Math STRATEGY needed for Graphing

[truethat]

The last two weeks have been a total nightmare and I've gone through three Math teachers who are supposed to be teaching a MATH STRATEGY class who simply refuse to teach strategy and keep trying to teach MATH.

As I've mentioned before, I teach test prep, and now we have a new test where my students (adults in a graduate program) are being required to score an 86.6 on a Math Test.

Most of the students have not done Math in years and so they are completely confused when it comes to Math. I have a created a series of strategies that work for quickly coming up with the answer on easier questions, but it seems that my Math Teachers will not SHAKE OFF the need to teach the entire concept of Graphing instead of just looking at the questions and helping the students quickly, using process of elimination get the answer.

The difference between Math Strategy and Math is that you don't need to show your work, you just need to get the right answer.


The problem that I am facing is teaching people how to quickly understand a graph and get the right answer. It's a multiple choice test, so I need some strategies that can be helpful.

At this point I'm in need of hiring a consultant to do it MY way because every time I've tried to explain this to a Math Teacher they insist on doing it their way.

I work with several colleges and the other day the Dean of one of colleges called to tell me that the students are complaining about the Math Teachers section of the class even though they understand mine. So I know I'm correct in my observation.

Help. PM me if you are interested in consulting.

 

[Lothian]

I am totally against what you are trying to do (if I understand you correctly)

We have too many teachers who teach kids to pass exams rather than give then the basic foundations and true understanding of the subject.

 

[truethat]

I am totally against what you are trying to do (if I understand you correctly)

We have too many teachers who teach kids to pass exams rather than give then the basic foundations and true understanding of the subject.

A. I don't teach Kids I teach graduate students who don't use math at all in their jobs. If they did, then I would agree with you but they don't, especially not this level of Math.

Compare it to a Sanitation Worker not being allowed to move up in a pay grade unless he passes a test on algebra and scores high in the test.

B. The Standardized test given to my students has recently been reassesed as not being a fair test and also having nothing to do with the persons' job. So there is actually a law suit going on right now about it.

On XXXXXX the Court found that the XXXXXXXXXXXX failed to establish, as required by federal law, that the XXXXXXXX was related to the job of XXXXXXX The XXXXXX was an exam created and administered by the XXXXXXXXX Test takers were required to achieve a passing score on the XXXXX in order to receive their License

The Court also found that because the XXXXX was not shown to be related to the job of XXXXX, the XXXXXX had violated Title VII by requiring plaintiffs to pass the XXXXin order to receive a XXXXX license.

Relief Rulings

C. That's all well and good, but these people are going to get stuck in a quagmire of backlogging because of this and the simple and most direct way to get through it just to pass the test.

This is my job.


Just to clarify. Several years ago I tried to teach a strategy course for people who DID need to do it for their job and I refused to do it after a while. I agree with you. But that is not what this is about.

 

[psionl0]

The problem that I am facing is teaching people how to quickly understand a graph and get the right answer. It's a multiple choice test, so I need some strategies that can be helpful.

This is too vague for anybody to offer any useful help. We would need to see samples of the graphs in question and what answers were required.

Other than that, the quickest way to acquire knowledge is drilling. Give them lots of multiple choice questions and let them practice ticking the right answers.

The problem is that such knowledge doesn't stick (unless used) and no real understanding comes from it but if all you want them to do is pass a multiple choice test then that's the way to go.

 

[truethat]

Basically what I need is for someone to explain a BASIC description of what a Graph is. IOW WHY are you being asked this question. It's not so much that I want to just give them the answer, I could easily do that. It's that I want them to have an understanding of what they are being asked to do.


So for example I asked my teachers to start of by explaining to the students WHAT IS A GRAPH.

I got three different replies. One was highly convoluted answer, the other was a snobby "they should know what that means" type answer, the other said "I would have to teach them all these equations for them to understand it.

So for example, in my part of the class I teach the different types of charts, bar graph, pie chart etc etc etc.

It makes sense to them because they can visualize it.


I am not good at this so I've been trying to come up with some ideas. For example I thought that perhaps if we took a jar in the front of the class and started filling it with water and then talked about the rise over rate, it would be a VISUAL for them.

