# Fragile Knowledge- Can it be prevented?

Frzn
I took the following excerpt from "Surely You're Joking, Mr. Feynman!"

I often liked to play tricks on people when I was at MIT. One time, in
mechanical drawing class, some joker picked up a French curve (a piece of
plastic for drawing smooth curves -- a curly, funny-looking thing) and said,
"I wonder if the curves on this thing have some special formula?"
I thought for a moment and said, "Sure they do. The curves are very
special curves. Lemme show ya," and I picked up my French curve and began to
turn it slowly. "The French curve is made so that at the lowest point on
each curve, no matter how you turn it, the tangent is horizontal."
All the guys in the class were holding their French curve up at
different angles, holding their pencil up to it at the lowest point and
laying it along, and discovering that, sure enough, the tangent is
horizontal. They were all excited by this "discovery" -- even though they
"learned" that the derivative (tangent) of the minimum (lowest point) of any
curve is zero (horizontal). They didn't put two and two together. They
didn't even know what they "knew."
I don't know what's the matter with people: they don't learn by
understanding; they learn by some other way -- by rote, or something. Their
knowledge is so fragile!
I did the same kind of trick four years later at Princeton when I was
talking with an experienced character, an assistant of Einstein, who was
surely working with gravity all the time. I gave him a problem: You blast
off in a rocket which has a clock on board, and there's a clock on the
ground. The idea is that you have to be back when the clock on the ground
says one hour has passed. Now you want it so that when you come back, your
clock is as far ahead as possible. According to Einstein, if you go very
high, your clock will go faster, because the higher something is in a
gravitational field, the faster its clock goes. But if you try to go too
high, since you've only got an hour, you have to go so fast to get there
that the speed slows your clock down. So you can't go too high. The question
is, exactly what program of speed and height should you make so that you get
the maximum time on your clock?
This assistant of Einstein worked on it for quite a bit before he
realized that the answer is the real motion of matter. If you shoot
something up in a normal way, so that the time it takes the shell to go up
and come down is an hour, that's the correct motion. It's the fundamental
principle of Einstein's gravity -- that is, what's called the "proper time"
is at a maximum for the actual curve. But when I put it to him, about a
rocket with a clock, he didn't recognize it. It was just like the guys in
mechanical drawing class, but this time it wasn't dumb freshmen. So this
kind of fragility is, in fact, fairly common, even with more learned people.

I'd just like to know if this kind of 'fragile knowledge' as Feynman puts it, is preventable? Or can only certain gifted minds do this sort of thing? I don't consider myself able to do this kind of thing at ALL, so is it too late for me to gain these skills as I go into physics next fall?

winowmak3r
I don't consider myself a Feynman by any means but I do have my moments. I find myself learning things for a class just to get an A but then never really applying them outside of class or making those real world connections until much later, if at all.

When I was taking a French class I would often try and figure out if I could say something I just said in French after I said it. I'd do this randomly throughout the day and just think about it to myself. I wouldn't be able to completely turn every sentence I said into French but I would get closer and closer the more often I did it. It got to the point where I could hold a modest conversation in French without even thinking (which is pretty good for a beginning foreign language class). What I'm trying to say is if you want to get better at recognizing the relationship between what you're begin taught and what's going on in the real world you have to think about it outside the classroom. The more you do this, the more you'll find those relationships.

I don't think the "fragile knowledge" is preventable. I think we all go through it at some point, it's just others make the connection between the classroom and the real world sooner than others. It's never too late to make those connections.

twofish-quant
I'd just like to know if this kind of 'fragile knowledge' as Feynman puts it, is preventable?

There are some teaching methods that are intended to prevent fragile knowledge, and a lot of the methods that are used to teach physics are intended to prevent this sort of thing. One way is to focus on problem sets. You are given a problem that requires going through a number of steps to figure out, only no one shows you the steps. After going through the a lot of problems, you start noticing patterns, and develop a "feeling" for these problems.

This type of teaching is very different from the type of instruction which most people get in high schools which emphasizes direct instruction.

Or can only certain gifted minds do this sort of thing? I don't consider myself able to do this kind of thing at ALL, so is it too late for me to gain these skills as I go into physics next fall?

It's a matter of environment. If someone just throws a dozen problems at you each week, you'll pretty quickly get a feel for them.

twofish-quant
I don't consider myself a Feynman by any means but I do have my moments. I find myself learning things for a class just to get an A but then never really applying them outside of class or making those real world connections until much later, if at all.

