# Should we teach simple content quickly or slowly?

The title might be a bit not specific, so let me give an example.

In China, Japan, Korea, etc, there is a type of problem about chickens (or crane, or anything with two legs) and rabbits (4 legs) in the same cage:

Some chickens and rabbits are in the same cage. There are a total of $$10$$ heads and a total of $$28$$ legs. How many chickens are there and how many rabbits?

This is the same as solving the system $$\begin{cases} x+y=10, \\ 2x+4y=28. \end{cases}$$

Now, we can teach this in the following two ways:

1. Teach how to solve a system of linear equations first, and write down the equations for this problem. Solve the equation.
2. Spend some time with trial and error, before teaching the equations.
3. Teach a method with does NOT involve any algebra. Example: assume all of them are chickens, then we get $$28-2\times 10=8$$ extra legs, which means there are $$4$$ rabbits.

Method 1 is the most efficient way, in the sense that it doesn't waste any time, and gives students the maximum amount of content. However, some argue that 2 or 3 is better.

Those who support 2 argue that it helps students understand why the equations should work and that it could help develop the idea of deduction (Modus ponens, Gen). However, those reasons are farfetched. It is hard to come up with the idea of equations after doing trial and error for days. Mathematicians would also agree that this problem is about calculations rather than deduction and proofs.

Reasons to support 3 is more simple - it would be nice and easy to avoid teaching new concepts like equations.

Back to the title: should we teach this problem in fast ways (1 and 3) or the slow way (2)?

• Again, as with other questions of yours, I do not believe there is only one answer to your question here.* One size does not fit all in math ed, and to try to make it so ends up doing some topics and some students a disfavor. A better question, I think, would be to ask when, in what circumstances, and with what level of student, each approach may be a viable option, perhaps? – amWhy Dec 26 '19 at 14:06
• Did you mean to write "following three ways?" – Bladewood Dec 26 '19 at 16:43
• Method 3 is arithmetic, method 1 is algebra, method 2 is a non-method and should never be used. – Rusty Core Dec 27 '19 at 0:12
• Is "simple contents" supposed to be "simple concepts"? – Barmar Dec 27 '19 at 14:30
• @Rusty Trial and error to find curious patterns is a common way for mathematicians to investigate hard problems. – Voo Dec 27 '19 at 14:59

(In this answer I'm taking 2 to mean that the students spend some time on trial and error, trying to work out how to solve the equations before they're told the recipe.)

I think for many secondary school teachers there isn't enough time to choose 2.

Personally, I would never choose 3. It teaches them a trick to solve a single question, but doesn't develop their mathematical ability. Some similar style might be applicable when preparing students for maths competitions. Or letting the students find that for themselves would help them develop their problem solving skills. But telling them outright would, I believe, never be a good choice for the majority.

1 is probably the standard choice for most schools. It has the advantage of working relatively well, and just about fitting into the time available. It's also what the majority of students are likely to want.

But by preference I try to use 2. I believe it helps students to understand why the methods work, and that makes it easier to remember what to do, and potentially to adapt the method yourself to new contexts, rather than being reliant on the algorithm taught. Also, I believe that it would give students a better understanding of what maths is. Most leave school thinking that maths in about running a fixed algorithm to get a number. But that's not really maths, it's being a computer. The experimenting yourself and trying to find a solution is more what maths should be about.

As I said, most school teachers don't have the time to make this decision, or at least don't think they do (some people have suggested that actually building the students' understanding buys back the time later). But this is a real tension in designing a university level curriculum, where there is more freedom to choose how much content gets covered. Personally I believe very much in keeping the content low enough that there is time to develop the students' understanding and problem-solving skills. There are many people who advocate including lots of content. I think in many cases these people don't realise how little of what gets 'taught' actually makes it into the students' heads.