The strategies I want are something like

Why do we make a graph


I've been watching some videos myself now and one good example that I picked up on is that if you write the linear equation along the line, then it kind of helps them see that this is the equation of the LINE but a line has an infinite number of points.


That's why I think I need a consultant.


The key I've learned over the years is that you have explain the reason for the CONFUSION before you jump into it. It's different than starting off with someone who has no understanding of Math at all. Most of them have done these equations in their youth but it's been almost a decade so they FREAK OUT when they see it because it's all scary to them.


Telling them that the quadrant and the slope yadda yadda yadda. It just freaks them out even more.

What's a slope? What's the difference between linear and non linear. and most important WHY DO WE MAKE THE GRAPH IN THE FIRST PLACE>

 

[truethat]

Here's something I like and what I mean. I want them to just explain the terms without doing Math. You see how in the beginning he says "I'm not going to do any Math"

I want them to do this BEFORE teaching any Math.

I'll see if I can find some sample questions

https://www.youtube.com/watch?v=-Rdst52PuTw

 

[Minoosh]

So for example, in my part of the class I teach the different types of charts, bar graph, pie chart etc etc etc.

I can't visualize what your role is in this whole process, let alone what you seek in a "consultant." If you are teaching students about graphs, what is it that you expect the math teachers to do differently?

Most of the students have not done Math in years

They've never followed a budget, never figured out how far they can get on a tank of gas, never realized that if you are paid by the hour, more hours = more pay?

Here's something I like and what I mean. I want them to just explain the terms without doing Math. You see how in the beginning he says "I'm not going to do any Math"

He is doing math; he just says he's not.

 

[truethat]

I can't visualize what your role is in this whole process, let alone what you seek in a "consultant." If you are teaching students about graphs, what is it that you expect the math teachers to do differently?



They've never followed a budget, never figured out how far they can get on a tank of gas, never realized that if you are paid by the hour, more hours = more pay?



He is doing math; he just says he's not.

Aha! Yes see those are the things I use to get them to understand that they DO math they just don't realize it.

One example I use is Christmas shopping. I'll say "You have all been in situation where you are in a store and you see a bunch of items on sale that would be good for you get for colleagues that you feel you must by for. You'll stand in the aisle and in your head do this equation "I need to buy for 10 people and these items are $45 on sale for 35% off. And in two minutes you will figure out the minimum it would cost for you get them all one of them."


So it's that kind of thing and I've just figured it out. I'm so STOKED right now!!!!! Yayyyyyyyy

How weird is this. While I'm researching Math Strategies for finding the slop and plotting linear coordinates on a graph, I'm playing this video in the background just half listening.

Turns out that the pilot flew his plane into the side of a mountain. And they couldn't figure out what went wrong. At 30:36 you can see they figured out that they put the wrong number in that shifted the angle of the plane 7 degrees and then as it kept flying the plane was going down at a "slope" steeper than they realized.

So this is the kind of thing I need. I kept saying I need something where they can look at the graph in a different way. Why do we use Graphs? Why do we need to know the slope? So we don't fly our plane into the side of a damn mountain.

I'll start them off reading about this plane crash. This will tap into a visceral feeling and bring it from the theoretical into REAL LIFE.

Then we'll go into the Math.


Oh and I figured another thing out. We need to always use graph paper when explaining it to them. The teachers have been doing it on regular paper and I think it is easier to understand when you see the actual squares.

Here's the video I was watching. This is exactly how we need to start it off

https://www.youtube.com/watch?v=nKJijdwDsZ0


Sorry, I'm so excited! But this is the kind of thing I need to do in all the classes. Take the theory and make it something that is REAL to them in real life, not just sketches and numbers on a piece of paper.

 

[truethat]

This is so so perfect, I can't believe that I have watched these videos for months, and the one time I needed to figure out a math strategy regarding slopes, this is the one I'm watching.

