Sources of students'  frustration in prerequisite mathematics courses ("PMC")
with frequencies higher than 39% in at least one of the groups of respondents
A. Sierpinska, G. Bobos, Ch. Knipping
Research on students' frustration in prerequisite mathematics courses
based on data collected in 2003-2004
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Mature:= 21+ years old and having spent some time away from formal education
 
Category of the source of frustration Description of the frustration Percent of All respondents
(N=96)
  Percent of mature respondents
(N=63)
  Percent of non-mature respondents
(N=33)
Deception by one's interpretation of a rule
1
Expecting the pace of the courses to be the same as that of math courses in secondary school or in college and being overwhelmed by the fast pace of PMC
(Drawing incorrect conclusions from the rule: the content of PMC is secondary school or college level math)
92% 94% 88%
Disappointment with a strategy
2
Expecting to be able to learn by following teacher's examples and instructions and not finding support for this strategy in the instructor's teaching approach (Frustration of a strategy) 71% 70% 73%
Deception in one's interpretation of a rule
3
Expecting not to have to change previously acquired ways of thinking and being deceived
(Drawing incorrect conclusions from the rule: the content of PMC is secondary school or college level math)
66% 65% 67%
Disappointment with the nature of the task
4
Being enthusiastic about coming back to school but frustrated with the study of mathematics: disliking or disregarding the concern with truth in mathematics 54% 56% 52%
Disappointment with the nature of the task
5
Being enthusiastic about coming back to school but frustrated with the study of mathematics: disliking, disregarding or having difficulty coping with reasoning in mathematics
(Frustration with the nature of the task)
50% 51% 48%
Disappointment with the PMC rule
6
Being enthusiastic about coming back to school but frustrated with having to take prerequisite math courses 49% 51% 46%
Deception in one's interpretation of a norm
7
Expecting that there would be feedback on one's performance or solutions whenever needed and being deceived
(Expecting too much from the norm: academic support will be provided)
48% 38% 67%
Disappointment with the nature of the task
8
Taking PMC for credits and grades but finding an obstacle to success in the concern of mathematics with truth 45% 37% 61%
Deception in one's interpretation of a norm
9
Expecting teachers to be encouraging and understanding of students' difficulties to manage life and school duties and being deceived
(Expecting too much from the norm: moral support will be provided)
44% 40% 52%
Disappointment with actual outcomes of one's actions
10
Taking PMC for credits and grades and being disappointed with one's achievement in the courses
(Frustration of goals, plans)
42% 44% 36%
Disappointment with the nature of the task
11
Taking PMC for credits and grades but finding this outcome difficult to obtain because of difficulties with reasoning in mathematics
(Frustration with the nature of the task)
41% 33% 55%
Disappointment with a strategy
12
Not liking to memorize rules and finding that to succeed in PMC it was necessary to memorize many rules
(Frustration of a strategy)
31% 22% 49%
1
Expecting the pace of the courses to be the same as that of math courses in secondary school or in college and being overwhelmed by the fast pace of PMC
(Drawing incorrect conclusions from the rule: the content of PMC is secondary school of college level math)
Estimation of the number of respondents who might be affected by this kind of frustration was calculated based on the union of respondents who agreed with statements in items 20, 37,  54, and explained their poor achievement in a course in item 68 by saying that the course was "too much material, too fast".

Here are these items:

Item 20. I usually want to understand every single detail.
80%, 83%, 76%

Item 37. In the course, the teacher was going too fast.
22%, 30%, 36%

Item 54. I learn better if I can do it at my own pace.
67%, 71%, 58%


Item 68. If you didn't do as well as you had hoped in a course, complete the sentence: I didn't do well in the course because...

Examples of responses to 68 mentioning the fast pace of PMC:

Respondent #39: "... because my teacher had a barrier with the English language (couldn't explain in more than one way if you didn't understand), I was already not that excited about the class from the beginning and the load of what to cover in a semester is way too much!"
Respondent #46: "... because it needs more time"
Respondent #64: "... because there was too much material to be covered in such a short time"
Respondent #85: "... because I had hard time learning the material, we would learn a new chapter almost every week which didn't give me time to let it all sink in"

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2
Expecting to be able to learn by following teacher’s examples and instructions and not finding support for this strategy in the instructor’s teaching approach

Counting students who appeared to believe in the strategy of learning by following teacher’s examples and instructions was based on taking the union of sets of respondents who agreed with the statement in item 48 or in item 49 (we label this set  A):

Item 48. I high school, I used to follow teacher’s instructions
71%, 71%, 70%

Item 49. In high school, teachers were usually clear about rules and methods to apply
70%, 70%, 70%

There were 85%, 84%, 88% students in the set A.