If the goal of this problem is to introduce students to systems of equations, I would avoid introducing notation and computation without first building up to it. One of the biggest problems that I have with my (first year college) students is that they are very good at symbol manipulation, but they have no idea what the symbols mean or why they are manipulating them. Students need a much stronger basic understanding of why they are doing the things that they are doing. That being said, I don't think that the three alternatives given in the question are the only possible alternatives.

It seems to me that the main idea is to somehow capture the relationship between rabbits, chickens, heads, and feet. As such, a good way to start is with the inverse problem: I tell you how many rabbits and chickens I have, and you tell me how many feet and heads there are. With small numbers, this isn't too difficult (you can count on your fingers and toes!), but if there are 103 chickens and 83 rabbits, it is a bit harder.

The idea is to get students to the point of understanding that there is a relationship between these variables: \begin{align} \text{heads} &= \text{rabbits} + \text{chickens} \\ \text{feet} &= (4\times\text{rabbits}) + (2\times\text{chickens}). \end{align} Note that there is also a secondary objective here: get students to grok the idea of a function of two variables (of course, to a midddle- or high-school audience, we might not use the notation and language of functions, but the idea is still there).

Before actually asking them to solve the problem, I would introduce other similar kinds of setups (A theater sells tickets on the ground floor for \$80 and balcony tickets for \$55 dollars; how much money do they make if they sell $$m$$ ground floor and $$n$$ balcony tickets? Hot dogs are sold in packages of 12, while hot dog buns are sold in packages of 8; if you buy 4 packages of hot dogs and 6 packages of buns, how many of each (dogs and buns) do you have? Follow-up question: how many hot dogs can you serve (you need a bun for each hot dog)?).

Once they are comfortable answering these kinds of questions (which, in an early secondary setting, should take one or two 50 minute lessons, with some homework for extra practice), we can start talking about solving systems. I would let the students struggle to figure out how to solve the problems on their own before introducing too much notation.

Finally, after some struggle, we can abstract away the references to rabbits and hot dog buns, and introduce notation. For example, in the original problem, we have \begin{align} &\text{heads} = \text{rabbits} + \text{chickens} &\implies H &= R + C \\ &\text{feet} = (4\times\text{rabbits}) + (2\times\text{chickens}) &\implies F &= 4R + 2C \\ &\text{there are 10 heads} &\implies H &= 10 \\ &\text{there are 20 feet} &\implies F &= 28. \end{align} If the students have determined general methods for solving these problems, we can help them to see how the notation enables this and makes it faster. If they haven't, then this is a good time to introduce general methods.

• As a teacher of both physics and maths I think you are missing some very important "number of"s in your equations. You just can't add rabbits and chickens, the "units" make no sense at all. You can however add the number of rabbits to the number of chickens to get the number of heads. – Jasper Dec 27 '19 at 17:09
• @Jasper I am thinking of "rabbits" and "chickens" as variables (I even say so before using the notation for the first time), not units. It is an abbreviated and informal notation which is not meant to be rigorous. Another approach here would be to treat "rabbits" as a unit, rather than a variable---this is likely even pedagogically appropriate to the level of a middle- or high-school algebra student, and could lead to an interesting discussion of unit analysis. However, this is not how I would structure the lesson in my own teaching. – Xander Henderson Dec 27 '19 at 17:15
• @Jasper Your comment, I think, could be expanded into another answer---I would be interested to see how you would structure the lesson in greater detail. – Xander Henderson Dec 27 '19 at 17:16

I largely agree with Xander, but wanted to add my own lesson/unit plan.

First off, solving two equations in two unknowns is not a simple concept and is worth devoting time so that students understand how it works and how to apply it in situations where they aren't just handed two equations to solve.

So, yeah, start off with some inquiry. Since my students were really well-behaved in the last unit and deserve a treat, I'll even give them some modeling clay and tiny pegs to play with. So they'll be building people (figures with two pegs for legs) and horses (figures with four pegs per legs). (I'm switching from chickens and rabbits to people and horses because I think they're easier to model with clay.) I'll divide the class into groups -- half of the groups are trying to find all the possible ways to create a scene with 10 heads and the other half are trying to find all the possible scenes with 28 legs. Fortunately, the unit before solving systems of equations is graphing linear equations, so each team has to create a chart and graph expressing their findings. When they're done, we superimpose the two graphs.

