[Blindmath] FW: Solving systems by Graphing

Tue May 27 17:56:47 UTC 2014

Actually, blind people in the working world don't use graphs that they have to make themselves. They tied points into programs like Excel and make graphs on the computer. As long as you can solve the equation and you have the values, you don't have to worry about the graph. You can either have another member of your team make it, or use a computer program. Further, most people who are taking higher level math courses aren't doing so to become mathematicians. You're doing to major in other areas where they wouldn't have to make graphs. If you're going to say that the blind student has to make every single grass and higher level math, explain what would be a time efficient way to do so. Note that if the blind student makes the grass on the computer, it is solely for the benefit of decided teacher and won't help the blind student at all in addition to taking more time so when you have a suggestion, it needs to be a graphic the blind person can feel as well. Graphing is a representation of a problem. Not the only way to get a job done. This is why I get so frustrated when interacting mathematically with other sighted people. People who can solve graphically, but not algebraically, don't get nearly as much flack from educators, and that isn't right. To wrap this up, I think the blind people should have that accommodation especially in higher level math and not as much and lower-level math. And lower-level math it will be much easier for the blind student to make the graphs, and it will be necessary to understand the concept of graphing.

> On May 27, 2014, at 12:20 PM, Mike Jolls via Blindmath <blindmath at nfbnet.org> wrote:
> 
> Susan
> 
> I totally agree that a possessing a calculator does not equate to understanding.  The student or professional needs to understand what's going on behind the scenes and then simply uses the calculator or the computer as a tool to perform the mechanics of solving what he or she already knows how to do ... or rather knows the theory of how to solve the problem.  I will say that I have seen ... in years past ... how some high school aged children were working fast food jobs and when the calculator or the cash register at the restaurant was broken or the batteries dead, they couldn't figure up the bill.  That really hit home the sad state of affairs about how kids weren't learning math.  As I say, a calculator doesn't provide understanding.  But .. I got a little sidetracked there since you mentioned calculators.
> 
> Regarding graphing, I would say that if a student can demonstrate that they can correctly and consistently generate a solution set of points for an equation, then they have solved the equation.  Perhaps the solution is not a graph, but the student knows how to solve the problem.  The blind student has, in my opinion, solved the equation with the solution set of points.  I think we can say that graphing does not have to stand in the way of the student learning the material, providing that the alternate solution is acceptable to the educator, the course the student is taking, and ultimately the industry that will eventually hire the mathematician.  This, however, does raise a fundamental question.  Is it acceptable to provide this sort of accommodation to a blind student?  To allow them to pass every math class that is taught that a student could take that employs graphing?  I'm speaking here of college math majors that are going to go out into the working world to do mathematics in industry professionally.
> 
> Certainly, I think accomodations should be made to lower level courses such as Algebra, Geometry, Trig, maybe even pre-Calc that are taught at the High School level.  Students are required to have a certain level of competency in mathematics, so we do need to provide high school students (blind students) with the ability to pass those courses and learn just like the sighted counterparts.  And I think we can agree that there will be a great number of students (blind and sighted alike) that will have to take some of these courses that have no interest in pursuing scientific coursework that will lead to professional careers in mathematics.  So accomodations need to be made for blind students in High School ... with respect to mathematics that involves graphing so that the blind student can graduate high school and not feel short-changed.
> 
> However, I wonder if the same accommodation should be made at the collegiate level.
> 
> let's say that a blind student is taking mathematics in college (i.e. Intermediate Algebra or in College Algebra or even Calculus or beyond) where graphing problems are taught and let's further say that the student is blind and that they want to eventually get a job as a graphics programmer who works on 3D flight simulators.  The conventional requirement is that this job requires that the applicant must possess knowledge of advanced mathematics and graphing so that they can successfully design and program 3D shapes into the computer.  This of course is a highly visual job.  Normally sighted workers would likely use their eyesight to visually verify that the software is functioning properly.  I'm sure they'd also use analysis of numerical data to do this to, but some visual verification would likely be required.  Being able to see what is happening visually is ... in my opinion ... almost mandatory.  Perhaps not 100% mandatory, and I think I can compromise on this point by saying that if the person desiring the job could generate a solution set in points and could know that their software was correctly generating the images that were required, then they could probably perform the job.  But this brings up an important question.  Should a student at the college level be able to pass the class without performing the visual graphing?  The teacher has no way to know what type of job the student will eventually seek and if the student is taking an advanced class such as Calculus or Differential Equations, an advisor would need to consider the goal of the student.  If that job was eventually the 3D flight simulator job, and if visual methods were required as part of the job, would the school be doing the student a favor to allow them to use the alternate method to demonstrate understanding and pass the class?  Would they be doing the industry a favor?  Certainly software could be written in industry to accept the alternate solution.  But would it be prudent for the college to allow the student to pass the class without being able to demonstrate that they can perform the visual skill ... graphing equations in this case?  If the student graduated the college in a degree such as mathematics and eventually got a job in such a field, the company would assume that the student could meet the challenges that the company was going to throw at them.  In a graphics job, this would be a visual environment.
> 
