How much symbolic calculations before plugging in actual values?

Question

My son is in high school (France, 2^nde) and I was watching how he solves math exercices. This led me to the following question: when are students expected to plug in actual values in their calculations?

As a background, I am a physicist by education and was always of the opinion that the symbolic calculations should be dragged as far as possible. The actual numbers are used at the very end and it is not the most important part.

It seems that the way math is taught today is different.

I will take a concrete example:

We have three points $M(7;-2)$, $N(0;t)$, $P(3;1)$. Find $t$ so that they are aligned.

I would have done it by generalizing the points ($M(x_M;y_M)$, $N(x_N;y_N)$, ...) and finding co-linear vectors with the calculations being done on those generalized points. It is only at the very end, having a general formula, that I would have used the actual values of $M(7;-2)$, etc.

My son is telling me that they do the calculations directly.

Which approach is expected at a high school level? The example below is a simple "training" exercise, but when the problems get longer I see that he is still using the numbers early on, without any generalization.

Answer 1

Calculate when you want an answer. Solve algebraically when you seek patterns.

If the problem was "show that for any two points, you can find a third point along the y-axis that is collinear with them," then symbolic logic is the right way to go. But if you have the points, just plugging them in simplifies the problem dramatically and makes it easier to visualize by drawing a picture. Drawing a picture of the generic case is hard.

Doing calculations opens up the door to simplifications that may not have been there in the symbolic algebra. There may be some very ugly $ab^2\ln(\sin c)$ term that causes so much ugliness, but was really nothing more than a combination of constants that equals 2. There may be shortcuts here. Sometimes the numbers provide very clean relationships, such as when two numbers turn out to be pretty much a multiple of 2 pi from each other. You can leverage symmetries that only appear in the special cases.

There may also be shortcuts because you can avoid pathological cases which make the math monsterous. If you'll note, in my first paragraph, I suggested somethign to show which was false. I didn't include any wording to account for there being a vertical line between the two points that never crossed the y axis. Nor did I consider the case where both points are on the y axis, such that every third point qualified.

You have to be careful. Consider this famous "proof":

• let a and b be equal nonzero values: $a = b$
• multiply by a: $a^2 = ab$
• subtract $b^2$: $a^2 - b^2 = ab - b^2$
• factor: $(a + b)(a - b) = b(a - b)$
• divide by $a-b$: $a + b = b$
• Substitute b for a, as they are equal: $b + b = b$
• Combine: $2b = b$
• Divide by b: $2 = 1$ QED

This is a rather famous false proof. One has to remember that division by $a-b$ is only defined if $a-b$ is nonzero, and in this construction $a-b$ is always zero. If we used numbers:

• Let a and b both equal 3: $3 = 3$
• Multiply by a (which is 3): $9 = 9$
• subtract $b^2$ (which is 9): $0 = 0$
• factoring is a bit irrelevant with the individual numbers
• Divide by $a - b$ (which is 0): $\frac{0}{0}=\frac{0}{0}$

Wow, we found that problem fast!

The way we do math has changed over the years, due to the quantity of data that is available and the amount of processing power we have available. The focus has indeed drifted away from the symbolic manipulation. That mirrors the way everybody does math in real life these days, with calculators literally a tap away.

Myself, I'm not a physicist. I'm an engineer. We'll take the symbolic logic just far enough, and then plug it into a simulation. Why? You really don't want to try to solve some of our equations these days by hand. In fact, most interesting ones are recognized to not have a closed form solution.

The approach I advocate today is to bring in symbology when you want to use it as a shortcut. For example, consider the finding of roots of quadratic equations. Teach students to complete the square, with numbers. Teach them to do this over and over. When they get tired of it, then you point out that you can put on your symbolic algebra hat, sharpen your pencil (no mistakes!), and prove that you can solve these problems by deriving the quadratic equation once and then merely plugging in values from there!

More than once, while Engineering, I relied on symbolic manipulation to show that some pathological case we were spending lots of money on can never possibly occur because the math shows it can never occur. I did that because it was the shortcut.

