My Journey With Mathematics

[Otis]

Math Equation Dividing Internet …

Content like that is often created (and then reposted) to instigate arguments, not to teach math.

See these threads, if you're trying to figure it out:

Help me resolve a stupid facebook argument

I just saw on Facebook a post

We have some discussions about the expression

 

[harpazo]

This is related to math. This is the ONLY reason why I posted this information from yahoo news.

 

[topsquark]

There are others here who feel that such a question does not have a well defined answer due to differing versions of order of operations. I respect those responses because these are people who know more than I do.

On the other hand I'm a big PEDMAS fan.
Do all parenthesis first:
\(\displaystyle 8 \div 2(2 + 2)\)

Now work left to right:
\(\displaystyle 8 \div 2(4)\)

\(\displaystyle 4(4)\)

\(\displaystyle 16\)

Done.

-Dan

 

[Otis]

This is related to math … from yahoo news.

Yes, it's an example of cursory blogging about a poorly-presented math topic (definitions), while at the same time fostering the spread of bad math "instruction" for the masses. Your reference claims that "no one can agree on an answer" (which is not true, by the way) and then goes on to contradict itself! It's not news, harpazo. It's a brief blog post written by Sage Anderson at Mashable that first soft-handles the topic (eg: "I was thinking that some dashing math majors would sweep in and put an end to the madness, but even they can't stop this viral fight…") and then invites readers who wish to continue the fight to add fuel to the controversy (via Mashable's comment section).

Sage is just one of many bloggers who have all created posts about the same math expression. Those people blogged about it because they saw it spread throughout social media. It spread mainly because there are a lot of users on facebook, twitter, et al who love controversy over clarity. Various "news" outlets are now reposting some of the blogs, trying to take advantage of the frenzy. Thanks for contributing!

 

[Otis]

… [no] well defined answer due to differing versions of order of operations …

You nailed it. People need to define an order of operations before asking the question.

… On the other hand I'm a big PEDMAS fan.

Thank you. That's primarily what we use at this site. It's primarily what is taught in school. It's primarily what software for students uses.

The reason I'm posting is because I wouldn't want any student becoming misinformed from following that silly controversy.

?

 

[harpazo]

I also dislike order of operations. So many people get the wrong answer all the time, including math tutors and teachers. It's easy to make an error in algebra and math in general.

 

[Otis]

I also dislike order of operations …

In addition to what?

Your dislike of ordering operations notwithstanding, do you understand why mathematics requires the concept?

\(\;\)

 

[harpazo]

In addition to what?

Your dislike of ordering operations notwithstanding, do you understand why mathematics requires the concept?

\(\;\)

1. I dislike order of operations in addition to others who dislike it.

2. Order of operations is important because it states which part of an equation should be solved first. Otherwise, we would get several different answers.

Note: 5 + 10*3

If I go left to right, the answer is 45. Using order of operations, the answer (the correct one) is 35.

3. Why do you think order of operations is a useful tool?

 

[Otis]

1. I dislike order of operations in addition to others who dislike it.
2. Order of operations is important …

It's a shame that you dislike the subject (because we can't get very far in mathematics without defining an order), but I'm glad that you can agree it's important.

Without a definition in place, the never-ending saga continues …

?

 

[harpazo]

It's a shame that you dislike the subject (because we can't get very far in mathematics without defining an order), but I'm glad that you can agree it's important.

Without a definition, the never-ending saga continues …

?

1. I do not hate order of operations. I just like other areas of math more.

2. My central goal in terms of math is to "master" or at least becoming very good at creating equations from given applications.

3. To read a math word problem and be able to solve it by forming an equation or formula is what truly separates math students from math professionals.

 

[harpazo]

Professor Wildberger has something to say about set theory. I want your input on what he stated in this video clip.

 

[Otis]

… I do not hate order of operations …

That's correct. You used the word 'dislike'. You dislike the order of operations. It's not clear why you told me that; perhaps it's not clear because I'm still not sure what you mean by it. If there's something about the order of operations that you're avoiding, that might be one of the reasons for why you've been going in circles with algebra for so long. (I'm wondering whether you skim over or skip sections you don't like.)

Are you able to articulate what it is about the order of operations that you dislike?

\(\;\)

 

[topsquark]

I watched most of it. I suppose if he does the job right he can lay claim that teaching Mathematics starting with natural numbers is the way to proceed. In fact this is the way it is taught in most secondary schools, so far as I know. However his intuitive way of structuring Mathematics is a bit too restricted. He claims that many topics are "too hard" to teach in the usual way but doesn't really seem to replace what he is complaining about.

In short, after doing some research on him I have decided that my opinion of him is that he isn't acknowloging just what the Mathematical revolution (for lack of a better term) started in the Renaissance and up to now is all about. The foundations of set theory are a very important and it gives modern Mathematics a structure similar to Euclid's The Elements. I see no reason to "throw the baby out with the bath water."

-Dan

 

[harpazo]

I watched most of it. I suppose if he does the job right he can lay claim that teaching Mathematics starting with natural numbers is the way to proceed. In fact this is the way it is taught in most secondary schools, so far as I know. However his intuitive way of structuring Mathematics is a bit too restricted. He claims that many topics are "too hard" to teach in the usual way but doesn't really seem to replace what he is complaining about.

