Can someone explain proofs to me?

[Laura Corin]

Although I did algebra, I don't think I ever did proofs. Calvin is coming across them in LOF and we are both a bit puzzled. Do they have a practical purpose or are they a 'the beauty of maths' kind of endeavour?

 

Thanks

 

Laura

[TravelingChris]

Well they have two purposes. The first is in math in that higher math (like majoring in math) is a proof based subject. But on a more practical note, it demonstrates understanding of the subject. It helps with logical thinking. it is a way most students actually do logic. A = B, A + B =C, therefore 2B=C. That sort of thing.

[Laura Corin]

Thanks, Christina. I think that it's the jargon that is offputting. As far as I can tell, in UK maths at this level, the pupil has to explain the process s/he is going through to solve an equation, but only in terms of, for example, 'divide both sides by 2' or 'subtract 3 from each side.' The example you give is fine - no jargon and entirely familiar to me. It's all the 'reflexive property of equality' stuff that is a bit of a surprise.

 

Laura

[kiana]

For "why do proofs", I really like this article (it's not mine :/)

 

http://www.math.sc.edu/~cooper/proofs.pdf

[Laura Corin]

For "why do proofs", I really like this article (it's not mine :/)

 

http://www.math.sc.edu/~cooper/proofs.pdf

 

It's a good article. I just wish I could see what his proofs are like. He mentions 'even if it's just a series of equations'. That's normal algebra to me, rather than this jargony thing Calvin is dealing with. He also talks about his proofs 'not being what you were taught in 10th grade' so I wonder what they are.

 

Laura

[forty-two]

I think that it's the jargon that is offputting. As far as I can tell, in UK maths at this level, the pupil has to explain the process s/he is going through to solve an equation, but only in terms of, for example, 'divide both sides by 2' or 'subtract 3 from each side.' The example you give is fine - no jargon and entirely familiar to me. It's all the 'reflexive property of equality' stuff that is a bit of a surprise.

 

The "jargon" is meant to explain *why* you are allowed to divide both sides by 2 or subtract 3 from each side. In other words, why that is a mathematically valid thing to do.

 

An example from Modern Algebra: A Logical Approach:

 

Find the solution set for the equation |x| + 7 = 9; x is a real number:

 

|x| + 7 = 9 [Given]

(|x| + 7) (-7) = 9 + (-7) [Addition property of equality - why you can add -7 to both sides]

|x| + [7 + (-7)] = 9 + (-7) [Associative property of equality]

|x| + 0 = 2 [Rules for addition of real numbers]

|x| = 2 [Addition property of zero]

x = 2 or x = -2 [Meaning of absolute value]

 

|2| + 7 ?= 9; |-2| + 7 ?= 9 [substitution property of equality]

2 + 7 ?= 9; 2 + 7 ?= 9 [definition of absolute value]

9 = 9; 9 = 9 [rules for addition of real numbers]

 

The solution set is {2, -2}

[Susan C.]

We just hit the proofs......yuck. DD didn't do too well on the first proofs in the test.... I did go to dummies.com and downloaded the proof ideas from their Geometry for Dummies, it is all online. Let's all hope it helps us! My mom said she took geometry in college, why, why, why do we have to do this in high school.....

[Laura Corin]

The "jargon" is meant to explain *why* you are allowed to divide both sides by 2 or subtract 3 from each side. In other words, why that is a mathematically valid thing to do.

 

 

But when the student writes 'divide each side by 2' he's not doing it on a whim: he's following a rule that he has internalised about how algebra works. He knows why - he went carefully through that when the rule was first introduced. It is, as you say, a mathematically valid thing to do.

 

This may turn out to be one of those two-countries-separated-by-a-common-language problems. I grew up internalising and using the rules; you grew up writing them out and then using them.

 

Laura

[Rosie_0801]

Am I way off here, or are these proofs basically footnoting your maths?

 

Rosie

[Laura Corin]

Am I way off here, or are these proofs basically footnoting your maths?

 

Rosie

 

The footnoting I grew up with explained what you were doing (divide both sides by 2, for example). The proofs that Calvin is being asked for are technical statements of the principles on which the decision to divide by 2 is based.

 

Laura

[tex-mex]

:001_huh: Reading this and dreading next year's Geometry already... lol!

[mamato3 all-boy boys]

 

This may turn out to be one of those two-countries-separated-by-a-common-language problems. I grew up internalising and using the rules; you grew up writing them out and then using them.

 

Laura

 

The algebraic proof looks more like what we did in geometry than in algebra. I don't ever remember having to write down algebraic laws. Like you, Laura, it was just something we all internalized (hs math for me was in the early-mid 80s).

 

I wonder....is it a new trend in math?

[Laura Corin]

The algebraic proof looks more like what we did in geometry than in algebra. I don't ever remember having to write down algebraic laws. Like you, Laura, it was just something we all internalized (hs math for me was in the early-mid 80s).

 

 

We'd have to mention, 'angles at a point on a line add up to 180 degrees', or 'angles of a triangle add up to 180 degrees.'

 

I did algebra and geometry in the late 70s in England.

 

Laura

[LoriM]

The algebraic proof looks more like what we did in geometry than in algebra. I don't ever remember having to write down algebraic laws. Like you, Laura, it was just something we all internalized (hs math for me was in the early-mid 80s).

 

I wonder....is it a new trend in math?

 

No, it is not a new trend in math. :)

 

Maybe this will help...I liken proof to the art of argument. While one never counters in a debate with the logical "rule" being applied (for example, if you say, "The ad hominem attack of my opponent..." you sound like a doofus!), you need to recognize an ad hominem argument and counter with the appropriate valid and sufficient evidence to negate it if you hope to convince your audience.

