Why has there not been any progress in the way we teach?
The latest buzz word in education is grit. Grittier students learn better. Well maybe, but shouldn’t there be a discussion of better ways of presenting the material? We have been teaching and learning since before we were human and, near as I can tell, pedagogy has remained the same.
We can do better. My particular interest in math. Here is a way of teaching the formula for arithmetic series that I have on my Web page.
Consider the numbers 1, 2, 3, 4 and 5. Without doing any calculation, take a guess what the average is for these numbers. Most people, would guess 3, because it is in the middle. Assuming that 3 is the average, how could we use that to find the sum. Easy, just multiply by 5. Add the numbers to verify.
How do we find the sum from 1 to n? What is the average? Some students may initially say n/2, but the example shows that is not right. We take the average of the first and last numbers, (n+1)/2. It then follows that the sum is n(n+1)/2.
This of course is not a proof. In an advanced class, the next step would be to try to find a proof. Inspired by what was done, someone may notice that if we write the numbers 1 to n and then write the numbers underneath from n to 1, we get n+1 in each column. The important point is that we were able to guess at a formula using intuition. Who’d have thought, mathematics and intuition? At the highest levels of math, it happens all the time, both in determining what to prove and in finding a method of proof.
I am certainly not the first person to see the formula for arithmetic series in this way, so why is the formula simply taught without explanation? Not only is the process more interesting, but the formula becomes easier to remember. Sometimes it seems that the teaching method is chosen so as to make the material seem as dull as possible.
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8 Answers
Me too & math was my favorite subject!!!
We have better tech. Also no more chalk bords.
Actually, schooling children is a fairly recent practice.
For a majority of human existence, children were taught only basics of survival and rules of community etiquette. The ancients who held learning sessions did so only with adult males.
Teaching children has had a very short evolution cycle. It began with apprenticeship. Children were taught one specific occupation.
Unfortunately, population makes changing the schooling process extremely difficult.
Longer lifespans also play a role. If we change things quite a lot for children, how do we make them compatible in the workplace with adults? Refresher courses? Would that be free, or an unexpected expense?
While it is true the school system needs a major overhaul, who is prepared to face such a challenge?
That’s why.
^There is a lot of work being done on designing computerized learning systems. While I am skeptical of the idea of replacing teachers, the computerized systems have the advantage of allowing students to work at their own individual pace. It still leaves open the question of how the lessons should be programmed.
For those who could not follow the my explanation, let me fill in the details.
The definition of average is: average = (sum of values)/(number of values), which we can write as a = s/n. Multiplying both sides of the equation by n gives na = s. For the values 1,2,3,4 and 5, n=5 and we guess the average as (1+5)/2=3, giving s = 5*(5+1)/2 = 15. In general, for the values from 1 to n we get, s = n(n+1)/2.
My granddaughter is 12. On occasion I have to work with her on her homework, including mathematics. What I have noticed over the years is that the way she is taught to solve various equations differs from the way I was taught. It may reach the same conclusion but it is not what I learned. Is it any better? I don’t know. The way I learned has worked for me for all these years. Is it quicker? Not that I can tell. Is it more intuitive? Hard to judge. She has as much trouble as I remember having but I cannot unlearn the way I know to learn the new method. I can, and have, learned the new ways in order to assist but knowing the old way makes it easier for me in some ways.
I remember learning by rote in such things as spelling, grammar and multiplication tables. They don’t seem to stress it as a learning tool much these days.
Teaching methods change over time. Sometimes they work, sometimes they don’t. Unfortunately, if they don’t then it is the kids that suffer. The stakes are high.
@rojo, I have heard that there is supposed to be more emphasis on understanding why things work the way they do, but I don’t know how this is done in practice.
I remember when New Math was being taught, which introduced set theory in elementary school. It was generally considered a disaster.
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