A Hybrid, Small-Group, Guided-Discovery Method of Instruction

** **

**A Rigorous, Non-Extremist Approach**

** **

**by Jerome Dancis**

I am a
proponent of using a mixture of traditional lecturing, together with my
variation on Neil Davidson's Small Group Discovery Method, to engage (mostly
engineering) students in the discovery and development of simple mathematics.
This is strongly motivated by the tradition of R.L. Moore, the biggest
proponent of and by far the instructor most successful at using discovery
learning. Students learn that they
can develop and discover much of the mathematics themselves, which is
empowering as well as good for their self-esteem. I still prove the bulk of the theorems. My classes do not cover less material
than a class taught in the regular lecture style; actually my classes cover an
enhanced syllabus.

In my
math classes, group learning is a
non-graded, low stress, nurturing learning system which trains students to
solve problems and to do mathematics. It also exposes and then deals with the
variety of misconceptions about mathematics that students have collected during
previous mathematics classes. I
use my interacting with the groups to steer the students away from
time-wasting, inefficient as well as dead-end approaches to problems.

For
example, the problem set which led my campus (not math) honors calculus
students to
developed/discovered-with-guidance the rules for dx^{r} /dx, when r
is a rational number, is presented in the appendix: "Formulas and discovery learning.
-- From product rules to rules for (rational) powers".

Orian L. Hight wrote : "In the
reform recommendations, the issues of teaching and learning are directly
addressed by two guiding principles:
(1) instructional activities develop out of problem solving situations
and (2) students learn by construction, not by absorption. In [Prof. Dancis' Applications of
Linear Algebra] course, we were engaged in instructional activities such as
exploration, representation, conjecture, proof, application, problem solving,
and communication. In other words,
we were "doing" mathematics which brought vigor and vitality into our
classroom".

As an
example, my *students discover and teach themselves* the
method of "back-substitution" (that is "back-solving"
"triangular" systems of simultaneous equations by starting at the
bottom equation and working one's way *backwards* to the top
equation) by doing a few problems like Problem #1 below. Problem 1 is assigned without
instruction.

__Problem
#1.__ Our team is designing a
"widget" for a space ship.
The boss says that we must calculate the width w of the
widget with 99% accuracy. If it is off by more than 1%, the space
ship may blow up. The proper
width L of the widget is connected to the height h
of a gadget by these equations:

L +10u
= 11

u +10x
= 11

x +10y =
11

y +10z = 11

z = h = 1.

We
measured "h" as
carefully as we could and found it
to be 1.00, that is: .99
< h < 1.01. Using h = 1.00, you may calculate L in these four
equations? Answer: L = 1=u=x=y=z.

But
then Mr. Wiseguy said that we should send the gadget to the Monopoly Measuring
Company which would use its special electronic equipment to measured "h" with accuracy of 99.99% The Monopoly Measuring Company charges $10,000. Would this $10,000 be well spent or
wasted? To find out,
calculate w again, this time using The Monopoly
Measuring Company's measurement of
h = 1.001.

Answer: L = 11 which is nowhere near 1, u = 0
and the $10,000 was well spent.

Thus
the students discover that a small 0.1%
error in the data or measurement of one variable can completely change
the answer. Problem 1 foreshadows
my later lecture on error analysis.
The board–work and homework often foreshadow and provide examples
and background for the next lecture.

Problem
1 could and, in my opinion, *should* be used in Prealgebra classes (after
students have become fluent in solving equations like 2x+5=9).

Problem
1 epitomizes Guershon Harel's Necessity Principle [H]. It is used as motivation for the next
problem:

__Problem
#1B.__ (a) Observe that Problem
1 may be written in matrix vector form as
M**v** = **w**, when h=1, and as M(**v**+**∆v**) = **w**+**∆w**, with ** ∆w**=(0,0,0,0,0,
.001)T, when h=1.001.

(b) Using the equations of part (a), show
that **∆v** = M^{-1}**∆w**.