Counting students who observed, in their classes, teaching strategies that could make learning by following teacher’s example and instructions difficult was based on the union (labeled  B) of respondents who agreed with items 39 or 41 or disagreed with item 42.

Item 39. In the course, the teacher did not provide us with clear and easy methods
23%, 27%, 15%

Item 41. In the course, the teacher expected us to do problems that were not discussed in class
27%, 25%, 30%

Item 42. In the course, the teacher did not avoid challenging problems
65%, 64%, 67%

The set B counted 83%, 86%, 79% students.

The intersection of sets A and B, representing students expecting to be able to learn by following teacher’s instructions and clear and easy methods and being disappointed, counted 71%, 70%, 73%students.

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In the descriptions of sources of frustration below, frequencies are given in the following format: A%, B%, C%, where A represents the percentage of all 96 respondents; B - the percentage of  the 63 mature students, and C the percentage of the 33 non-mature students.
 
3
Expecting not to have to change previously acquired ways of thinking and being deceived
(Drawing incorrect conclusions from the rule: the content of PMC is secondary school or college level math)


PMC focus on algebraic techniques, even in calculus (simplification, expansion, factorization, substitution, solving equations). In principle, they do not require ways of thinking in mathematics that would go beyond those supposed acquired in high school or college. It is, therefore, natural to assume that students would not expect changing their ways of thinking in the courses. The discovery that changing one’s ways of thinking might be necessary could therefore be a source of frustration for the students. We took the union of the sets of students who agreed with the following statements:

Item 43. In the course, the teacher wanted me to completely change my way of thinking:
4%, 6%, 0%

Item 44. In the course, I was not allowed to use whatever methods I liked:
27%, 27%, 27%

Item 60. In the course, I had the impression that my thinking was different from the teacher's:
29%, 27%, 33%

Item 63. The math in this course was very different from what I've seen so far:
31%, 32%, 30%

The union counted 66%, 65%, 67% and so about 2/3 of the respondents could have been frustrated with having to change their ways of thinking in mathematics.

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4
Being enthusiastic about coming back to school but frustrated with the study of mathematics: disliking or disregarding the concern with truth in mathematics

Students’ dislike of the concern with truth in mathematics or their disregard for truth in mathematics was evaluated based mainly on responses to item 75, where students were asked to express their preference between two solutions to an absolute value inequality, one of which was incorrect.

We also looked at students’ reasons for not liking math in item 66, their attribution of achievement in items 67 and 68, their descriptions of mathematics in item 76 and the interviews with students. But only two students referred to the true/false aspect of mathematics in their reasons for not liking math and nobody referred to this aspect in their attribution of failure (or even success). In item 76. Math is… two people referred to this aspect of math, both in a positive sense, not as an obstacle for them:

Item 66. I don't like mathematics because...
Respondent #64: ... there is only one right answer
Respondent #94 (KaWa):... I feel I cannot express myself because of the concrete answers.

In the interview KaWa explained what she meant by "concrete answers".
"Interviewer: So you are saying you are not intelligent in math.
KaWa: Oh no, not at all, and I'll admit it.
Int.: Why do you think you are not intelligent in math?
KaWa: Because I failed twice... because I always felt you could wiggle your way out in anything else, like, unless it's science, or mathematics, or statistics, something that's solid, that's been proved, that this is what it is. [In social science] you can always have a theory, you can always have your own theory, this is your opinion because so and so and so. In math I could never do that."


Altogether, 68%, 59%, 85% students were counted as having a dislike or disregard for the concern with truth in mathematics. The difference between ms and nms is striking here.

The intersection of the set of these respondents with the set of those who agreed with
Item 9. I was enthusiastic about coming back to school
represented 54%, 56%, 52% of the respondents.

It is on this basis that SoF, being enthusiastic about coming back to school but frustrated with the concern with truth in mathematics, was ranked 4th


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8
Taking PMC for credits and grades but finding an obstacle to success in the concern with truth in mathematics

The basis for identifying the set of students who considered the concern with truth in mathematics an obstacle for them was explained in the description of SoF 4 above.