In short, after doing some research on him I have decided that my opinion of him is that he isn't acknowloging just what the Mathematical revolution (for lack of a better term) started in the Renaissance and up to now is all about. The foundations of set theory are a very important and it gives modern Mathematics a structure similar to Euclid's The Elements. I see no reason to "throw the baby out with the bath water."

-Dan

Wildberger's teaching style is not for me. I saw a few minutes of his Rational Trigonometry video series and walked away more confused than ever before. Trig does not have to be taught the RATIONAL way.

 

[harpazo]

Most students fear word problems. I myself have been struggling with word problems for most of my life. There are gifted people that do not have a math degree but can solve a word problem by simply reasoning their way to the right answer. I am not one of those individuals.

Pretend you are taking a math test and come across a word problem that you've never seen before. What would you do to find the right answer? Take me through the steps, the process leading to the right answer. Of course, you need a problem to work with.

Here is one:

Two numbers add up to 72. One number is twice the other. Find the numbers.

Note: I know how to solve this problem. However, I would like to see different ways of solving this problem STEP BY STEP. Keep in mind that you have never seen this problem before. Explain.

 

[MarkFL]

If I were taking a timed test, I would immediately write:

[MATH]x+2x=72\implies x=24\implies 2x=48[/MATH]
Obviously, \(x\) is the smaller number, and \(2x\) is the larger.

 

[harpazo]

If I were taking a timed test, I would immediately write:

[MATH]x+2x=72\implies x=24\implies 2x=48[/MATH]
Obviously, \(x\) is the smaller number, and \(2x\) is the larger.

I know what you did here but can you explain the steps, the process PRETENDING to never have seen this problem before?

My Breakdown:

Two numbers add up to 72.

Let a = one number
Let b = the other number

So, a + b = 72.

One number is twice the other.

a = 2b

I see two equations.

a + b = 72...Equation A
a = 2b...Equation B

Find the numbers.

I would then proceed to use the substitution method.

a + b = 72

2b + b = 72

3b = 72

b = 72/3

b = 24

To find a, I can plug b = 24 in EITHER equation A or B.

What do you say?

 

[JeffM]

This is exactly how I teach those I tutor to approach word problems in algebra, but it is a very basic technique that applies only to certain kinds of problems in algebra. It is not going to work for a problem in differential equations or constrained optimization.

Here is a more general approach. What am I trying to do and is there a mathematical discipline that will help me do it? If there is a mathematical technique that I think will work, I then ask myself what is already known and what must be discovered. Next I begin translating from English into the appropriate mathematical notation.

Now what you did follows that general approach.

What do I want to do? In your example, you want to find numbers that satisfy given conditions.

Is there a mathematical discipline that answers many such questions? In your example, the answer is algebra.

With respect to algebra problems, I have a standard technique: I assign a symbol to each unknown, and then I look for equations that equal the number of unknowns. And that is exactly what you did.

But if I was trying to solve a problem in optimizing a differential function subject to constraints, I'd use a different standard technique.

Ultimately, solving word problems cannot fully be reduced to rule. It involves "seeing" that certain mathematical techniques may apply and then having a systematic way to translate the problem into the form required by the technique. Once you get enough experience, you may see short cuts for certain kinds of problem.

 

[harpazo]

This is exactly how I teach those I tutor to approach word problems in algebra, but it is a very basic technique that applies only to certain kinds of problems in algebra. It is not going to work for a problem in differential equations or constrained optimization.

Here is a more general approach. What am I trying to do and is there a mathematical discipline that will help me do it? If there is a mathematical technique that I think will work, I then ask myself what is already known and what must be discovered. Next I begin translating from English into the appropriate mathematical notation.

Now what you did follows that general approach.

What do I want to do? In your example, you want to find numbers that satisfy given conditions.

Is there a mathematical discipline that answers many such questions? In your example, the answer is algebra.

With respect to algebra problems, I have a standard technique: I assign a symbol to each unknown, and then I look for equations that equal the number of unknowns. And that is exactly what you did.

But if I was trying to solve a problem in optimizing a differential function subject to constraints, I'd use a different standard technique.

Ultimately, solving word problems cannot fully be reduced to rule. It involves "seeing" that certain mathematical techniques may apply and then having a systematic way to translate the problem into the form required by the technique. Once you get enough experience, you may see short cuts for certain kinds of problem.

What an informative reply. Thank you, JeffM.

 

[Otis]

… Pretend you are taking a math test … a word problem that you've never seen before … What would you do …

I'm not sure what you're thinking there. For example, you can't possibly expect to have seen every variation of a distance/rate/time situation. On the other hand, if you're talking about a problem involving math that you've never seen before, then the first thing I would do is wonder, "Why is this question on the test?" I would skip questions like that, and, if I had time after finishing what I could, I would then use all remaining time to show some effort on the rest.

If you're talking about a problem involving a topic or methodology taught in class, then I would use knowledge and experience gained from instruction and lots of practice to recognize it, followed by doing what I'd been taught.

If you're talking about not recognizing something on a test that you're supposed to know, then I would say, "Darn it", let the chips fall where they may, and revisit the material afterwards.

Most students fear word problems …

I don't believe that.

\(\;\)