(c) Calculate the difference of the answer
vectors in Problem 1, label it **∆v**;
calculate M^{-1}; multiply M^{-1}**∆w**;
observe that these calculation check that
**∆v** = M^{-1}**∆w.**

** **

Later, for matrices M=(m_{ij})
and N=(n_{ij}), the "absolute value" of a matrix is
defined by
| M | = ( | m_{ij }| )
and "≥" for
matrices by M ≥ N if each m_{ij} ≥ n_{ij}. Then it is easily proven that M**v** = **w** implies that |M| |**v**| ≥ |**w**| and
that "≥" for vectors is a transitive
relationship. This completes the background needed for the simplest
perturbation problems associated with a set of linear algebraic equations M**v** = **w**, when M is an invertible matrix:

__Simple
perturbation problems__.
Given M**v** = **w**, and M(**v**+**∆v**) = **w**+**∆w**, when M
is an invertible matrix.
Suppose that M, **v** and **w** are known or are easily calculated.

(a) If bounds on |**∆v** | are known, then bounds on |**∆w** | may be
calculated by

|**∆w** | ≤ | M |
x |**∆v**|.

(b) If bounds on |**∆w** | are known,
then bounds on |**∆v** | may be
calculated by

|**∆v** | ≤ | M^{-1} | x |**∆w**|.

Now,
students can examine the more difficult problem when the entries of matrix M
are not known exactly.

__Problem
#2.__ Suppose there is an
error matrix E associated with the invertible
matrix M. We are given these two
matrix vector equations:

* Calculations Equation: M**v** = **w** and * (Unknown)
Reality Equation: (M+E)(**v**+**∆v**) = **w**.

Show
that **∆v** = - M^{-1 }E(**v**+**∆v**).

Oops,
we have just solved for **∆v
** in terms of itself; not a satisfactory situation. Using this equation as a basis, a tight
and the simplest bound on **∆v
** is presented in my educational paper [D].

I teach
mathematics at the University of Maryland, a big state university. I mostly teach the first and second
courses in matrix algebra to students majoring in Engineering and Science. My classes have 15-30 students without
discussion sections. I allocate
10-20 minutes at the end of a 50-minute class period for group work about twice
each week during the 1st month and about once a week thereafter. This is not enough, but the time for
group work comes at the expense of lecturing and I must strike a balance between
them; after all we meet three
times a week for only 14 weeks.

Students
enter my class with quite varied educational backgrounds, and even students
with similar backgrounds will have forgotten or garbled different knowledge or
skills over the summer. Some
significant side effects of group work are *
students learning new things from each other, * students
filling in each other's gaps,
* students reviewing topics
for their teammates and
* students correcting math misconceptions collected in previous classes.

Following
Neil Davidson, I have my students divide themselves up into teams for group
work. The students are free to change groups at any time. My input into team
membership is minimal. If, due to
absences, there is a group of only two students, I will send them off to join two
groups with three students each.
Also, I will split up a team composed entirely of weak students.

My
instructions are simply these:
"Form teams of three or four students each. Introduce yourselves. Lay claim to a section of the
blackboard by putting all your names on it. As a team work through the
problems. When someone makes a
mistake like writing '3 + 4 = 8', respond in an __adult__ manner by saying
that the __mathematics__ is incorrect or wrong and then correct it. Do __not__ say "That's
idiotic". Do __not__ say the person is wrong or dumb or stupid. Do __not__ make a high-school level
cutting remark. Do __not__
make any personal remarks. Ask
questions when you do not fully understand what is happening. Answer each
other's questions and explain the material to each other."

During
group work, I give much *semi-personal* attention (tutoring) to the
students in the form of "one-on-four" instruction. Classes with more than 20 students (5 groups) suffer from the
fact that I cannot run around the room fast enough to provide the needed
"one-on-four" instruction. This is contrast to Dorier's class where
each group solved the problems without outside help, [Do].

Orian
Hight wrote "We helped each other and the instructor referred to this as
"team work". We were
given an opportunity to learn how to explain the problem to other team members,
to spot mistakes in the work of others, to accept criticism in an adult manner,
and to defend our work
when it was correct."

The
group problems are not drill problems.
They are usually one to two levels more difficult and/or interesting
than the normal homework problems.
Ideally, they are problems that an average student will have difficulty
with, but a team of students can solve, or problems that the team will have
difficulty with, but the team, with minimal interaction with the instructor,
can figure out in 15 minutes.