> Don't get me wrong.  I don't want to throw water on someone who has the ambition to do such a task.  I'm just asking at what point is it a requirement to be able to perform the visual task of graphing?  Is there such a point?  Or, do we lobby for accomodations in industry (for example) where the company needs to come up with the alternate method so that blind people can assume a role that leverages their God given talent.  This is a difficult question since we want the blind person to succeed, but we also want to ensure that the blind person can fulfill the job that they ultimately want to perform.
> 
> My personal belief is that if we can come up with a solution that allows the alternate method to be easily integrated into the workforce, then using that alternate method would not be a problem and that a blind person would simply be allowed to use a different technique.  However, industry changes slowly sometimes.  It might be difficult to convince them to change their systems of doing things.
> 
> What do you think people?  Should we require at some point that blind people should be able to perform the sighted task in order to attain a certain level?  It's a difficult question because you want to be fair to the blind person and the job.
> 
> Let's see how much discussion this generates.
> 
> 
>> To: blindmath at nfbnet.org
>> Date: Sat, 24 May 2014 14:37:16 -0600
>> Subject: [Blindmath] Solving systems by Graphing
>> From: blindmath at nfbnet.org
>> 
>> Just because someone is able to push the buttons on a calculator or to use 
>> math software does not mean they understand math.  The new way of teaching 
>> math that relies on the use of these tools makes it easier for sighted 
>> students and harder for blind students to get good grades in a math class 
>> but it also makes it harder for everyone to gain an understanding of math. 
>> There is definitely something serously wrong about the way math is being 
>> taught now in the United States and it sounds as if we aren't the only ones 
>> with a problem.  It seems to me that with proper teaching blind students 
>> should be able to understand math just as well as sighted ones.
>> 
>> Now I want to write about the subject of this email.  I am going to try to 
>> give an introductory explanation that I think blind students should be able 
>> to understand.  I would appreciate comments about anything that is hard to 
>> understand in order for me to get a better appreciation of the problems some 
>> of you are experiencing.
>> 
>> When studying for a test it is good to memorize definitions.  I like this 
>> definition of a system of equations from a 1945 Intermediate Algebra book. 
>> "When two or more equations are considered together, and a common solution 
>> for them is desired, this is called a system of equations."
>> 
>> This book goes on to state that a set or system of equations which make 
>> contradictory statements is called an inconsistent system. An example would 
>> be if the system has two equations and the first one states that x plus y 
>> equals zero and the second one states that x plus y equals one.  Since both 
>> equations cannot be true at the same time they are inconsistent.  In other 
>> words, they don't have a common solution.
>> 
>> Whether or not you understand the reason why you can at least  memorize the 
>> statement that one way to tell if equations are inconsistent is to graph 
>> them.  If the graphs don't cross yes they are inconsistent. So you could at 
>> least get this fact correct on a true false test.
>> 
>> Now let's back up and try to understand this statement about graphs crossing 
>> with some simple examples.  In algebra you first start with solving simple 
>> linear equations in one unknown.  Solving means finding the value of the 
>> unknown. An example is x plus three equals zero where x represents the 
>> unknown.  A standard way to solve such an equation is to rewrite it so the x 
>> is on one side of the equals mark and the numbers are on the other.  So x 
>> plus three equals zero can be rewritten as x equals minus three. You can 
>> check this result by going back to the first equation and substituting the 
>> minus three for x.  Minus three plus three does equal zero!
>> 
>> Now lets consider a simple equation with two unknowns. x plus y plus three 
>> equals zero. There is not just one right answer. Typically this type of 
>> equation is rewritten with the y on one side of the equals mark and 
>> everything else on the other side. In this case we would get y equals 
>> negative x minus three. The value of y depends on the value we choose for x. 
>> If we choose x equals zero then y = -3. If we choose x = 1 then y = -4. An 
>> alternative to graphing that everyone should be able to do is to make a 
>> table or chart. You could do this in Excel or other math software but you 
>> should be able to do this by hand.  You can make a simple braille chart 
>> using a slate and stylus. Put some choices for x in one column and then what 
>> y turns out to be for that particular x in another column in the same row. 
>> The chart is easier to read if you choose sequential values for x such as 
>> minus one, zero, one, two, three and so on in the column for x.
>> 
>> You can use this chart method as an alternative to graphing  All a graph 
>> does is uses markers to represent each pair of values of x and y that are in 
>> the same row of the chart.
>> 
>> Now lets see how to use a  brailled chart with three columns to illustrate 
>> our earlier example of two inconsistent equations.  The first equation is x 
>> plus y equals zero and the second  equation is x plus y equals 1.  Now make 
>> a chart where you put a value for x in the first column, the corresponding 
>> value for y according to the first equation in the same row of the second 
>> column and the corresponding value for y according to the second equation in 
>> the same row of the third column.  Then add some more rows with different 
>> values of x. Now see if you can find a row where both values of y are the 
>> same for the same x.  If you can't that means that if you had graphed these 
>> two equations they wouldn't cross.  Of course you need to put enough rows in 
>> the chart to see the pattern to be sure you will never get to a row where 
>> both values of y are the same. No matter how many rows you add for this 
>> example you won't find a row with the same values for x because these two 
>> equations are inconsistent.
>> 
>> I am hoping that if you have trouble understanding what I've written here 
>> you will take the time to figure out where you got lost.  Then maybe we can 
>> work together to begin developing some suggestions for teachers who want to 
>> do a better job helping all their students really understand math. One of 
>> the wonderful things about math is that once you understand it no one can 
>> take it away from you by changing the rules.
>> 
>> Best wishes,
>> SusanJ 
>> 
>> 
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