Who is good at seeing equations, but bad at graphing?

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My daughter has hit a road block, and I am trying to help her get it. She has always had math come so easily to her. But now that she had to find systems of solutions and can't see the answer in her head, she is becoming frustrated. She even confuses the X and Y axises. I have told her that number lines (the ones she has done) are X, so the line going the same way on a graph is also X, but she gets confused. She literally has bad dreams involving graphing. It is odd, because I see the world of math as a graph (a large multi-dimensional graph). She sees math as rules, numbers and formulas. The two of us together would make an unstoppable force in the world of mathematics, but since I see the world graphically, I don't know how to explain it to her so that she understands.

She knows the steps to making a line. She knows point-slope, she knows the form for the equation of a line, but when given a system, she freezes. She cannot "see" the solution and it frustrates her. She seems to understand that the solution to the system is where the two equations meet, but it seems to be more of a fear than a real misunderstanding. I think what started to confuse her was when I just started sketching rough graphs using an x-axis, a y-axis and tick marks. She could not see the point I drew. She could not even estimate it. I have since bought her graphing paper so that she can count and do the mechanics, but she really needs to know these for when she starts trig and Calc. She won't always have graphing paper. I know if I can find a way to explain it to her, it will click and she will be fine, but I have no clue how to explain it without using the graph as an example. Besides, how do you fix mixing up the X and Y axises?? (She does have a visual processing disorder). I have labeled the axises for her, but that just confuses her more, because "It looks like science (which she gets), but she refuses to see that the math she is doing in Algebra and the science she is doing are the EXACT SAME THING!! Maybe I should talk to her physical science teacher...

Anyone have any ideas? I have a week to come up with a plan. She is leaving for band camp and I get to write her next lesson plans.

what is her age? (not that it has any real implications beside how to explain it)

sometimes shifting focus completely away from the math and using other concepts like for an example her school days and the hours she has each day, not to learn the math but to learn the graphical relation the axis have towards eachother(very crude but usually once the first two pieces are in place the rest of the puzzle is simply a matter of time)

i think that seeing some data she can already relate to instinctually might nudge those pieces along

am i getting this right that it is actually the relation between the graphical and mathematical problems she is having trouble with?


(disclaimer i am going of the top of my head as in a brainstorm)

She is almost 16, and is entering 10th grade. This is Algebra I work. The science class she just took was Physical Science. They covered some geology, physics and chemistry. She had the harder time with physics, and waves were very difficult for her to grasp. They are not numerical (at least they didn't look that way to her), and PLUS, they are on a graph! (sin waves). Yes, she is having issues relating the graphical representation to the math, but I also think the larger problem for her is that mathematics has always come so easy to her. All she has to do is look at the question (as long as it is numerical) and she sees the question, the work and the solution in her mind. She says the work it all there in her head. She sees it. One of her intervention specialists told her that was impossible, but I believe her. That's how I am with graphs. Now, when she was trying to see the solution to the system of linear equations, she broke down into tears because she could not see where the lines intersected in her head. She could work out the problem mathematically, but using only the graphing technique, she couldn't do it (in her head). I think it is good that she came to this road block (makes her realize that she always has things to learn), but now that she has hit it, I need to help her get around it. I also think t hat if she gets geometry (I am nervous for her on that one), then she may be able to relate trig and geometry to graphing. I hope, at least. For me, I relate the math to the grapg and get it. It is when I don't know what the graph of some rule looks like that I have trouble with it.

As far as her age, I think what is more important is where her mind is. She is capable of doing advanced calculus work, if only she knew the rules of algebra. I have already covered the basics of limits, derivatives and integrals with her (but when it came to factoring, she had not yet learned this, so I didn't go any further, plus she has no clue about trig yet. She is just entering geometry and Algebra 2 this year). She sat in on one of my Calc II classes on sequences, and afterwords the teacher asked her if she understood. She just looked at him "like duh" and said she did. She also sat in on a few of my Discrete math classes and understood better than half the class (including me) did. I wish math came that easily to me.

Teach her Gaussian Elimination. Faster and more efficient than graphical approaches anyways.

If she's able to do the math symbolically and numerically, the graphical stuff doesn't really matter much.

orwell has a point, graphical representation is a tool to promote understanding of mathematics and not math itself.
that being said if it's for school work where grades are involved she probably have no way of avoiding graphs.

im sorry but i am kind of stumped, if i think of anything i will update, (darn been writing this for 20 minutes, deleting every idea i had because it didnt hold up, )

True Orwell, but she is going to be tested by the school on graphing (she has to pass this test to go to Algebra 2 - her dad placed her in Algebra 1a, where she had absolutely no business being - I am teaching her the second half this summer), so even though it is a better way and much easier, she still has to learn the way the school says she has to learn it. Plus, if she does not understand graphing, it will make some of the higher level math classes more difficult.

What exactly do you mean by higher level? My math professors have often gone out of their way to ensure that their problems would be excessively difficult to solve graphically, without making them any harder to do using the normal symbolic method.

If graphing is a weakness, then perhaps she could use the normal algebraic techniques (like Gaussian elimination) to work out the relevant information and then translate that into a graph, rather than trying to do the actual work on a graph.

I'm the same way with math. Equations make sense, and I could never show my work because it didn't break down that far.

I learned graphing (quite tediously) by memorizing tables of rules for graphing, and learning to dissect the equations. Perhaps you could completely set the graphs aside, and begin by defining equations for all of their elements. Once that is reasonably grasped, then work on translating it into the visual representation.