The interaction between the teams and
the instructor is crucial. The help may be a polite reminder to review some
topic learned earlier. It may be a
short explanation or a 2-minute mini-lecture on the point at which they were
stuck. The help may be a
"What is this object?" type-of-question. Often, the help is as simple
as reminding the students to write all the hypotheses on the board. I usually point out errors and
misconceptions by pointing to an equation on the board and (following R. L.
Moore) asking the students to justify or explain how an equation follows from
the preceding one. While trying to
answer my question, they often discover their mistake. (The interaction between R. L. Moore
and his students was also crucial.
R. L. Moore provided students with the minimum amount of help, albeit
indirectly, deemed necessary for the student to then be able to complete the
problem. Moore got to know his
students extremely well, which enabled him to tailor the help to the individual
student [M].

Orian Hight wrote "The instructor
moved around to each team, observed each team's work at the blackboard, asked
questions that required elaborate, thoughtful responses, gave supportive
comments, and encouraged the sharing of ideas among team members, all of which
helped to promote our confidence in linear algebra."

The
purpose of the group work may be to get the students started on the right road
to a solution; to get them past the question "How can this new, unusual
and/or unexpected type of problem possible be solved?". This is quite important for problems
that they have __not__ been programmed to do. For a highly computational problem, the teams may figure out
an algorithm for doing the problem in class, and then do the calculations,
possibly individually, for homework.

It is
common that I will assign several problems on a Monday that are due on Friday;
and have the students work as a team on the harder problems at the board on
Wednesday. This way the students
are not starting the team work "cold", they have had 2 days to at
least become acquainted with and/or start the problems. It also provides each
student with an opportunity to solve any problem by themselves before seeing
what the other team members do. As
such this is a hybrid of R.L. Moore's and Davidson's methods.

One of
my students said: " Also, group work is new for me and I think its
good. Because if a student doesn't
want to seem like a nut, then he has to study, to prepare to
participate." I do not know
how common this attitude is, but it is consistent with the views expressed by
Harvard students that they studied real hard for foreign language classes in
order not to appear ignorant when called upon in class [L].

A basic
pedagogical question: if students are having considerable difficulty with the
standard problems, how can giving them harder problems help instead of
frustrate them?

The
answer: first of all, in doing interesting hard problems students get practice
in all the skills they need to review.
For example, doing division exercises gives students plenty of practice
with both subtraction and multiplication.
In the course of trying to do hard problems on the board, the sources of
the student's difficulties become clear, and the instructor can address the
weaknesses.

The
second part of the answer is that by working together in teams, the students
help each other over all sorts of rough spots, as they fill in each other's
gaps, correct each other's misconceptions. Four students pooling their knowledge and working together
can and will solve many problems that they would all give up on if working
individually. I can't
emphasize enough how important it is that students get help with their mathematical
misconceptions in *ungraded* situations.

As four
of my MATH-ED students said (emphasis added):

MATH-ED
student: " The group
work allows students to get together and bounce ideas off each other to solve
the problem. In my group, K might
know how to do part of the problem and then S, A, or I usually jump in to
help. Dr. Dancis's class provides
a opportunity for students to really get their "feet wet" with
applications of linear algebra. He
provides a unique classroom environment that *allows students to work
together to make sense of linear algebra*.."

MATH-ED
student: "More than in any
other course, I found the group work, both in and out of class, to be
essential. *Working with
different people allowed me to see several approaches to a concept, or problem,
that I would not have considered were I working independently.* I was able to
learn from other people and I hope they were able to learn from me. Brain storming and
struggling to prove identities/theorems with other students was very helpful."

MATH-ED
student: "*Finally, what
helped me the most was the group work.
Working together with other people made me more aware of my flaws and
mistakes, my mathematical judgment.
Some concepts that were not as clear to me became easier for me to do
with the help of my study buddies.* It is also good to study with
study groups after class sessions like my group did."

MATH-ED
student: "*Group
work was my favorite part of the class.
I found it extremely helpful.*

**Student
Proofs.** College seniors arrive in my second semester matrix
algebra class with math proof phobias. It is especially absurd that Math Educ.
majors who will be teaching high school math the following year are proof
phobic. When asked to prove a
matrix identity, they start with the "To Prove"; which works
wonderfully well for trigonometric identities but often produces false proofs
for matrix identities. I have the
groups prove many of the theorems of the course. Frequently, I would assign an exercise with several
equations listed as given and the problem was to prove another equation or
matrix-vector identity. At the
next class, I would present a new theorem and note that its proof was largely
yesterday's student work (See Exercise 5A and Theorem 5B below). Sometimes a problem is a
"generic" theorem, that is, a theorem, with specific numbers
replacing the variables. Then the
proof would be to have a "word processor" simply replace those
numbers by letters in yesterday's calculations.

