New answers tagged primary-education
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I agree with @Carser: Talk to the teacher. It sounds like there is a
disconnect between the simple sums and, e.g.,
"various arithmetic 'games'"—They can be quite difficult!
You could challenge your son by asking him to answer the too-simple
sums in his head, especially the "much larger numbers, eg 367 + 58." That can be
quite challenging. ...
3
Most of the answers assume, as does the OP, that the purpose of proof is verification. In fact "establishing the truth of a proposition" is only one of the reasons that mathematicians prove things.
The most-cited paper on this subject is probably Michael de Villiers' "The Role and Function of Proof in Mathematics" (Pythagoras, 1990, pp. ...
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Now, he asked me why you'd need proofs for anything ever, since it's
enough to see that an equation works after testing it for a few values.
So let me give my take on a practical example which does not involve numbers or formulas, so it should be relative simple to understand.
Having a mathematical proof for some calculation is like having a map.
Let's say ...
2
Let me be provocative.
Your friend is right. Nobody needs proof except mathematicians. The reason you want absolute proof is that you have a mathematical mindset. For all other people (farmers, carpenters, engineers, physicists) overwhelming evidence is sufficient.
It is telling that most non-trivial statements about actual nature cannot be proven to be ...
3
If you've got a sum you're keeping track of (e.g. you're adding up your purchases in a shop), you can quickly check whether your final answer is right by just repeatedly adding up the digits, ignoring any 9's, until you get a one-digit number, and then checking whether the sum of the digits of your sum is equal to the sum of the digits of the summands after ...
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Keep this person as a friend and admit -- if asked -- that you have no good explanation for someone unfamiliar with mathematics. (In general, spouses of mathematicians never ask!) If your friend plays bridge or chess, discuss the strategies of the game. You could then say that proofs give mathematicians the guarantee that their strategies will work. ...
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It is possible to tile (tessellate) a plane with any triangle.
It is possible to tile a plane with any quadrilateral.
It is possible to tile a plane with some pentagons (such as 'houses' where the sides all have equal length, the base is a square, and the top is an equilateral triangle).
It is possible to tile a plane with some hexagons (such as regular ...
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Put 100 red marbles in a box and 1 white marble. When you blindly pick a few marbles and they are all red, it does not mean all marbles in the box are red.
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Money, even imaginary money is a good motivator
Much mathematical thinking historically, came from gambling and card playing. If you want to avoid losing your shirt, you need to understand the mathematics. It is a very practical use of the subject.
A simple experiment is to throw two dice. Ask your friend to bet on which sum will come up most often (use a ...
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Here's a surprising fact I've learned today on this very site:
Have your friend raise with a calculator
$1.5^{1.5^{...}} $.
At some point it goes to infinity.
Now let him do the same with
$1.4^{1.4^{...}} $
Funnily it stop growing. That raises some questions about the previous one : did it actually stop growing? Or was just the calculator screen to small and ...
1
since it's enough to see that an equation works after testing it for a few values.
Would they be willing to bet their life on that if that equation was being used for an autopilot in an airplane or some other piece of equipment? Or if the equation takes millions of possible combinations of values?
Suppose I make an engine. I test it and it works. Great. ...
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Does your friend get bothered by multiplication? You say $+$ and $-$ are okay but division is problematic, so you skipped over multiplication.
If your friend is okay with multiplication, and knows about breaking up numbers into prime factors, here's something to claim: every number has at most three different prime factors ($45 = 3 \cdot 3 \cdot 5$ has just ...
2
There is quite an amazing collection in response to the
11 yr-old MathOverflow
question, Examples of eventual counterexamples.
(This a precursor to the already cited 9 yr-old MSE question,
Examples of patterns that eventually fail.)
Here's one (among many) I like (posted by Gerry Myerson)
The numbers $12$, $121$, $1211$, $12111$, $121111$, etc., are all ...