After ANDLO the VOR/DME approach profile calls for a 5.5% slope (3.3deg angle of descent) to the Strasbourg VORTAC. While trying to program the angle of descent, "-3.3", into the Flight Control Unit (FCU) the crew did not notice that it was in HDG/V/S (heading/vertical speed) mode. In vertical speed mode "-3.3" means a descent rate of 3300 feet/min. In TRK/FPA (track/flight path angle) mode this would have meant a (correct) -3.3deg descent angle. A -3.3deg descent angle corresponds with an 800 feet/min rate of descent. The Vosges mountains near Strasbourg were in clouds above 2000 feet, with tops of the layer reaching about 6400 feet when flight 148 started descending from ANDLO. At about 3nm from ANDLO the aircraft struck trees and impacted a 2710 feet high ridge at the 2620 feet level near Mt. Saint-Odile. Because the aircraft was not GPWS-equipped, the crew were not warned.

http://aviation-safety.net/database/record.php?id=19920120-0

If anyone would like to rephrase that paragraph in laymans terms I'd much appreciate it.

It's a perfect solution. Vertical versus horizontal clarification, slopes being off by just a bit and then it completely changed the line.

They will get it.

 

[AlaskaBushPilot]

A graph illustrates relationships.

We can find answers to many problems simply by looking at a graph, like how to reach a maximum or minimum of something (e.g. maximum output; minimum cost).

It would help to know what graduate study programs you are teaching to so that subject matter could be chosen that they relate to. I am a pilot and the problem you posed might be interesting to me, but not to someone in, say, social work.

By far the worst students I had in graphing were from a school of social work. They had extremely bad attitudes towards math, yet it was pretty easy to find examples they could relate to.

So what subject matter are these graduate students studying? It remains remarkable for a lot of people to see this fear and loathing in graduate studies. My goodness the students in Asian countries are doing this kind of thing in middle school.

 

[AlaskaBushPilot]

edited by request.

I will probably post this very relevant material in a new thread later.

 

[TubbaBlubba]

Basically what I need is for someone to explain a BASIC description of what a Graph is.

Given a (real-valued) function f(x), a graph is the set of all ordered pairs (x, f(x)), and this is often represented by a figure in a Cartesian coordinate system (x,y). Thus, by locating some x in the coordinate system, we can find f(x), and vice versa, provided the function is well-behaved.

Without a somewhat intuitive understanding of what a function is, it will be very hard to explain what a graph is (the issue isn't helped by the fact that in introductory calculus, the definition of a function is typically extremely close, but not equivalent, to the above definition of a graph.)

 

[Earthborn]

So for example I asked my teachers to start of by explaining to the students WHAT IS A GRAPH.

A graph is just a visual way of doing the maths so you don't have to use all the cryptic Greek letters and nonsensical squiggles. They represent the same thing, but the graph is easier to understand and the squiggles are more precise.

So for example, in my part of the class I teach the different types of charts, bar graph, pie chart etc etc etc.

The nice thing about graphs is that they represent the underlying maths in a way that makes it look like an abstractified version of something people already know from everyday life. A pie chart looks like a pie, makes it a lot easier to see who gets the biggest slice of the pie, literally or figuratively. A bar chart is just a bunch of stacks, perhaps of coins if the values used represent money. A line graph can be seen as a hilly landscape. Draw in a stick figure walking from left to right for good measure.

Why do we make a graph

Mainly because it is easier and more fun than to deal with the squiggles, and because it is easier to grasp what is going on. Even (or rather: especally) the greatest mathematical minds use graphs to make sense of things. If some mathematician invents/discovers some new superformula, the first thing s/he is going to do is draw it into a graph and say "Aha! So that's what it does!". If that graph forms a pretty picture, someone at PIXAR is going to get rich from it. If it is a picture that looks a bit boring, the mathematician willl realise that the formula can be radically simplified, and everyone around the world will look up to him//her for coming up with such a elegant solution.

Mathematicians are often very visually oriented people. Some of them can even imagine just what the graph will look like, just looking at the squiggles on the blackboard. That does not mean the squiggles are more "maths" than the graphs.

What's a slope?

Surely everyone who has ever walked up OR down a hill knows what a "slope" is. It is just how steep the hill is. Draw in a stick figure walking from left to right and let people imagine what s/he is going to find along his/her way.