The set of students who took the courses for credits and grades was estimated based on responses to item 65 quoted below. We counted students who chose option (c) or said "it was a prerequisite" in option (f): we obtained: 67%, 65%, 70%.

Item 65. I took this course because
(a) I'll need the math in my profession
(b) I thought I'll need the math
(c) The academic advisor told me to take it
(d) Math helps developing my analytical skills
(e) I want to get a better paid job
(f) Other (explain)


The intersection of the set of students who disliked or had no regard for truth in mathematics with the set of those were taking PMC for credits and grades counted 45%, 37%, 61% respondents. The difference between ms and nms is remarkable here.


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5, 11
Rational character of mathematics perceived as an obstacle
                                               to keeping up one’s enthusiasm about school (source of frustration 5)
                                or
                                              to success in PMC (source of frustration 11)


“Rational” here is used in the sense of based on reasoning, proving. Respondents’ dislike or difficulty with the rational character of mathematics was inferred from students’ responses in the questionnaire and in the interviews rather than from their explicit expressions. Only five students made explicit statements to this effect: #26, 34, 41, 44 and JesTia (in items 66, 76, and the interviews). However, students’ preference for procedural solutions over reasoned deduction in items 74 and 75 (see Table 74 & 75) suggests a widespread reluctance to making explicit the theoretical basis and reasons for the calculations made, this reluctance being much more frequent among nms than among ms. Altogether, dislike or difficulty with the rational character of mathematics was found in 63%, 56%, 76% respondents. Let’s call this set R.  

The intersection of R with the set of those who agreed with

Item 9. I was enthusiastic about coming back to school

counted 50%, 51%, 48% respondents. (source of frustration 5).

The intersection of R with the set of those who were taking PMC for credits and grades, i.e.
chose options c or said "the course was a prerequisite" in option f ("other") in

Item 65. I took this course because...
(c) The academic advisor told me to take it
(f) Other


counted 45%, 37%, 61% respondents. (source of frustration 11).
                                                                                                                                  
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Details follow.

We first look at  items 74 and 75.

Item 74
Given a problem: Solve |2x - 1| < 5.  Which solution do you like best?
Why?

       Solution a
|2x - 1 | < 5
2x - 1 =5             neg.         2x - 1 = -5
x = 3                                     x = -2
Answer:  -2 < x < 3

      Solution b
We use the theorem:  |a| < b <=> -b < a < b
|2x - 1| < 5 <=> -5 < 2x - 1 < 5  <=>  2x - 1 > -5  and  2x - 1 < 5 
         <=> x > -2  and x < 3
Answer: -2 < x < 3


Item 75
Given a problem: Solve |2x - 1| > 5. Which solution do you like best?
Why?
       Solution a
|2x - 1 | > 5
2x - 1 =5             neg.         2x - 1 = -5
x = 3                                     x = -2
Answer:  3 >  x  < -2
       Solution b
We use the theorem:  |a| > b <=>  a < -b  or  a > b
|2x - 1| > 5 <=>  2x - 1 < - 5  or  2x - 1 > 5  <=> x < -2  or x > 3
Answer:  x < -2  or  x > 3

The frequencies of answers to items 74 and 75 are represented in the following table:

Choice of solution in items 74 and 75 % All
(N=96)
% Mature students
(N=63)
% Non-mature students
(N=33)
74 a
procedural solution
69% 65% 76%
74b
reasoned solution
19% 21% 15%
75a
procedural solution
62% 52% 79%
75b
reasoned solution
20% 24% 12%
74a and 75a 58% 49% 76%
74b or 75b 24% 29% 15%
 
Very few respondents made reference to reasoning in their reasons for choosing one or the other of the solutions.

In choosing 74a, the reasons were that Solution (a) was
      simpler (15 all, 12 ms, 3 nms),         (ms := mature students;  nms:= non-mature students)
      easier (8 all, 4 ms, 4 nms),
      no need to use any theorems (5 all, 3 ms, 2 nms),
      clearer (4 all, 3 ms, 1 nms),
      less confusing (4 all, 2 ms, 2 nms),
      more efficient (2 all, 1 nms, 1 nms),
     “that’s the way I always solved it” (1 all, 1 ms, 0 nms),
     “because I understand better from top to bottom than from left to right” (1 all, 1 ms, 0 nms).

Here is an example of a response to "Why" in choosing 74a:
 
Respondent #35 I prefer the solution (a), we don't really need the theorem in this case. We have understood the first solution since we were in high school.