As for the axis issue, perhaps you could rename the axes and work on defining them by their orientation. I think, then, that having the tests labelled with her axis labels could be considered reasonable, although it will add extra interpretation when the variables are given representative x/y terminologies.

Once a student gets into higher order equations (quadratic and higher), differential equations and partial differential equations, graphing is not a feasible method for solution. Rough graphing can express some qualitative aspects of the mathematical functions but are unsuitable for an kind of numerical solution. Why? Graphing is basically a two dimensional operation and most problems are in domains of three dimensions and more. In addition some of the manifolds on which the functions operate are not globally euclidean. Euclidean planes are dandy for cross sections and projections but do not fully capture the details.

ruveyn

Is she trying to make a visual graph in her head and then read off numbers?

Can she read off the coordinates of a point normally if it's a graph with tick marks but no lines nearby? When I do this, I kind of imagine a line going from the point to the axis in a perpendicular way, and if she's not so great with visualizing/visual processing maybe that's the hard part.


Relate it to something else. North/South/East/West if she gets that. If you put the paper on the wall, y is up/down, x is like the floor.


Math is the language of science. It looks the same because science is using math to talk about things.


What work does she have to show on a test? If she can easily do the symbolic calculation in her head, she'll get the right answer, then all she needs is whatever graphical work they want. She understands how to graph lines, so she could just find the answer, write it down, and then graph both lines, if that's all the work they want.

I had trouble with drawing the X and Y axis correctly in high school, I got them backwards frequently. To me they are kind of arbitrary but one thing that kind of helped me (besides always getting points off for doing it wrong and learning the 'hard way') was to think of equations in terms of functions. Basically instead of y = 2x + 3, I think in my head f(x) = 2x + 3. It is kind of a subtle difference but it helped me with graphing equations. When I use function notation it becomes obvious that x is an independent variable, and as a general rule I draw the independent variable on the horizontal axis.

For plotting I like to work with the slope. I always think of the slope as rise/run, so a slope of 2 is really 2/1, meaning go up two units and right 1... this is another easy way to figure out how to graph things. Just plot the y-intercept and then follow the 'built-in instructions' given by the slope to find another point on the line.

Estimating answers off of a graph always bugs me too, I would get hung up on whether the answer was 4.4 or 4.5 or something. However, there is one thing to keep in mind when it comes to homework problems and exams. They tend to design problems where the solution is a nice whole number or numbers with simple fractions, 1/2, 1/4, etc.

Humm..... very interesting. For me, I often see things in terms of the way in which things "should be" but in fact each point on a graph is a valid reference, which can be represented as a variable. Of course manipulating references does not change the number, but the reference. When we are thinking of solutions in the case of this problem, we aren't really interesting in all the references outside the intersection, but we are thinking about how linear operations on references will or will not lead to the same reference. In other words, I think of the line as a repated operation on a reference.

So back to the original problem. I once found a book I think, "A Journey to Genius" which had a proof where you could convert a rectangle into a square. It took me about 45 minute at starring at it, where my mind finally realized with excitement that I could treat the rectangle as the number I wanted the root to, and the square as the area with the side who is the square root of the number. In other words the height of the rectangle was 1 and the width was the number I wanted the root to. I could then apply the method to come up with the answer I wanted.

Let me tell you, I think some of this math stuff requires so much concentration and focus, that had I not looked at that book and refused to give up, I would never have had that realization on how cool the proof really was.

Perhaps your daughter will find even greater success if she can do the same... never give up.

Well, she is back from band camp and I think we have found the problem. She is way too precise. If I make a graph like this:

|
3--- *
|
---------|---
| 1
|

(Sorry for it being crude), she cannot see that the point is (1,3). She cannot see it because it is not exact enough. If I put tick marks that are in groups of 5 (she wanted groups of 10, but I convinced her that 5 was a nice rounding number too), she can see the imaginary lines that make the grid pattern, but just putting a tick mark for each number is beyond her head at this point in time. I will have to have them add it into her plan that at least for this year, she is to use graphing paper for all tests. She can experiment all she wants with the homework.

It really bothers her when something is not precise. She almost had a meltdown tonight over it (other things were bothering her too, but this was just too much for her to handle). One positive step was something she remembered her 5th grade teacher telling her about the X-Y coordinate plane. You must first w*k the ground before you can climb a tree. This was to get them to write the coordinates in the correct (x, y) format, but she is also going to try to use it to remember the axises as well. I also explained this to her (and I know it is not entirely accurate, but she will not be dealing with Z until later and she knows the basic concept). I had a pen on the table. I told her that when she first started solving for a variable, they used x (when not talking about lines). It is like a table. It is flat. They used number lines. They are flat like a table. Just one dimension. When they introduced lines, they added the y to the equation (no pun intended). This brought a second dimension, and I started to have the pen rise at an angle on the table. I said y kept it from being flat. And to do that, it needs to be up and down. Then I told her that Z make it 3-D (and made the pen face away from a piece of paper). I know that's not it exactly, but I wanted to show her how that worked (she would have asked if I did not say it). I also told her to remember them in ABC (or XYZ) order. X is first, making one dimension. Y is second making it 2-D and Z is third making it 3-D. Even if I explained it here in a confusing way, she actually got it and that's when she said the walk before you climb thing.

Now I get to drill her on graphing. She is so frustrated with her processing speed. She is very smart, but it takes a while to come out. I have to tell her over and over that I don't mind if she takes a while, as long as she understands the problem.

In the same para talking about precision too. /joke.

Thats what I've been trying to figure out lately, is how to bring from a graph to an equation, when the graph is differentiable.