That's one reason why people have fragile knowledge. People quickly figure out what they have to do to get ahead, and in most classes this involves learning the bare minimum that you need to get an A on the test, and then forgetting everything the moment the test is over. Something that I found interesting was that some cognitive scientists gave some professional physicists freshmen tests in physics, and they didn't do terribly well on them.

You have to look at the educational system not only from the point of view of the student, but also the school. The problem with imparting non-fragile knowledge is that it is extremely expensive both in money and in time, and a lot of the reason the system is designed in the way that it is is because we don't have infinite time and money. One reason rote education and memorization is used a lot is that it is quick and cheap. If you need 10,000 science teachers, then you can't find 10,000 Richard Feymanns.

When I was taking a French class I would often try and figure out if I could say something I just said in French after I said it.

If you are in a situation where you have to actually use French in order to eat and in which you are surrounded by French, then you'll learn pretty quickly. One weird thing is that people that have learned languages in this way can communicate, but they do surprisingly badly on academic tests of grammar.

Goldbeetle
I agree with twofish-quant. BTW: who's Richard Feynmann, anyway?

mal4mac
Timothy Gowers (Fields medallist, Cambridge Professor) in "Mathematics: A Very Short Introduction" encourages students not to worry if they have to go the "fragile" path.

“Suppose that a pupil makes the common mistake of thinking that x^(a + b) = x^a + x^b. A teacher who has emphasized the intrinsic meaning of expressions such as x^a will point out that x^(a + b) means a + b xs all multiplied together, which is clearly the same as a of them multiplied together multiplied by b of them multiplied together. Unfortunately, many children find this argument too complicated to take in, and anyhow it ceases to be valid if a and b are not positive integers.

Such children might benefit from a more abstract approach. As I pointed out in Chapter 2, everything one needs to know about powers can be deduced from a few very simple rules, of which the most important is x^(a + b) = x^ax^b. If this rule has been emphasized, then not only is the above mistake less likely in the first place, but it is also easier to correct: those who make the mistake can simply be told that they have forgotten to apply the right rule. Of course, it is important to be familiar with basic facts such as that x^3 means x times x times x, but these can be presented as consequences of the rules rather than as justifications for them... it is quite possible to learn to use mathematical concepts correctly without being able to say exactly what they mean. This might sound a bad idea, but the use is often easier to teach, and a deeper understanding of the meaning, if there is any meaning over and above the use, often follows of its own accord.”

twofish-quant
Timothy Gowers (Fields medallist, Cambridge Professor) This might sound a bad idea, but the use is often easier to teach, and a deeper understanding of the meaning, if there is any meaning over and above the use, often follows of its own accord.”

It's easier for Timothy Gowers to teach math this way, but it's certainly not easier for your average seventh grade algebra teacher that has a bachelors degree and barely knows algebra themselves. You can scream at that teacher for being unprepared, but at that point you have the problem of 1) teaching the teachers and 2) that you have several million seventh graders that are waiting to learn algebra *now* and by the time you finish teaching the teachers, they will all be college age.

As far as easier for the student. If you are a math major at Cambridge university, then yes going into abstract concepts may be a better way of teaching, but most people aren't math majors at Cambridge or math majors at all, and I've found that in teaching people that *aren't* math majors that the easiest way of getting them to have any grasp of the problem is to break it down into small steps with different parts so that they can memorize it.

The problem with going abstract is that if you go too abstract and then the student looks at you with the glazed look saying that they don't have any clue what you just said, you are stuck, whereas if you break the problem up into steps, you can find the crucial piece that the student does not understand.

Yes, it's likely that they won't get any appreciation for the deep abstract beauty of math, but they mostly aren't interested in that. They just want something practical so that they can pass the class and make more money.

twofish-quant
One other thing, I'm not a math-geek, I'm a physics-geek, and I have problems following abstract arguments myself. What I do in order to have people correct that mistake is to put some simple numbers in for x, a, and b, and you quickly find that it doesn't work. x=1, a=1, b=1. oops

Also to show people the general rule, I rely on counting arguments to show where the rule comes from.

DukeofDuke
Sorry man, but to get your knowledge up and above that fragility, you either have to be very very smart and just look at things and get them, or spend a LOT of time OUTSIDE of class trying to understand the material.