4
get him to draw a triangle (maybe even freehand), and measure the internal angles.
point out all the things in the world built with triangles - electricity pylons, eiffel tower, maps etc - triangles are important, and its important in order to build our world to understand triangles completely.
what was the sum of the internal angles? (in my experience it ...
1
To some extent, your friend is totally right. If you want to check if an equation always holds, then just checking several values is often a nice and practical way to do that.
But proofs can certainly come in handy sometimes. Here's a simplified version of a problem I encountered at work this week.
We had 8 cables of various lengths and 8 conduits of various ...
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An example that directly contradicts the idea of "seeing that an equation works after testing it for a few values:"
Theorem (not): The polynomial $x^2-x+41$ is prime for every integer value of $x$.
This works after testing it for every integer $x = 0, 1, 2, \dots, 40$.
So "obviously" (according to your friend!) its value is a prime number ...
15
Most answers describe proof as an improved correctness checking tool.
But, I don't think this gets at the core of the issue. The goal of math isn't to check off theorems, but to understand them and consequently nature (broadest possible meaning of the word).
Proofs are a way to describe insights into a subject in a clear manner. Impossibility proofs tell ...
4
Try solving Pell's equation by trial and error. For example:
$x^2-61y^2=1$ has a whole-number solution $x=1$, $y=0$. Are there any others?
Try a few values for $x$ and $y$, and it will appear not. Is that a proof that there are no more solutions?
It turns out that the next smallest solution is $x=1766319049$, $y=226153980$, which would take quite a while ...
11
Maybe you can use some non-mathematical examples...
Black swans. Europeans assumed that all swans were white, because all swans in Europe are white, to the extent that a black swan was considered impossible--like a "flying pig". When they finally got to Australia and discovered that there are black swans, it was a big shock.
Suppose someone tells ...
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Because mathematicians want to be really, really certain.
There are lots of ways you can be convinced that something is true, and be wrong. By requiring proofs before you are convinced, you can make that amazingly less likely.
A proof requires that someone lay bare all of the weaknesses in their claim, and show how those weaknesses are not a problem. ...
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One thing that can't be shown without proof techniques is the impossibility of something. One reasonably concrete example of this is the impossibility of constructing a square and a circle with the same area using a straight edge and a compass. Individual cases can be shown to not work and to demonstrate that it's a hard problem, but there's a world of ...
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From a practical point of view, your friend is right. One just has to be careful to test enough numbers. Engineers don't believe calculus because it has proofs based on analysis which is based on set theory; engineers believe calculus because lots of engineers before them used calculus to build bridges and very few of the bridges fell down. We have ...
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More fun than equations are patterns that seem to hold. Put a dot on a circle, connect it to all the other dots (none yet), there is 1 region. Second dot connects to first, two regions. Third dot connects to first two, there are now 4 regions. It sure looks like the regions are doubling. In fact, they are not. It's a fun problem, and takes you by surprise. ...
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Theorem: Suppose you start with the number 1, and you're allowed to multiply it by 2, 3, and/or 5, as many times as you like. In this way, it's possible to get any whole number.
Proof: Check this for 1, 2, 3, 4, 5, and 6. Also true for other random examples I think of, such as 10, 100, and 96. It's clearly true.
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I would suggest taking it easy to start but maybe something like base numeration is a way in here. There is no reason we have to use base 10 and students use of different bases can be very important to understanding operations that require regroupings, and that weird alignment under the typical presentation of multiplication algorithms.
One example to check ...
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Forget New Math, it does now work with young kids, and it is not needed for learning arithmetics in elementary school.
Instead, first teach the concepts of "same as", "more than", "less than" by lining up real objects like apples or backpacks or horses one against another, making pairs. Then you ask, how do we figure out "...
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I highly recommend the book The Number Sense: How the Mind Creates Mathematics, by Stanislas Dehaene and published by Oxford University Press. Another book that comes to mind is The Language Instinct: How The Mind Creates Language, by Steven Pinker. Both books have had broad impact on scholarship related to your question.
As you try to refine your question ...
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