What's the difference between linear and non linear.

"Linear" just means it is a straight line. Someone walking up or down that hill will not find it easier or harder the further along. Non-linear just means not a straight line. Maybe the hill starts steep and flattens out near the top, or starts not so steep and gets increasily steeper.

WHY DO WE MAKE THE GRAPH IN THE FIRST PLACE>

Because nobody can make sense of the squiggles.

One example I use is Christmas shopping. I'll say "You have all been in situation where you are in a store and you see a bunch of items on sale that would be good for you get for colleagues that you feel you must by for. You'll stand in the aisle and in your head do this equation "I need to buy for 10 people and these items are $45 on sale for 35% off. And in two minutes you will figure out the minimum it would cost for you get them all one of them."

After which they'll just roll their eyes, as they realise that they never ever calculate percentages in their heads. (Almost) nobody does. People just notice that is cheaper than it used to and let the the cash register calculate the new price. Some clever individuals may notice that 35% is about a third, and taking a third from $45 dollars is a lot easier than taking off 35%, but that is about as much maths an ordinary person might use in everyday life.

Calculating percentages in one's head is hard, which is why there are so many scams using percentages.

We need to always use graph paper when explaining it to them.

I think that should go without saying. Teaching to graph without graph paper is a bit like teaching to add without an abacus. People can't see what they heck they are doing.

The teachers have been doing it on regular paper

Those teachers should be fired.

If anyone would like to rephrase that paragraph in laymans terms I'd much appreciate it.

The way to rephrase that paragraph in laymans terms, is making it into a graph; a picture of an airplane with an arrow showing where it is going (and maybe an arrow where the pilot thinks he's going) and a line depicting the slope of the hill. Any layman will be able to see that that isn't going to end well.

 

[truethat]

Wonderful post! Thank you so much. I love that you used the word "squiggles"

This is a breath of fresh air.

 

[TubbaBlubba]

Mainly because it is easier and more fun than to deal with the squiggles, and because it is easier to grasp what is going on. Even (or rather: especally) the greatest mathematical minds use graphs to make sense of things. If some mathematician invents/discovers some new superformula, the first thing s/he is going to do is draw it into a graph and say "Aha! So that's what it does!". If that graph forms a pretty picture, someone at PIXAR is going to get rich from it. If it is a picture that looks a bit boring, the mathematician willl realise that the formula can be radically simplified, and everyone around the world will look up to him//her for coming up with such a elegant solution.

Mathematicians are often very visually oriented people. Some of them can even imagine just what the graph will look like, just looking at the squiggles on the blackboard. That does not mean the squiggles are more "maths" than the graphs.

This goes a bit too far in my view. What you're saying is generally true up to the level of say, first-year college calculus, and in applied mathematics. But even then, any decent introduction to calculus textbook should show the many, many, many examples of how limited (pictures of) graphs are. Beyond that, the emphasis is typically on the need for strict and rigorous definitions, and on the drawn graph as nothing more than a mental aid at best. In many cases (say, multilinear algebra), graphs become so cumbersome and complicated that they quickly grow more detrimental than helpful, and generally devolve into flowcharts with at least as many "squiggles" as the formulae themselves. The mental images employed by mathematicians similarly grow more abstract, symbolic and less representable than the nice smooth curve representing some simple, real-valued function. Try drawing a nice picture representing the tensor product - it's not exactly easy.

So drawn pictures are important, but their importance should not be overstated, and they should not be relied upon as a substitute for the understanding of mathematical axioms, theorems and definitions in their native language, beyond the most elementary level.

 

[AlaskaBushPilot]

This goes a bit too far in my view. What you're saying is generally true up to the level of say, first-year college calculus, and in applied mathematics...

So drawn pictures are important, but their importance should not be overstated, and they should not be relied upon as a substitute for the understanding of mathematical axioms, theorems and definitions in their native language, beyond the most elementary level.

Yes, of course. But despite the students in question being in graduate school, the level of math they are having difficulty with is, according to common core standards, roughly second grade through at most middle school. They are a high risk pool for flunking the test. The others don't need to take this class.