Reasons for choosing 75a were quite similar, and students often just said “same reason” in response to “Why” in item 75. An unexpected answer was that of #34, who chose (b) in item 74 but then chose (a) in 75, claiming that (b) is wrong:

Respondent #34 If the solution includes the answer then I prefer a. The OR in b is not quite satisfying. I believe there will always be a part of the disjunct that is going to be false. The biconditional holds in 75b, but theorem 74b is much better.

The choices of 74b and 75b were justified by reference to reasoning only in perhaps two responses (#27, 76):

Respondent #27. [I choose 74(b)] because the train of thought is more logical and even if one does not know this theorem, each step can be easily deduced from the last
Respondent #76 [I choose 74(b)] since |2x-1| < 5, thus, 2x-1<5 and 2x-1>-5 not 2x-1=5 and 2x-1=-5

Interviews with students suggested that respondents rarely read and tried to understand the reasoning in items 74 and 75, even when they chose the (b) solutions. ParMa (female, mature, already holding a degree in linguistics from another country) was one of the few students who chose solutions (b) in items 74 and 75. In the interview, we asked her to give us reasons for her choice. She said she liked solutions (b) better because she was more used to this type of solutions; that’s how she learned to solve the absolute value inequalities; she didn’t choose them for their more “reasoned” or theoretical character. And in general, she would rather look to some external authority to make sure her answer was correct:

Interviewer:  When you get an answer, how do you know it's correct?  In anything?  In Math?  You solve a problem, you get an answer.  How do you know it's correct?
ParMa:  I'm never sure.
Int:  You're never sure.  What do you do to make sure?  What kind of things do you do to?
ParMa:  If there is like a manual or something, so I look at...
Int:  the answers?
ParMa: the answers.  If not...
Int:  But do you care to be sure?
ParMa: Yeah, I do and if [I am] not [sure],  I have to ask someone.  If there is no one, then I do it again
or look at it and say okay, so maybe it's true, but yeah, it is important to know if I'm right or not.


In PMC, mathematical reasoning is not an explicit object of teaching. The focus is on algebraic and analytical techniques. In lectures, there is very little systematic exposition of theory and problems "to prove" are rare or non-existent on exams. Theorems are, most of the time, only stated, not proved, and they are immediately interpreted as formulas or methods to be applied in exercises. This may explain why mathematical reasoning was not evoked very often in students' responses or interviews.  In response to item 66. Do you like math?, for example, only one student referred to proving in math and he did it to explain why he liked mathematics:

Respondent #41 [I like math because] I like problem solving and that there is always a way to prove your solution

Attributions of success/failure did not refer to the “rational” character of mathematics, and neither did students’ responses to all other items except for item 76. Complete the sentence, 'Math is', where the reference was made in four responses. In these, reasoning in mathematics was evoked as something difficult. One respondent (#34) stressed that, although reasoning is difficult, it doesn’t make sense to avoid  teaching it and criticized the “rule-giving fashion” of teaching in PMC.

Respondent #34 (…).… Math is the tool of science.… But math can also be seen as a theoretical investigation of the most powerful formal system that man has created/discovered. However, math is being presented to students like a set of fixed rules. (…) I am surprised that most people don't care to ask why these rules hold. (…)  A few good instances will convince most people that a rule works (confirmation bias) (…)  So, math is so profound, that unless students are engaged in theoretical problems of mathematics, I doubt that the educational system will ever give them greater insights on the very tools they will be using everyday….

The theme of rationality appeared in all interviews with instructors. It appeared in three contexts, mainly:
- responding to students’ “why” questions, usually about the meaning of a rule;
- explaining a mistake to a student, usually during negotiations of a test grade;
- theory teaching.

Instructor XG (female, PhD student, having taught a prerequisite math course twice) gave the example of students asking why -(x-y) is -x+y and not -x-y.  She gave them a counterexample, and this type of argument - using concrete numbers - was well accepted. But, in general, she said that students were happier with justifications using diagrams (as those used in showing that (a+b)2 = a2 + 2ab + b2) or numerical examples than mathematical proofs. XG also spoke of students’ reluctance to learning definitions and reasoning from them, even in cases where it would substantially alleviate the memory load:

XG. I agree with you in that [students don’t want to reason from definitions about] those rules, [although] all the rules come from the definition…. It's very... is especially true when we learn those, I think, the seven rules of the exponential [function]. Sometimes, I just try to let them know that those rules, [it is enough to just] know four, or even three, if one knows the definition well. You don't need to put so much time on recalling all those rules in your mind.  But when I try to explain those things, they don't like it. They ask me 'Why, why you do this?'