As one
of my Math-Ed students noted:
" I think this class was unique in the fact that when we proved a
theorem we did not know we were actually proving a theorem. Dr. Dancis disguised the proofs in
exercises in the back of the sections.
In other math classes, whenever I had to do proofs I would get really
nervous and clam up. But in this
class I worked the problem as a exercise and then he told us it was a proof to
a theorem."

Mathematicians
often prove something with a seemingly complicated formula, which is then
simplified by the unexpected canceling of terms and/or factors. The empowerment and enjoyment of
simplifying complicated formula in surprising ways is missing from most math
classes. I purposely include
problems where this occurs. For
example Problems 3, 4 and 5.

__ __

__Problem
3__. Given a
"diagonalization" of a matrix
M = P^{–1}DP.

(a) Check that M^{5} = P^{–1}D^{5}P.

(b) Guess the corresponding formula
for M^{7}. Check your guess.

(c) State a general rule.

(d) Suppose that you know/given
matrices M, P and D, find a quick
way to calculate M1001.

__Comment__. The solution to the vector sequence
equation: vn+1
= M vn,
n = 0, 1, 2, 3, ..., where M is a (diagonalizable) matrix and {vn, n = 0, 1, 2, 3,
...} is a sequence of coordinate vectors, vn = Mn v_{0}. The results of
Problem 3 are used to easily calculate M^{n}. This is
one way that I use students' solutions to problems as data, examples,
motivation and/or useful information for later material.

I like
to give students simple, but non standard, problems. Frequently, their reactions are

(i) I (we) have **not** seen this type of problem before.

(ii) Therefore, we do **not** know to solve it.

(iii) If they are doing homework alone, many
students would simply give up quickly and go on to something else.
But when I have them "pinned" to the blackboard, they try some
things and often solve the problem.
Especially, since I assign many problems in which doing the most natural
calculations (or using the most natural guess) will lead to a solution. There is something of this in each
problem stated in this report. It
is especially true for Problem 3 and part (c) of Problems 4 and 5.

I like
problems which demonstrate how current material is used elsewhere -- mostly in
other mathematics courses. For
example, Problem 4 demonstrates how a fact from freshmen calculus (the
derivative of e^{4x}) is used in sophomore calculus (differential
equations). I also like problems
which combine current material with material from earlier chapters or
courses. This *livens* up my classes
as well as *broadens *my students experience with the basic material and
demonstrates the interconnectedness of mathematics.

__Problem
#4__. Let us discover which
functions y = f(x) satisfy this
equation of motion of a shock absorber:

y´´ + 11y´ + 28y = 0. (1)

(a) Is y = e^{–4t} a solution to Equation (1) ? Substitute in and find out.

(b) Which of these functions are also
solutions to Equation (1):
(i) y = e^{–7t}, (ii) y = e^{–2t},

(iii) y =√2 e^{–4t}
+ 97 e^{–7t}?

(c) For (iii), is there anything special
about the numbers √2 and
97? Can you find other numbers
which will also yield solutions? *Guess.* Check
your guess. Can you state a
general rule which describes many solutions?

__Answers__:
y = e^{–2t } is not a solution; all the others are solutions. (c) Nothing special, all numbers work.
y = A e^{–4t} + B e^{–7t} is a solution for all numbers A
and B.

I pair
the last problem with the next one which can be used alone in high school
classes which are studying geometric progressions.

__ __

__ __

__Problem
#5__. Let us discover which
sequences satisfy this equation:

a_{n+2 }= –11
a_{n+1 }– 28 a_{n}, n= 0, 1, 2,
3, ... .

or
equivalently

a_{n+2 } + 11 a_{n+1 }– +28 a_{n} = 0, n= 0, 1, 2,
3, ... . (2)

(a) Does the sequence {an} = {(–4)^{n}}
= {1, –4, 16, –64, 44, –45, ...} satisfy Equation (2)?
That is, is Equation (2) valid when a_{n+2 }= (–4)^{n+2}, a_{n+1 }= (–4)^{n+1} and a_{n} = (–4)^{n} ? Substitute in and find out.

(b) Which of these sequences are also
solutions to Equation (2):
(i) {a_{n}} =
{(–7)^{ n} },
(ii) {a_{n}} =
{(–2)^{ n} },
(iii) {a_{n}} =
{√2(–4)^{ n} + 97(–7)^{ n} }?

