As a child, I never experienced what I would now call mathematics teaching. (I revised my understanding of what teaching is, when, as an adult, I saw it in action on several occasions. Although I have no memory of having been taught that way, I guess it is possible that as a very young child I was.)

What I was presented with at school was instruction. The quality varied a lot, but looking back it was definitely instruction, not teaching. The teacher would explain some new concept or demonstrate to the class a method to solve a particular kind of problem, and then we would all work through several problems of the same type. And that was the procedure followed in all the math classes I can remember.

I quickly figured out how to play that game successfully – success in that case being measured by my being able to solve under exam conditions, problems like the ones the teacher had shown us and we had practiced in class and done for homework. Many of my fellow students did not master that game, and fell by the (well-populated) mathematical wayside.

The technique I mastered early for succeeding in that regime lasted me all the way through to calculus and on into university. Then things changed dramatically. Most of my professors provided little more than rapid summaries of new concepts and gave minimal instructions as to how to solve problems. I had to figure it out for myself afterwards, in collaboration with my fellow students, occasionally supplemented by going to the professor for help. It was at university then, when I discovered (collaborative) learning (the long sessions with my fellow students) and the power of real teaching (the activity that took place when I sat down with my professor to get help), and both were powerful and transformative.

As far as I can tell, most people in the US (and the UK) who last took a math class at high school have never experienced good mathematics teaching. Nor have many students who went on to take math classes at college level, but were not able to sit down one-on-one or in a small-group setting with the professor, as I did. All they have ever had is instruction. They often refer to it as teaching, since that is their only model. But it isn’t teaching; to call it that is to unintentionally insult the many thousands of good teachers out there.

Instruction is primarily one-directional, from an instructor (we should not use the word teacher here) to the student. Education in the instruction mode proceeds along the lines: first provide information, then give an opportunity to practice, then test.

Many students do learn to do well in this system. Some of the ones who do well actually learn

**what the course is supposed to be about**, though others (and I suspect most) simply learn how to pass the course tests. Case in point: I got straight A’s on all my high school calculus courses (“freshman calculus” in US terms), but only when I was a doctoral student in mathematics faced with running problem sessions for math undergraduates did I actually start to understand calculus. At school I had merely learned how to pass the tests. At graduate school, five years later, I finally

**learned**calculus, by way of trying to teach it.

The point is, unlike instruction, which is essentially unidirectional and provides no guarantee of learning that which is ostensibly being “taught,” teaching (the real kind) is bi-directional. In fact, you can’t separate real teaching from learning. They are simply two perspectives of the same human

**interactive**process. From the teacher’s perspective it is teaching, from the student’s perspective it is learning.

For anyone who has experienced real teaching, what I am saying is obvious. Unfortunately, someone who has not experienced it likely has no idea what I am talking about. So let me give some examples that most people are familiar with.

Compare your school math classes with learning to drive, taking tennis lessons, being taught how to ride a bicycle, being taught to play a musical instrument, or being taught how to ski or improve your golf. Unless you were being seriously ripped off or shortchanged, each of those was highly interactive, with your teacher watching your performance and guiding you toward improvement.

Along the way, your teacher almost certainly gave you some instruction. Indeed, you might have stopped the learning

**activity**and gone into a classroom where the teacher explained something at a whiteboard, or showed a video. Teaching and learning usually involve instruction. But giving and receiving instruction no more

**is**teaching/learning than bricklaying is architecture. One is just a part of the other. An essential part, to be sure, but still just a part.

Long after I left school, I found myself visiting math classrooms where real teaching takes place, and nothing could be more different from what I experienced. Though I have observed many different styles of good teaching, two things they all have in common is that they are highly interactive and there is learning going on at the same time, as part of the process.

The distinction between instruction and teaching/learning becomes significant when cash-strapped education districts look to technology for assistance. For whereas technology can provide instruction and can provide teachers and students with resources to assist them, what is cannot do on its own is teach them. (Whether you think that is an inherent limitation of today’s technology or a fact of nature likely depends on your view as to how far artificial intelligence can go. I stated my position in my book Goodbye Descartes way back in 1998 and it has not changed since. But let’s leave that to one side, since my focus here is on the educational world today.)

The magnitude of the problem facing any teacher is made clear when you carry out a simple test that is all too infrequently done. (Actually, although simple in concept, it is time-consuming and difficult to carry out well, which is probably why it is done so rarely.) You sit down and talk with the student to find out what she or he has learned.

You might think that this is what the end-of-the-course test does, but nothing could be further from the truth. The well known educational consultant Marilyn Burns is an expert in carrying out such tests. Take a look at the following example of the kind of thing she discovers.

**to those of us who do understand place-value**, her verbally articulated reasoning screamed out conceptual understanding. But as the subsequent interview made clear, at best she has a rudimentary understanding in a particular context, and if truth be told we really have no idea what she knows.

The fact is, the human brain is a remarkable pattern-recognizing device. It will discern a pattern – usually many patterns – in a random display of dots on a screen. But is it the “right” pattern? Cena clearly recognized some pattern. But it is not clear what it was.

This problem bedevils all of us who seek to develop educational software or technologically-delivered courseware, from the free-to-all Khan Academy and the online classes given by MIT and my own Stanford, to for-profit spinoffs like Sebastian Thrun’s Udacity. They can be good. (I am planning to give my own online Stanford class this fall, and I surely would not attempt that if I did not see value in it.) But what they can be good at is providing instruction. They don’t teach and they do not guarantee learning of the intended material. For that, you still need a teacher, and the instructional material should be a tool that the teacher actively uses.

The kinds of problem exhibited by Cena are surely less worrying for older students, particularly students who have already learned how to learn – arguably the most important goal of schooling. But still there are dangers.

For example, my fall course will include lots of video, and there are known, significant problems with video instruction, as this video (sic) shows:

It’s tempting to try to overcome the Cena-type problem by introducing an interactive component. But that too does not work, as was discovered in 1973. In what rapidly became one of the most famous and heavily studied papers in the mathematics education research literature, Stanley Erlwanger exposed the crippling limitations of what at the time was thought to be a major step forward in mathematics education: Individually Prescribed Instruction (IPI).

But already this column has gone on long enough. If you want to find out about Erlwanger’s findings, skip over to my personal blog, profkeithdevlin, where I discuss "Benny's Rules" in the context of my work on mathematics education video games.