I wanted to add one further point to the many good answers already given here: "black box" symbolic computation, in the absence of understanding the formal definitions, can work when everything goes right, but is very unstable with respect to student errors (which are sadly all too common). Knowledge of definitions provides a crucial extra layer of defence against such errors.

For instance, without knowing the formal definition of the derivative, a student could very easily start with a true formula such as

$$ (x^2)' = 2x$$

and do something like "substitute x=3" to obtain the false formula

$$ (9)' = 6.$$

(An example that I have actually seen: someone attempted to prove Fermat's last theorem by starting with the equation

$$ a^n + b^n = c^n$$

and then differentiating with respect to $n$. Ironically, a variant of this type of trick actually works when solving FLT over polynomial rings, but that's another story...)

Now, without bringing in the definition of a derivative (and of a function), how could you explain to the student what went wrong here in a way that the student will actually remember? Saying that one can use the law of substitution or the trick of differentiating both sides in some situations, but not in others, is likely to be recalled inaccurately, if at all (and may have the side effect that the student may view such basic moves as substitution as somehow being "suspect", thus avoiding it in the future).

I wanted to add one further point to the many good answers already given here: "black box" symbolic computation, in the absence of understanding the formal definitions, can work when everything goes right, but is very unstable with respect to student errors (which are sadly all too common). Knowledge of definitions provides a crucial extra layer of defence against such errors. (Of course, it is not the only such layer; for instance, good mathematical or physical intuition and conceptual understanding are also very important layers of defence, as is knowledge of key examples. But it is a key layer in situations which are too foreign, complicated, or subtle for intuition or experience to be a good guide.)

For instance, without knowing the formal definition of the derivative, a student could very easily start with a true formula such as

$$ (x^2)' = 2x$$

and do something like "substitute x=3" to obtain the false formula

$$ (9)' = 6.$$

(An example that I have actually seen: someone attempted to prove Fermat's last theorem by starting with the equation

$$ a^n + b^n = c^n$$

and then differentiating with respect to $n$. Ironically, a variant of this type of trick actually works when solving FLT over polynomial rings, but that's another story...)

Now, without bringing in the definition of a derivative (and of a function), how could you explain to the student what went wrong here in a way that the student will actually remember? Saying that one can use the law of substitution or the trick of differentiating both sides in some situations, but not in others, is likely to be recalled inaccurately, if at all (and may have the side effect that the student may view such basic moves as substitution as somehow being "suspect", thus avoiding it in the future).

I wanted to add one further point to the many good answers already given here: "black box" symbolic computation, in the absence of understanding the formal definitions, can work when everything goes right, but is very unstable with respect to student errors (which are sadly all too common).

For instance, without knowing the formal definition of the derivative, a student could very easily start with a true formula such as

$$ (x^2)' = 2x$$

and do something like "substitute x=3" to obtain the false formula

$$ (9)' = 6.$$

(An example that I have actually seen: someone attempted to prove Fermat's last theorem by starting with the equation

$$ a^n + b^n = c^n$$

and then differentiating with respect to $n$. Ironically, a variant of this type of trick actually works when solving FLT over polynomial rings, but that's another story...)

Now, without bringing in the definition of a derivative (and of a function), how could you explain to the student what went wrong here in a way that the student will actually remember? Saying that one can use the law of substitution or the trick of differentiating both sides in some situations, but not in others, is likely to be recalled inaccurately, if at all (and may have the side effect that the student may view such basic moves as substitution as somehow being "suspect", thus avoiding it in the future).

I wanted to add one further point to the many good answers already given here: "black box" symbolic computation, in the absence of understanding the formal definitions, can work when everything goes right, but is very unstable with respect to student errors (which are sadly all too common). Knowledge of definitions provides a crucial extra layer of defence against such errors.

For instance, without knowing the formal definition of the derivative, a student could very easily start with a true formula such as

$$ (x^2)' = 2x$$

and do something like "substitute x=3" to obtain the false formula

$$ (9)' = 6.$$

(An example that I have actually seen: someone attempted to prove Fermat's last theorem by starting with the equation

$$ a^n + b^n = c^n$$

and then differentiating with respect to $n$. Ironically, a variant of this type of trick actually works when solving FLT over polynomial rings, but that's another story...)

Now, without bringing in the definition of a derivative (and of a function), how could you explain to the student what went wrong here in a way that the student will actually remember? Saying that one can use the law of substitution or the trick of differentiating both sides in some situations, but not in others, is likely to be recalled inaccurately, if at all (and may have the side effect that the student may view such basic moves as substitution as somehow being "suspect", thus avoiding it in the future).