Using "Clickers" with Mathematica's Dynamic Interactivity

Bruce Atwood

Beloit College

Abstract

Scientists have frequently been able to do physical demonstrations in class – while mathematicians have rarely had the opportunity. "Manipulate" provides mathematicians with a tool where we can now easily do classroom experiments. This is particularly effective if combined with a classroom voting technology often called "clickers."

Classroom voting (using clickers) is a significant new educational technology of the last 10 years and is rapidly growing in popularity. How does it work? Using the classroom computer projector, I show students a multiple choice or true/false question in a Mathematica window. Each student has an assigned "clicker" that they use to answer the question. Their individual responses, anonymous to the rest of the class, appear collectively on a bar graph after a certain time period. (Software that comes with the clickers handles the administration of timing, collecting, and displaying student responses.) This gives both the student and teacher immediate feedback. Ideally some questions are sufficiently challenging that about half the class answers correctly. Then I tell the students "Convince your neighbor why your answer is right," and I poll the class again. Frequently the proportion of correct student responses increases dramatically. This makes for an engaged class where students are thinking and having fun.

Mathematica has always been a useful companion to clickers. Frequently I have made a plot or done a computation in class while discussing the answer to a clicker question. With "Manipulate" the marriage of clickers and Mathematica has opened up new ways to enhance student learning. For example, ask the students "What will be the effect on A by moving the slider to change B?" This can then be followed up with another conceptual question: "Why did that happen?" Calculus examples will be presented showing concept questions based on classroom experiments made possible with Mathematica's dynamic interactivity. The Wolfram Demonstrations Project is a great source for materials that can be turned into classroom experiments.

What are clickers?

A good general reference is Clickers in the Classroom, How to Enhance Science Teaching Using Classroom Response Systems, Douglas Duncan, Pearson Education, 2005.

"Clickers" (classroom voting, classroom response systems, personal response systems) are one of the most significant new educational technologies of the last 10 years. How does it work?

Each student is assigned a transmitter (clicker) that they bring to class every day. The buttons on the clicker send a signal to a receiver that is connected to the instructor's computer. Each clicker has a unique identifier, so the computer stores how each individual answered for later grading if desired. Each student's response is unknown to the rest of the class, but a chart with an identifier shows whether their response has been received.

I use a laptop with the question I'm asking in a Mathematica window and the PRS interface in another window, both of which are projected on a screen. I give the students a minute or two to answer a question. At the end of the response period a bar chart shows how many students chose each possible answer.

If 90% of the class answers correctly I go on to the next topic. If few students answer the question correctly I back up. However, if only 50% answer correctly, I ask them to discuss the question with their neighbor and then ask the same question again. Typically the percentage of correct answers increases dramatically. One reason for this is that students who initially answered correctly have a more persuasive argument for their response than those who answered incorrectly. Another reason may be that students new to a concept often know exactly where the difficulties are and are better able than the professor to offer a quick clarification. This use of concept questions for peer instruction is discussed in detail by Mazur (Peer Instruction: A User's Manual, Prentice Hall, 1997)

Fluid statics example

ConcepTest question from Peer Instruction, A User's Manual, Eric Mazur, Prentice Hall, 1997.

A boat carrying a large boulder is floating on a lake. The boulder is thrown overboard and sinks. The water level in the lake (with respect to the shore)
	1) rises.
	2) drops.
	3) remains the same.

Answer

Answer (2): When it is inside the boat, the boulder displaces its weight in water. When it is thrown overboard, it only displaces its volume in water so the water level of the lake with respect to the shore goes down.

Why are clickers useful?

Classroom attention/retention

Violin diagram used with permission from Wikipedia under the GNU Free Documentation License.

Carl Wieman and Katherine Perkins (Physics Today, November 2005) tell the following story:

We brought a violin into class. We explained how, in accordance with the physics we had just explained, the strings do not move enough air to create the sound from the violin. Rather, the strings cause the back of the violin to move via the soundpost, and thus it is the back of the violin that actually produces the sound that is heard. Fifteen minutes later we asked

The sound you hear from a violin is produced
	a) mostly by strings
	b) mostly by wood in back
	c) both equally
	d) none of the above.

The response was only 10% correct:

Some advantages and disadvantages

Advantages

Clickers are an effective learning technique.