(c) For (iii), is there anything special
about the numbers √2 and
97? Can you find other numbers
which will also yield solutions? *Guess
* and then check your guess. Can you state a general rule, which describes many solutions?

Answers: {a_{n}} = {(–2)^{ n}
} is not a solution; all the
others are solutions. (c) Nothing special, all numbers work. {a_{n}} = {A(–4)^{ n}
+ B(–7)^{ n} } is a solution, for all numbers A
and B.

__Technical
notes on the wording of these two problems__: Instead of the numbers
√2 and 97, I originally used the numbers 5
and 300. This mislead some students to correctly
write "5 divides 300" as
their answer to the inquiry:
"What is special about
5 and 300?"; which resulted in their
missing the general pattern. Then
I tried the numbers 5 and 97 but this
lead some students to say that
5 and 97 are both prime.
Later, I tried the numbers
5 and 6
but this resulted in some students noting that 5 is prime and 6
is not. Finally, I settled
on the numbers √2 and 97, which does lead to any "side patterns". The advantage of using an awkward
number like 97 is that students are unlikely to multiply out and they keep
the 97 as a factor, which makes it easier for them to see the
general pattern.

This
demonstrates the importance of trying out the problems on live students and
then modifying and improving the problems in response to how students attempt
to do them. This helps
future students to work though the problems efficiently, avoiding various
unintended traps and inefficient approaches.

I use the results from the last two
problems as data for my later lectures, when I define "linear
combinations" and "linear transformations"; it is also
motivation for Problem #5A.

Jean-Luc
Dorier ([Do] Page 180 line 5) states
"... a problem that students could start solving ... where the
concept to be taught would be the right and unique tool to finish the solution. In this way, the concept would be
taught in a process of problem solving as the right tool to answer the
question". Later [Page 184]
he writes that for introducing a unifying and generalizing concept, that
"... one starting situation is usually not adequate". In Problems 4 and 5, "linear
combinations" is such a concept presented in two quite different contexts.

Dorier
([Do] on Page 177 Line -7) states: The unifying and generalizing concept of
vector subspaces was "created ... to make the solution of many problems
easier or more similar to each other". This is exemplified by Problems 4 and 5 here; in sharp
contrast to the contrived problem [Page 188], Dorier presents to
"motivate" abstract vector spaces.

__Problem
#5A__. Given that L:V--->W is a linear transformation. L(v) = 0 =
L(w), for vectors v
and w Î V. A and B
are numbers and u = Av + Bw.

To
Show: L(u) = 0.

__Theorem
#5B.__ All linear combinations
of solutions to a homogeneous linear equation are more solutions.

__Theorem
#5B Alternate.__ The
general solution to a homogeneous linear equation is a subspace.

The day
after the students do Problem #5A, I state Theorem #5B and then show them how
to translate it into Problem #5A.

A
problem (from my matrix algebra class) that appears difficult until an easy
solution is found follows.

__Problem
6__. Suppose that there are
three 7x7 matrices A, B and C,
such that AB=I and I=CA. Show
that B=C. Note to students who know about
inverses, you may not assume that
any of these matrices have an inverse.

Even
though many of my students have seen and used matrices in previous courses, it
still takes a group of students about 15 minutes to do this problem; the first
10 minutes is mostly spent trying things that do not work. I need to suggest to some groups that
they start with one of the given equations and then multiply it by something
(unspecified). Some students are
prone to combine the two equations into
AB=CA and try to work with just this single equation. To discourage this, I precede this
problem with one in which there are matrices, A, B and C, such that
AB=CA, but B≠C. Some groups have to be
reminded about this. At the next
class, I quote Problem 6 as the bulk of the proof that matrix inverses are
unique.

One
reason for the group work at the beginning of the semester is to ensure that
all students make an acquaintance of at least two other students in the
class. This ensures that everyone
knows someone that he/she can ask for class-notes if he/she is absent and that
everyone knows other students that they can invite to form a study group or
someone they can simply talk to if they happen to meet on our large campus.

I
accept homework with up to four names on it. This results in * many less errors for me to correct
since the students will find many of each other’s mistakes, * less papers for me to grade,
and * encourages students to work
together outside of class. Since
half the students commute to my campus, this occurred much less than I wanted.

