In honor of my recent election to the Hillsboro, Oregon School Board, I thought it might be nice to discuss an education-related topic that's been popping up a lot recently. Back in the 90's there was a fad for what became known as "discovery-based" K12 math curricula. This means that rather than teachers directly telling the students how to do standard operations, such as adding n-digit numbers or simplifying equations, the students would be directed in small groups to experiment and discover such ideas themselves, with some guidance from the teacher. These were widely criticized and debunked-- see the "Mathematically Correct" website linked in the show notes-- and largely fell out of favor. But in recent years, connected with the Common Core movement in the U.S., new forms of these type of programs seem to be coming back. It seems like there is an inherent conflict between two approaches to math education: should students be helped to discover the basic principles of math, or should they be presented with and drilled in standard algorithms? There are some basic criticisms of these discovery techniques that have a lot of weight: the readiness of children to derive standard algorithms, and the internalization of math basics.
As a motivating example, I have a link in the show notes to a video of a young girl adding some 3 and 4-digit numbers, using the methods she learned in a "modern" school. She tries two methods. First she draws pictures representing the thousands, hundreds, tens, and ones in each of the numbers, and after a tedious 8 minutes of counting pictures, gets a wrong answer. Then she uses the standard method (which her mom taught her), of writing the numbers on top of each other, adding in columns, and carrying when needed-- and in less than a minute has the correct answer. The whole process of drawing the pictures seems rather absurd, and it's clear that the girl really has no understanding of how the pictures really relate to place-based notation, or of how the standard algorithm is really just an advanced abstraction of her picture-method. Several online commentators pointed out that the picture method was directly analogous to roman numerals, which no sensible person would use today for nontrivial calculations.
The first question to ask is: are children in these classes ready to derive the standard algorithms that took mathematicians thousands of years to create? Going from simple direct methods, like the little girl's pictures which were analogous to roman numerals, to the modern algorithms, requires some clever abstract reasoning. As I discovered when I tried to clarify the higher-dimensional concepts from the Flatland movie to my 6 year old, there are some types of logic and abstraction that a child may not be ready for. Many psychologists talk about Piaget's stages of cognitive development: the "symbolic function" stage at ages 2-4; the "intuitive thought" stage at ages 4-7; the "concrete operational" stage at ages 7-11; and the "formal operational" stage beyond that. You can see more details at the link in the show notes. The exact ages vary from child to child of course, but the key point here is that at the first three stages, a child's ability to develop abstract algorithms from concrete examples is severely limited; and the capacity for true abstract reasoning isn't really developed until high school for a majority of kids. This is probably one reason why the girl in the video had trouble seeing the connection between the picture-based method and the standard method of addition. But at these younger ages, kids are much better at picking up and internalizing rote facts and procedures, which would seem to make it an ideal time to teach them standard algorithms.
We also need to ask whether these new types of lessons, assuming they focus on both the historical derivation and the current best-known-methods, will actually result in students learning the standard algorithms. Here there is reason for concern. Often the modern math lessons have many fewer practice problems in the homework than traditional math, and encourage the use of calculators and computers to do basic calculations. Unfortunately, these are missing a very imporant aspect of all this drilling-- it helps people to really absorb basic mathematical algorithms, and make them instinctive. Back in episode 70, "Number Nonsense", I discussed my frustration with a fast food cashier who could not recognize that 2+2=4 without a calculator. And during my recent school board campaign, I came across a mom who was upset that her child took out a calculator when asked the difference between 6 percent and 600 percent. More important than these anecdotes, though, is that in order to have any hope of success in advanced science and engineering classes, kids really need this basic number sense. Even if the hard problems are ultimately crunched by computer programs, it's simply not possible to have an initial discussion of an engineering problem if every rough estimate needs a pause to get out a calculator. The little girl in the video was lucky that her mom took it in her own hands to work with her at home and make sure she had a good understanding of the standard algorithm.
Our standard methods, like the one that enabled a young girl to add 4-digit numbers in less than a minute, were developed after thousands of years of thought by very smart people. And since then they have been used regularly by a huge population of businessmen, scientists, and engineers, the majority of whom have never given a thought to the details of how these were derived or why they work. A lot of proponents of the new teaching methods criticize the "drill and kill" of traditional math, the many exercises given to practice and memorize standard algorithms, which they consider boring. But the advantages of all this drilling is that using these techniques becomes easy and instinctive: how often do we stop to ask ourselves why we are able to do simple addition problems? Well, maybe Math Mutation listeners do, but most people can add just fine without thinking about the historical development of place-based notation. Being able to do basic math opens the doors to advanced concepts of science, engineering, and computers. And students who have NOT mastered these basic foundations will find these important topics forever beyond their reach.
So, am I advocating that we throw out all these modern and Common Core math programs, and go back strictly to traditional methods? Not necessarily. A properly designed program which teaches and drills the standard algorithms, while also emphasizing problem-solving and using the history of their derivation to provide some motivation and color to the class, might very well be a great success. But in the urge to eliminate the drill-and-kill method and make math more fun, we seem to be constantly rediscovering the lesson that Euclid taught to Ptolemy over 2000 years ago, that there is no "royal road" to mathematics. We need to be very careful of the tendency of education efforts every few decades to come up with silver bullets that will somehow make math easy for everyone-- internalizing the basics algorithms of math usually requires a lot of concentrated thought and hard work. We've been here before, as you might recall: back in episode 145 I discussed the New Math movement of the 1970s, similarly based on professors promising that adding theoretical foundations to elementary mathematics would somehow make it easier to learn, and ending in disaster. Many aspects of math really can be fun, as I hope most of my podcast episodes have shown, but to understand the subject overall you really need a solid mastery of the basics.
By the way-- I know from listener emails that many of you out there are actually math or science teachers. I'd really like to hear from you about experiences, positive or negative, with these new mathematics programs. Please send emails to email@example.com. Thanks!
And this has been your math mutation for today.
http://hillsboroerik.com : My school board / education blog.
http://en.wikipedia.org/wiki/Mathematically_Correct : Mathematically Correct
http://www.youtube.com/watch?v=1YLlX61o8fg : Video of young girl doing addition
http://en.wikipedia.org/wiki/Piaget%27s_theory_of_cognitive_development : Piaget's theory of cognitive development
http://wheresthemath.com/curriculum-reviews/math-textbook-reviews/testimonies-about-programs/ : article on problems with discovery math
http://www.cpm.org/ : Discovery-type program being adopted in my district