Natural origins or learned habit: Why do students skip concepts before applications?

Question

When teaching elementary mathematics, it takes a lot of time and effort to teach students that our goal is not to learn the examples, but to learn the concepts first, and then apply them to specific examples or problems. But students have the tendency to just try to learn the examples and skip the concepts. You see this issue when students attempt to do their homework without having learned the concepts first, and then if there is a homework problem that they have not seen before, they get stuck, even if it can be solved by a simple application of definitions.

Example. Consider the concept of continuity, for instance (although my question is broad and is not related to this particular concept). You give your students a problem to determine whether a certain function is continuous at a given point or not. They get stuck and come to you for help. You ask them if they can tell you what it means for a function to be continuous at a given point and you realize that they don't know. It is very strange to see students trying to solve a problem about continuity without knowing what it means for a function to be continuous. But apparently it is not strange to them.

The way most students in elementary courses try to learn mathematics is by looking at solved examples and then mimic the solution to solve other similar examples, without necessarily knowing or understanding the concept that is involved.

My questions are:

1. Is there a name in the literature for the phenomenon that I described? Has this been studied?
2. Does this phenomenon have natural origins, or is it a habit that students develop because of how they have been taught in prior mathematics classes? By "natural origins" I mean does it stem from internal characteristics of human learning?

Addendum

I received fascinating responses to my question. I would like to make my question more precise by adding one more example.

Example. Suppose I want to teach my calculus students that if they want to verify whether the limit of a function $f$ at a number $x=a$ exists, they should check to see if limits from the left and the right of $f$ at $x=a$ are equal or not. I don't have any issues with learning through examples or inductive learning. We can demonstrate this concept through as many examples as necessary. At the core of my question lies the observation that many students seem to focus on the appearance of the functions in the examples that we solve, rather than what needs to be checked. In the end, when I test my students, I want to see what they have learned: have they learned that to verify whether the limit of a function $f$ at a number $x=a$ exists, they must check to see if limits from the left and the right of $f$ at $x=a$ are equal, or do they only remember the steps that were taken in each appearance of functions in the examples that we solved? My experience is that, it is often the latter, and the number of examples solved doesn't seem to help. For instance, we solve many types of examples (I emphasize, many, not just this) including

$$f(x)=\left\{\begin{array}{lcl}\frac{x^2-4}{x+2}& \mathrm{if} &x>-2\\ 2&\mathrm{if}&x=-2\\x^2+x-6&\mathrm{if}& x<-2\end{array}\right.$$ When we solve this example, it seems that students don't see the solution as checking the equality of one sided limits of $f$ (of course, I emphasize that multiple times). Instead, they look at the appearance of the function $f$ and see the solution as follows: first take the first expression $\frac{x^2-4}{x+2}$ and find the limit, then take the last expression $x^2+x-6$ and find the limit, then see if the limits are equal to each other. In other words, they create "rules" based on the appearance of functions in each solved example, not realizing that regardless of the appearance of the function, in all these examples we are just checking the equality of the one-sided limits. Consequently, if I give them a similar piecewise-defined function in the test, they can answer the question, because they remember the appearance of this function and their own "rules" for it, but if I give them a function that has a new appearance, which they have not seen before, they cannot solve the problem. For instance, if I give them this question: "Given that $$f(0)=2,\: g(0)=-2,\:\lim_{x\rightarrow0^-}f(x)=-1\:\lim_{x\rightarrow0^+}f(x)=0\:\lim_{x\rightarrow0^-}g(x)=1\:\lim_{x\rightarrow0^+}g(x)=0,$$ does limit of $f(x)+g(x)$ as $x\rightarrow0$ exist?" they cannot solve it, because they do not attempt to check the equality of one-sided limits. That's not what they took away from the examples we solved. They just remember the "algorithmic rules" that they created based on the appearance of each function in each example, and those don't help for this new function.

So my question was, is this the nature of human learning, or is it a learning technique that is encouraged by the way we test them (e.g., solve an example involving a piecewise-defined function, then test them by giving another piecewise-defined function)?

Answer 1

A relevant idea from the literature is the theoretical framework of "concept image and concept definition." It has been very popular since it was introduced, and you can find studies that use it to analyze students' reasoning about all sorts of mathematical concepts. Tall and Vinner (1981) introduced the framework by discussing examples of students' reasoning about limits and continuity. Here's an excerpt from that paper.

The human brain is not a purely logical entity. The complex manner in which it functions is often at variance with the logic of mathematics. It is not always pure logic which gives us insight, nor is it chance that causes us to make mistakes. To understand how these processes occur, both successfully and erroneously, we must formulate a distinction between the mathematical concepts as formally defined and the cognitive processes by which they are conceived.

Many concepts which we use happily are not formally defined at all, we learn to recognise them by experience and usage in appropriate contexts. Later these concepts may be refined in their meaning and interpreted with increasing subtlety with or without the luxury of a precise definition...

We shall use the term concept image to describe the total cognitive structure that is associated with the concept, which includes all the mental pictures and associated properties and processes. It is built up over the years through experiences of all kinds, changing as the individual meets new stimuli and matures...

The definition of a concept (if it has one) is quite a different matter. We shall regard the concept definition to be a form of words used to specify that concept. It may be learnt by an individual in a rote fashion or more meaningfully learnt and related to a greater or lesser degree to the concept as a whole. It may also be a personal reconstruction by the student of a definition... In this way a personal concept definition can differ from a formal concept definition, the latter being a concept definition which is accepted by the mathematical community at large.