One of
my graduate students in MATH Educ. said:
"Group study was encouraged inside and outside of the class. It allowed us to work on difficult
problems together, prepare for tests, and make friends with people we may
otherwise not talk to."

As
another student said: "The
group work made the atmosphere in class very friendly, *less stress*. It also
helps students have friends which in turn encourages them to work in groups
out–side of class and the group work is usually more successful than
individual work."

Many
students are initially uncomfortable with group work because it goes against
the American spirit of self-reliance and rugged individualism. Also they are at
an age when it is uncomfortable and embarrassing to make minor mistakes in
front of their peers. It
takes time and practice for students to learn to work effectively in
groups. A skill that will serve
them well later in life, both on and off the job.

Dr.
Elizabeth Shearn of the U. MD Studies Skill Center has noted the antagonism of
freshmen students (in a remedial math class at U. MD.) toward participating in
study groups. As one student told
her "Juniors and seniors work together all the time, but you can't expect
freshmen to do it!" Dr.
Shearn credits the compulsory group work both for the fact that one class of
students scored especially high on the "uniform" final exam and that
these same students gave their instructor an especially low evaluation. If an instructor does not "fully
program" the students on how to solve every problem (the expectation of the
many students) because the instructor wants the students to learn to figure out
some things for themselves, then it is "natural" for the students to
claim (on the teacher evaluation sheets) that the instructor is not explaining
the material clearly.

A
survey of Harvard undergraduates found that the students who thrived
academically were those who regularly discussed course material with someone
else; sometimes with a professor, but often it was with fellow students in an
informal study group [L]. This was
especially true for science students and "doubly especially" true for
female science students. The usual
rules were that the students did the homework before coming to the study group
and then they discussed ramifications of the course work. The usual rules for papers were that a
participant could bring a second draft of a paper for criticism.

Group
work by itself is not a panacea.
Group work is being treated as the current fad in some school
systems. Sometimes this means
having students work together in small groups on straight-forward skill
practice problems while receiving occasional guidance from the instructor. Technically this is group
learning and is much better than individual seatwork, but this group busywork
bears* little substantive* resemblance to the group discovery
learning described in this article.

One of my MATH-ED students said that
group work worked well for her in my class. She had been in favor of group work in theory,
but not in practice since she had been taught its virtues in her education
classes, but group work had not worked for her (as a student) in previous
classes.

Having
noted that R.L. Moore was, by far, the instructor most successful at using
discovery learning, why not use his method? The Moore method has math majors, in an advanced calculus or
later course, as a class, prove all the theorems as competitively, individually
done homework. In the 1960s, I
taught an undergraduate point-set topology course using the Moore method (or my
variation on it which did not include any month long problems that Moore's
students so fondly reminisce about).
A decade later I tried to do it again, but quickly realized that my
class of 5 students was not qualified to prove theorems. I used the same list of theorems, but
had the students largely proving them as a group in class. Of course, what works for
classes of math majors, in an advanced calculus or later course, does not work
for classes of engineering students who have not learned advanced
calculus. My teaching style is my
reasonable major adaptation of Moore's method for classes of engineering
students. With gratitude, I acknowledge that I learned how to divide
theorems and problems into student-appropriate bite size pieces from my major
professor, R.H. Bing, who learned it from his major professor, R. L. Moore.

**Zealots**. There are
Group learning zealots who advocate that *all* learning be group learning. There are Group learning super zealots who insist that *all* learning be group discovery learning. There are education zealots typified by
their slogan: "Teacher speaks - BAD, student speaks - GOOD." Their more polite and mildly less
arrogant team members say "Be a guide on the side, *not* the sage on the stage". I wish to distance myself from such ideological
positions. I believe that
there is much value in traditional lecturing. This paper advocates a judicious mixture of group learning,
discovery learning, student development of the subject together with
lecturing. In many courses as in
sports, it is useful for the instructor and coach to be both a guide on the
side *and* the sage on the stage

**A
good syllabus.** Of course, having a good
syllabus makes it easier for students to learn the material in any class. But it is especially important in a
discovery learning situation where the students are participating in the development
of the material; it reduces unproductive work and wasted time when students are
stymied.

