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When teaching multidimensional differentials (I'm assuming the students grasped the one-dimensional case), there are many useful parallels relating to spatial imagination.

For example, when explaining that the gradient of a function is the direction of the steepest slope and properties of other related vectors, it is inspiring to let them imagine a mountain path which keeps its height. Then, the path is in the direction perpendicular to gradient, and if we look up perpendicular to path, then we would often see the peak.

Another example might be to explain Lagrange multipliers and related optimization methods using an orange and the thickness of its peel.

However, this does not goes through to students, which have poor spatial imagination. They are unable to see themselves in the mountains looking at the slope.

The questions are:

  • Are there any good, non-spatial parallels on multidimensional differentials? E.g. for single-dimensional case there is position/velocity/acceleration, but this is much harder with multiple dimensions.
  • Are there any methods to make spatial analogies easier to students? E.g. making a model could be one solutions, the problem is, it is often unfeasible.
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  • $\begingroup$ Out of interest, can you explain how the Lagrange multipliers relate to the orange peel? $\endgroup$ – DavidButlerUofA Jul 14 '14 at 19:58
  • $\begingroup$ @DavidButlerUofA Think of finding a spot with an extreme (minimal or maximal) thickness of the peel. $\endgroup$ – dtldarek Jul 16 '14 at 2:58
  • $\begingroup$ @dtlarek So it's "maximise thickness, subject to being on the surface of the orange"? I can see you could use Lagrange multipliers to do this, but does the Lagrange multiplier itself have a physical meaning on the orange? $\endgroup$ – DavidButlerUofA Jul 16 '14 at 6:01
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In my experience, being too tightly wed to geometry was actually a hinderance to me when I was learning multivariable differential calculus.

In my mind, the essential feature of the multivariable derivative is that it gives a linear map between tangent spaces.

I strongly suggest giving your students enough linear algebra to at least understand what a linear map is, and how you can associate a matrix with a linear map, before starting to teach them the multivariable derivative. To do otherwise would be like teaching the single variable derivative before understanding the slope of a line.

If you are interested, Jim Fowler and I are writing an online multivariable calculus course which takes this approach here. We attempt to get the student to construct all of the mathematics through engaging exercises, with hints. We will also be covering the rudiments of multilinear algebra to have a convenient algebraic package for higher order derivatives.

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  • $\begingroup$ Indeed, the Fréchet derivative definition is of the form "the matrix is the best (whatever that means) linear approximation of the given function" and this my favorite intuition. On the other hand, sometimes it is too high-level (e.g. it is great to think about permutations as bijections of type $A \to A$ for some finite set $A$, instead of, for example, orderings of numbers; yet, at first it is often too much). In such case I can give some spatial explanations, and the part that gets it have unfair advantage above the rest. $\endgroup$ – dtldarek Mar 14 '14 at 10:00
  • $\begingroup$ I think if you devote enough attention to the linear algebra prerequisites, and drive home that the derivative is the best linear approximation, not only will you reach students how are not great visualizers, but you will enable them to understand differentiation in high dimensional contexts where we really do not have good ``visual'' intuition. $\endgroup$ – Steven Gubkin Mar 14 '14 at 10:50
  • $\begingroup$ Yeah, but, for example, the intuitive connection between shape, principal curvatures and eigenvalues is priceless. Also, visualizing how the matrix norm, the shape of transformed unit ball and the largest eigenvalue relate to each other is impossible to substitute with something else. I know, that that's not about derivatives, but I wanted to say that spatial intuitions are very helpful. Why refuse to use them if there's little to lose and much to gain? And so, if there is a way, why shouldn't I help students with underdeveloped spatial imagination learn that too? Hence, the OP. $\endgroup$ – dtldarek Mar 15 '14 at 22:35
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    $\begingroup$ @dtldarek One of your questions was "Are there any good, non-spatial parallels on multidimensional differentials?". I was answering this part of the question. I am a highly visual person, often to a fault. Just trying to offer another perspective in this answer. I really do not know how to help someone "visualize the mountain". I do keep playdough in my desk when I teach multiV so I can sculpt with students during office hours... $\endgroup$ – Steven Gubkin Mar 16 '14 at 0:47
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    $\begingroup$ I'm not saying that your answer is wrong, on the contrary, I like it very much! I've just got an impression that you suggested that visualizations are not important, which I see, after you last comment, is a misunderstanding. $\endgroup$ – dtldarek Mar 16 '14 at 10:43
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If you have a piece of paper, you can model a 2-dimensional surface! Tape a piece of graph / grid paper on to a desk with an object underneath (a cell phone, face-down, perhaps?), and you can talk about the slope at different points on the paper and how it changes as you move.

A few questions you could ask:

  • Put your finger on the paper and follow a line to the left. Is your finger moving closer to the ceiling, or further away from it?

  • Trace a line around the [object you put under the paper]. If you follow that line, does your finger move up or down? Why or why not?

  • Put your finger on the paper at the edge of your object. If you move to the left, does your finger move up or down? What if you move forwards or backwards?