Instructor WB (male, PhD student, had taught PMC several times) was telling us that he would do very little theory in class, replacing proofs by diagrammatic or graphical representations, and giving meaning to theorems, formulas and methods by historical anecdotes:

WB: Actually, to be honest, I don't do much theories, or proof or anything like this, you know... in such classes. I try to avoid it as much as I can, but let's say about the integral thing that I just did, I filled out all the rectangles with colors, and then I told them this is fun, then I put the definition of definite integral, then I put a remark: 'In fact this is a theorem, you know... and it was proven by Riemann... '. Then I told them about Riemann a little bit, they were happy, and that he still has problems and it's worth many dollars to solve this. So I made the mood and then I moved to fundamental theorem of calculus, just the statement without proof, without anything you know, then start giving examples. Yeah, because the... I don't think they like proofs because in a proof you cannot put numbers or anything, you have to do it abstractly and this they hate. Yeah, they don't like this.

While the instructors XG, Wb and SI (male, PhD student, having taught PMC several times) - all three PhD students - used the grade negotiations with students in their office after an exam as an opportunity for engaging in mathematical reasoning, instructor CS (male, college instructor hired part-time at The University to teach PMC, at least 20 years experience) refused to assume that students wanted to understand their mistakes. For him, students were playing the bargaining game, trying to negotiate some marks.  If students wanted to understand their mistakes, he would tell them to compare their solutions with model solutions which he made available to them. He would refuse to talk about the mistakes with students.   His conversations with students seemed like “arguments” in the sense of quarrelling.

Interviewer: After a test, suppose a student is not very happy about the grade, will they come to your office and try to argue about it?
CS: Many of them know me, and the ones who know me, will not challenge me at all. The ones who don't know me will try, and say, 'You mark too hard!', and they'll give me, 'You took out five points for this question'. 'So? This is the way I mark'.
Int.: Do you explain the mistake to them?...
CS: Oh, they have the solutions…
Int.: So, they know what's wrong. Right?
CS: Yes. There's no problem with that, they know what's wrong, and they just try to get away with points. And they'll compare with their friends. 'You took two points off him for this, how come you took three off for me?'. I said 'Is it exactly the same error?' And they say 'Yeah.' 'Then you copied the same paper then'. .... And then, of course, not necessarily for the mature ones, but for the immature ones, they'll come 'I need this course, that's why I want it, to take on another course. So, can you pass me? Please'. And, then comes the whining. The whining, I've come to a point where I just, I follow what some professors do in other departments. Two of my friends, what they do, is they simply, when a person wants to re-evaluate, they'll say: 'Ok, here's your exam paper, sit down, here's a piece of paper, here's a pen, write down whatever questions you have, don't even talk to me, write down whatever questions you have, and just give me the paper back, I'll take it home, and if there's a change, I'll let you know by e-mail. But, there's no talking'.… But that's all I’d do, that's as far as I go. I won't discuss anything with them.

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7
Expecting that there would be feedback on one's performance or solutions whenever needed and being deceived 

Students who expected academic help were counted as respondents who agreed with statements in items 55 or with 56 or disagreed with the statement in item 38: 87%, 84%, 91%.
Here are details:

Item 55. I need feedback on my solutions in order to learn
77%, 76%, 79%

Item 56. I need the teacher to tell me if I am right or wrong
67%, 65%, 70%

Item 38. It should not be assumed that at the university one is expected to be an autonomous learner
10%, 11%, 9%

These students' agreement with the statement in

Item 45. In the course, there was little feedback on my performance,

could mean that they were frustrated with the lack of feedback. 

The total number of students who agreed with the statement in item 45 was 52%, 44%, 67%.

The intersection of this set of students with the set of those who expected academic help counted
48%, 38%, 67% students.

Thus an important number of students could be disappointed with the level of academic support they were receiving.
                                                                                                                                     
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More details follow

Respondents rarely mentioned feedback on their performance or solutions among factors that contributed to their success or lack thereof in the courses (items 67, 68). One student (#54) wrote that he did well because

I did all the exercises from the course outline and I procured myself the elaborated answers from the teacher to each of the questions for feedback”.

