Examples Of Problems To Use In Active Learning For Mathematics Teaching

"Active learning" has now come to the fore in much post-secondary mathematics teaching. This has been referenced in a recent article in the Notices of the American Mathematical Society ('What Does Active Learning Mean For Mathematicians?, Vol. 64, No. 2, p. 124). The definition given is:

"Active learning refers to classroom practices that engage students in activities such as reading, writing, discussion or problem solving, that promote higher order thinking."

In fact, a major component is inquiry -based learning. This entails "students spending class time working on problem sets individually or in groups, presenting solutions and or proofs to the class, and receiving feedback from peers and faculty"

An important point to note, is that unlike inquiry-based learning in physics (as I discussed in my March 27 post), the courses integrating AL are not based on pure, unguided student discovery. Rather, faculty design a series of carefully structured activities for individuals, pairs, or small groups - as well as some for whole classes.  Below I give some ideas for individual and paired AL problem solving I myself have used over the years in assorted college math classes I've taught:

1) Abstract Algebra:

Let (G, o) and (H, o) be groups. Then a homomorphism of (G, o) into (H, o) is a map of the sets G and H which has the following property:  f(x o y) = f(x) o f(y)

Example:

(G, o) = (R1 +)

(H, o) = (R*, ·)

Take f = the exponential function, so f(x) = exp (x), f(y) = exp(y)

Then: f(x + y) = exp(x + y) = exp(x) exp(y) = f(x) f(y)

Or:

H = R* = {x Î R: x  not equal 0}


And: exp R -> R* so exp(x + y) = exp(x) exp(y)


Def.: Isomorphism: An isomorphism of G onto H [(G, o), (H, o)] is a bijective homomorphism.


Example: H = P = {x Î R: x > 0}         (P, x)


Let G = (0, 1, 2, 3) for the operation (o) which is addition in Z₄

Let H = (2, 4, 6, 8) for the operation (o) which is multiplication in Z₁₀

Problem: Prepare the respective tables for the isomorphism and give specific examples in terms of the function φ, i.e. show specific mappings.  (Where: φ(x) φ(y) = φ(xy) for example)

2. Discrete Mathematics:

i) Show that (p ® q)  «  (~q ® ~p)  is a tautology.                                       

    (ii) Let x  Î { 2, 3, 4} and y Î {12, 16}.  Let the propositional function

P(x, y) be the statement “x is a factor of y”.  Write the following  propositions using conjunctions and disjunctions and determine the truth value of each.

            (i) "x $y P(x, y)              (ii) $y "x P(x, y)                                          

(iii)   Show that log n! = O (n log n).                                                                

(iv)     Let f(x) = 2x³ + 3x –1 and g(x) =  log x. 

Find the least integer n such that:
 (fg)(x)=          O(xⁿ)                                                                                          

(v)Use mathematical induction to prove that

       (1x2) + (2x3) + (3x4) + … + n(n + 1)  =  n(n +1)(n +2)/3.                     

       (vi) Prove that if n is an integer  and n³ + 5 is odd, then n is even using:

                        (a) indirect proof         (b) proof by contradiction.   

3) Linear Algebra:

I) Show that the matrix M =

(1 + i.....2)
(2..........5i)

is not Hermitian

ii) Determine whether the matrix Y =

(1.....(1+ 1i).......5)
((1- i).......2... ...i)
(5..........-i...........7)


is Hermitian or not.

iii) Determine whether the matrix, X =

(-i...1)
(1.....i)

is unitary or not.

iv) Let A and B be 2 x 2 Hermitian matrices. Show that (A + B) is Hermitian.

4) Analytic Geometry:

Two ellipses are graphed on the same axes as shown below:

Obtain the analytic equation for each ellipse, and in particular, show how they are related to one another.

5) Calculus:

i) Find the residue for f(z) =  3 exp (z)/  z ⁴

ii) Find all the residues at those singular points inside the circle  ÷ z  ÷     =    2  

For:  f(z) =   z ²    / ( z ⁴  - 1)

iii) ∫ _-¥  ^¥  (1 + x² )  dx / 1 + x⁴     =   ?

iv)  ∫ _-¥  ^¥    x  dx / (x²   - 2x + 2)  =  ?

6) Differential equations:

i)  100 gallon tank is full of pure water. Let pure water run into the tank at the rate of 2 gals/ min. and a brine solution containing 1/2 lb. of salt run in at the rate of 2 gals/min. The mixture flows out of the tank through an outlet tube at the rate of 4 gals/min. Assuming perfect mixing, what is the amount of salt in the tank after t minutes?

ii) A block of mass m = 2.0 kg rests on a smooth horizontal surface attached to a spring. The spring has the property that it is stretched 0.05 m by a force of 10 N. If the block is displaced 0.05 m from the equilibrium position and released, find: the frequency and period of the motion.

iii) Consider the diagram of a rocket's trajectory shown below:

Where the launch angle is denoted q_o, and the final x, y- coordinates (e.g. at impact) are shown as the terminal point of the path.

Use appropriate differential equations to find the  x- and y-coordinates of the terminal point on the trajectory for a rocket launched at an angle of 80 degrees with an initial velocity of 100,000 f/s if the air resistance is 0.01mv. Also find the value of x and y after 10 seconds.

Note: The layout for such active math learning was generally in a lab setting using tables at which students could work collaboratively, with me and at least one assistant in attendance to provide insights if needed.  Generally, the students (based in Barbados) were able to work through each numbered set or problem in one hour or less. After, there was time for discussion - given a 90 minute total class time - with followup inquiry problems to be worked out.

The same layout can also be used to conduct Math Olympiads, with three or four student teams competing against each other. Many other permutations on active learning are discussed in the AMS Notices article, and ought to provide lecturers much room for more fully engaging their students.