As Daniel Collins points out in his answer, mathematicians often assume that students understand the role of definitions in mathematics. This is an expert blind spot. If we want students to reason using definitions, they must be taught how to do so. When students practice applying the formal definition, it becomes incorporated into their concept images.

It is not the case that mathematical thinking requires the replacement of concept images with formal definitions. Tall (1991), in an article on "advanced mathematical thinking," says the following about how a psychologist might describe the mathematicians' notion of "intuition vs. rigor."

One may view the usual mathematician's dichotomy between intuition and rigour as one between the holistic, visual thinking characteristic of the right brain, and the rigour of the sequential, logical thinking characteristic of the left. But the psychologist sees the possibility of more sophisticated (secondary) intuitions arising from refined concept images which can include the mental imagery of logic and deduction. Thus aspects of logic too can be honed to become more "intuitive" to the mathematical mind. The development of this refined logical intuition should be one of the major aims of more advanced mathematical education.

So when we ask students "What does it mean for a function to be continuous at a point?" and they struggle to answer in words, it does not mean that they are "skipping concepts." It means that they do not have a concept definition retrievable from long-term memory. They very likely have a strong concept image of "continuity," but their concept image does not help or may even actively conflict with the task at hand.

One effective response to this, as some other answers have advised, is to test students on recall of formal definitions, model your own reasoning with definitions, and provide worked examples that show how to use definitions. You mention in a comment,

Assume that definitions have been stated and explained and a reasonable number of concrete examples have been solved and demonstrated and in each example, the teacher has emphasized how the example is solved by referring back to the definitions. Now the teacher tests students' learning by giving them a new problem that they have not seen before, but nevertheless can be solved by referring back to definitions... such "new" problems are difficult for most students and they cannot solve them, because they still try to approach the problem by searching their inventory of solved examples.

It seems here that students still do not have a notion of "referring to the definition" as a strategy, even though you may feel that you are emphasizing it. If students have indeed seen a variety of worked examples that involve first referring to the definition, then it may be fruitful to have them compare these examples and synthesize this strategy from the comparison. With worked examples, students often attend to surface-level details and not to the deeper structure that experts can see. There is research on this as well, though I don't have time to add it to the answer right now.

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Here are some references on how novices and experts perceive "deep structure" and how this relates to instruction and examples. There have been studies on this in various domains, but I'll focus on references from the mathematics education literature.

One early study is Silver (1979) on "Student Perceptions of Relatedness among Mathematical Verbal Problems." He had eighth-graders sort various word problems into groups. The problems varied in both problem context and mathematical structure. He found that "the tendency to sort problems on the basis of mathematical structure was significantly positively correlated with all of the general verbal and mathematical ability measures."

Schoenfeld and Hermann (1982) in "Problem Perception and Knowledge Structure in Expert and Novice Mathematical Problem Solvers" asked mathematics professors and undergraduates to sort problems that were designed to have a mathematical "surface structure" and a mathematical "deep structure." For example, a problem might have a surface structure of "prime numbers," but the deep structure is "contradiction." They found that "the sorting by novices appears to depend more on surface characteristics of problems, while the sorting by experts depends on the deep mathematical properties of the problems."

Atkinson et al. (2000) synthesize research in "Learning from Examples: Instructional Principles from the Worked Examples Research." They suggest that good design for teaching with examples involves varying surface features to encourage students to search for deep structure, and to encourage students to self-explain deep structure.

Rittle-Johnson and Star (2009) discuss the effect of comparing examples in "Compared to what? The effects of different comparisons on conceptual knowledge and procedural flexibility for equation solving." They state that "Moderately similar, rather than highly similar, examples should help people ignore irrelevant surface features and abstract a more general underlying solution structure... The current findings support this prediction for a problem solving task and extend it to outcomes not previously considered – procedural flexibility and conceptual knowledge."

Lee et al. (2017) in "Embellishing Problem-Solving Examples with Deep Structure Information Facilitates Transfer" had participants study examples of "algebra-like" problems, then had them attempt "transfer problems" that required adjustment of their procedures. They found that "students transferred better when they studied with examples that emphasized problem structure rather than solution procedure."

Answer 2

Students have the tendency to just try to learn the examples and skip the concepts. Is there a name for this phenomenon? Has it been studied? Does it have natural origins, or is it taught in prior math classes?

I'm not familiar with the literature, but I do know that "math people" tend to severely overestimate the degree to which the rest of the population reasons from first principles. Most people exclusively reason by analogy:

Is $f(x)$ continuous? Well, based on the examples I've seen, continuous functions look "normal" and discontinuous functions look "weird." In this case $f(x)$ looks more normal than weird, so $f(x)$ is probably continuous.

Side Note: Students who exclusively reason by analogy generally don't see it as problematic that they don't know how to proceed in the absence of similar examples. When that happens, they view it as a "data" problem, not a "model" problem, so they "fix" the problem by obtaining a similar example from someone/something that seems trustworthy and adding it to their dataset of examples for future decisions.

This isn't just in math. It's across life in general. Consider stereotypes:

Is X true about thing Z? Well, based on the examples I've seen, thing Z seems similar to the type of things for which X is true. So X is probably true about thing Z.

To me, this would suggest that overreliance on analogical reasoning has natural origins (even if the specific decision rule for a particular case of analogical reasoning might be learned from other people).

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Response to OP's Addendum:

. . . they [students] create "rules" for each appearance of functions in each solved example, not realizing that regardless of the appearance of the function in all these examples, we are just checking the equality of the one-sided limits . . . is this the nature of human learning, or is it a learning technique that is encouraged by the way we test them (e.g., solve an example involving a piecewise-defined function, then test them by giving another piecewise-defined function)?