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  • $\begingroup$ Did you try this in practice? $\endgroup$ – dtldarek Mar 14 '14 at 2:33
  • $\begingroup$ One-on-one, yes - but I think it would scale pretty easily. I've added a few questions you could ask to my answer. $\endgroup$ – Andy Mikula Mar 14 '14 at 2:37
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    $\begingroup$ I tried this one semester, and based on the comments I got in course evaluations, it left an impression. Not, however, a positive impression. $\endgroup$ – Henry Towsner Mar 14 '14 at 3:39
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Geometric models

Students who have trouble imagining often need to see it with their own eyes in order to imagine. So a model is often the best approach to support their imagination, and it's not too infeasible.

Using playdough

You can buy commercial playdough, or make your own if you want a really big blob. (Recipe below)

I like to draw an x and y axis on a piece of paper, and then put the playdough on top in some sort of shape. To mark points on the surface, you can poke a tiny hole with a sharp pencil. A sharp pencil can also be used to scratch lines in the surface to make level curves.

If you pick a line and slice it with a knife, you can show the exposed face and note that it makes an ordinary curve, which has slopes of its own. Marking an axis on the paper along the knife-cut helps to point out that direcitonal derivatives require a new axis with points marked by a unit vector.

Slicing parallel to the x-axis or y-axis helps you point out the meaning of the ordinary partial derivatives.

If you are in a lecture theatre with a document camera, you can very successfully do this on the document camera -- just remember to lift and tilt it every so often so they can see the three-dimensional nature of it. If you have tutorials with a smallish number of students, you can get them to stand around one table.

PLAYDOUGH RECIPE (from McKenzie's Cream of Tartar box):
Ingredients:
2 cups plain flour, 1 cup cooking salt, 4 tbsp cream of tartar powder, 2 tbsp cooking oil, 2 cups water, food colouring
Instructions:
1. mix the dry ingredients in a saucepan.
2. add the wet ingredients including the colour and mix well.
3. cook over medium heat stirring continuously until the mixture congeals.
4. tip out and allow to cool.
5. knead until smooth.
6. store in a plastic bag in an airtight container.

Using a sand-box

I haven't tried it myself yet, but I think a clear container with slightly wet sand could do a similar job to playdough, though it is messier.

Cardboard for tangent planes

When talking about tangent planes, students often have real trouble imagining them. You need to have a proper model of the plane. Paper is too floppy and so doesn't work. You need something thicker. Any cardboard should do. You can draw vectors on the plane to represent the x-slope and the y-slope. A playing card would be a good thing to use -- you could even use pins to attach several of them to the playdough at the points they are tangent to.

Water levels for level curves

If you can get your hands on a big clear container and a heavy object, then you can explain level curves using water. Put the object into the container and start pouring water slowly. The edge of the water describes a level curve on the surface. I reckon it would be even more striking with coloured water, especially if the object was a very different colour, though I haven't tried this.

Potato printing for level curves

Take a potato (or other vegetable/object) and cut it. Dip the end in paint and press it down on a piece of paper. The shape left is the shape of the level curve. Slicing it again at another height and redoing it can show how the curve changes as you move up the potato. Changing colours as you go can highlight a sequence of level curves.

Slightly less geometric models

The temperature or smell analogy

Instead of trying to visualise the function, you can appeal to the students' other senses. I often use the analogy that at each point in a room, there is a temperature -- each point "feels" hot or cold. A gradient in a particular direction is how quickly the temperature changes when moving that direction. For example, the temperature will increase when you move towards the heater.

Another alternative is smell. You can imagine a smell being stronger in one place in a room than in another.

A visual version in 2D is to use colour -- possibly red-to-blue gradients to match with the temperature.

These three reinforce the idea that the function doesn't have to represent a height, and that you can think of it like the function value "lives" at each point -- if you inspect the point itself you can make it reveal its function value.

Colour-coded level curves

One way of thinking about a function is that at each point there is a value right there for the function. If you colour in all the points with the same function value, you will make curves. If you use a gradient of colour for changing values (say blue-to-red), students can see the speed at which colour is changing.

If you space your curves evenly by function-value, then you can pick any point and any direction and draw a line, noticing how far you have to travel to get to the next function value. This is a representation of the rate of change. When lines are close together you pass by a lot of them very quickly so the function value is changing quickly.

It is important to colour the lines so that you can tell whether it's going up or down.

Before-and-after

One way to think of functions is as input-output machines. A two-variable function takes a point in 2D as input and produces a number as an output -- which can be interpreted as a line. So you can actually draw two separate pictures -- a plane for the input and a line for the output.

You can mark points in the plane, and then mark the matching points on the line. Colour-coding can help to identify which point produces which output. If you move an input point along a line, you can investigate how quickly the output point moves.

Geogebra should allow you to do this by having two graphics windows open at once. Make a point A in one window. In the second window turn of the x-axis and make a point B = (0, x(A) + y(A)^2) [or whatever function]. Then try moving the point A and see what happens to the point B. See a quickly-made example here

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  • $\begingroup$ Just added another example. $\endgroup$ – DavidButlerUofA Jul 25 '14 at 6:51
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You can make 3D images that can be manipulated by mouse and put them on the web. There are a bunch of examples at http://neil-strickland.staff.shef.ac.uk/courses/MAS243/pics/. These are done with a Java applet called JavaView (http://www.javaview.de/) which is a little unfortunate as browsers are getting rather aggressive about applet security policies these days. At some point I should convert to some kind of HTML5 version that does not require any plugins, but I am not teaching the relevant course at the moment so I have not done that.

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