# Teaching math by serving it as games with rules first, not intuition?

Just now I'm thinking a crazy reversed idea: Is it good to teach math to elementary students without starting from its real world motivation or making it intuitive first, but rather by realizing that it is a game with some rules which revealed later to have real uses?

I am thinking that if we start with the intuitive ideas of some concepts, then we are required to make almost every concept we introduce intuitive..which is not quite obvious for a lot of students. Thus, sometimes we have to skip introducing some concepts intuitively, but we still introduce only some easily intuitive concepts. Doesn't this affect students' motivation, making them ask: "I can't grasp this. What's the real world example?" when there is not an easy connection..

I mean, since math is abstract, although many ideas in it are intuitive, (many are not)..if we start by introducing a concept as a game with rules and students have to master this game, how would this impact the learning process?

Example: When teacher teach about fraction, they usually start with dividing cake or pizza or something, and then put it in the mathematical notation of fraction that we know well. I used to (and probably still) think this is a reasonably good way to teach math. However, as implied in my question, can the teacher not start with this real world motivation, but instead start immediately by introducing the notation for fraction, and give some of the rules of summing, multiplying, etc, and the challenge for the students is to apply this rule..which sounds pretty much like playing a game of fraction?

Hence, I don't mean that we use games when teaching math, but I mean math itself is made as collection of games. Then, at some point, the teacher relates that fraction to real world uses, if any,..such as dividing pizza, etc.

I don't assume this idea is new. In fact, I ask whether some have considered this method or not. I just think that this way, students won't rely too much on real world uses of math before studying it, hence not being overly dependent on that..since some math concepts such as negative times negative, or even the whole integers, or square root, or even some geometrical objects, are quite abstract in the minds of elementary students, right?

I am asking this only relating to elementary math, not higher math, since I know the higher we go, we have motivation before abstraction..and at some point math students will learn that..but can we start by giving "the rules of the game" and "play" for elementary students..

• It seems that you are thinking of the philosophy of mathematics called formalism. Sep 22 '18 at 7:11
• Very broad and open question. Also seems to be a little implication that this is a new idea to the world, when it is just a new idea for the questioner. FWIW, I think there is a place for play in learning math but that it should not be the only method (you don't learn music or a sport by scrimmaging and play only). Also agree that application justification is probably lost on young children. Probably "you need to learn this for the next subject" has more justification than "you need to learn this to get a good job". That said, lower level math has clear application with personal finance. Sep 22 '18 at 22:13
• @Joel I indeed mean to relate this to formalism.
– bms
Sep 23 '18 at 3:37
• @guest I do not think this idea is new at all. That is also part of the question, which can mean to ask whether this idea in my head has been there or not. I will add examples to my question by editing.
– bms
Sep 23 '18 at 3:39
• I was taught much formal math in physics courses. The "real world" intuition was manifest, yet the actual math in play was usually obscured through a mixture of inattention to detail and/or adherence to antiquated notation. For kids learning arithmetic, I think you have a point. Formal games are easier to teach and we have more teachers who can teach a game than can explain a full system of mathematics where the proper reasoning behind various operations is fully communicated. So much we see in social media amounts to teachers now teaching motivating things as unmotivated games. Sep 23 '18 at 18:57

You may be overestimating how much students learn from symbolic rules. Most textbooks already teach symbolic rules first then put word problems last. This is backasswards.

Most students succeed at exercises such as:

Compute the exact missing value. $$\frac{7}{9}\times39=\_\_\_$$

But, if you do not show them that prior question and instead begin with this, most students bomb horribly:

Without calculating, determine if the missing value is greater than 40, less than 40, equal to 40, or if it is not possible to tell without calculating exactly. Explain your answer, then rate your confidence as certain, fairly high, medium, or low. $$\frac{7}{9}\times39.884536=\_\_\_$$

Not only do many students not know the meaning of multiplication with fractions, they will fight you when you try to tell them they should be able to do the 2nd exercise trivially. They will fight you when you tell them they don't understand what it means but need to. They will fight you if you do anything other than tell them how to compute the exact missing value.

Parents will generally take the student's side.

[If you don't believe me, start asking your students my 2nd question and ask gentle probing questions to elicit their thinking aloud.]

Why?

Because the first type of question - which can be answered by symbolic rules - is always on the test. The second type of question - which requires meaning - is almost never on the test. Testing symbolic rules - i.e. rote learning - has already convinced them that math is just rote rules and nothing more.

These kinds of issues are especially prevalent with fractions because many students never learn that a fraction is a number, so it never occurs to them to compare $$\frac{7}{9}$$ to $$1$$ whole.

So, no, do not teach math as a bunch of symbolic rules. Make math meaningful.

• Hmm. I don't know about elementary math. But the best results (regarding student understanding) of calc I, I've ever seen was taught starting with derivatives as a black box computation rules (linear functional on polynomials extended through lots of hand waving). The meaning, epsilon delta etc. only came later. Still the kids internalized it by far the best of the 12 odd courses I've seen.
– DRF
Oct 23 '18 at 16:15
• What do you mean by "black box computation"? That makes me think of the kid who sees $f(x)=x^2, f'(x) = 2x$ and has no idea how the equations relate or that calculus has anything to do with rates of change, but is satisfied with the check mark he received on a quiz. Is that what you meant by black box? Oct 23 '18 at 20:21
• I'd also be hesitant to make big inferences from this given that it was definitely not a controlled experiment, nor was it likely to have a random sample of students (since it's a calculus class at one institution and most students, at least in North America, do not take calculus). Oct 23 '18 at 20:44
• Oh it certainly wasn't a controlled experiment and I'm not trying to draw heavy inferences. It just convinced me (a staunch supporter of "math is about proof and understanding, rote memorization and rules are the wrong approach" that this might not be as true I thought. To be honest I haven't seen (and you don't cite either, that would probably help the answer) any experiments/studies saying that going rules->understanding is worse than going understanding->rules.
– DRF
Oct 24 '18 at 14:12
• As to black box computation I mean they got the rules $\frac{dx}{dx}=1$, $\frac{df+g}{dx}=\frac{df}{dx}+\frac{dg}{dx}$,$\frac{da f}{dx}=a\frac{df}{dx}$ and $d{fg}{dx}=f\frac{dg}{dx}+g\frac{df}{dx}$. Assuming $a$ is a constant and $f,g$ are functions. Given some extra work this is enough to characterize the derivative. This was all they pretty much knew about the derivative (no limit definition, no rate of change definition, though I think that might have been mentioned in passing a short while later) until a fair way into the semester.
– DRF
Oct 24 '18 at 14:27

The approach you envision aligns with that of the Bourbaki group. They espoused a formal, self-contained, and abstract approach to mathematics. Their thinking influenced the School Mathematics Study Group of the early 1960s, which produced the infamous “New Math” curriculum reform in the US. This reform was for both primary and secondary education.

So yes, your idea has been considered and even implemented at the elementary school level. Examples at this level included teaching of set theory and numeral systems other than base-ten. The reform movement was criticized by many, including influential thinkers such as Richard Feynman. In his 1965 article “New Textbooks for the ‘New’ Mathematics,” he criticized this approach and emphasized that "we do not want to just teach words,” and “subjects should not be introduced without explaining the purpose or reason.”

Here is a more visceral reason why we have to ensure math is meaningful and that it makes sense.

This video shows student responses to the following exercise.

And reactions like this happen quite often. Students aren't making mathematical sense of problems, they're just overwhelmingly brainwashed into thinking they need to jump into calculation procedures.