torquetum | Entries tagged with proof

Redish, E. & Smith, K.A. (2008). Looking beyond content: Skill development for engineers. Journal of Engineering Education, 97(3), 295-307.

Redish and Smith summarise key findings in cognitive and neurological research on how learning occurs and is manifested in the brain. They cite Pellegrino’s three main threads of educational research: constructivism, knowledge organisation and metacognition, and his three components of educational practice: curriculum, instruction and assessment. They proceed to link Pellegrino’s summary of educational research with cognitive research into learning, giving what I found was a really useful summary of various key cognitive findings. Their theoretical framework for learning is based on the concepts of activation, association, compilation and control.

Activation refers to the activation of neurons, becoming entrained and working together in clusters. Activation is related to the differences between long term memory and working memory and how working memory can only handle about seven “chunks” of knowledge at a time. This essentially constructivist relationship of neurological activation to learning supports the idea that student misconceptions are not rigid but can be changed.

Association refers to entrained neurological pathways becoming associated with one another through repeated practice to create schemas, “skeletal representation[s] of knowledge abstracted from experience that organizes and guides the creation of particular representations in specific contexts” (p. 298). Associational paths link to Pellegrino’s knowledge organisation. Association is highly context dependent: change the context and different associations are made. This can be disadvantageous in educational contexts as correcting a student’s misconceptions in one arena can fail to transfer to another due to different associational paths: “students build up alternative associational paths; one set of knowledge is activated specifically for a physics class but the other intuitive knowledge is not erased but remains for activation in all other situations” (p. 298). The authors report that this idea links to work in cognitive science on “conditioning, attempts to eliminate conditioning (extinction), and its reemergence (relapse)” (p. 299).

Compilation is related to the idea that “items that are originally weakly associated may become very strongly tied together” (p. 299). This is the process by which chunks are created. Experts have many actions strongly tied together, reducing the cognitive load of certain types of problem solving, while novices experience greater cognitive load. Experts have to “reverse engineer” how certain types of problems are solved in order to teach solution processes to students. It can help to observe students solving problems, to see where their difficulties lie.

Control refers to executive control and metacognition. We are constantly (and usually unconsciously) deciding what input from the external world to pay attention to. These decisions are made by control schema which have developed over time and experience. “These control schemas have three important consequences: they create context dependence, they give us a variety of resources for building new knowledge and solving problems, and they control which of these resources we bring to bear in given circumstances.” (p. 299). Context dependence: We all have developed “expectation schemas” which consider available input and decide what is important and how to react.

Epistemological resources: we understand that knowledge can be transmitted and also created. Some knowledge is outside our control and other is within our control.

Epistemological games, or e-games, is the term given to “a reasonably coherent schema for creating new knowledge by using particular tools” (p. 300). Such e-games include an understanding of beginning and end, what information to use, what structure to impose, etc. such as the typical process of solving a physics problem. Choice of e-game is crucial to successful problems solving.

Having provided a summary of cognitive research into learning, the authors take a look at the role of mathematics in engineering and how the differences between how maths is typically taught and how maths is used by engineers can create some serious obstacles to learning. In engineering, students are expected to learn the “syntax” of mathematics, but also what it means and how to use it effectively in engineering contexts. How a mathematician or engineer interprets symbols can differ widely. To a mathematician the letters chosen to represent the variables might be arbitrary, but an engineer they carry meaning. “we [engineers] tend to look at mathematics in a different way from the way mathematicians do. The mental resources that are associated (and even compiled) by the two groups are dramatically different. The epistemic games we want our students to choose in using math in science require the blending of distinct local coherences: our understanding of the rules of mathematics and our sense and intuitions of the physical world.” (p. 302). Differences between how maths is used in maths and in engineering include (p. 302):

Equations represent relationships among physical variables, which are often empirical measurements
Symbols in equations carry information about the nature of the measurement beyond simply its value. This information may affect the way the equation is interpreted and used.
Functions in science and engineering tend to stand for relations among physical variables, independent of the way those variables are represented

Students can fail to make the connection between symbols and their physical meaning and their relationship to empirical measurement.

The authors provide a schematic describing modelling: define your physical system, represent the system mathematically, process that mathematical system as appropriate, interpret your results. Later they report that one view of modelling is that it is inseparable from problem solving. Redish and Smith, however, feel that modelling is easier to teach than problem solving and that, in practising modelling, much problem solving is learned along the way. They support this view by framing problem solving in similar terms to their schematic of the modelling process. I am a proponent of the Polya framework for problem solving, which differs somewhat from the Redish and Smith framework. There are similarities, however, and a fruitful comparison of the two could no doubt be made.

While the role of proof in engineering mathematics is not a focus of this paper, the authors do have a few things to say which support a relatively high presence of proof.

“Often what our students learn in our classes about the practice of science and engineering is implicit and may not be what we want them to learn. For example, a student in an introductory engineering physics class may learn that memorizing equations is important but that it is not important to learn the derivation of those equations or the conditions under which those equations are valid. This metamessage may be sent unintentionally.” (p. 297)

“Often, in both engineering and physics classes, we tend to focus our instruction on process and results. When we teach algorithms without derivation, we send our students the message that “only the rule matters” and that the connection between the equation we use in practice and the assumptions and scientific principles that are responsible for the rule are irrelevant. Such practices may help students produce results quickly and efficiently, but at the cost of developing general and productive associations and epistemic games that help them know how their new knowledge relates to other things they know and when to use it. As narrow games get locked in and tied to particular contexts, students lose the opportunity to develop the flexibility and the general skills needing to develop adaptive expertise.” (p. 303)

There are some “bad habits” which can be learned in maths class. Giving algorithms without derivation gives the impression that the assumptions and scientific principles behind the rules are not important (see quote above). Another bad “maths” habit is substituting numbers early in the problem-solving process. This tends to hide associations between variables and inhibits reflecting back on the process. Another habit learned in maths class is to elevate the “processing” part of the modelling process above the others, thereby hindering transfer of the skills to other courses more based on the whole modelling package.