I've had countless similar experiences working with students who were flat-out resistant to first-principles reasoning. You go through a zoo of different-looking examples, emphasizing that while there are different "rules" for each example, the rules all stem from the same "first principle" -- but some students just ignore the first principle and instead prefer to memorize a special rule for each special case.

Why does this happen? Isn't it easier to just remember the first principle and apply it as needed? It's less to remember, and it's guaranteed to cover the complete space of problems. The resistance to first-principles reasoning is especially weird because memorizing special cases takes more effort, and at the same time, the students who do this are typically the ones who are trying to minimize the amount of effort that they put forth to get a desired grade in their class.

I think what's happening is that these students are reaching an "abstraction ceiling" beyond which they find it exceedingly difficult to engage in first-principles reasoning -- so difficult, that it's actually less effort to memorize a bunch of special cases.

On the surface, the idea of an "abstraction ceiling" might seem weird. Doesn't a ceiling imply that there's a single topic marking the dividing line between the abstractions that a student is or isn't able to grasp? Is there really a level at which one is suddenly incapable of learning math? Yeah, I don't think that makes sense either. But that's not what I mean by the abstraction ceiling. What I mean is a "soft" threshold at which the amount of time and effort required to learn math begins to skyrocket due to friction in the learning experience.

There are numerous sources of friction that come into play as you move up the levels of math:

• Maybe the new information isn't being broken down and scaffolded enough, and it's overwhelming your working memory capacity.

• Maybe you're missing prerequisite knowledge.

• Maybe you've covered the prerequisites, but you didn't get enough practice to hammer in sufficient mastery.

• Maybe you mastered the prerequisites at one point, but you didn't get enough review to retain that information, and consequently, you forgot it.

• Maybe you've lost motivation / interest due to feeling like the math is no longer relevant to your future.

• Maybe you've lost motivation due to being in a class where everyone else is just as good at math as you, and math is no longer the thing that makes you "special."

And not only do these sources of friction increase as you climb higher in math, but they also compound. The more prerequisite knowledge you're missing, the harder it is to learn new material, and bigger your knowledge gaps grow. The more motivation you lose due to no longer being the best among your classmates, the worse you get relative to your peers, and the more motivation you lose.

As you climb the levels of math, these sources of friction conspire against you and eventually throw you off the train. And one of the first warning signs is when you stop understanding things at the core, and instead try to memorize special cases cookbook-style.

The unfortunate part is that, while some sources of friction can be mitigated through effective pedagogy, individual differences can stack the deck against you. For instance, it's known that working memory capacity varies between individuals. Relatedly, it's also known that people differ in their ability to generalize patterns from data, and even in their rate of forgetting. We all have cognitive differences, just like we have physical differences. It's just that the physical differences are easier to see.

We're getting into a really touchy subject, and this may be an uncomfortable (maybe even unpopular) take. But surely there's no issue accepting that, even with years of practice, not everyone can really understand Topological Quantum Field Theory at its core, right?

Say there's some math genius who takes some number of hours to master Topological Quantum Field Theory starting from high school calculus. Say a best-in-the-university math student takes twice as long, an exceptionally talented math student takes twice as long as that, a very talented (but not exceptionally so) math student takes twice as long as that, and so on. Where does it end? Keep on going and you eventually hit an infeasibly large time commitment, and keep on going after that and you eventually hit a time commitment so large that it exceeds the sum total waking hours in the average human lifespan.

And if you accept this for Topological Quantum Field Theory, then how is any different for, say, high school calculus?

In fact, it would be weird if there were a single level of math that marked the dividing line between "everyone who's learned the prerequisites can do it" versus "reserved for geniuses only." It's likely a continuous thing that starts in fairly elementary math -- the higher the level of math, the fewer the number of people are capable of really understanding it at its core, given a reasonable time commitment. Each student eventually hits their "abstraction ceiling," the amount of time and effort it takes for them to succeed in math class begins to skyrocket, they find other subjects that they enjoy more (which may or may not use the math that they've learned so far), and they take an "off ramp" into those subjects.

Of course, this is in no way an argument for giving up on students who are struggling. You support them as much as is feasible given the context of the class (see Daniel R. Collins's answer for a great example of this). But at the end of the day, when you're reflecting on how well you've done as a teacher, you need to have realistic expectations.

If you run a summer basketball camp, you might have a couple really impressive kids who make you think "wow, maybe I'll see them play professionally someday," but you're not going to turn all the attendees into future pro ball players. Most of the kids probably aren't even going to get a basketball scholarship to college. You're definitely not going to get the 5'2" kid to dunk. And that's okay. Hopefully, you'll get each kid to have fun and become the best basketball player that they have the potential to be.

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Addendums to This Answer

Addendum 1: Commenters (ryang & Passer By) have pointed out that analogical reasoning is also known as inductive reasoning and first-principles reasoning is also known as deductive reasoning. You may have better luck using those terms when searching the literature.

Addendum 2: This is not to claim that analogical reasoning should be avoided entirely. Analogical reasoning generally does a good job of supplying you with a quick decision that is probably correct (or at least more likely correct than incorrect). In most situations in life that's all you need.

As so elegantly stated in Yakk's insightful answer, even professional mathematicians "have heuristics and patterns . . . which they are confident they can reduce recursively to first principles."

A typical professional mathematician doesn't determine if a function is continuous by doing an epsilon-delta test. Given $\sin(x)+\cos(x),$ they will know it is continuous without thinking; and if asked to verify, it is the sum of two continuous functions (again, not epsilon-delta).

