The Reflective Educator

Education ∪ Math ∪ Technology

Menu Close

Day: January 18, 2011

Reward Innovators with Responsibility

A problem with education is that we have too many "best practices" and not enough innovation. Once you establish a procedure as a best practice there’s no room for argument about whether or not it works. We should call it a "current practice" instead. Now we have the freedom to explore this practice and confirm whether or not it is actually working, and find new innovations in education.

We need to encourage more innovation in education and explore a wider range of what is possible. People need to be rewarded for their innovation by being given the responsibility to implement it, provided that they can show solid evidence that it works for other educators too. We need more peer review of what we do as educators.

Imagine a teacher has an idea for improvement in their teaching. They try it in their classroom, bounce the idea off of a friend and collect evidence that it works. They convince a colleague to do the same, and show that whatever it they are doing differently works for both teachers. They submit their idea to a peer review panel, which then would make recommendations on what ideas have wider merit. The teacher who had the original idea would then be given the responsibility and the authority to try out their idea with a wider group of students or teachers. Teachers would be rewarded for innovation with the responsibility to see their idea through, and spreading it to other schools where it could also work.

We would end up with a lot of benefits from this system. First, the amount of collaboration between educators would increase. Next, school improvement ideas would spread out of the school in which they are initially implemented. There would be encouragement to share one’s ideas rather than jealously guard them. Furthermore, educators would be encouraged to innovate. It would lead to improvements in education by greatly increasing the pool of talent to produce possible innovations. So many educators come up with awesome ideas that never make it past the door of their classroom.

Let’s turn our system which just uses "best practices" as determined into researchers into one which includes innovations from educators in the field.

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?

What works in education

Let’s suppose the picture below represents the possible states schools can be in, with the peaks being "good" places to be and the valleys being bad places to be. We don’t really know yet what variables we are even representing with this picture, in fact it is likely that the picture itself would be better represented in 20 or 30 dimensions, as there a huge number of factors which affect how successful schools are.

Peaks and valleys

(For those of you who are interested, the equation of this curve is z(x, y) = sin(3x) + cos(2y) + 1.5sin(x) – 3sin(0.5y) and it was created using GraphCalc)

The first thing which is clear from this picture is that there are a lot of ways to be a good school. No one formula works, no single arrangement of the variables corresponds to the best solution. Similarly there are a unfortunately lot of ways to be a bad school. 

The next thing I notice about this picture is that it’s not easy to be at the top of one of these peaks. Make a small nudge in policy, lose a key teacher or administrator at your school, and you can quickly move from a peak to a valley, and your students suffer. The valleys are hard to move out of because of the inertia of a bad school and the energy required to overcome that inertia. Educators can literally feel like Sisyphus, pushing their school out of a valley only to see it collapse back down again with a change in district support, funding, or policy.

Worse, this picture changes over time and the peaks and valleys don’t remain in the same places as external pressures push on schools. Outside pressures from society can have huge influences on school. Think about what schools would look like if the Internet had never been invented. Would they need to change that much? What about if the civil rights movement hadn’t happened?

It is also equally possible that two people from different schools could be standing on different peaks and not recognize that each person works in a wholely different but equally successful school. Or that two people could be in different valleys, and what works to improve one school is completely unsuccessful at another school.

The key message here is that there is no one single formula to improve a school. All standardizing curriculum and increasing accountability in schools will do is shift schools in the same direction on this graph. This will work for some schools, and fair at others, and can potentially push successful schools out of a peak.