# Counteracting Destructive Student Misconceptions of Mathematics

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## Abstract

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## 1. Introduction and Background

- Students’ conception of mathematics as a set of basically meaningless disconnected procedures that have to be memorized, and the consequences such a perception may have once the students begin to invent their own rules concerning mathematical procedures in order to create at least local meaning in what they are supposed to do.
- The interplay between students’ everyday experiences and conceptions and their conceptions regarding mathematics, and in particular what happens when everyday conceptions interfere with and influence their mathematical work and reasoning in a negative manner.
- Students’ conception of mathematical instruction as a decade-long training of what they perceive as essentially the same procedures and methods used over and over again, and the potential consequences such a conception may have for students’ self-efficacy.

## 2. Theoretical Constructs Related to Student Beliefs

Psychologically held understandings, premises, or propositions about the world that are thought to be true. Beliefs are more cognitive, are felt less intensely, and are harder to change than attitudes. Beliefs might be thought of as lenses that affect one’s view of some aspect of the world or as dispositions toward action. Beliefs, unlike knowledge, may be held with varying degrees of conviction and are not consensual. Beliefs are more cognitive than emotions and attitudes.[2] (p. 259)

**Beliefs about mathematics education**: (a) beliefs about mathematics as a subject; (b) beliefs about mathematical learning and problem solving; (c) beliefs about mathematics teaching in general**Beliefs about the self**: (a) self-efficacy beliefs; (b) control beliefs; (c) task-value beliefs; (d) goal-orientation beliefs**Beliefs about the social context**: (a) beliefs about the norms in their own class ((a1) the role and the functioning of the teacher; (a2) the role and the functioning of the students); (b) beliefs about the socio-mathematical norms in their own class [4] (p. 28)

- Mathematics problems have one and only one right answer.
- There is only one correct way to solve any mathematics problem—usually the rule the teacher has most recently demonstrated to the class.
- Ordinary students cannot be expected to understand mathematics; they expect to simply memorize it and apply what they have learned mechanically and without understanding.
- Mathematics is a solitary activity, done by individuals in isolation.
- Students who have understood the mathematics they have studied will be able to solve any assigned problem in five minutes or less.
- The mathematics learned in school has little or nothing to do with the real world.
- Formal proof is irrelevant to the processes of discovery or invention. [7] (p. 359)

The DC is the set of reciprocal obligations and ‘sanctions’ which each partner in the didactical situation imposes or believes to have imposed with respect to the knowledge in question, explicitly or implicitly, on the other; or are imposed, or believed by each partner to have been imposed on them with respect to the knowledge in question. The DC is the result of an often implicit “negotiation” of the mode of establishing the relationships for a student or group of students, a certain educational environment, and an educational system.[12] (p. 54)

## 3. Methodological Aspects

## 4. First Case: Mathematics as Disconnected Procedures

**Christian**, in the technology/science stream (called htx) of Danish upper secondary school [15]. The teachers knew Christian from a course in programming, where he had shown a clear understanding of the logical aspects involved. However, from Christian himself and from a colleague who taught him before, they learned that he was close to a failing grade in his mathematics course. This led the two prospective maths counsellors to further investigate the case of Christian. They wanted to find out what he thought of mathematics, both as a school subject and a discipline, and what his perceptions of logic and mathematical truth were. Hence, they designed a diagnostic interview in order to uncover some of his mathematics-related beliefs that might have acted as stumbling blocks and destructive myths in his learning of mathematics. The teachers found that Christian felt alright with mathematical topics which he immediately could see as having a practical application, e.g., calculation with percentages, whereas he saw no relevance of trigonometry. When performing simple arithmetical computations on specific numbers, e.g., involving addition, multiplication, and parentheses, Christian experienced no difficulties but, when presented with a simple algebraic task in a symbolic domain, he exclaimed: “I hate algebra. Because it’s full of ‘cheaters’” [15] (p. 147). Asked about the history of mathematics, Christian showed some knowledge concerning counting and computations as he replied “... counting with stones, later the abacus” [15] (p. 146). When asked about the hierarchy of arithmetical operations, he suggested that “... someone had invented it” and provided the following account of it: “[when] multiplying, you calculate with larger numbers to begin with. Then you want to reduce them first, so you take the simple part and pluses and minuses afterwards. Parentheses are often like… what’s inside them is more important. You always ‘frame’ what’s more important” [15] (pp. 146–147). On the basis of the interview, the two prospective maths counsellors diagnosed Christian as having a conception of mathematics as a subject “consisting of an incomprehensible swarm of weird and incoherent mnemonic rules that make no sense” [15] (p. 147). Because of this, he seemed to have convinced himself that he could not understand or do mathematics at the level required and might as well give up. In other words, he was under the spell of a destructive myth.