This approach is consistent with a story I was told from how they did apprenticeships in German machine shops. Obviously the Germans are known worldwide for their precise machining with powerful tools that drive tolerances as close to 0 as one can get. When an apprentice is brought into one of these shops, they are handed a length of square bar stock. They are handed a file and told "make it circular." And the plug away at the problem, learning how to make a circle. Then they are invited to learn what circular really means, showing them the ridiculous tolerances one may have, and hand techniques that can be done to make them circular. Then, once they have a circular bar that passes standards, they are told to make it square. And they file away for a bit, before they are taught what square really means. Only then are they introduced to the lathe and the mill, which are machine tools designed to solve circularness and squareness for all sorts of circumstances.

Answer 2

Speaking from an American perspective, your son's approach strikes me as much more natural. For instance, to solve your problem

We have three points $M(7;-2)$, $N(0;t)$, $P(3;1)$. Find $t$ so that they are aligned.

my students would start by observing that the slope (gradient) of $\overline{MN}$ would have to be the same as that of $\overline{MP}$ and from there go straight to $$\frac{t-(-2)}{0-7}=\frac{1-(-2)}{3-7}\\\frac{t+2}{-7}=\frac{3}{-4}\\-4(t+2)=-21$$ and finish solving that equation to find the value of $t$. By contrast, your approach would look more like

$$\frac{y_N-y_M}{x_N-x_M}=\frac{y_P-y_N}{x_P-x_N}\\(y_N-y_M)(x_P-x_N)=(y_P-y_N)(x_N-x_M)$$ which has us multiplying out two pairs of binomials and then rearranging the subsequent eight terms in an effort to solve for $y_N$ in terms of the other five variables and then turning it from there into a relatively complex arithmetic problem. Not only is it easier to multiply numbers than bimonials, it is far easier to avoid or at least spot the simple arithmetic or bookkeeping errors.

---

This was driven home to me in my calculus class a few weeks ago. We were working on a problem along the lines of

Find the slope of the tangent to the curve $x^3+y^3=9xy$ at the point $(4,2)$.

So the first step is to implicitly differentiate both sides with respect to $x$ to get $$3x^2+3y^2y'=9xy'+9y$$ My expectation was then that students would do some algebra to get to $$y'=\frac{9y-3x^2}{3y^2-9x}=\frac{3y-x^2}{y^2-3x}$$ and then plug in the variables. But a student piped up and suggested that it was much less effort if you substitute in $x$ and $y$ immediately after the differentiation, and he was absolutely right.

---

This is not to say that there isn't an important place for the strategy you describe. We wouldn't have valuable tools like the quadratic formula or Heron's formula if we didn't stop and say "Okay, just for once let's solve this problem where all of the values are arbitrary variables instead of specific numbers." But I always have to really slow down when I do those derivations to make sure that I'm not making mistakes along the way.

Answer 3

I don't think this is a nationalistic difference (I'm in the US), but I also don't think your example is optimal. As an example where the correct technique is more well defined, let's say we have a physics problem like this:

A bug starts from rest and accelerates with constant acceleration for 0.53 s, traveling 1.37 m. Find the bug's acceleration.

I would consider it wrong wrong wrong to solve this by first writing down the equation $x=(1/2)at^2$, then plugging in numbers, then solving for $a$. As you say, a competent person will do the algebra first and then plug the numbers in at the last step. Below is a section from my syllabus for a physics course where I present this kind of thing, including this example.

The unfortunate reality today is that many physics instructors use computerized grading of homework, and they implement this in such a way that they never actually see their students' written work. Therefore their students never get feedback on this kind of thing.

Answer 4

You can't generalize a single, isolated question, which is what your high school question about collinear points is.

Of course you can invent a set of similar problems, create a general solution for the whole set, and then solve the particular problem given - but why do all that unnecessary work? If your child's homework set contains several similar questions, would you expect them to create the same (or slightly different) general solutions for each one?

This is the same as a general principle in computer software design, known as YAGNI - If "you ain't gonna need it," don't waste your time dealing with it.

The situation is different if you already have a general tool that you can use. For example, if you already know the formula for solving a general quadratic, plugging in the known values to solve the equation in front of you may be quicker than completing the square, or factorizing the equation by inspection - or it may not, depending both on the equation and on the individual student.

