[Blindmath] FW: Solving systems by Graphing

[sabra1023 via Blindmath]

I would say that it's like if you had a child with autism. You would point out to them directly social rules they didn't understand. I think it's the same with spatial disorientation. You need to directly point out cause and effect relationships and the environment that you think should just be intuited. Someone with spatial disorientation won't into it does things though. I used to have that a lot when I was traveling, and I was able to make some of it go away. For me, I think a lot of what caused it is that I didn't have a lot of opportunities to travel and move independently when I was young. For instance, you could go to a shopping center with a lot of stores. You could go to one store, then identify with your children where the other stores are from that location. Then you could select a different store and do the same thing. You could do the same thing with aisles and the grocery store. Once they get better, you could take them to a random store or random aisle and they have to identify where they are and then find the other stores again. You can also do the same thing with streets. If your city has an address system that can make the street layout make more sense, you could use that as well. It also helps me to write down information about an area and study it later if I have trouble remembering. You can also do what I described above in an unfamiliar room. You have to focus on how the locations of different things relate to each other. When you start out, just identify the location of one thing, and then identify the location of two other things in relation to that thing. Another thing you could do is to explore when you throw something, what does it do when where does it end up? Be it would also help that when you're driving in the car, you could describe what you're doing what streets are passing, and how you know where you are. Maybe your children won't be able to use the same landmarks you do, but they can at least understand the concept of identifying landmarks and use what you do to identify some of their own. Also, being aware of what your bodies doing is important. A lot of times, I feel like I'm walking in a straight line when I'm not. Let's say I'm at point A. I know that if I walk straight from point a, I will get to point B. from my point of view, I did walk straight from point a, but point B didn't show up. Now, I would know that I probably didn't walk in a straight line, but when I was younger, I just thought that things kept on moving around. Used to also turn without realizing it at all. I felt like I was walking straight. A way to fix this is to use landmarks and your environment to help you. For instance, you should have your children focus on the street and that street should always be in the same place when traveling. If the street is supposed to be on your right, but somehow the street is behind you, you know that you turned by accident. You can use the sun as well, but it's bad to get into the habit of using the grass or trailing walls because the grass can curve around and make you fear into the parking lot, and in addition to making you look weird and childish, constantly trailing walls can make you turn down hallways that you didn't mean to. Also, you want to make sure that the methods you use it will also work for going through open spaces. For instance, you might need to get to the other side of the parking lot and you might need to walk in a straight line.