Lack of success was attributed to bad teachers (words such as “horrible”, “obnoxious”, “treating students like s....”, “lacking teaching skills”, “lacking language skills” were used) by 9%, 11%, 6% students. Lack of feedback was not explicitly mentioned. Students did not complain about teachers’ mathematical incompetence, but pedagogical and/or linguistic incompetence, in the sense of some teachers’ inability to express themselves clearly in English. This, students wrote, made it difficult for teachers to answer students’ particular questions or “explain more than one way if you didn’t understand”. Thus, lack of language skills could make also the feedback on students’ solutions quite difficult for the teachers.

In response to item 76 (“Math is…”), nobody said anything explicit about feedback. But there were remarks about the need for the teacher to be “helpful” (#10), or “patient” (#14), and able to answer students' questions, which is more difficult if the teacher has a heavy accent or poor mastery of the language of instruction (#17).

This issue of academic support took quite a bit of space in the interviews with both teachers and students.

Teachers had the impression that students did not use the opportunities for getting feedback that were made available to them: classes, extra problem-solving sessions, office hours, and the Math Help Centre, where graduate students, retired college teachers or faculty members were available for help.
Some students don’t come to classes, where they could get feedback during class discussions. In lower level courses, teachers said, students’ absence could reach 30%- 50% of the class. This is confirmed, to some extent, in responses to the Questionnaire: 47%, 48%, 46% agreed with the statement in Item 27. I sometimes couldn’t come to class. Instructors SI (male, PhD student) and CS (male, part-time, college instructor of many years)  were particularly adamant about students’ not using the learning opportunities offered to them.
SI said:

SI: … many students are not coming to the class. I have 55 students in this course, I’m seeing maybe thirty, thirty-five, something like that, so the rest of the students I do not know why they have registered for this course.They are spending money. They could drop the course, if they don’t like it, if they think it’s not good, or the course is very hard for them. They could drop the course, but they are still registered, and they are not coming to the class. And what is the reason? For some of them, they have work, fulltime work, they are working, so they are tired, they don’t feel well enough to come to class at night and I’m teaching at night. Some of them, maybe, don’t like the teacher. [Another thing is that] I have two hours office hours, and I don’t see students coming to my office and they are, they are not doing well [in the course]. I don’t know why they are not coming, maybe they are busy, they are working or maybe they think that they don’t need to come…. The frustration,… for some students, they [owe the] frustration [to themselves]:… not studying,… not coming to the class,… not coming to the office hours. If [a student] is [not doing well], if he doesn’t understand, why doesn’t he come to my office? Why doesn’t he go to the Math Help Center? Maybe he is not working, also. He has time to come to Math Help Center, to come to my office, but he is not coming. Maybe, he is not serious…. He is getting frustrated from himself, you know…. So, if they don’t understand the basic steps of some problem, how [can] they solve the problem? So,… they will be frustrated more and more and more…

Instructor WB (male, PhD student) was organizing problem-solving or revision sessions before tests. But students generally preferred being given solutions to past finals and study at home, occasionally asking the teacher a question by e-mail, if they had trouble understanding a solution. WB was very much against giving students solutions. He claimed that many students just uncritically learned the solutions by heart, without understanding, and the solutions were not always correct. He was giving this example:

W: The other day, one student came to my office… It was about interest rate.... In one of the questions, one part was about interest compounded continuously and the other part was annually. And then she said “You are using this formula, the discrete formula, but here I have the solution and they use the other”. “So what? [I said] It’s wrong, what they did is wrong”. Then she insisted, she wanted to show me [the solution]. I waited for her, then she showed me, I said “Wrong”. Then she told me, “Why?” I told her, “Because it’s saying annually, you know, annually is a discrete thing”….  I showed her the two formulas, I told her that “Here you have the discrete time, here you have the continuous time and here it says, once it says annually or monthly or whatever is discrete, you have to use this formula”…. I don’t know if she was convinced but she stopped arguing. But you know, this one thing, you know, which I feel… is good in some sense and it’s not good in some [other] sense: to give solutions [to students], where they are full of mistakes and errors and then the students they just take these solutions, they don’t solve the problem, they just read the solution and they believe whatever they read, you know, this is like…. But… students… got used to it, so they ask whenever, let’s say, I’m giving a sample test, “Do you have the solutions?”, I say “No, we’re organizing a problem-solving session and we’re going to…you solve it at home, then we come and we solve some of it and you come at the board and you solve some, and this way…”. Yeah, they usually like to just get everything [solved] and they don’t want to come to the university anymore, just do it at home.