^{2}= a

^{2}+ b

^{2}+ 2ab, again supported by figures showing the corresponding squares and rectangles. In the third session, he had to work with the power conventions and rules. Upon being asked to deduce that a

^{0}= 1, Christian uttered: “Oh, I’ve seen that on the Internet, you just have to accept that it is 1. I once asked a third year student about it, and he said that the proof was like really long” [15] (p. 177). As the math counsellors explain, once Christian was told that in less than two minutes he would have made this proof himself, he looked rather suspicious. Yet, applying the previously deduced power rules, he was indeed able to do so. At the end of the session, Christian was asked to respond to the fact that he had just deduced all the power rules and that the only mnemonic rules he had applied were some basic axioms and the hierarchy of the arithmetic operations. He was rather baffled—although obviously also proud of his accomplishments.

I have come to view the solving of mathematics tasks in a different way. The same is true of programming. I’ve begun considering what possibilities I have for getting help, when there is something I can’t figure out—as opposed to giving up, which is what I would have done before. I still prefer to solve the tasks by myself. When something is difficult, I have now learned that I can address the task by simplifying it—place things in a sequence and take them bit by bit, instead of looking at everything in one big hotchpotch. [...] I found out that mathematics is simpler than I thought prior to the intervention. I can now remember how the formulae [rules] are connected. Actually, I’m going to have some math as part of my chosen higher education program, where I’ll be studying ICT technology. [...] I’ve always known that there was some kind of logic in mathematics, only I hadn’t found it. Now I have. [...] After the intervention, I have gained the courage to pursue a higher education degree, where I have to apply mathematics. Before, I believed that I suffered from dyscalculia. You made me see mathematics in a completely different way. Now I know that I’m not innumerate.[17]

## 5. Second Case: Everyday Conceptions in Mathematics

**Student A**, as they called her, closely throughout her three years in the technical branch of upper secondary school where she took mathematics and physics at the highest possible level, Lomholt and Larsen being her teachers in these subjects [18,19]. They specifically intervened towards her learning difficulties during the last semester of her second year and the first semester of her third year.

^{2}but not that d⋅d = d

^{2}. When examining or making statements involving formulae, she had a strong preference for working with specific numbers rather than with symbolic expressions. She was neither able to interpret, nor convinced by, conclusions obtained by manipulating formulae or expressions. When asked whether it is true that the volume of a cube is multiplied by 8 if all edge lengths are multiplied by 2, she wrote V = s⋅s⋅s and then said, “I don’t think that the new volume is eight times as large” [18] (p. 94). She maintained this response when prompted to write V’ = (2s)⋅(2s)⋅(2s) = 8s

^{3}. When asked whether it is true that if one halves the diameter of a circle then both the area and the circumference of the new circle are half those of the original circle, respectively, she was convinced that the answer is “yes”, even though she was able to correctly state the formulae for the area and circumference of a circle. The teachers invited her to make drawings representing doubling the edge lengths of a square and of a cube, which she was hardly able to do. After having been assisted by the teachers in making the drawings and then asked to interpret them, she was not convinced that they show that the area and the volumes are multiplied by 4 and 8, respectively. As the teachers wrote: “She is algebraically convinced but remains uncomfortable with drawings, at least when chequered or diced patterns are involved.” [18] (p. 95) In other words, A was a strong believer in the “law” that if something in a context is doubled, then everything is doubled, which is a marked instance of the over-generalization of proportionality, as previously studied [20,21].

## 6. Third Case: Long-Standing Training of Procedures

- Rikke: This is pretty linear!
- Teacher: Why does this surprise you?
- Rikke: Because normally, when we have something that grows per time unit, then it is an exponential function. [23] (p. 228)

- Rikke: I think it is because there are so many different ways to calculate it, and then you begin to doubt [...]. I often think about if what I do is actually correct. [...]
- Teacher: Is it the same in other subjects, i.e., that you feel this insecurity, or is it peculiar to mathematics?
- Rikke: In biotechnology, for example, we also calculate and solve problems sometimes. But there you get insecure because it is difficult, and the others think so too. So you are more in it together, finding it hard. And then we can ask the teacher. But in maths, which is a very important subject, and I have it at A level… there are other factors involved, I think.
- Teacher: But biotech is also important and at A level.
- Rikke: Yeah, I know that, but, in biotech, we are all working together on tasks that are pretty hard, and which we haven’t tried before, whereas we’ve had mathematics since forever. Biotech is new in a completely different way, and we don’t have the same background knowledge as we do in math. In biotech everything is new, and we all have to learn the same, from the same ‘beginning’, if you can say that.
- Teacher: Is it that you feel it is worse in math because you ought to be able to do it, since you’ve studied maths for so many years?
- Rikke: Yes, I think so, and then I get insecure, if there is something I can’t figure out. Then I think, ‘fuck, it’s going to affect my grades’ or something. I feel stupid.
- Teacher: In math, the pressure of expectations is greater in some way?
- Rikke: Yeah, a little I think. It seems like the expectations are a little greater, and that you should be able to do everything in math, I think.
- Teacher: Are these expectations from the teacher? The classmates? Or the textbook? Where do they stem from?
- Rikke: All of the above, kind of… Now, you are in upper secondary school and you are supposed to be able to do it.
- Teacher: A final question. In the final national written exams in mathematics, B level and A level, how many students do you think fail these?
- Rikke: Not very many.
- Teacher: Make a guess. In percentage.
- Rikke: Maybe 5%.
- Teacher: No, actually it is a lot higher. I’m not saying this to scare you, but rather to deal with your idea that ‘everybody is able to do maths’ and ‘we’ve had it for such a long time, so we ought to be on top of it’. Math is hard and it is very different in upper secondary school. So, you need to view it in the same way as you view biotech. [23] (pp. 238–239)