At university level, it is much more likely that students do know (or should know!) some general tools that are relevant - Newton's laws of motion, Lagrangian or Hamiltonian dynamics, conservation laws, special or general relativity, calculus of variations, or whatever. So it is more likely that it is worth carrying a "symbolic" solution further, especially if the question is about getting insight into the general way the system behaves for a range of parameter values, not just a numerical value for one set of conditions.

At any level, there is a trade off between pure computation and mathematical theory. For example, understanding the solution of a set of linear equations using determinants and Cramer's rule can give valuable theoretical insight, but calculating a numerical solution to any system of order > 2 by Cramer's rule is the wrong way to do it, because it is less efficient than simple elimination. And the apparently smart idea of calculating the solution of a system of equations by writing it in matrix form and explicitly inverting the matrix numerically is also the wrong way, because factorizing the matrix numerically is almost always more efficient and more numerically stable than inverting it - though sadly, I have met dozens of PhD level students who think they are experts using MATLAB or Mathematica, but have never learned those basic facts of numerical life!

"If the only tool you have is a hammer, every problem looks like a nail." That is not what students should be learning!

Answer 5

I think the difference between the two approaches is the goals. Your solution is exactly what I would do, but I'm an engineer.

In high school, the goal of learning is usually more about understanding the general concept and gaining some practice with the mechanics. I would think that plugging in the values early on is "easier" to grasp the ideas than solving algebraicaly for the general case.

However in first year university the goals shift and you're no longer expected to just loosely grasp the ideas but you are expected to have a deep understanding. At that stage I would say that solving for the general case would be more expected.

In retrospect after years of having done similar problems in the general case solving the singular case becomes a trivial matter, but you (and I) have years of experience doing this. In high school they may have only solved a handful of problems.

When you're first introduced to a topic many like to see it work, algebra tends to abstract the result away. It's not until you've gained som experience and practice that you really see the elegance in it. Solving the problem numerically the first few times demonstrates the concept, then the next step could be for the teacher to show how the process can be abstracted into a general form.

A really good teacher (in my opinion) would show first how the problem might come up in real world use (showing that there is an application). Then let the students struggle with solving it on their own, not showing them the algebra but allowing them to try it on their own. For a practical problem that you only need once, numerical solutions are typically the go to first step. Then after they have struggled and gotten familiar, showing them that if you take a step back from the specific case and look at it generally you can solve it more elegantly.

I consider this to be a better approach as it allows the students to see why they should care and then why it's worth while to not rush into a problem. But also allows them to both struggle on their own and then also guides them to a general and more elegant solution. The process can then be applied to many different subject areas. Math, physics, engineering, programming etc.

Answer 6

My own experience in math classes (as a student in Germany) and tutoring my peers would lead me to the following conclusion regarding your question:

Many students have problems with the generalized formulas, because they find it rather unintuitive to calculate with "letters" rather than numbers and they would frequently ask for real world examples and applications for these mathematical problems. They had an easier time following examples with real numbers/values/problems. I can totally see where you are coming from and I would say that there is much value in teaching the generalized formulas, but they were very daunting to most of my peers and did throw many of them off of maths.

I think the main job of highschool maths is to get the students interested in solving these kinds of problems, seeing the value of learning maths (which in my experience was pretty low among my peers) and equipping them with the tools to solve more complicated problems, though the extend of the last point was always debated among the teachers at my school, because it was very easy to scare people off that were not interested in the first place or to just lose their attention during class.

So it is a fine balance for our education system between stimulating the interest of pupils and giving them in-depth/generalized solutions to those kinds of problems I'd say. And more in-depth courses for pupils that are either very apt at maths or just interested in learning more should always be an option.

But for tutoring I always seemed to have to most success in starting off with real world examples/real numbers and providing the generalized formulas afterwards, most of my "students" got the gist of it with this approach.

Also Diesel has a good point that coming from real world examples and letting pupils figure the more general approaches/formulas out themselves will give them the chance to develop analytical skills on their own, which can be very gratifying and motivation plays a big part in learning.