> On May 28, 2014, at 12:43 PM, Lynn Reed via Blindmath <blindmath at nfbnet.org> wrote:
> 
> Steve-I want to help both of my legally blind kids with exactly what you were talking about. Spacial awareness. They are 12 & 14. We are a busy family but I love to throw in useful things when I can. Other than the things you talked about before,what kinds of other things should I try to reinforce the concepts?
> 
> 
> Peace and Love to all!
> 
> 
>> On May 28, 2014, at 9:08 AM, Steve Jacobson via Blindmath <blindmath at nfbnet.org> wrote:
>> 
>> From what I have experienced and observed, the ability of blind persons to get useful information from graphs 
>> varies a great deal.  I know that there are differences between people, blind or sighted, in how easily such 
>> things as graphs and maps are interepreted.  We likely have all encountered sighted people who can't make much 
>> sense of road maps.  <smile>  Yet, in my case, I have found I can usually picture a graph even though I have not 
>> ever seen a graph except as a tactile representation.  If someone is good at generally describing a graph, it can 
>> give me useful information more quickly than I would derive from looking of tables of coordinates, even though 
>> tables of coordinates would also give me that information.  One of my math professors was fascinated by my ability 
>> to handle materials on the revolution of a solid, where one had to grasp the concept of what sort of a solid 
>> object would result from rotating a two-dimensional graph on an axis.  He concluded that my ability to picture it 
>> in three dimensions gave me an advantage over some other students who had trouble with representing in on two-
>> dimensional paper.  I don't honestly know how true that was, because as we have all experienced, some people are 
>> amazed that we can do anything.  
>> 
>> While I believe some of this ability is present to a greater degree in some people, I also think that exposure to 
>> the representation of spacial concepts when we are young can have an impact as well.  When I was a kid, I was very 
>> curious about maps because looking at maps was often a part of our vacation planning.  I had a puzzle map of the 
>> United States which taught me a lot of things that were not obvious to me as a child.  My mother made a little map 
>> of our neighborhood for me by carving streets into a bar of wax, not high tech or fancy, but it worked.  I think 
>> all of these things caused me to picture representations of larger areas which was later translated into an 
>> ability to derive useful information from tactile graphs.  I would like to see more kids have those opportunities.  
>> I own APH's talking maps and I love them for checking out the routes and understanding where streets go.  Still, 
>> they don't give me the kind of picture that a tactile map gives me.  It is harder to know that two parallel 
>> streets might almost come together in spots, etc.  I hope that technology will help to imprint some of this on 
>> kids and not make it such that the need is no longer recognized.  What I do not think is often considered is that 
>> sighted people get exposure to spacial concepts in a way that we don't all the time.  Kids are seeing seating 
>> charts, floor plans, and even photographs than convey this information daily, while we tend to have to seek out 
>> such information, and if we are not curious about such things we don't get the information.  Recognizing that 
>> given the same input, we won't all derive the same benefits, I feel the effort to understand maps and graphs at 
>> least to some degree will also make us better travelers.  Nevertheless, if one did not have some of the 
>> opportunities that I had and if they find that audio graphs or refining ones skills to make generalizations from 
>> tables of coordinates works, that can work as well.  I'm glad there are alternatives. 
>> 
>> Best regards,
>> 
>> Steve Jacobson
>> 
>>> On Tue, 27 May 2014 23:24:03 -0500, sabra1023 via Blindmath wrote:
>>> 
>>> That's why I'm saying that for the lower level maths, a blind person should be making the graphs, but by the time
>> you get to higher-level maths like calculus, you start to know what shape agraffe is based on the equation and you 
>> can adequately describe it. The equations are so exact that you can't make it on the board. For instance, you 
>> could write down points that would be on the graph, asymptotes, intercepts, and transformations without having to 
>> physically make the graph. I like audio graphs better than tactile grass anyways because I can always replay the 
>> sound in my head, but I can never remember what it Tactel graph feels like once it goes away. Realistically though 
>> what I would do if I needed to make a graph and the workplace would be to write down the points I needed, use them 
>> to make a table in Excel, then specify what graph I wanted. For instance, I could tell XL to turn my table into a 
>> pie graph, a bar graph, and so on. Also, when you make a PowerPoint, you can insert a graph, it opens up an Excel, 
>> and then you can modify the provided Templet so that your values are in the right place. Once you close XL, a 
>> picture of your graph is in the PowerPoint. A graphs shape can tell me something simple like if data has an upward 
>> or downward trend, but ultimately, graphs mean very little to me. I don't think of information is being in a 
>> physical shape. For instance, the sign graph and the cosine graph are both just waiting lines to me. When I was 
>> taught about them using tactile graphs, I thought they were exactly the same, which Innoway they were. They are 
>> exactly the same shape, they are just in different positions so their points are different. If I had been started 
>> with a table of values, and then used that table to understand the shape of the graph rather than the other way 
>> around, I would've understood it much better. Unfortunately, this didn't happen for me until college. A lot of 
>> grass feel the same to me, and they are hard to read without the little squares underneath. Looking at equations 
>> or tables of values before the graphs helps me know what their differences are so I can actively look for them. I 
>> made a list for myself of differences to look for, but if a difference isn't on that list, I will miss it. For 
>> instance, one graph might have twice the exponential growth of another graph. Say that I have felt both graphs, 
>> gone through every difference on my list, and that wasn't one of them. Then I won't be able to tell how both 
>> graphs are different the matter how hard I try. They could have lots of differences, but figuring that out without 
>> any type of guidance causes my brain to process too much unnatural information and shut down. I can sometimes 
>> bypass this by using real objects like moving rubber bands around on a pegboard to add vectors rather than having 
>> to feel it on a piece of paper. Another example is using a pulley to learn about Lanier and angular speed. When I 
>> was younger, I even had problems like a ladder leaning against a tree. I couldn't understand that that was 
>> supposed to be a triangle. Scatterplots are the graph I hate the most. Because the dots are just thrown on the 
>> paper everywhere and there are no lines connecting them, it's really hard for me to get trends in the data from 
>> that. My hand will get lost if it has to go through an open space to read a graph, which is why I need the 
>> squares. I had training to read graphs, and learning to make a graph did help me read the better, but they still 
>> can't be tiny and they can't have lots of lines. Also, I have to have one graph per page. If there to grafts on 
>> one page, they both get jumbled together and I can't separate them in my head. Also, I can't have a picture of a 
>> three-dimensional object like a box or cylinder. I have to have an actual box or actual cylinder, or just get a 
>> verbal description. For instance, instead of giving me a picture of a cube with a height of 3 inches, you could 
>> just tell me that the cube has a height of 3 inches and that I need to get its volume. If I know the height of the 
>> cube or have an actual Cuban measure it, I can get the volume, but if I have a picture, you might not know whether 
>> I can get the volume or not because more than likely, I will get it wrong because the picture won't make sense to 
>> me. It just looks like a bunch of squares, little diamonds, big diamonds, solid lines, and dotted lines. For a 
>> rectangular box, I was able to teach myself at one point to count the little diamonds on the top, the big diamonds 
>> on the top, and the big diamonds on the side, but that is all I was ever able to do.
>> 
>>>> On May 27, 2014, at 7:41 PM, Lynn Reed <iamlvr at yahoo.com> wrote:
>>>> 
>>>> I would say that blind student should dictate intervals,axis,title etc to sighted adult who would then create
>> the graph on a cork board with tacks that rubber bands could then be attached to. If you "truly" think a blind 
>> student would benefit from this,where is the tactile model??? You can't seriously think a blind student could 
>> create this alone, without prior knowledge of how graphing works, right?
>>>> 
>>>> Peace and Love to all!
>>>> 
>>>> 
>>>>> On May 27, 2014, at 10:56 AM, sabra1023 via Blindmath <blindmath at nfbnet.org> wrote:
>>>>> 
>>>>> 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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>>>>>> 
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>>>>> 
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