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  dxr /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  Mv = 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=(mij)  and N=(nij), the "absolute value" of a matrix is defined by                | M | = ( | mij | )  and  "≥" for matrices by   M ≥ N  if  each   mij  ≥ nij.  Then it is easily proven that   Mv = 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  Mv = w,  when  M  is an invertible matrix: 


Simple perturbation problems.  Given  Mv = 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:  Mv = 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–1DP.  


(a)  Check that  M5 = P–1D5P.         


(b)  Guess the corresponding formula for  M7.  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 v0.  The results of Problem 3 are used to easily calculate  Mn.   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 e4x) 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:


                                   an+2  = –11 an+1 – 28 an,          n= 0, 1, 2, 3, ... .


or equivalently


                                 an+2   + 11 an+1 – +28 an = 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   an+2 = (–4)n+2,   an+1 = (–4)n+1  and    an = (–4)n ?    Substitute in and find out.


(b)  Which of these sequences are also solutions to Equation (2):  (i)  {an} = {(–7) n },  (ii)  {an} = {(–2) n },  (iii)  {an} = {√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:  {an} = {(–2) n }  is not a solution; all the others are solutions.  (c)  Nothing special, all numbers work.  {an} = {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  Π 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  Rn  with basis  a  to  Rn  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.