They can probably generate an epsilon-delta proof . . . But very few mathematicians will look at $\sin(x)+\cos(x)$ and generate a first principles proof that it is continuous and use that to answer the question.

There's a difference between using analogical reasoning and over-relying on analogical reasoning. When you use analogical reasoning, you need to have a sense of its confidence (i.e. probability of correctness) and how that measures up against the stakes of the situation you're in. If the confidence does not measure up against the stakes, then you need to fall back to first-principles reasoning (provided that you have enough time to do so). People who over-rely on analogical reasoning do not do this.

Addendum 3. Some hardline pure mathematicians might argue that the entire point of math classes is training in deductive reasoning, so inductive reasoning has no place. While that may be at least partially true from the perspective of some pure math classes, especially those really niche classes focused on the axiomatic foundations of math, this answer interprets math in its most general sense, and in most areas under the gigantic umbrella of math, inductive reasoning has some place. For instance, throughout applied math there's a saying "all models are wrong but some are useful." And in machine learning specifically (an extreme case for illustrative purposes), the whole point is to use math to build automated systems that use inductive reasoning to make decisions that are correct enough to be useful.

Answer 3

An observation that I've made in the past is this:

Math is hard because by its nature it's taught at TOO HIGH a level compared to other classes.

Consider: Does the curriculum make time and space to discuss and directly test whether students have first learned the definitions? In most cases, no; rather, we're usually running application exercises mere seconds after providing a definition. So this directly signals to students that the definitions are ignorable boilerplate, and we expect them to jump right into applications.

There's a bit of a toxic mix of (a) mathematicians assuming everyone knows how critical knowing the definitions are, and that they'll study/reflect/test themselves on it, and (b) knowing how few people ever learn the definitions, thinking it's a hopeless cause and just not bothering to spend time on them.

If we take Bloom's taxonomy as a model, in a math class we generally hand-wave the level 1-2 issues (simple factual recall) as trivial, and jump directly to levels 3-4+ (applications, analysis, etc.). Whereas instructors in almost any other discipline I know constantly bemoan that they can't get students beyond the level 1-2 memorization issues, in math we do the opposite trick and basically spend no time on them at all.

I'm teaching college algebra for the first time in several years, and I'm having to dig deep to remind myself to stop on the definition slides, direct students to carefully copy them down, and reflect about what they're telling us. Quick example this week: the standard form of a linear equation in two variables is $Ax + By = C$. Here's a whole slew of questions I can pose to my students as we reflect on that:

• What operations are used here?
• Are there any exponents or roots?
• What side of the equation is the $x$ on?
• What side of the equation is the $y$ on?
• What side of the equation is the constant on?
• Are there any fractions here?
• (In any examples) Is this a linear equation?
• (In any examples) Is this in standard form?

Here's the thing: My students take in literally none of that information on first sight. They even, as a group, get some of these trivial questions wrong on first inquiry (e.g., after saying the $x$'s are on the left, by symmetry they'll mistakenly say the $y$'s are the right, even while looking directly at the definition). I'm re-familiarizing myself that these structural "forms" communicate absolutely nothing to my population of students; every bit of information has to be explicitly called out and highlighted.

(To say nothing of a whole slew of key facts that the textbook never bothers to explicitly say: e.g., linear equations have lines for graphs; solving an equation isolates a variable on one side; two points identify a line; solving a linear system requires as many equations as variables; etc.)

I think in an ideal world we would model learning definitions and applying them logically by actually spending time in class reflecting on them, and maybe half of our assessments would test for direct recall and identifying example cases -- that is, we'd have a more uniform distribution of time spent on the various levels of Bloom's taxonomy. But most textbooks, curricula, and standardized tests don't provide time or space for that, so most of us are handcuffed to this rather bad model that actively sets an example of skipping over the foundational definitions and principles.

Answer 4

Have you heard the story of Von Neumann and the train problem?

The Train problem is a classic one. You have two trains heading towards each other, one going at 30 km/h the other going at 45 km/h, starting 371 km apart. When they start moving a fly takes off from one train. It flies to the front of the other train, flying at 150 km/h, lands, takes off instantly and flies back to the first. It continues to do so until it is squashed between the two trains.

How far does the fly travel before it is squashed?

(Actual values don't match the traditional version; it was in miles per hour)

The joke is that Von Neumann answered this question instantly. The person asking the puzzle was disappointed, and said "oh no, you knew the trick". Von Neumann answered "what trick?", as he had just generated the infinite series and summed it in a fraction of a second.

(The trick is you can note the trains crash in 371 km/(30+45 km/h) time, and the fly flies at 150 km/h constantly, so the fly travels (150/75) * 371 = 742 km.)

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Using first principles all the time is a bit like always summing the infinite series. It works, but it is often much easier to see patterns and skip on the extra work.

And to the best of my ability to determine, most professional mathematicians don't reason from first principles either.

They have heuristics and patterns that they apply, and they are capable of verifying their logic using first principles logic (and often not even that; they verify using other patterns which they are confident they can reduce recursively to first principles).

A typical professional mathematician doesn't determine if a function is continuous by doing an epsilon-delta test. Given $\sin(x)+\cos(x)$, they will know it is continuous without thinking; and if asked to verify, it is the sum of two continuous functions (again, not epsilon-delta). They can probably generate an epsilon-delta proof that the sum of continuous functions is continuous; they may even be able to generate a proof that sin and cos are continuous.