Hi.I’d like to thank you for the opportunity to participate in the intervention. It was a way for me to come to think about and view math in a different way than what I’ve done before. [...] in the beginning it did in fact put my self-confidence on trial, but that has changed, of course. [...] I truly feel that my motivation for raising my hand in various subjects has grown. I am now more daring, even if what I say sometimes turns out to be wrong. In principle, I have come to trust myself more than I did before. It’s not that my attitude towards math has changed as such, since I’ve always liked maths. But the sessions with you guys made me desire to be taught more math, simply because I now feel more confident in the subject. I’m also more confident now in my own approach to solving tasks—if you see what I mean. I don’t wait for my classmates to confirm that they have used the same approach as I did—now I actually trust myself, what I do, and what I know. Cooperation with my classmates in maths—I feel—has also changed to my advantage, if I may say so. The methods you provided me with in the sessions made me trust myself, without all the time needing to seek confirmation that what I’m doing is right. [...] You told me that maths has a high failure rate—this has made me aware that I’m not the only one who finds maths hard sometimes, and that I’m not alone in this.Since I now have some tools and other ways of seeing mathematical relationships, I also have more appetite for doing cranky tasks on my own. I’ve learned to turn my insecurity into something positive, so I look at [mathematical] relationships from different angles and use my calculations to arrive at results that I myself believe in. An important point, I think, is not always just restricting yourself to the basics [rules, formulas, etc.], but to do experiments, make some calculations. For example, for exponential functions there is a certain formula, a certain way the graph has to look, and some requirements that have to be fulfilled. Previously—as I believe you also noticed—I restricted myself to considering a mathematical relationship solely based on the information given [in the task] and the basics and used that to decide whether a function is exponential, linear, and so on. This didn’t always work so well for me [...]. You taught me that with different tools and methods you can easily determine the relationship, without initially being able to see it on the basis fof the given information. [...] I can definitely use these experiences and my newly acquired tools and methods from the intervention in my everyday maths classes. And I want to say that I’m quite positively surprised at what I’ve actually gained from all of this.I hope you can use my answer for something.Rikke[23] (pp. 256–257)

## 7. Analysis of the Three Students’ Beliefs

**Christian**was a student who had come to believe that mathematics is a subject that is out of reach in terms of understanding, at least for him. His own encounters with the subject had failed, also according to his previous teachers, and he accepted this and in fact agreed. He acknowledged that it is potentially possible for others to learn and do mathematics, but unfortunately not for him. He believed to be operating under a didactical contract saying that most of mathematics needs to be memorized and learnt by rote, because the subject makes no sense from a logical point of view. In particular, things related to algebra and geometry appeared almost as “abracadabra” to him. Still, he did accept that some aspects of mathematics can be of practical importance.

**Student A**, it is difficult to say whether her beliefs and the myths she were under primarily concerned “mathematics as a discipline” or “mathematics education”, although we lean towards the latter. This was not least due to the fact that she considered mathematics to be a subject dealing mainly with computations. In fact, she expressed almost fear—or “math anxiety” [24]—when having to perform any type of mathematical reasoning or algebraic manipulation. From a didactical contract perspective, Student A knew that she was required to do this as part of mathematics, but she was very reluctant to embark on such undertakings.

**Rikke**was a student who was embarrassed by the fact that she did not do as well in mathematics as she, according to her own beliefs, thought she ought to, since she had been learning mathematics for 10 years already. Furthermore, she believed that she was one of only a very few students who was performing somewhat poorly. These beliefs mainly correspond to the edge of the tetrahedron connecting the vertices “beliefs about mathematics education” and “beliefs about the self”, although there are also links to the vertex “beliefs about the social context”. Rikke was concerned about her academic appearance and worked hard to avoid exposing her difficulties to other students and to her teachers. She was also concerned—in fact even nervous—about her grades in mathematics, which was probably one reason why she was rather reluctant to participate in the intervention in the first place.

## 8. Discussion of the Efficacy of the Interventions

## 9. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Jankvist, U.T.; Niss, M.
Counteracting Destructive Student Misconceptions of Mathematics. *Educ. Sci.* **2018**, *8*, 53.
https://doi.org/10.3390/educsci8020053

**AMA Style**

Jankvist UT, Niss M.
Counteracting Destructive Student Misconceptions of Mathematics. *Education Sciences*. 2018; 8(2):53.
https://doi.org/10.3390/educsci8020053

**Chicago/Turabian Style**

Jankvist, Uffe Thomas, and Mogens Niss.
2018. "Counteracting Destructive Student Misconceptions of Mathematics" *Education Sciences* 8, no. 2: 53.
https://doi.org/10.3390/educsci8020053