This gives us the opportunity to reinforce the fundamental concept of analytic geometry (which fortunately is another standard we needed to cover this year): the solution set of an equations in two unknowns can be graphed and there is a correspondence between points on that graph in our domain (in this case, just non-negative integers) and individual solutions to the equation. In this case, we get to add the new lesson that the intersection points of two different graphs correspond to solutions of the system of equations.

From here, we can leverage this problem throughout the rest of the unit to discuss strategies for solving systems of equations if we didn't have modeling clay or graph paper. The substitution method can be introduced by noting that the equations of the two lines in slope-intercept form are $$H=-P+10$$ and $$H=-\frac 12P+28$$ and we can equate the two RHS and solve for $$P$$ to find the common solution. And, with any luck, one of the students came up with your method 3 in the initial lesson, which is noting that we can subtract $$2H+2P=20$$ from $$4H+2P=28$$ to get $$2H=8$$, which is the elimination method when you express it in algebraic language.

Reasons to support 3 is more simple - it would be nice and easy to avoid teaching new concepts like equations.

On the contrary, "teaching new concepts like equations" is our job. If we don't train introductory algebra students to think about systems of equations because simple problems can be solved with arithmetic, then what will they do next year when the third variable is introduced? No, method 3 is attractive because it allows us to observe that our intuitive strategies can be expressed in the algebraic language in ways that will continue to work when our intuition fails us.

• Given how much emphasis I put on geometric understanding in my own classes, I am kind of sad that I didn't think of this aspect of the problem initially. (+1) – Xander Henderson Dec 27 '19 at 17:53

Well, for me, it depends on what you are aiming to give to your students with this problem. That is, are you using this as an introduction to linear systems? Then, for sure go for a more in depth treatment that lets the students see the necessity of introducing linear systems - however, this problem may be just a part of this introduction.

If, on the other hand, you aim just to present this problem to your class, then you may want to let your students find their own way of solving the problem. In general, problem solving aims to allow students explore their own capabilities in finding ways to solve problems, so, from my perspective, there's no reason to restrict them to just one way - always depending on their age and previous knowledge.

Summing up, it's a matter of what your goals are as a mathematics instructor alognside with the level of the students you have in your class.

As a non-mathematician and not a teacher (but parent of two children,) I'd like to suggest you avoid "trial and error."

German math classes have this thing they do. Kids in second or third grade (about 7 or 8 years old) are given assignments that can easily be solved with a system of equations, but which the kids can only "solve" by guessing. I saw this with my kids, and the children of friends. I assume the idea is to make practice adding and subtracting less boring, but what it does is to cause the kids to hate math - and it generally drives the parents batty, too (average parents don't know algebra.)

The task given to the child looks like this:

The number in each "brick" is the sum of the two bricks it sits on. The children have to fill in the empty bricks to make all the sums fit. The given numbers are not always in the corners - any three bricks can be used, but the three corner task is the worst.

The solution to the example looks like this:

Kids can only get that solution by repeatedly guessing or systematically trying numbers in the empty blocks until they get an answer that fits. They get frustrated, and it gives them the idea that math is just randomly doing things until you get an answer.

You can systematically solve it, but you need concepts that children at that age haven't seen yet.

If you must use trial and error, do it yourself in front of the class as a demonstration. Make a couple of attempts. That shows how inefficient guessing is. Then tell them that it's a bad idea to repeatedly guess at possible answers, and introduce the systematic way to solve whatever type of problem you are working on.

Explain how to get from the description of the task to a mathematical equation. That is often the "missing link." When I was a kid, most kids hated the "story problems" because they couldn't see how to get from the description to the equations we had learned. From my own children, I know that they are never explicitly taught how to break down a task into a solvable equation - you either manage it on your own, or you go through school failing on story problems and hating math.