But very few mathematicians will look at $\sin(x)+\cos(x)$ and generate a first principles proof that it is continuous and use that to answer the question.

We shouldn't expect students learning mathematics to do this either.

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Look at the concepts. Learn some examples. Go back to the concepts, and see how the concepts relate to the examples. Look at more examples.

A feedback loop, instead of just memorizing concepts.

The thing is, the space of proofs is insanely large. The intuition you gain on what the next step in a proof should be is heavily based on pattern-matching.

How do you know how to prove $f(x) = x$ is continuous? Because you have seen that proof before. And you've used the techniques to prove it in a myriad of situations that are similar. You don't even think about those situations - you just see the options of where you can go in your proof, and have a good idea which of those lead closer to your goal.

On top of that, you nearly instantly see the next step after doing a certain step in a proof. When you have your epsilon and delta picked, you instantly see the conclusion. A person more new to this proof has to go and check that the epsilon and delta portions match, and doesn't get it instantly.

What more, if they are even less practiced, they can't keep the entire epsilon and delta part in their head all at once because their brain hasn't learned to chunk it into appropriate pieces. So now they have to literally go back and compare parts of each section to see if they match.

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Coming up with a new proof technique is insanely hard! Piles of mathematics is done using known proof techniques combined in new and interesting ways. Mathematicians learn those techniques from others, and apply them in new areas - and even that is a hard task.

By expecting your students to start with the definition and generate proofs de novo, you are effectively requiring that they come up with new proof techniques. Sure, to you they are old hat - but they haven't seen it enough to be able to see the patterns.

If you have a 5 line proof, and at each step there are 10 different ways to manipulate the previous line, the search space of possible proofs is 100000 large. Knowing which route to take towards the conclusion is something the student isn't going to know.

And in reality, there are way more than 10 ways to go in a proof, and proofs are often longer than 5 lines.

Prove $f(x)=2x$ is continuous.

Choice 1: We pick an epsilon.

Choice 2: We pick a delta. Which delta? Well, we guess. As someone experienced, we probably guess right!

Choice 3: We pick a point $x_0$. Note that I was lucky; $f(x)$ is uniformly continuous. So picking a point afterwards is safe; if I wasn't lucky, my proof fell apart.

Choice 4: I describe the points within delta of $x_0$.

Choice 5: I mention that $f(x)$ is monotonic, so $\min(f(X)) = f(\min(X))$ for a set $X$, and similar for $\max$.

Choice 6: I describe the points reached by the min and max of the points within delta of $x_0$.

Choice 7: I go back and modify my definition of delta (set it equal to half epsilon actually) and show that $f( \text{points within delta of } x_0 )$ are within epsilon of $f(x_0)$.

Without knowing where I am going, finding those 7 steps is simply insanity. We find it easy, because we have done similar proofs thousands of times.

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Don't expect the student to come up with new proof techniques. Instead, teach them proof techniques through examples, and expect them to mimic. Then teach them how to recombine multiple proof techniques. Expect them to mimic the recombining. Do this in a pile of different domains, so the student knows how to use many proof techniques and how to recombine them. Teach them how to steal proof techniques from one domain and use them in another.

As this builds up and if this works, the student will no longer seem to be merely mimicing you.

The worst part of this is the trap of the good student. Often you'll have students who have no problems when you skip steps. This is usually because they have been good mathematician students in the past, and they can chunk things better (so their retention is better), and they even sometimes read ahead (so they have seen the new techniques already).

Between better chunking (lower load) and reading ahead (or having seen advanced techniques in the past and half-remembering them), they will appear to be able to answer your challenges - when you ask them to solve a problem you haven't taught them and they haven't seen before, they come up with a proof!

It is unlikely they came up with the proof technique on the spot. What you are detecting can be is that some of your students know more than you have taught them, not that good students don't have to be taught proof patterns.

Answer 5

I second Justin's comment and want to add to it that if one hasn't been raised with Smullyan's or Carroll's books, one often has trouble comprehending a statement with 2 quantifiers while a typical definition in undergraduate mathematics has 3 to 5 (the definition of continuity at a point $x$ has 3: for every $\varepsilon$ there is a $\delta$ such that for every $y$ that is $\delta$-close to $x$, we have $|f(y)-f(x)|<\epsilon$). It is just something rather incomprehensible for many beginners, so even if they try to reason from the first principles, they need to go over many examples to understand what that first principle really is and even then there is a danger that instead of understanding the principle itself, they'll just memorize the algorithm of the continuity/discontinuity verification (which is something very different). The Greek letters traditionally used in the definition don't help much either unless you are teaching in Greece though they can add to the fun if you understand the concept itself rather well. So, what might possibly compensate for that?

When I was a student, my undergraduate analysis teacher tried to restate that continuity definition in many ways using just the layperson language. I still remember some analogies like a good and a bad shower temperature control handle, feeders and traps: if a muzzle of an animal $y$ enters the feeder ($x-\delta,x+\delta)$, its tail $f(y)$ gets caught in the trap $(f(x)-\varepsilon,f(x)+\varepsilon)$ (this is pretty much how a humane mousetrap like "mice cube" works, BTW, so the $\varepsilon-\delta$ question becomes "how close to the back wall should you place the cheese to make sure that the mouse tail won't prevent the trap door to shut fully?"), etc.. In addition he drew the rectangles in which graph should be contained near the point, and whatnot, just to convey one simple idea that formally is fully described by the standard definition. Note that he was very careful to keep all those layperson language descriptions exactly equivalent to the actual definition (if you interpret them correctly, of course). That was all before we have formally checked the continuity of $1$ and $x$ (which was done in all of those languages too).