Indeed, if badly used, “model solutions” could deprive students of the feedback they were yearning for: they are not sufficient for understanding what exactly was wrong with their own thinking about the concepts or techniques.

In interviews with students, the theme of feedback came up with four of the six interviewed students JesTia (female, mature), ParMa (female, mature), MiKa (female, non-mature), KaWa (female, non-mature). MiKa complained about not being able to see one of her teachers, who, when she tried to reach him, would not return her calls or answer her e-mails. This teacher was giving solutions, but she preferred the “individual approach” rather than just looking at the model solutions. But she also said she didn’t go to see her teacher in another course she was taking, although her classmates were saying that he was always available and explained well. So she might have been one of those students that did not use the opportunities for feedback that were offered to her. JesTia stressed her need for good teachers and complained a lot about not being able to get any help with her problems in learning math. But she expected being told exactly what the teacher wanted her to produce rather than try to produce something on her own and then discuss it with a teacher.

The really formative feedback could come from the weekly assignments, but, in PMC, these assignments are marked by a “marker”, a graduate student hired to do the job, not the teacher. The marker doesn’t see the students, doesn’t meet them in person, and usually marks only in a right or wrong fashion (a check mark for correct and a cross for incorrect). ParMa complained a bit about the lack of more personal feedback on her assignments, but she said she understood that this would be very difficult because there are so few markers and so many students in the courses.

ParMa and KaWa also complained about the intimidating behavior of the teachers, which made them afraid to go and ask questions in the office hours.  ParMa was depicting a very temperamental teacher, whom she was afraid to ask questions.

ParMa: [The teacher] doesn’t really care that much for the questions… He asks, ‘any questions’, but if you ask a question… he says, ‘oh, don’t jump ahead; I haven’t come to that point yet…’. And he’s so mad, you know, he’s not behaving like a teacher… He’s getting mad so fast, and… Well, there is a problem with some students who are sitting in the back and sometimes talking and have their cell phones ringing, and coming and going all the time, and he gets mad, too, because of that, and he was throwing chalk and things like that….
Interviewer: Can you go and ask him questions during his office hours?
ParMa: Well, I have never been able to do that, because right before that I have a class, so I don’t.
Interviewer: Can you take an appointment outside of his office hours?
ParMa: I wanted to do that, but I was afraid because he’s always so mad… Even some days, I am really scared to raise my hand if I have a question….

KaWa also gave a story of a very intimidating teacher: 

KaWa: [The calculus course I failed]  totally discouraged me from taking any Math courses afterwards.  It was horrible.  I think it was my winter semester, my teacher was just haywire, he was everywhere, he wasn't structured at all.  You would ask him questions, he would yell at you and he was very intimidating, so…I was always used to having teachers where I have a question, let me talk to you, let me ask you because I don't understand what's the derivative of 'x' and why is it 1…I never…it never clicked to me and he was very defensive all the time.  He had the tendency of the first forty five minutes always lecturing us on other things than Math and discouraged me horribly, then when I failed…
Interviewer:  But did the teacher explain why the derivative of 'x' 1?  Did he prove it somehow?  Did he explain it?
KaWa: In my winter semester, no, because at one point I remember asking him a question and then he goes 'so what's the derivative of 'X'?  And I said I have no idea, and so then he started screaming and yelling that if you don't do your homework and you don't do the book, and so he never really explained it.  He just lectured me on how I should have known it, and I don't remember in the summer the teacher explaining it, but me knowing already, so I would never asked him because being screamed at for not knowing it prior…in the other semester, okay, I know it's 1 so okay, I don't have to ask you.


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6
Being enthusiastic about becoming a member of The University and being disappointed with having to take the pre-requisite math courses

The vast majority of respondents were enthusiastic about studying at The University:
84%, 95%, 64%  agreed with

Item 9. I was enthusiastic about coming back to school.

But many -  59%, 54%, 70% - also agreed with

Item 13. I’d rather NOT take this course if I had the choice:

The intersection of these two sets can be seen as representing disappointment with the PMC rule:
49%, 51%, 46%

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9
Expecting teachers to be encouraging and understanding of students' difficulties to manage life and school duties and being deceived

Among the participants in our study, the expectation that the university administrators and teachers will be caring and encouraging in their struggle to fit their studies within their busy schedules as workers or parents, was quite widespread among ms, but also very frequent among nms, which may be due to their holding part-time jobs, so that they can rent an apartment, have a car, etc., even if they don’t yet have a family.