Common Core grade 8 functions and graphing expectations can be found here:

http://www.corestandards.org/Math/Content/8/F/

But the level of explanations asked for in this thread are far earlier in the academic expectations for children - like what a graph is and why we do it. An example of grade 2 bar chart graphing and line plots for example can be found here:

http://www.eduplace.com/math/mw/background/2/03/te_2_03_overview.html


We are talking about a level where you have to physically grab their fingers and start from the X-axis, move their finger up to the line, then over to the y-axis in order to get through to them that this is what the purpose of the graph is. It is not enough to show them. I did this in a special program for remedial students. I also had them line up in groups for bar charts. That is what it took to get them to see that a longer bar meant more people.

The ISIS format works pretty good too though. If you ask a question and someone gets it wrong, you chop off their head. You'll find this in the psychology literature under "negative reinforcement".

 

[SezMe]

I am not good at this so I've been trying to come up with some ideas.

Then maybe you shouldn't be trying to teach it.

Turns out that the pilot flew his plane into the side of a mountain. And they couldn't figure out what went wrong. At 30:36 you can see they figured out that they put the wrong number in that shifted the angle of the plane 7 degrees and then as it kept flying the plane was going down at a "slope" steeper than they realized.

So this is the kind of thing I need. I kept saying I need something where they can look at the graph in a different way. Why do we use Graphs? Why do we need to know the slope? So we don't fly our plane into the side of a damn mountain.

I'll start them off reading about this plane crash. This will tap into a visceral feeling and bring it from the theoretical into REAL LIFE.

I seriously doubt that. Even people who have flown (as passengers) don't really get a feel for glide path, slope, rate of descent, etc.

If you want to give them a visceral feeling, use an example that is second nature to them: driving a car up or down slope. Or, as Earthborn suggested, even walking. EVERYBODY knows the difference between and gentle hill and a steep one. Build on that.

 

[SezMe]

Given a (real-valued) function f(x), a graph is the set of all ordered pairs (x, f(x)), and this is often represented by a figure in a Cartesian coordinate system (x,y). Thus, by locating some x in the coordinate system, we can find f(x), and vice versa, provided the function is well-behaved.

Without a somewhat intuitive understanding of what a function is, it will be very hard to explain what a graph is (the issue isn't helped by the fact that in introductory calculus, the definition of a function is typically extremely close, but not equivalent, to the above definition of a graph.)

You make two mistakes here. According to truethat, she is dealing with mathphobics. Give them your first paragraph and they'll either be asleep or walking out the door.

Second you confuse a graph with a function. They are NOT the same thing.

 

[TubbaBlubba]

Yes, of course. But despite the students in question being in graduate school, the level of math they are having difficulty with is, according to common core standards, roughly second grade through at most middle school. They are a high risk pool for flunking the test. The others don't need to take this class. '

I'm aware, I was responding to Earthborn's assertions about mathematicians.

You make two mistakes here. According to truethat, she is dealing with mathphobics. Give them your first paragraph and they'll either be asleep or walking out the door.

Yes, but when working out a simplified definition, it helps to start from an actual definition.

Second you confuse a graph with a function. They are NOT the same thing.

I do not, in fact I adressed this point in my last paragraph. A function f is generally defined (semi-informally) as a set of ordered pairs (a,b) where you have (a,b) and (a,c) iff b = c, with some additional restrictions wrt domains. A graph is defined based on a function f as all ordered pairs (x, f(x)) with x in the domain of f and so on

From Wikipedia:

In mathematics, the graph of a function f is the collection of all ordered pairs (x, f(x)).

Furthermore:

In the modern foundation of mathematics known as set theory, a function and its graph are essentially the same thing.

Because you could just plot the (a,b) discussed above. However, it helps to keep them separate, because when you draw an actual representation of the function rather than its graph, you're typically drawing something like f: R -> R.

 

[psionl0]

Yes, but when working out a simplified definition, it helps to start from an actual definition.

I'm with SezMe here. It is more useful to do it the other way. Start off with real life experiences where they read graphs and gradually introduce the jargon as required.