Can easily incorporate peer instruction.

Increase student attention and engagement – make students think in class!

Shy students, students that don't speak English well and others reluctant to participate in class discussions can answer in a safe (anonymous) environment.

Class participation from 100% of the students.

It's fun! Adds drama to class.

Immediate feedback to students and teacher.

Students think about their own learning.

Can generate class discussion.

Could be used to test whether students did the assigned reading.

Graded questions increase attendance.

Will particularly help teachers who are trying to rely less on lecturing to make a more active classroom.

Students make positive comments about clickers on evaluations.

Disadvantages

Making up or finding good questions is very time consuming.

Setup and takedown of the equipment takes approximately five minutes at the beginning and end of class.

Other teaching techniques can solve the same problems, so there is no point in introducing clickers if they aren't needed.

Cost and administration.

Less time in class for other things.

Types of questions

Simple questions

Practice, review, gives feedback to student and teacher

Self discovery

"Leading" questions that move students toward a definition or formula, a key step in a derivation or proof

Address misconceptions

"Classical" misconceptions, common mistakes on homework or exam

Harder questions

Make students think hard!

Fun questions

Jokes, etc.

Experiments

In the physical sciences one can often set up a demonstration or experiment in class. These can be used for very effective questions. Alex Pines, Chemistry, U.C. Berkeley: 1) What do you predict will be the outcome of the experiment? 2) Why did that happen?

Classroom math experiments with Mathematica

It isn't often that we can do meaningful physical experiments in math class. We could do animations in earlier versions of Mathematica, but now Manipulate and the Wolfram Demonstrations Project have opened up a world of opportunities for experiments and demonstrations in math.

Period of sine

Manipulate[Plot[Sin[ω t],{t,0,2 π},AxesLabel->{"t","sin(ω t)"}],{ω,1,5,1,Appearance->"Labeled"}]

How does the plot of sin(2t) compare to that of sin(t)?
	1) Oscillates twice as fast.
	2) Oscillates half as fast.

It t has units of seconds then sin(ω t) has period
	1) (ω/(2 π)) seconds
	2) (ω/(2 π)) radians
	3) ((2 π)/ω) seconds
	4) ((2 π)/ω) radians

Squeeze theorem

The manipulate was inspired by an animation by Selwyn Hollis. The questions are from the Good Questions Project at Cornell University.

Manipulate[Plot[{(x^n) Sin[(1/x)],Abs[(x^n)],-Abs[(x^n)]},{x,-(10^(-i)),(10^(-i))},PlotStyle->{Black,{Thick,Red},{Thick,Blue}},MaxRecursion->8,ImagePadding->10],{n,0,3,1,Appearance->"Labeled"},{{i,0,"Zoom"},0,3},FrameLabel->"\!\(\*Cell[TextData[Cell[BoxData[
FormBox[
RowBox[{
SuperscriptBox["x", "n"], " ", 
RowBox[{"sin", "(", 
FractionBox["1", "x"], ")"}]}], TraditionalForm]]]]]\)"]

The reason that (lim_(x->0)) sin((1/x)) does not exist is
	1) because no matter how close x gets to 0, there are x's near 0 for which sin((1/x))=1 and some for which sin((1/x))=-1.
	2) because the function values oscillate around 0.
	3) because (1/0) is undefined.
	4) all of the above.

(lim_(x->0)) (x^2) sin((1/x))
	1) does not exist because no matter how close x gets to 0, there are x's near 0 for which sin((1/x))=1 and some for which sin((1/x))=-1.
	2) does not exist because the function values oscillate around 0.
	3) does not exist because (1/0) is undefined.
	4) equals 0.
	5) equals 1.