I didn't need any of those restatements, so I'm not qualified to judge how efficient they were for the purpose of conveying the message, but what I have derived from it was that you don't understand the concept fully until you can restate what it means in pretty much every conceivable situation using almost arbitrary underlying objects. So I'm still preaching to my students that mathematical thinking is mainly just common sense and you absolutely don't need to talk about numbers, functions, equations or geometric shapes to speak "Mathematiese".

You can try this technique too and see if it works for you. Warning: It takes extra time!

Just my two cents :-)

Answer 6

Because examples are not only easier to think about than abstract concepts, but learning examples is a necessary precursor to learning the concept which generalises them.

A lot of maths educators have observed this from many perspectives, which many other answerers have discussed already. I'd like to add two more perspectives that don't seem to have been covered yet. The first is from Prof. Tim Gowers, who calls it the "examples first" principle:

if a general definition is at all complex, then you will have quite a lot to hold in your head. This can be difficult, but it is much easier if the various aspects of the definition can be related to an example with which you are familiar. Then the words of the definition cease to be free-floating, so to speak, and instead become labels that you can attach to bits of your mental picture of the example.

The second comes from the APOS theory of learning, set out in a 2001 paper by Dubinsky & McDonald:

The theory we present begins with the hypothesis that mathematical knowledge consists in an individual’s tendency to deal with perceived mathematical problem situations by constructing mental actions, processes, and objects and organizing them in schemas to make sense of the situations and solve the problems.

The behaviour described in the question, where a student attempts to solve problems "by looking at solved examples and then mimic[king] the solution to solve other similar examples" is roughly the 'action' level in this theory ─ the student's conception of e.g. continuity is limited to a set of steps to be followed in a set order. Those steps aren't given in the definition of continuity, but the student can learn the steps by seeing examples of the definition being applied.

In the question, it's said that students do this "without necessarily knowing or understanding the concept that is involved". In terms of APOS theory, we would say that there are different levels or stages of understanding the concept of continuity ─ corresponding to different conceptions or mental constructions which the student possesses ─ and a student who is able to reliably follow the steps according to the given examples does understand continuity at the 'action' level. The next goal is to get them to understand continuity at the 'process' level, where they might

• Understand how the steps for checking whether a function is continuous relate to the definition,
• Know why each step is needed (or when certain steps aren't),
• Have a variety of strategies for checking whether different kinds of functions are continuous, e.g. from first principles or by applying a theorem.

Crucially, the theory predicts that students must reach an 'action'-level understanding before a 'process'-level understanding. So it's no surprise that many students who don't demonstrate the latter, may still be able to demonstrate the former.

Answer 7

I think there's an issue of time efficiency. Looking at the examples and working the problems is the most bang for the buck. Of course to have 4.0 ability, you should do both. But people are lazy, have other time demands, etc. Also, prioritization applies to curriculums and advancement paths also.

This is not meant to cut but to make a point. I think the above argument should have already occurred to a teacher. I'm continually amazed how people with such strong intellect and math subject knowledge are so babylike new to the world in terms of pedagogy or even basic human psychology.

Note that many math people tend to think the definitions are sufficient. As if students were calculating machines. But human brains are imperfect computers. They require practice to remember. In some cases need practice to comprehend.

Really one should have some recursion from definitions to problems and back. Because we are imperfect learners. However even here, realize that there is not unlimited time. But also realize that belaboring defintions and side cases too much before doing some drill can lose the class and waste time also.

For those who want immediate feedback in terms of reciting definitions, I would recommend to look at programmed learning textbooks. They tend to introduce a definition snd then immediately test restatement of it by a fill in the blank problem. Very simple snd easy snd immediate. Almost sounds babyish. But works. They seem to do this well and are very time efficient and neophyte accessible. However, they seem to be disdained by the education establishment as illiberal.

Answer 8

I suggest that this is just human nature, and not immaturity as some of the other answers suggest. Most people learn things because they want to do things with the knowledge. So the most natural way to approach a problem is to try to do something with the knowledge immediately.

I would offer as an example the way that computer programmers learn new technology. Every new product will have a “Quick start” guide that allows you do something with the technology right away—e.g., create a simple application. That engages with people's brains, much more than a technical explanation of how the code works would do.

Speaking for myself, I used the Pythagorean Theorem for years successfully, without any concept of why it worked. Eventually I did go back and learn that, and that was intellectually satisfying. But I suspect most people are just content to know that it does work, not how.

(I wonder how many people could give really rigorous definitions of derivatives and integrals—even engineers who use the concepts daily.)

Answer 9

Undergraduate engineering student here. I like math, but at the end of the day my job is to apply it to a situation. I currently have a professor who just talks theory, and I end up trying to jump through a lot of mental hoops trying to figure out why some theorem or definition is important, how I'll need to use it later on, or just trying to figure out what the point of an hour long lecture was. But I agree that just examples aren't a great way of going about things either, the theory is definitely important as well.

Personally I've found what works best for me is to start with a problem, one that has some feature that can't be solved with the current tools I've learned in that class. Then explore why we can't solve it, look at ways we might solve, and then generalize that definition and eventually find a theorem or proof that can be applied to any situation (and then apply it!). In this way, I've been given a reason to care about the theorem, and know why it's important, while still learning the theorems and then being able to apply them to problems I haven't seen before.

To go back to the continuity example, where the students ever given a reason to care about continuity? Were they given a reason as to why continuity is important? Were they given any examples where ignoring discontinuity would lead to incorrect results? Do they have a reason to believe it has actual applications and isn't just a cheap party trick?