 

But if you don't know what ad hominem is...you might counter with "NUH-UH! YOUR MOM!" or some other superintelligent response. Grin!

 

So, the idea in math is that you learn the "rules" and you learn the names of the rules, and you learn why they are true for real numbers, but maybe not true for rational numbers, or whole numbers. Because, honestly, the math we do before high school graduation is limited in the number theory to the most boring numbers imaginable (LOL), but once you recognize there are lots of other number systems and properties of number, you want to know what rules are applicable. Just like if you were on a different planet, you'd want to know what the properties of matter were, and whether or not if you stomp your foot you are likely to shoot off into space. :)

 

For some people, the art of proof is just that--an art form. Personally, I love to see an elegant proof. It is soothing to me to read an algebraic statement that is validated by the properties of the operations applied. When they use good set notation, and demonstrate closure under the operation, it makes my heart sing.

 

But frankly, we don't ask beginning to students to do much more than be able to recognize a methodical, step-by-step solution, and to apply the definitions of the properties being applied at each step. It is not exactly elegant. It's not unlike diagramming a sentence to see the component parts of the grammar. It doesn't make the sentence more true to know which word is the subject and which word is the predicate nominative. But it does make the reader more aware, and the nuance of the language more real. It gives the writer more power ultimately, to *know* the language so well.

 

So, the idea of the kinds of proof that a high school student does in Algebra and Geometry is simply diagramming the structure. It is about labeling. It is about really knowing the rules. And yes, most of it is memorizing...just like we memorize the alphabet, the prepositions, the vocabulary from Biology or the generals from a particular battle in our history. It is so that we *know* it better.

[mamato3 all-boy boys]

 

So, the idea of the kinds of proof that a high school student does in Algebra and Geometry is simply diagramming the structure. It is about labeling. It is about really knowing the rules. And yes, most of it is memorizing...just like we memorize the alphabet, the prepositions, the vocabulary from Biology or the generals from a particular battle in our history. It is so that we *know* it better.

 

I guess my wondering was more "is it novel these days to explicitly write out a proof for algebra?" We did proofs every day of geometry, but not in algebra. I totally understand the reasoning for it (I *loved* geometry proofs), but I just don't remember doing in in algebra.

[LoriM]

I guess my wondering was more "is it novel these days to explicitly write out a proof for algebra?" We did proofs every day of geometry, but not in algebra. I totally understand the reasoning for it (I *loved* geometry proofs), but I just don't remember doing in in algebra.

 

Oh, in that case...No. Not new at all...we did buckets of proofs in Algebra in high school (early 80's also), and I have my mother's high school "Foundations" book (PreCalc) which is loaded with proof. If anything, I would guess we do less proof now than ever before.

[Jane in NC]

Oh, in that case...No. Not new at all...we did buckets of proofs in Algebra in high school (early 80's also), and I have my mother's high school "Foundations" book (PreCalc) which is loaded with proof. If anything, I would guess we do less proof now than ever before.

 

Seconding the notion of the proof (or at least justification) in Algebra "back in the day". My high school texts from the 1970's were the Dolciani books that I continue to admire.

 

Laura, I wanted to refer you to an older thread that was created by Myrtle (whom we miss dearly on these boards) and contains contributions from her husband, Charon, who enjoyed any chat on math or philosophy.

 

Many non-math people associate proofs with the two column proof of geometry. A good basic algebra course should include some proofs or at least what I will call justifications, i.e. statement of what one is doing and reasons why one is allowed to do so. The two column proof of high school is not the sort of proof that mathematicians use in everyday life. It is a hand holding methodology to help students piece together logical flow of one idea to the next.

 

Non-math people often say math is about getting the right answer. To me math is about existence, uniqueness and the logical process used to demonstrate these things. The "answer" is far less important than process even in a high school course .

 

One highly useful method of proof employed regularly by mathematicians is that of Proof by Contradiction. You might want to google the basic proof that the square root of two is not rational using this method, also called indirect proof. It is a useful logical device (Reductio ad absurdum) which Calvin might enjoy.

 

Jane

[Nan in Mass]

Geometry proofs are where I learned to really think. That's what it felt like at the time, anyway. It was the only class in high school where I was required to really think. It was the proofs that did that. I came away from that class different. I can apply that skill to any murky-looking real-life problem when my usual "I just feel like doing it this way" method fails me. Usually, my brain does the feel way fast and well and well, but occasionally it doesn't work or I have to explain to someone else why I'm doing what I'm doing. Then, the practice with proofs comes in handy. By the way, I also did lots of proofs in algebra. -Nan

[Laura Corin]

Laura

[fractalgal]

Although I did algebra, I don't think I ever did proofs. Calvin is coming across them in LOF and we are both a bit puzzled. Do they have a practical purpose or are they a 'the beauty of maths' kind of endeavour?

 

Thanks

 

Laura

 

Proofs (using logic and language) are intended to demonstrate that a statement is true in all cases, without a single exception. Once a statement has been proven, it can be used as a basis to prove further statements.

[Testimony]

I know that my son will do this stuff in a few more months, but I just wanted to add (it might have been said) that doing proofs is the first step into science. I know that when I studied proofs and theorems, they frustrated me. I discovered that this is where math and science meet. It truly is like doing a science experiment or a logic problem. Math, especially higher level math, is used in engineering, physics, chemistry, biology, astronomy (scientists found planets using math). Proofs and theorems are the stepping stones into these types of sciences.

 

Blessings,

Karen

http://www.homeschoolblogger.com/testimony