Math shouldn't be that hard. It's just applying rules and following them to their conclusion. Poor explanations and reliance on trial and error make it harder than it has to be, and makes kids hate it.

There is a general solution for the bricks assignment. It isn't taught to the kids at any point.

I wrote it out and discussed it with my daughter's math teacher when it came up. The teacher wasn't interested - the assignments came out of the book, and that was all the teacher was interested in. Kiddos do what they're told, and get the correct answers after playing guessing games, or fail the assignment. My daughter still hates math.

• I disagree with your point of view, and I think it's clear why. You say maths is 'just applying rules and following them to their conclusion'. That is not what maths is. The sort of question you describe is precisely aimed at developing the sort of maths skills that you have dismissed. Instead of randomly trying numbers until they find one that works, your children should have been thinking about what a better method would be, or at least how to make intelligent guesses for the numbers. – Jessica B Dec 27 '19 at 12:50
• You do not learn problem solving by being told the solutions to problems. To learn problem solving, you need to try and solve problems to which you have not been given the answer. That is what those sorts of problems should be used for. – Jessica B Dec 27 '19 at 16:28
• That question is not beyond the ability of an 8-year-old, unless they've been taught to think that it is beyond them. If they've been taught to think, which I admit is rare, they can certainly make a decent stab at it. – Jessica B Dec 27 '19 at 16:31
• I don't know about you, but I guess all the time. I guess about the form of an integral (maybe I can make a useful change of variables? oh, nope... that didn't work... what about this change of variables?); I guess about the form of bounds (maybe this function is bounded by some other nice function... oh, nope, that thing blows up... what about this nice function? oh! that works!); I guess all the time. My guesses are just more sophisticated, and have less to do with direct computation, but they are guesses all the same. – Xander Henderson Dec 27 '19 at 18:31
• Additionally, you assert that mathematics "is about learning how to calculate answers using methods you have been taught". No. I cannot disagree more. Mathematics is about solving problems with whichever tools get the job done. The mentality that students should just learn how to apply specific methods for solving particular kinds of problems is immensely damaging, and leads me to teach remedial classes to college students who have to unlearn a lifetime of rote memorization without comprehension. – Xander Henderson Dec 27 '19 at 18:33

I think best learning happens, when students discover the rules on their own.

But students shouldn't be forced to re-invent the wheel, so to say. Because this is too difficult and frustrating for them. Original discoveries in math took people hundreds or even thousands of years to do. So, it's unreasonable to expect ignorant students to do something like this in a few minutes, or hours at most.

Students can discover the rules for themselves by looking at several solved examples. And then they can become competent in using these rules by solving a lot of similar problems and checking if they get the answer right or not.

I think starting with specific examples and discovering abstract rules in them mimics how original math discoveries were made. It's more natural and easier for people to learn this way. But if you learn abstract ideas first and then try to apply them to specific examples, then this is like thinking backwards in terms of learning and discovering. It's much more difficult and awkward for people to learn this way.

And practicing with immediate feedback about how well or poorly you've done is probably the most effective way gain competence is solving a class of problems. Because without immediate feedback, students often deceive themselves into believing that they've learned it. And they only discover to their dismay on a test, that they haven't learned it nearly as well as they thought they did.

True knowledge is when you know both what you know and what you don't know. It's meta-knowledge and not just knowledge. And only practicing with immediate feedback provides this kind of knowledge.

Solving these sorts of problems is extremely common in algebra, of which students get two years--three if you count pre-algebra, and then in later math classes and in many science and engineering classes (particularly chemical stoichiometry). Given this commonness, I absolutely think students should experience this sort of problem in MANY different ways. I.e. absolutely take the time (there really is a slew of time versus content in "algebra one"). Seeing a problem like this in many different ways is helpful just as it is good to picture a function in many ways...it helps us come to terms with it.