If I just wanted a random stream of theorems and proofs, the internet has been able to do a pretty good job of that for at least the past decade, and textbooks for at least the past couple centuries. The way I view the job of the teacher is to guide me through that stream of human knowledge, to give me a reason why I should care about certain things and to give meaning and reason to a curriculum greater than the individual proofs and theorems it contains.

I don't think it's that the students don't care about theory, I think they might have just never been given a reason to care. Hope that helps :)

Answer 10

In fact this is well studied in the literature and it has to do with concept formation. There are two ways we do this: one is by collecting exemplars and doing a sort of nearest neighbor search to see if we know an example we've seen to help with the new input. This is why you see students making bonkers assumptions they trying to tell you "I thought it was X" when it clearly isn't. A good example of this was from my graduate studies days. It instructor jokingly said she would give us "proof by special example", by which she meant "here is an example, if you want the proof is longer than the time we have". However a student in the class later on, when asked to do a proof, asked "can we use proof by special example here?"

The other way we gain concepts is by abstracting those examples into something more like what you are looking for. However that only really happens if the concept is very simple or you've seen lots of examples.

You can see this when experts solve problems. They essentially "prune the tree", ignoring irrelevant knowledge and examples to figure out a smaller subset of things that need to be considered.

So your students are doing exactly what they need to do: all they have to go on is what they've seen so far, so the best they can do is relate a new problem to an old one and hope for the best. The concepts come later through lots of exposure to examples and discussion, but it almost never happens that we get a concept first then magically can use it.

Answer 11

One thing helping me in teaching is homework like: 1. Write down the definition.... apply it to prove the following examples. 2. the announcement: In the next test I will ask not only for proofs or examples but also for the following definitions.

Answer 12

1. One name that is used in the literature is "generalizations".

2. Students (people) naturally use a number of different heuristics to solve problems and make decisions. Reasoning from examples is one of them.

The problem of lack of generalization is both natural and taught. You can teach your students to just match examples, and you can teach your students is such a way that they just match examples, and they can naturally just match examples.

But teaching and learning efficiency is demonstrated by teaching "from the concrete to the abstract" -- normal people are much more efficient at learning abstract concepts when they are grounded in previous concrete examples. An attempt to 'teach the concepts first' is sometimes successful, but contrary to research and experience.

On the one hand, a common problem for non-teachers is that, once they learn the concepts, the concrete examples just get in the way, and they want to teach for optimum post-learning understanding, rather than for optimum acquisition.

On the other hand, just teaching examples may never lead to useful generalisation.

Students have to learn concepts. You have to teach concepts. Concrete examples are part of the process, but not the end point.

Answer 13

The quick answer is that children's brains are still growing and until their early 20s their brains don't function like adults.

The longer answer is that yes this has been studied, and you will mostly find the answers in psychological literature, education literature, and more recently neuroscience literature. In psychological literature, district period of brain growth were hypothesized almost a century ago, and have, during the past few decades, been verified through research. Conceptual understanding and task completion go hand in hand with respect forming "knowledge." For elementary school students and even into middle school consider that they don't even have a real concept of time. When an event occurred in the past they will say things happened variously like "yesterday" or "a week ago" without any hesitation. They aren't making things up or out and out lying- their brains haven't finished the process to fully develop logic. If you search the literature for structural brain development and mathematics education you will find a multitude of studies on this topic.

Even when they learn definitions and can connect the skill, they don't really understand in the sense that "continuous" is both a time concept as well as a symbolic concept.

At this point many people have asked me: then why try to teach concepts? The answer is that you are building the logic structure with mathematics. Fundamentally, there are 4 very different periods of brain development marked by an explosion of neuron growth and then a pruning of the least efficient neuronal pathways. Mathematics in early and middle childhood is really for teaching kids how to think, with the what being peripheral. This is why the is a big push to stop teaching algebra before the 9th grade- because for the average learner, the ability to think symbolically is just started to emerge.

I think that's likely enough info for now, and I think it's great you both noticed this and are looking into it. I think child brain development should be an essential emphasis for teachers- less frustrating for us and the students. One more thing: if you have a younger student that can grasp concepts that other kids can't, then you are likely looking at a student with a higher IQ. It's not a certainly, but it's certainly a big indicator asking with the quickness at which they "get" the skill you are teaching. These are also the kids that will tentatively grasp conceptual information in general- and those kids may sometimes need to know the "why" along with the "how". For the bulk of the students, however, the "why" will only be parroted until about 9th grade. The great part is that at that point the concepts will be familiar and that makes it easier for the brain to assimilate and accommodate. If you want more in depth, check out the bodies of literature I discussed above and look up Jean Piaget as well. His work is seminal and laid the foundation for our understanding of childhood brain development. Good luck!

Answer 14

Learning is neither learning by analogy or learning the definitions.

It is by being exposed to both examples and definitions and then re exposed to both that we come to learn what the definition means.

Answer 15

Yes, it's natural. Action is prior to conception phylogenetically, ontogenetically and logically, even if entwined temporally.

You might class things learned into concepts and skills. A skill is a way of acting in the world, at some level of competence (systematic chance at success), vis-a-vis certain objects and goals. Those goals are most often 'tangible' in the sense of not open-ended, even if the object realm is abstract.

Skill is learned through guided repetition. Learning anything entails learning skills, both regarding learning itself and items figuring in it, and the subject matter. At the same time skill development is likely to require some mental structuring by outwardly presented formal or folk concepts or ones the learner makes up.