The best way I've found to do this, is to free-read textbooks that are not assigned by your class, but that are on the same material as your class is. See, when learning things the class way, you automatically turn your deeper, stronger mind off (because it takes up so much time to work) and you turn on the part that looks at a formula, grabs it without letting it sink, and then applies it for that shiny A.

Read the Feynman Lectures alongside your freshmen physics courses. It will take a lot of time, but there will be enough of a distance between your class and your private reading that you can learn it reeeeal gewd.

jgm340
It's easier for Timothy Gowers to teach math this way, but it's certainly not easier for your average seventh grade algebra teacher that has a bachelors degree and barely knows algebra themselves. You can scream at that teacher for being unprepared, but at that point you have the problem of 1) teaching the teachers and 2) that you have several million seventh graders that are waiting to learn algebra *now* and by the time you finish teaching the teachers, they will all be college age.

twofish-quant, allow me to expand on your statement:

[LOTS OF WORDS] It's not so much that there aren't enough qualified people to teach 7th grade algebra, but rather everyone who is even slightly over-qualified for such a job does something else instead which makes more money or comes with more recognition. The best teachers I have ever had all could have been out doing much more prestigious work.

My mother actually teaches at a not-so-great public high school. She went to Columbia undergraduate and University of Chicago for her masters (she was going for a PhD, but started our family instead - wise choice if you ask me!). However, she has helped so many kids. She started a huge program to help failing kids, and she puts so much work into her teaching. A lot of the people she works with, however, are incompetent. They meet the minimum requirements, basically, and pander to the administration until they get tenure.

The problem seems to trace its roots to the massive pressure society puts towards over-qualification. Any high school graduate these days is essentially forced into a 4-year college. Besides the forced pace and debt, this is not so bad, because people should develop themselves intellectually as far as they are reasonably able. However, what is bad is the accompanying pressure to attain the most prestigious job you can for the qualification you just worked to earn. In other words, there is pressure to be the worst worker at a better job rather than the best worker at a mediocre job. Still, someone has to sweep the streets, stock the shelves, teach the kids. The result is that the people doing these jobs are qualified in a technical sense (a degree), but have no actual ability. They are under-qualified in the only sense that matters, and that is of being up for the job.

Anecdote time: at one of the dining halls at my university, there is this one old man who works serving food. He wears a tie and a nice, white button down shirt. He has a very sophisticated sense of humor, and could clearly be doing something much more glamorous. The fact that he is there, however, makes my day! It makes it worth it to be in that awful place. There is also another guy who makes the best burgers. The absolute most deliciously tender burgers you could ever imagine. And he's working in a dining hall instead of some restaurant making more money? You know what, I'm glad he is!

Basically, the world would be better if people working at McDonald's were going home and talking philosophy. You don't get this by forcing the people who would be working at McDonald's to study harder so they can work at IBM. You get this when people who like the to think about philosophy realize it's okay to work at McDonald's. [/LOTS OF WORDS]

*********

Now, to address the question of methods of teaching/learning math/physics:

As far as easier for the student. If you are a math major at Cambridge university, then yes going into abstract concepts may be a better way of teaching, but most people aren't math majors at Cambridge or math majors at all, and I've found that in teaching people that *aren't* math majors that the easiest way of getting them to have any grasp of the problem is to break it down into small steps with different parts so that they can memorize it.

The problem with going abstract is that if you go too abstract and then the student looks at you with the glazed look saying that they don't have any clue what you just said, you are stuck, whereas if you break the problem up into steps, you can find the crucial piece that the student does not understand.

Yes, it's likely that they won't get any appreciation for the deep abstract beauty of math, but they mostly aren't interested in that. They just want something practical so that they can pass the class and make more money.

I disagree. The abstract parts of math are actually the easiest parts. What makes math difficult is the language: the method of communicating such a complex idea. When kids are having trouble with math, I don't think it's because they are incapable of understanding the deeper concepts. In fact, I think that anyone not bright enough to understand the concepts with only fail more miserably trying to memorize the mathematical procedures, because the procedures which result from the simple concepts in math are far more complicated than the concepts from which they are borne.

Anecdote: and this applies not only to math. I am taking an introductory psychology course. The professor who teaches it is overqualified (he researched at Yale, back in the day), and he spews a bunch of information at us. However, what is important to understand in order to understand the study of psychology is just a few basic concepts. I am convinced that the professor doesn't even know these concepts exist. He focuses so exclusively on a lot of the unimportant details of psychological experiments, and presents them in a way which demonstrates little understanding for the conceptual importance of these experiments. My teaching assistant, who lacks the qualifications, in one day has taught me more than all the lectures. The pedestal of position is an awful thing if you won't step down from it.