Horizontal asymptotes of rational functions

Manipulate[((Expand[Numerator[f]/(x^n)])/(Expand[Denominator[f]/(x^n)])),{f,{((2 (x^2)-x+1)/(4 (x^2)-3 x-1)),((2 (x^2)-1)/(4 (x^3)-5 x-1)),((4 (x^5)+3 (x^2)+1)/(2 (x^4)+(x^2)+3))}},{n,0,5,1,Appearance->"Labeled"},FrameLabel->{{None,None},{None,"Divide numerator and denominator of f by (x^n)"}}]

To find the limit as x->∞ for the second example what would be an appropriate choice for n?
	1) 1
	2) 2
	3) 3
	4) 4
	5) 5

For the third example what is the limit as x->∞?
	1) 1/3
	2) 2
	3) 0
	4) ∞

Newton's method

Adapted from "Two Geometric Views of Newton's Method" from The Wolfram Demonstrations Project
 http://demonstrations.wolfram.com/TwoGeometricViewsOfNewtonsMethod/

newton[f_,x0_,a_,b_,n_]:=Module[{list=NestList[#-((f[#])/(f'[#])) &,x0,n]},Grid[{{Plot[f[x],{x,a,b},ImageSize->5 72,Epilog->{{Arrowheads[Small],Arrow[Most[Flatten[Map[{{#,0},{#,f[#]}} &,list],1]]]},{PointSize[0.015],Point[{x0,0}]}}]}}]]

Manipulate[newton[1-(E^(-((#)^2))) &,x0,-2.5,2.5,n],{{x0,0.5},-1.529,1.529},{{n,0},{0,1,2,3,4,5},ControlType->SetterBar},SaveDefinitions->True]

True or False: As |x| exceeds some number Newton's Method will fail to converge.
	1) True
	2) False

Going further

f[x_]:=1-(E^(-(x^2)))

g[x_]:=Simplify[x-((f[x])/(f'[x]))]

FindRoot[g[g[x]]==x,{x,1.528}]

Solve[g[g[x]]==x,x]

Accumulation functions

Manipulate[Plot[{(t^3)+0.365(t^2)-1.47 t,0.735 (a^2)-0.121667 (a^3)-((a^4)/4)-0.735 (t^2)+0.121667 (t^3)+((t^4)/4)},{t,a,x},PlotRange->{{-1.789,1.514},{-2,2}},Filling->{1->Axis},AxesLabel->{"t"}],{a,-1.79,1.514999},{x,-1.78999,1.515}]

The accumulation function F(x)=(∫_a^x)f(t)dt is shown in red and f(t) is in blue. Moving a to the right 
	1) has no effect on F(x) on the domain (a,x).
	2) causes F(x) to shift up or down.
	3) causes F(x) to shift left or right.
	4) changes F(x) to an entirely new function.

The accumulation function F(x)=(∫_a^x)f(t)dt is shown in red and f(t) is in blue. Moving x to the left 
	1) has no effect on F(x) on the domain (a,x).
	2) causes F(x) to shift up or down.
	3) causes F(x) to shift left or right.
	4) changes F(x) to an entirely new function.

The accumulation function F(x)=(∫_a^x)f(t)dt is shown in red and f(t) is in blue. Moving a to the right 
	1) has no effect on F'(x) on the domain (a,x).
	2) causes F'(x) to shift up or down.
	3) causes F'(x) to shift left or right.
	4) changes F'(x) to an entirely new function.

Level curves

Manipulate[Show[{Plot3D[(x^2)-(y^2),{x,-2,2},{y,-2,2},RegionFunction->(((#1)^2)+((#2)^2)≤4 &),MeshFunctions->{#3 &},Mesh->{{z}},MeshStyle->{{Red,Thick}},AxesLabel->{"x","y","z"}],Graphics3D[{PointSize[Medium],Point[{0,0,0.1}]}]}],{z,-4,4,Appearance->"Labeled"}]

The surface f(x,y)=(x^2)-(y^2) is shown. When z (the height) is 0
	1) the level curve is not defined since level curves can't cross one another.
	2) the level curve is two straight lines that intersect at (0,0,0).
	3) the level curve is two curves (not straight lines) that intersect at (0,0,0).
	4) the level curve is just the single point at (0,0,0).

Why did this happen?
	1) The equation (x^2)-(y^2)=0 has no solutions.
	2) When you solve (x^2)-(y^2)=0 you factor to get (x-y)(x+y)=0.
	3) Either z=(x^2) or z=(y^2).
	4) When you solve (x^2)-(y^2)=0 you get x=y=0.

Conclusions

"Clickers" are a useful teaching tool.

Mathematica's dynamic interactivity has made it easier to do experiments and demonstrations in the math classroom than ever before.

We can help our students learn by marrying these two technologies.

This is a great way to leverage off of demonstrations at the Wolfram Demonstrations Project. Don't just show the students a demonstration – make them think about it!