Orian Hight observed that: "The use of a broad selection of
examples and problems to motivate the theory and to provide the opportunity to
solve many problems (both computational and conceptually) extended throughout
the course since one way to learn linear algebra is by solving problems. The purpose of so many problems was to
show the various applications of the theory, to provide problem solving
experiences with non routine problems, and to provide the opportunity to learn
a specific content area in more depth when it is first introduced. Also, the problems helped us to make
connections among related concepts.
*By developing a linked network of ideas, we began to perceive linear
algebra not as a collection of disjoint procedures for rote learning but as a
coherent body of knowledge to be understood.* In addition, the problems promoted
student centered investigations which furnished the motivation and context for
problem solving, reasoning, and communication." (Emphasis added)

"The
content was the springboard for the exemplification of the principles
established in Standards 1,2,3 and 4 (mathematics as problem solving,
communication, reasoning, and mathematical connections) of the __Curriculum and Evaluation Standards for School
Mathematics __ (National Council
of Teachers of Mathematics, 1989) and of the principles outlined in __Moving
Beyond Myths__."

"The
instructor set good examples of several recommendations contained in the __Professional
Standards for Teaching Mathematics (Professional Teaching Standards__) (National Council of Teachers of
Mathematics, 1989). He showed
"a deep understanding of mathematical concepts and principles, connections
between concepts and procedures, connections across mathematical topics..., and
connections between mathematics and other disciplines" (p. 89) and
encouraged mathematical discourse so that we could gain the same type of
understanding. This style of teaching is recommended in Standard 4. He also provided mathematical
activities (in class team work) and many opportunities for discourse, both of
which are required for problem solving, reasoning, and communication (Standard
5)."

As
another one of my students wrote: "[The college matrix algebra course] was
all taught and organized in such a way that I know I'll be able to* remember
it all for years to come*, unlike most
other, less sensible classes" (emphasis added).

After
each lesson, the exercises assigned not only provide practice with the ideas
taught, but also

* provide repeatedly practice in
reasoning. This makes me a strong
proponent of and practitioner of Guershon Harel's Repeated Practicing of
Reasoning Principle.

* provide practice in how the ideas are
used (elsewhere) in math;

* demonstrate or use connections with
previous lectures and other math courses, thereby reviewing previous material;

* foreshadow later lectures.

This
helped developing the linked network of ideas In Hight's quote above.

**Other
Basic Principles for a good syllabus.** I am a strong proponent of and
practitioner of the Donald Duck Cartoon Principle, which overlaps the commonly
known "Keep It Simple
Student" ("KISS") slogan.
My version of the Donald Duck Cartoon Principle is: If there is a small but steep mountain
to be climbed, it is the instructor job to guide the students up one of the
easier paths, not to leave them to waste time floundering on a difficult
route. Following the other principles
listed below helps me to do this. "Remembering Algebra 1 a year
later" is one of the mountain tops in [D2]; this paper shows that the common instructional technique is
leading students to a difficult or near impossible route to the top, in contrast
to the easier route that I suggest in [D2]. Realizing
that the students have not remembered Algebra 1, it is common for the bulk of
the Algebra 1 to be repeated in Algebra II, this corresponds to giving up on
the mountain top and only taking students part way up.

Also, I
am a strong proponent of and practitioner of Guershon Harel's Necessity
Principle [H]. For both these
reasons, I do not inflict on my undergraduate engineering students the concept
of a matrix which represents a linear transformation from R^{n} with basis a to R^{n}
with basis b. Avoiding such things saves my students
much frustration and avoids unproductive class group work where students are
stymied.

**Pick
your Students' Struggles**. Some purists believe that the students
should struggle through *all* the
material. My method has the
instructor choosing the syllabus (with some consideration of where the students
are at) and deciding how much time to allocate to student struggle; then the
instructor judiciously, picks struggles/problems/theorems from the syllabus,
which will fill up the allotted time for student struggling/problem
solving/theorem proving. The instructor presents the remaining material. The
instructor sets the pace not the students. In this way *the syllabus gets covered*. My students
struggle more than those in pure lecture classes, they also are required to be
more independent (of the instructor).

**Constructivism**. The discovery aspect described above is part of what is
now called "constructivism".
A current "constructivism" fad is to have the students
"develop" *all* knowledge themselves, often with little
or no guidance from the teacher and often when the students do not have
sufficient background. This often results in much floundering and frustration
on the part of students. One text
asks the students to discover for themselves why "Pascal's Triangle"
is named after Pascal instead of the Chinese mathematician who had discovered
it centuries earlier [A]. There is
*no* way for the students to know or discover the answer. (The
answer in the teachers’ manual is *wrong*.) Often the"constructivism" fad has students
"construct" knowledge by making conjectures and than checking their
conjectures on several examples, *without*
deductive proofs. Again, this type
of discovery learning bears* little substantive* resemblance
to the rigorous learning with emphasis on deductive proofs described in this
article.