Klockan3
Sorry man, but to get your knowledge up and above that fragility, you either have to be very very smart and just look at things and get them, or spend a LOT of time OUTSIDE of class trying to understand the material.
Is there really no way? I always wanted to believe that there was something that people just missed, that there is an easier and better way for them. I mean, it can't be that hard to just understand can it? But I am starting to lose faith in it. It seems like people needs a huge amount of time to properly grasp even the simplest of concepts properly and it would not be feasible to give that much teaching.
I disagree. The abstract parts of math are actually the easiest parts. What makes math difficult is the language: the method of communicating such a complex idea. When kids are having trouble with math, I don't think it's because they are incapable of understanding the deeper concepts. In fact, I think that anyone not bright enough to understand the concepts with only fail more miserably trying to memorize the mathematical procedures, because the procedures which result from the simple concepts in math are far more complicated than the concepts from which they are borne.
You would think that, but it isn't true. Abstract algebra is a perfect example of a course that feels totally trivial after you learned it properly but the first time you see it everything just feels incomprehensible.

Also I have experienced that people who go through that route without going through the heuristic computational route first often have extremely poor understanding of the material, so it don't really help either. They know the theorems and such and maybe memorized some proofs, but that is a perfect example of fragile knowledge.

What you want to do is to make the rules they get as easy to adhere to as possible or they wont understand anything at all. In that regard minimizing the amount of rules they have to learn by making them exceedingly abstract is counterproductive, you want to go there eventually with maths students but it doesn't help to take that shortcut.
Anecdote: and this applies not only to math. I am taking an introductory psychology course. The professor who teaches it is overqualified (he researched at Yale, back in the day), and he spews a bunch of information at us. However, what is important to understand in order to understand the study of psychology is just a few basic concepts. I am convinced that the professor doesn't even know these concepts exist. He focuses so exclusively on a lot of the unimportant details of psychological experiments, and presents them in a way which demonstrates little understanding for the conceptual importance of these experiments. My teaching assistant, who lacks the qualifications, in one day has taught me more than all the lectures. The pedestal of position is an awful thing if you won't step down from it.
That is a classic example, overqualified people usually makes for bad teachers. Like we had one of my countries best mathematicians teaching our calculus course and it was the first time he was teaching undergraduates. What happened was that he started by going through the Greek alphabet, then the normal mathematical proof notation and then he started proving things about limits and continuity using those. In a computational calculus course... Like the second time he showed us a proof he had done himself on the mean value theorem.

After that almost none went to his lectures. He probably thought that he was doing us a favor by skipping the trivially boring computational parts but it doesn't really work that way.

The funny thing is that this was probably what happened at your psychology course. Your professor thought that it would be best to throw you into the deeper material instantly while the assistant went for the heuristic explanations that makes sense. Your mind do want those heuristic explanations, when they teach you that 1+1=2 by holding apples it is since it makes sense to people. That is the most powerful way to learn. Skipping that is like building a house without a foundation, it looks good on the outside but you can't use it for anything and it will probably break down pretty soon.

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Gold Member
Sorry man, but to get your knowledge up and above that fragility, you either have to be very very smart and just look at things and get them, or spend a LOT of time OUTSIDE of class trying to understand the material.

I don't buy this. I seem to notice the people who have knowledge about this 'fragile knowledge' are the ones who can put the connection together between their education and reality. Most people I know in my department have probably never even wondered how an engine works or a simple drill or what have you works. It seems to be as Feynman says, people can't put 2 and 2 together between what they learn and reality, or at least don't try to. One of my professors once told me that he had a math major in one of our freshman/sophmore physics courses who was a senior himself and that only after taking the physics course, did he realize what a first derivative actually meant.

jgm340
You would think that, but it isn't true. Abstract algebra is a perfect example of a course that feels totally trivial after you learned it properly but the first time you see it everything just feels incomprehensible.

Also I have experienced that people who go through that route without going through the heuristic computational route first often have extremely poor understanding of the material, so it don't really help either. They know the theorems and such and maybe memorized some proofs, but that is a perfect example of fragile knowledge.

I haven't taken algebra yet, so I can't speak to that.

