The Reflective Educator

Education ∪ Math ∪ Technology

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Day: January 13, 2011

A Fundamental Flaw in Math Education

Here’s what math curriculum looks like in most schools.

Math computation at the centre

The computations we want students to be able to do are chosen, and then we find problems that match these computations, and if we are able, we find some real life connections to the computations. Generally the real world connections are an after thought, and many times teachers are responsible for finding these connections when the textbook problems they are given are really examples of pseudo-context rather than a real connection. It can be difficult to attach real world context to the curriculum we are expected to teach and many times teachers aren’t able to do it.

There is a serious flaw in how this curriculum is constructed. The mathematics that is chosen has no motivation in the minds of the people learning it. If you have wondered why so many people hated mathematics class, it was because they couldn’t see the point of it. That’s because there was no point. People who hate mathematics were learning apparently meaningless algorithms for the sake of the algorithms themselves rather than for the processes in our world where the algorithms describe.

What if we flipped this process? What would happen if we changed the diagram so the real world problems were in the middle of the diagram, and we chose what mathematics to look at based directly on how it helped us understand the real world? The diagram would look like this:

Real world at centre

I need to be careful here to define “real world.” I do not mean that students will spend their time learning how to solve the math problems that may arise in their life like an endless stream of super-market math. What I mean by “real world” is that the problems students would work on would have a shared context to which students will understand. This context might be a real problem that students or their teachers find, it might be part of an interesting challenge given to them by a mentor, or it might be simply exploring ‘what if’ in a puzzle.

The thing is, when you place context at the centre of the curriculum an immediate shift happens. Now the mathematics itself has immediate relevance, since the applications are focused on something which has meaning for the students. You also gain the ability to shift away from a computational focus (since that’s what should be for) and look at problem formulating and solving. Gone is pseudo-context. Gone is mathematics which has no relevance in the lives of our students.

Formulating the curriculum like this also makes finding connections to other areas where the students are studying much easier. Everyone can talk about the real world, and it can happen in every subject area. It becomes easier to turn our curriculum from caged subject areas into an open dialogue about life.

It also becomes easier to update the curriculum. The topics of a typical mathematics curriculum have changed very little in the past 100 years, and are nearly universal across the globe. However, in this curriculum the focus is on what is happening in real life, and it becomes easier to select what is important to teach as all of us experience the real world daily. In fact the curriculum itself could easily change from year to year and the actual mathematics that is taught could be different in each school. After all, what is important in mathematics is the process that students go through, not the end result.

There is also room in this model of curriculum for fairly advanced topics. You can see that calculus could be one of the branches of this type of curriculum since it deals intimately with understanding many complex phenomena in our world. If a student wants to extend themselves and get excited by the mathematics itself, we can still give them that opportunity. We can be more flexible in how we plan our lessons and give students more choice in how they approach problem solving.

What is not easy to represent in this diagram is what the arrows, which are common to all of the different possible branches of mathematics and their representation to the real world content. In my mind, these arrows represent part of the process of translating a real world problem into mathematics. Students have to know how to formulate problems, develop criteria for establishing what pieces of information they have are useful, and determine if their solution makes sense. Since they know for certain that the solution represents a real world phenomena, it will be easier to judge a correct solution from an obvious false one.

This shift also makes it easier to talk about the big ideas of mathematics. Most of the time we spend our classroom time so focused on the minutia that we forget that there are some powerful ideas in mathematics that are useful tools for thinking for students.

Here is a presentation to explain some of what this shift looks like for me.

 

Mathematics education has to change. I have spent my life either feeling defensive about my love of mathematics, or commiserating with people who agree with me. People say that they hate mathematics because they do not see how it is relevant. Let’s change mathematics curriculum so that context (which does not necessarily have to be “real world” but should be meaningful) is king.

 

Update: In my current work as a curriculum developer, I’ve been working on an alternative to the proposal above in our curriculum work which embeds contexts where they are meaningful but does not flip the arrangement as above. The key shift in our curriculum work is viewing students as sense-makers and fore-fronting student thinking as much as possible while still making mathematics accessible to all students (through shifts in instructional practices rather than shifts in the mathematics itself).