When we humans talk about 'concepts' we mean something capable of expression and transmission by language, something only we can have. Yet many other animals' action is situated and adaptable, and some can be observed cogitating on a challenge (recognized by their going straight through a successful procedure afterwards). Now, the nature of animal conceptualizing is still unclear and there are those who dislike the idea. But the spectrum of specificity and flexibility in various animals' action together with human developmental psychology make palpable that structuring and acting on perceptions is prior to structuring and acting on notions.

Sorry I'm not a scholar of these things, but I believe none of the above is controversial.

Answer 16

The answer to your question is teachers want to rush the learning and associate student struggle with laziness. Students have to spend a sufficient amount of time in the knowledge phase. They need time with vocabulary and opportunities to associate the vocabulary with their acquired skills.

If teachers move too fast then student learning is compromised. Some students will be ready to take a deeper after a few days…some students may need longer. Respect the learning process and your kids will begin performing at higher levels.

If you teach where you wants kids to be, then many of your students will never get there, or will take them much longer than needed. This is my 2 cents🤷🏽‍♂️

Answer 17

Apologies up front, I don't have references to the studies I'm about to mention.

Psychologists have studied how the brain processes language and math, and found that language and math are processed in two different places in the human brain.

Language is processed like a picture. As children learn language, they learn the sounds of words first, then the letters, the the individual words, followed by sentences. The reason why children progress through this system and increase their speed is that their brains are memorizing patterns of how the words look and matching them to the sounds for the words they heard. Inititally, each letter was a picture, but after 1000's of letters, the child can pick out the letter A no matter how stressed it is. The same pattern follows with words, like CAT, and eventually books containing stories about CATS.

When it comes to Math, which is processed in a different region of the brain, the language system passes the picture to the math system and say process this. Here's where things get interesting. The math portion of the brain has to break the picture back into it's constituent parts "1 + 1 = 2", and then it has to ask the question .. Is this correct? .. The Math portion of the brain is a difference engine constantly testing the accuracy of the mathematical pictures it's given.

The effect that this has on children / humans in general, is that we never know if the mathematical statement we are looking at is true or false until we have enough examples under our belt to say it's true. This is why mathematical statements like "Continuous Functions" don't immediately register. The students see the definition, read it, understand what it's saying, but can't apply it, because they have no mental picture that says .. true or false. They have to build that picture by seeing 100's of examples and working through 100s of problems specifically tailored to that concept.

Now, consider the problem for someone with Dyscalculia. They see "1 + 1 = 2", "2 - 1 = 1", "1 - 2 = -1", "1 + 1 -2 = 0" and are asked, are all these statements identical? The answer is .. NO! .., but not for the reason you are thinking. The answer is .. NO! .. because they all look different. As a picture, they are all different. To be processes mathematically, the brain has to break the picture down, pass it to the math processor and ask the question, are these all mathematically identical? Then, and only after having seen these problems for 100's of times, does the brain maybe say .. Yes! ..

Remember, as a difference engine, if garbage data gets in, it will muddle the responses. The student may be recalling instances where the answer was not correct. Or, were told that the answer was not correct. These bits of garbage have to be filtered out through careful retraining. This can be esp frustrating to students who don't trust their notes, and don't have access to resources to check their notes, home work, understanding, .. Which is why I turn to sites like Slader.com to work through problems, at least I can see how someone else solved the problem, reason as to if the problem was solved correctly, and then see if I get the correct answer.

So, when considering teaching a topic, look for as many ways to cement the concept into the student as possible ..

Discuss a specific topic

• use 100's of Examples
• use Text
• use Video
• use Interactive Demos
• use Pictures / Diagrams
• use Graphs
• use Practice Problems
• use Spaced Recall
• use Physical Experiments
• check Terminology
• review Strategy

The focus is to build the students confidence in the specific topic

For higher educational classes .. University Classes .. Have the students

• identify which section of the book discusses this problem
• write their strategy before solving the problem.
• write all the constants needed for the problem
• identify any equations that seem relevant to the problem
• draw graphs and diagrams .. I do this for sine and cosine ..
• transform equations before using them (solve for x)
• do gap analysis -- what variables are unidentified? -- do the units match up? -- how is the correct answer different from their answer? -- does a mathematical transform make sense?

When I look at problems for Physics, Math, and Engineering, I always ask, how close is the problem in pages to the content that describes the strategy to solve the problem. This distance is one of many factors I use to determine the difficulty in solving the problem, as it indicates how much effort will be involved to remember / recall where the problem was discussed but what else is built on top of this discussion.

Another problem that I'm seeing is how the question is worded. Some questions introduce new terminology that is not discussed in the text. Some questions attempt to teach the student something not discussed in the text. This confuses the student / me as I'm simply looking for the question .. Solve for X .. Instead I get a page and a half description about various components of the problem to be solved, and no real clear definition of what I'm solving for. Or, worse, the question doesn't seem to match what their expectations of the answer are at all.

Example: Prove that (E) must be real by replacing (E) with (Eo + iT), and show that for Eq 1.20 to be true for all (t), (T) must be zero.

Another thing to consider that confuses Dyslexics, is that common letters in math look the same or are distinctly different .. w and W .. are the same when drawn on a board with no other reference points to discern them. Where as Δx, dx, and ∂x will all mean the same thing until 100's of examples differentiate them.

Hopefully this helps inform your instructional practices and gives you ideas on how to approach learning gaps / deficits in math.

One Educational Psychologist used the following terms .. Encoding, Generation, Gap Analysis, Recall ..