Memorizing a theorem is memorizing words. Memorizing a proof is memorizing words. It's completely meaningless unless you *actually* know what it means. Just because you know the words and the grammatical structure doesn't mean you understand the poem. Take, for example, the sentence preceding this one. You first understand the basic grammatical structures and little idioms, "Just because", "grammatical structures", etc., which make up the sentence. Then you begin to understand the explicit meaning of the sentence, and you can rewrite it, if you like, in a form that makes more sense to you such as "(understanding words) =/=> (understanding the poem)". This is where most people stop in math. They can use it in a proof, so that's enough. But this isn't even the concept! To understand the concept, you must first understand what the sentence *really* means. The sentence is not meant to be a statement of truth. It's meant to be a demonstration of a concept, which the author of that theorem intends you to be able to see. It's like an analogy.

Or, more explicitly stated, a theorem is valuable not as a tool for solving problems, but as demonstration of how one mathematician has gone about finding truth.

Anyway, as you say, doing computations is necessary. I never said it wasn't. However, the aim of doing those computations is to get at the concept! If the computations are so far removed from the concept (for example, the physical act of row operations on a matrix is very removed from the concept of a vector space), then no amount of work will get you to the concept. Mathematicians have figured these concepts out before, and a good student doesn't spend time trying to figure them out on his own through brute force by doing computations hour after hour.

What you want to do is to make the rules they get as easy to adhere to as possible or they wont understand anything at all. In that regard minimizing the amount of rules they have to learn by making them exceedingly abstract is counterproductive, you want to go there eventually with maths students but it doesn't help to take that shortcut.

Math is not about "following rules". "Abstract" is not more difficult at all. "Abstract" is the natural way people think about things. Exceedingly abstract is exceedingly easy. Rules are very specific and (seemingly) arbitrary, and thus exceedingly difficult to make sense of. The advantage rules have is that if you make them simple enough, there is no possibility of mis-communicating (no, Billy, I told you for the thousandth time to put x in the denominator!). The main disadvantage is that you aren't actually communicating anything.

That is a classic example, overqualified people usually makes for bad teachers. Like we had one of my countries best mathematicians teaching our calculus course and it was the first time he was teaching undergraduates. What happened was that he started by going through the Greek alphabet, then the normal mathematical proof notation and then he started proving things about limits and continuity using those. In a computational calculus course... Like the second time he showed us a proof he had done himself on the mean value theorem.

After that almost none went to his lectures. He probably thought that he was doing us a favor by skipping the trivially boring computational parts but it doesn't really work that way.

The funny thing is that this was probably what happened at your psychology course. Your professor thought that it would be best to throw you into the deeper material instantly while the assistant went for the heuristic explanations that makes sense. Your mind do want those heuristic explanations, when they teach you that 1+1=2 by holding apples it is since it makes sense to people. That is the most powerful way to learn. Skipping that is like building a house without a foundation, it looks good on the outside but you can't use it for anything and it will probably break down pretty soon.

My professor is psychology is NOT throwing us into deeper material. He is stumbling around on stage focusing on trivial things (and I don't mean subtle). He spent an entire class explaining the grade distribution of our exams. He spends a considerable amount of time each lecture establishing his "lineage" in academics (who he has worked with, etc).

When a teacher holds two apples up to demonstrate "1+1=2", the teacher is creating an analogy between "Apple + Apple = Two Apples" and "1+1=2". This works brilliantly because children at that age already have a concept of quantity, and a concept of language to express concrete quantities.

When little Johnny is trying to learn derivatives, then, why do we insist on showing him all this stuff about limits and such first? All this complete non-sense which has nothing to do with anything he's ever seen before, just to build a foundation to understand something so simple and intuitive as a derivative? And then we start talking about epsilon and delta? Really? That is not holding up two apples.

You don't build an understanding of math by building up computational ability. You build it by fleshing out concepts.

Klockan3
rstanding of math by building up computational ability. You build it by fleshing out concepts.
I think you misunderstood my post. And as I read back on these posts it seems like I misunderstood your post as well, I thought that it was you who posted that thing Frzn posted.

I've just started doing education units (only a few weeks in) but in reflecting about what I have learned in math I come to see a few things:

The first is that knowledge today is highly synthetic: that is it is highly refined and polished to a point that an attempt is made to cram in such an enormous amount of knowledge into the most optimal presentation of that knowledge to each student.

