[Blindmath] FW: Solving systems by Graphing
Lynn Reed via Blindmath
blindmath at nfbnet.org
Wed May 28 21:43:09 UTC 2014
Again,Love it! Thank you!
Anytime.
Peace and Love to all!
> On May 28, 2014, at 1:46 PM, Steve Jacobson via Blindmath <blindmath at nfbnet.org> wrote:
>
> Lynn,
>
> There have already been some good ideas put forth, but one that I found interesting and useful when I was growing
> up was to understand what was on the other side of the wall when in one room of the house, or what is directly
> above or below if there is a second floor or a basement. This requires relating points that cannot be directly
> verified. Sighted kids don't always know what is on the other side of a wall, so such an exercise could involve
> blind and sighted kids. Which part of the yard is under a particular window is another variation. I realize we
> are straying a bit from math here, though, even though there is a connection.
>
> Best regards,
>
> Steve Jacobson
>
>> On Wed, 28 May 2014 10:43:28 -0700, Lynn Reed via Blindmath 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
>>>>>>>>
>>>>>>>>
>>>>>>>> _______________________________________________
>>>>>>>> Blindmath mailing list
>>>>>>>> Blindmath at nfbnet.org
>>>>>>>> http://nfbnet.org/mailman/listinfo/blindmath_nfbnet.org
>>>>>>>> To unsubscribe, change your list options or get your account info for Blindmath:
>>>>>>>> http://nfbnet.org/mailman/options/blindmath_nfbnet.org/mrspock56%40hotmail.com
>>>>>>>> BlindMath Gems can be found at <http://www.blindscience.org/blindmath-gems-home>
>>>>>>>
>>>>>>> _______________________________________________
>>>>>>> Blindmath mailing list
>>>>>>> Blindmath at nfbnet.org
>>>>>>> http://nfbnet.org/mailman/listinfo/blindmath_nfbnet.org
>>>>>>> To unsubscribe, change your list options or get your account info for Blindmath:
>>>>>>> http://nfbnet.org/mailman/options/blindmath_nfbnet.org/sabra1023%40gmail.com
>>>>>>> BlindMath Gems can be found at <http://www.blindscience.org/blindmath-gems-home>
>>>>>>
>>>>>> _______________________________________________
>>>>>> Blindmath mailing list
>>>>>> Blindmath at nfbnet.org
>>>>>> http://nfbnet.org/mailman/listinfo/blindmath_nfbnet.org
>>>>>> To unsubscribe, change your list options or get your account info for Blindmath:
>>>>>> http://nfbnet.org/mailman/options/blindmath_nfbnet.org/iamlvr%40yahoo.com
>>>>>> BlindMath Gems can be found at <http://www.blindscience.org/blindmath-gems-home>
>>>
>>>> _______________________________________________
>>>> Blindmath mailing list
>>>> Blindmath at nfbnet.org
>>>> http://nfbnet.org/mailman/listinfo/blindmath_nfbnet.org
>>>> To unsubscribe, change your list options or get your account info for Blindmath:
>>>> http://nfbnet.org/mailman/options/blindmath_nfbnet.org/steve.jacobson%40visi.com
>>>> BlindMath Gems can be found at <http://www.blindscience.org/blindmath-gems-home>
>>>
>>>
>>>
>>>
>>>
>>> _______________________________________________
>>> Blindmath mailing list
>>> Blindmath at nfbnet.org
>>> http://nfbnet.org/mailman/listinfo/blindmath_nfbnet.org
>>> To unsubscribe, change your list options or get your account info for Blindmath:
>>> http://nfbnet.org/mailman/options/blindmath_nfbnet.org/iamlvr%40yahoo.com
>>> BlindMath Gems can be found at <http://www.blindscience.org/blindmath-gems-home>
>
>> _______________________________________________
>> Blindmath mailing list
>> Blindmath at nfbnet.org
>> http://nfbnet.org/mailman/listinfo/blindmath_nfbnet.org
>> To unsubscribe, change your list options or get your account info for Blindmath:
>> http://nfbnet.org/mailman/options/blindmath_nfbnet.org/steve.jacobson%40visi.com
>> BlindMath Gems can be found at <http://www.blindscience.org/blindmath-gems-home>
>
>
>
>
>
> _______________________________________________
> Blindmath mailing list
> Blindmath at nfbnet.org
> http://nfbnet.org/mailman/listinfo/blindmath_nfbnet.org
> To unsubscribe, change your list options or get your account info for Blindmath:
> http://nfbnet.org/mailman/options/blindmath_nfbnet.org/iamlvr%40yahoo.com
> BlindMath Gems can be found at <http://www.blindscience.org/blindmath-gems-home>
More information about the BlindMath
mailing list