We estimated the number of students having these expectations based on items 30 and 50. We denote by E the union of sets of students who agreed with the statements in these items.

Item 30. I have many commitments outside of studying.
66%, 67%, 64%

Item 50. I expect my teachers to be supportive even with those who cannot keep the pace.
81%, 84%, 76%

Quite a few students found teachers’ moral support lacking. We based this estimation on “disagree” responses in items 46 and 47. We denote the union of these two sets by D.

Item 46. Not true that, in the course, the teacher used to encourage us.
30%, 27%, 36%

Item 47. Not true that, in the course, the teacher understood my difficulty to manage life and school duties
38%, 32%, 49%

One ms (#12) commented, sarcastically, “Yes [the teacher encouraged us] with things like, 'you don't want to work at McDonald's for the rest of your lives, do you?'”

The intersection of E and D can be seen as representing students feeling deceived in their expectations of moral support from the teachers: 44%, 40%, 52%.

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10
Taking PMC for credits and grades and being disappointed with one’s achievement

Many respondents (over 2/3) seemed to be taking PMC mainly for the academic credits and grades: 69%, 67%, 73%. We note the higher frequency among nms. But many of these students were not happy with their achievement in the courses: over two fifths of the respondents took the courses for credits and grades and were not happy with their results: 42%, 44%, 36%.  Mature student were slightly more likely to face this contradiction than the non-mature students.

Here are some details.

The number of students who took the courses for credits and grades was taken from responses to Item 65, which was:

Item 65. I took this course because (you can choose several options):
(a)I’ll need the math in my profession
(b)I thought I’ll need the math
(c)The academic advisor told me to take it
(d)Math helps developing my analytical skills
(e)I want to get a better paid job
(f)Other (explain)


Students who chose (c) or wrote “prerequisite” in (f) were assumed to be taking the math courses “for credits and grades”. We denote the set of these students by A.

The number of students who were not satisfied with their achievement (64%, 68%, 55%) was calculated based on the union B of sets of respondents who agreed with statements in items 12 and 68:

Item 12.  I was not happy with my results in the course:
43%, 52%, 24%

Item 68. I didn’t do as well as I had hoped in a course
57%, 60%, 52%

The number of students who were taking the courses for credits and grades and were not satisfied with their results was calculated based on the intersection of A and B.

The high percentage of students apparently taking the courses mainly for credits and grades was consistent with the interviewed teachers’ impression of the goals of their students:

Instructor CS : Some mature students just want to finish a degree so they can get promoted in their jobs…. [they are saying] ‘I need the course, I need the credit, that’s all I want’

Instructor WB (male, PhD student): I’ve never seen any students [studying mathematics because they wanted to] because what we are teaching… are mostly business students. They just want to finish the course and they don’t want to see [math] anymore.

One of the teachers (CS, male, college teacher with many years experience) had the impression that the “credits and grades” attitude was less frequent among mature students than in non-mature students.

CS: the majority of [ms] are motivated, they want to learn, they know what it is to be the workforce… They work harder, they’re more inspired… They will do the work, they won’t come up with excuses, they won’t whine. Whatever happens, happens…. They know it’s not ‘If I cry, somebody’s gonna do something for me’. They know it’s not going to happen…. No matter how much coursework you put on them, they’ll do it… and even if they don’t do it, they’ll keep quiet about it…
Interviewer: So by contrast you’d say that nms whine a lot and get more frustrated about having to take these courses?
CS: Yes, very much so, very much so.
Interviewer: But are [the mature students] the better students for that?
CS: In the long run, yes…. They will come out, and they will learn, they will learn the course much harder. They will ask questions, they will not just accept the formula, ‘Explain the formula to me, what does it mean?’… How it can be used? Where can I use it? What are the restrictions?....[About] revenue functions, cost functions, they ask questions, ‘Why? Where did you get this? How come I never see this where I am working?’


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12
Not liking to memorize rules and finding that to succeed in the course it was necessary to memorize many rules

The importance, for the respondents, of this source of frustration was estimated by taking the set of students who agreed with items 35 and 5 (intersection).

Item 35. To succeed in the course it was necessary to memorize many rules
70%, 64%, 82%

Item 5. I don’t like memorizing rules
46%, 38%, 61%

The intersection counted 31%, 22%, 49% respondents.

The proportion of non-mature students is over twice as high as in the group of mature students.

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