The Prisoner’s Dilemma as it Applies to Education

In the prisoner’s dilemma scenario, two prisoners are each interviewed separately and asked to cooperate with the authorities. Here is an except from Wikipedia on the dilemma.

"The classical prisoner’s dilemma can be summarized thus:

  Prisoner B Stays Silent Prisoner B Betrays
Prisoner A Stays Silent Each serves 6 months Prisoner A: 10 years
Prisoner B: goes free
Prisoner A Betrays Prisoner A: goes free
Prisoner B: 10 years
Each serves 5 years

In this game, regardless of what the opponent chooses, each player always receives a higher payoff (lesser sentence) by betraying; that is to say that betraying is the strictly dominant strategy. For instance, Prisoner A can accurately say, "No matter what Prisoner B does, I personally am better off betraying than staying silent. Therefore, for my own sake, I should betray." However, if the other player acts similarly, then they both betray and both get a lower payoff than they would get by staying silent. Rational self-interested decisions result in each prisoner being worse off than if each chose to lessen the sentence of the accomplice at the cost of staying a little longer in jail himself (hence the seeming dilemma). In game theory, this demonstrates very elegantly that in a non-zero-sum game a Nash equilibrium need not be a Pareto optimum."

So how does this apply to schools? Let’s replace the word "prisoners" with students and the word "authorities" with "school administrator or teacher." We actually replicate the exact circumstances of this dilemma in schools every time we "try and get the bottom" of a situation between two students.

Instead of the highly negative choices for the students being prison sentence, they are detentions, or reduction of marks, or a million other consequences we apply to children when they misbehave. Further, children will apply their own consequences for "snitching" so that even if the adult administering the punishment goes easy on the child who tattles, and acts harshly on the child apparently "responsible" for the dilemma, there is often a consequence for the child who tattled; they may be ostracized by their peers or even face physical violence.

A typical analysis of the prisoner’s dilemma indicates that the optimal choice for players of the game is to remain silent and accept the minor punishment, even if they haven’t committed the crime. When we place school children in the same situation, don’t they react exactly the same way? This is not what we want, simply because in order to help a child who has misbehaved, we need information in order to help them. We need to understand what is going on so it is not repeated over and over again, as is normal in unresolved disputes.

There are two solutions I see for resolving the prisoner’s dilemma in schools. One, don’t isolate the kids and have the conversation about the misbehaviour as a group. This breaks the conditions of the prisoner’s dilemma as now each participant is aware of the other’s response. Two, change how you define misbehaviour so that you don’t end up in this situation. My recommendation, if you are more interested in the second of these options is to check out Restitution, although I am sure there are other ways to do this.

A Democratic Education

Joe Bower posted this video on his blog.

Joe writes:

If we really cared about real accountability, we would first ask if kids like school, and then we would have to care how they answer. Until then, kids will continue to be victims of a system that cares more about sustaining its current self than serving students.

Here is my response which I left as a comment on his blog:

I love this video Joe. It frames what I believe to be true about education in a way which is catchy, interesting to watch, and (I feel) hard to argue with.

When we share these ideas with other people, they keep going back to the same argument: how will we know kids are learning?

So I think our challenge is to come up with an answer to that argument which parents will accept without having to make too much of a radical change in their life.

I mean, I know my son is learning because I spend a lot of time with him and I keep track of the changes I see. I know he’s learning because I listen to the questions he has. I know he’s learning because I catch him trying to read (he’s 4). I know he’s learning because I’m involved deeply in his life. I want this to always be true, although over time I also want him to really learn the skill of autonomy, so I want to be a participant in his learning, rather than the sole director of it.

Our system of education absolves parents of the responsibility of paying attention to their kids learning needs by giving them easy numbers and letters to look at which are substitutes for the measures we already have for learning. You will know your child is learning if you are a part of the process, and pay attention.

Our economic system requires that parents work harder than should be necessary to support themselves, often leaving them without the time to support their kids emotionally. Something is flawed with a society which devalues the relationship between parents & kids and suggests that only a stranger can best teach your child everything they need to know. 

The transformation in our education system will happen when we give students real choice about what they learn, when we take away the barriers to developing individuality, and when we stop pretending that it is possible to carefully select the curriculum students will learn from the millions of useful things they could be learning. One size fits all doesn’t work. There is no other industry I can think of which believes that every one of their "customers" has the same needs.