effect multiply.
There are many practical ways of doing programming, from visual languages to formal languages. Mathematica is itself a language - one with a very broad range of styles - that's used in computer science courses, amongst many others.
There's every reason to answer math questions with programs or even apps, not formulas. For a start, writing a program really checks understanding of concepts. Then you and the computer can hammer on it and find out more of the reality of when it works - just like real life!
In a sense, each of the knowledge apps on our Wolfram Demonstrations Project site is the program answer to a computation question. Although many are from professionals in a wide range of fields, some are student projects or a teacher's class material.
Learn.org You've pointed out that removing hand calculation from math education would allow educators to reorder the curriculum based on concepts. What do you think that a concept-based math curriculum might look like?
CW: We've started work on that now at ComputerBasedMath.org (do sign up!), but it's in the early stages. Calculus will no doubt figure much earlier, as will techniques on how to verify results effectively. Wherever possible we'll use real life as our guide with much more open-ended questions and much more realistic problems. Understanding and verifying complex existing models will be prominent alongside building new models. These are some of our early ideas.
Learn.org You recently asserted that the first country that uses computers to bridge the gap between school math and real world math for students will 'leapfrog others in achieving a new economy.' Can you talk more about this prediction - what new skills will students gain, and how will they have a transformational effect on the economy?
CW: I think there are three practical reasons for learning math: technical jobs, everyday living (e.g., working out which mortgage is best), and knowing how to reason logically (whether specifically for math tasks or other fields).
Taught correctly as the mainstream subject, computer-based math could improve all three simultaneously, both by attracting many more people to want to take the subject and also by making their conceptual understanding and practical application of it far greater. It could be much more fun too!
In the end, a mathematical way of thinking needs to be as second nature for everyone - an everyday way of understanding the world - as it is for a small elite now.
The biggest economic value of the future will lie in having most of the population creating or working with ideas rather than knowing information. Harnessing the power of math and computation could not be a more key ingredient to manifesting this creativity.
That's why I believe the first country to achieve widespread creative computer-based STEM education will gain a huge economic benefit and move from an era of 'knowledge economy' to an era where computationally working ideas out from the knowledge is democratized - what I call a 'computational knowledge economy'.
Learn.org Finally, I'd like to give you the opportunity to share anything you'd like about your own research and the use of computers in math education.
CW: Here are a few resources to learn more about computer-based math:
- Wolfram Demonstrations: Thousands of interactive examples to try out for free.
- Wolfram Alpha: Enter what you want to calculate or know about.
- Wolfram Course Assistants: An app for every course.
These aren't a new curriculum - we're just at the start of trying to build that - but they are all tools that allow you to experience math and play with it. Good luck!