There is no way that a student can build the appropriate context for that knowledge to have meaning in a variety of perspectives that makes it "fragile". I would agree that there are some methods that are a lot better than others that make it "less fragile" like two fish mentioned but personally I think that this "context" is a developed thing that can take many years to build up.

When I went out to dinner with a family friend, he told me that in the arts there is a 10,000 hour rule: that is it takes about 10,000 hours roughly to become highly proficient in your chosen field. There was some research which I was reading about which studied and analyzed a different variety of experts in a wide range of fields and found that on average it took about ten years before a certain mastery was demonstrated of their chosen endeavor by each individual. (I would cite the study but i can't remember the exact work or book that cited it).

What i have found so far in uni is that the more experienced professors have often added more context to learning than ones who have a more superficial understanding of the material. They are also the ones that are able to explain ideas to people with less mathematical inclination with ease as they can relay the concepts and appropriate ideas much more directly and succinctly where required where over-riding complexity is turned into simplicity.

The fact that information is becoming that much more refined and polished is through analysis and reflection of the material and related pedagogy is creating better forms of teaching, but in my mind I don't think anyone be it Feynmann or anyone else can be sure that context can be taught in semester long courses at university.

mal4mac
It's easier for Timothy Gowers to teach math this way, but it's certainly not easier for your average seventh grade algebra teacher that has a bachelors degree and barely knows algebra themselves. You can scream at that teacher for being unprepared, but at that point you have the problem of 1) teaching the teachers and 2) that you have several million seventh graders that are waiting to learn algebra *now* and by the time you finish teaching the teachers, they will all be college age.

As far as easier for the student. If you are a math major at Cambridge university, then yes going into abstract concepts may be a better way of teaching, but most people aren't math majors at Cambridge or math majors at all, and I've found that in teaching people that *aren't* math majors that the easiest way of getting them to have any grasp of the problem is to break it down into small steps with different parts so that they can memorize it.

The problem with going abstract is that if you go too abstract and then the student looks at you with the glazed look saying that they don't have any clue what you just said, you are stuck, whereas if you break the problem up into steps, you can find the crucial piece that the student does not understand.

Yes, it's likely that they won't get any appreciation for the deep abstract beauty of math, but they mostly aren't interested in that. They just want something practical so that they can pass the class and make more money.

In the quoted text from Gowers i don't think he is going 'too abstract'. Basically he is just saying - learn the rule x^(a + b) = x^ax^b. So, for school maths, going abstract is simply learning a few rules - which is, surely, easier for many students than putting in numbers, counting, whatever...

I was amazed to see the teacher in "The Wire" using this method with his inner city kids. 'Just follow da rule...' Pure Gowers. Just give all those algebra teachers a copy of Gowers, some of them should get the message...

Point taken about going 'too abstract', but I don't think Gowers is guilty of that. His 'very short' book is definitely not aimed at mathematics majors at Cambridge,. It's for everyone with some motivation to learn a little bit about mathematics (although even the Cambridge high flyers could benefit from reading it, I'm sure...)

mal4mac
Math is not about "following rules".

It is, partly, according to Gowers.

Rules are very specific and (seemingly) arbitrary, and thus exceedingly difficult to make sense of. The advantage rules have is that if you make them simple enough, there is no possibility of mis-communicating (no, Billy, I told you for the thousandth time to put x in the denominator!). The main disadvantage is that you aren't actually communicating anything.

You are communicating 'x^(a + b) = x^ax^b'. Nothing arbitrary about this. Apply this rule and you will get the correct answer in any of your physics calculations where you need to make this transformation. Might mean the difference between a pass and a fail. Seems like pretty powerful communication to me!

When a teacher holds two apples up to demonstrate "1+1=2", the teacher is creating an analogy between "Apple + Apple = Two Apples" and "1+1=2". This works brilliantly because children at that age already have a concept of quantity, and a concept of language to express concrete quantities.

What happens when teacher wants to demonstrate that 23 + 45 = 68? Get two barrels of apples and pour them on the floor? How does Johnny know if there are 68 or 97? Rules make it simple. Johnny learns the rules 3 + 5 = 8, and 2 + 4 = 6, and then teach tells him to put them together... No need to think about breakfast or try to visualise the impossible... So why not start by introducing 1+1=2 as a rule, as well as waving apples about? And then quickly leave the apples behind... maybe after 1 + 2 = 3...