**Chunking**. In my second
matrix course for engineers, the system of seven second order linear
differential equations which models the equations of motion for seven
horizontal blocks connected by 17 horizontal ideal springs, without outside
forces, is written as the standard matrix vector equation Mv´´ + Kv = b, where M is the 7x7
"mass" matrix, K is the 7x7 "spring constants"
matrix and v(t) is the
"position" vector. Of
course, these types of matrix vector equation occur commonly in engineering
textbooks. Lyn English and Graeme
S. Halford label this "chunking". In [E-H], they explain why it is very useful for students
learning. When I do it, I am
chunking, but when this is done repeatedly in the course and students are
required to do exercises, then the students are also chunking. In this way chunking is part of
the Donald Duck Cartoon Principle and KISS.

**Segmentation**. Problem # 5A
sets the stage for finding many solutions to all types of homogeneous linear
equations, namely find some special solutions (as in Problems #4 and 5) and
then take all linear combinations.
Lyn English and Graeme S. Halford label this "segmentation".
In [E-H], they explain why it is very useful for students learning. Segmentation is part of the Donald Duck
Cartoon Principle and KISS. When I
do it, I am segmenting, but when this is done repeatedly in the course and
students are required to do it in exercises, then the students are also
segmenting.

The
ability to chunk, segmentize and KISS is a crucial ability that sets
mathematicians apart from the general population. Our textbooks do these things, but often not as much as
desirable.

**Avoid
the Pedantic** aspects of the subject. Avoid abstraction when only one (or
two) concrete examples are included in the course. Leave it for a course where it is useful. Avoid/minimize concepts and results not
useful in the course; again leave it for a later course.

Following
these pedagogical principles makes it easier for students to learn the material
and participate in the development of the math.

**Summary**. In summary,
using a Small-Group, Guided-Discovery Method enables students to discover and
develop some mathematics. They
participate in the presentation of the mathematics by proving formulas and by
working out examples and counterexamples, which foreshadow, motivate and
provide a basis for later lectures.
The teams solve problems that individual students would give up on. This results in increased ability and
self-confidence to tackle more difficult problems as well as decreased math
anxiety. Students learn that they
can discover and develop some of the mathematics, which is empowering as well
as good for their self-esteem.
The instructor is crucial as he/she provides much *semi-personal*
tutoring in the form of "one-on-four" instruction.

** **

[A]
Richard Askey, Presented at the Jan. 1998 MAA annual meeting in Baltimore.

[D] Jerome Dancis, The effects of
measurement errors on systems of linear algebraic equations, International
Journal of Mathematics Education for Science and Technology, (1984) Vol. 15, Pages 485-490.

[D2] Jerome Dancis, Toward Understanding and
Remembering Algebra 1 (Unpublished)

[D-D] Jerome Dancis and Neil Davidson, "The Texas Method and the Small
Group Discovery Method",
Legacy of R. L. Moore Project, Center for Amer. History, Univ. of Texas,
Austin.

[Do] Jean-Luc Dorier, "Meta Level in the Teaching of
Unifying and Generalizing Concepts in Mathematics", Educational Studies in
Mathematics Vol. 29 Pages 175-197 (1995).

[E-H] Lyn English and Graeme S. Halford,
Mathematics Education: Models and Processes, Cognition and Cognitive
Development Ch. 2. Lawrence Erlbaum Assoc. (1995)

[H] Guershon Harel, Two Dual Assertions: The
first on learning and the second on teaching (or vice versa), The American
Mathematical Monthly, Vol. 105 Pages 497-507 (1998).

[L]
Richard J. Light, Explorations with students and faculty about teaching,
learning and student life, The Harvard Assessment Seminars (Second Report),
Kennedy School of Government, Harvard University (1992)

[M]
William S. Mahavier, What is the Moore
Method?, Legacy of R. L. Moore Project, Center for Amer. History, Univ.
of Texas, Austin. (1998)

The Donald Duck Cartoon Principle was found as a side of a Donald Duck Orange Juice container.