The authors close their paper with a discussion of using cooperative learning in a course teaching modelling. They give some interesting examples of relating physical reality to mathematical models and vice versa.

I thoroughly enjoyed reading this paper. Some papers I can summarise in a paragraph. This one took me two pages simply to summarise and I also have four pages of notes! I already knew a lot of the cognitive findings, but some was new to me, such as epistemic games and conditioning. This is the sort of paper one reads and rereads many times.

Do not treat this blog entry as a replacement for reading the paper. This blog post represents the understanding and opinions of Torquetum only and could contain errors, misunderstandings or subjective views.

Varsavsky, C. (1995). The design of the mathematics curriculum for engineers: A joint venture of the mathematics department and the engineering faculty. European Journal of Engineering Education, 20(3), 341-345.

Varsavsky and colleagues ran a survey amongst the convenors of 130 engineering courses at their institution (Monash, Caulfield) of what maths topics were needed, by whom and when. The paper was interesting for its broad overlaps with my local situation: the maths taught is similar and the challenges of diversity are similar, as well as the lack of communication between involved parties. The survey found, as I have found in a similar survey, that the overall maths needs are large, but most courses use very little, or only basics. The paper closes with some sensible suggestions for setting up an engineering maths curriculum. I am interested to see that, once again, proofs don’t get a mention.

Mustoe, L. (1995). Industry expects – but who should determine the curriculum? European Journal of Engineering Education, 20(3), 313-323.

Mustoe asks “How far should an undergraduate education in engineering aim to meet the needs of industry as perceived by industry?” (p. 313). He proceeds to come at this question from multiple directions: the requirements of industry, of engineering academics, of mathematicians, of professional bodies. He finds that the views of what maths is necessary and desirable varies widely across these groups and that it is impossible to meet everyone’s demands. For example, does the engineering mathematics lecturer teach mathematical concepts in context or not? In-context problems at the first-year level are hard to find as real-world problems are messy and ill formed. In addition, the students might not yet be familiar with the engineering context and confusion rather than a sense of relevance will result. Mustoe cites a view that concepts are easier to study in the abstract without confusing contexts, however, many clamour for relevance in the maths curriculum and show how students lose interest due to no clear links between maths class and other courses.

Throughout Mustoe’s literature review of a wide variety of viewpoints, there is an emphasis on basic manipulative skills as well as modelling. “Five general areas of competence worth aiming for are: a ‘feel’ for the magnitudes of quantities, an ability to handle fractions without fear, an appreciation of the use of approximations, the ability to manipulate formulae confidently and the skill to cope with calculus.” (p. 320) Mustoe has little patience for long integration problems: “Other than training in patience, persistence and concentration, there is little to be gained from an exercise in integration, for example, which requires three successive substitutions and the use of a trigonometric identity for good measure.” (p. 321). There is not much mention of the role of proof in engineering mathematics courses within this paper, except for a citation of Craggs (1978): “Craggs was concerned about the level of rigour in the mathematics taught to engineers. His conclusion was that the aims for teaching were to encourage accuracy in manipulation, to state theorems with proof where it was easy and with heuristic justification where it was not, to illustrate the possible gains and losses if work was undertaken outside the domain of these theorems and to respect the additional expertise of the trained mathematician.” (p. 318).

Mustoe concludes that curricula should be dismantled and rebuilt from the ground up with a clear view of what the role of mathematics is in the degree programme. (Note that this is a 1995 paper, so hardly current.) I loved this paper for its variety of views and broad literature review.

Almeida, D. (2000). A survey of mathematics undergraduates’ interaction with proof: some implications for mathematics education. International Journal of Mathematics Education in Science and Technology, 31(6), 869-890.

The author ran a large (N = 473) quantitative study of students’ perceptions of proof and then a smaller qualitative sub-study of proof perceptions and practices. He lists several tensions: between novice and professional, between informality and formality, between state of becoming and state of being. “From an educational point of view the prototypical proof practices of beginning undergraduates, despite being imperfect, ought to be initially accommodated simply because the learner believes in their veracity—for the student it is a proof. A tension also arises in the process of convincing the student of the necessity of making the transition to formal proof” (p. 870). Hmm, reference to Ruthven and Coe (1994). I should dig out work by these two if I do a study of my own; it looks intriguing. This study has a nicely described methodology. The qualitative part included students judging the efficacy of proofs across a range of representations, such as visual or abstract. There were also interviews. There is lots of detail on the qualitative stuff, I won’t summarise it here. One point: he finds the 2nd year students have a deeper understanding of proof than the 1st years, which is an encouraging result. The qualitative results show that most students have poor proof perceptions and practices despite apparent agreement with the principles encountered in the questionnaire. The author speculates that at least part of the problem is the difference between how maths is taught and how original mathematics is constructed. The author closes with a suggestions related to development of concept images. The paper has a detailed Appendix which could be of use of one wanted to replicate the study.

Almeida, D. (2003). Engendering proof attitudes: can the genesis of mathematical knowledge teach us anything? International Journal of Mathematics Education in Science and Technology, 34(4), 479-488.

The author “reports on how proof attitudes could be inculcated in students by offering them a course design that is faithful, to some extent, to the historical genesis of modern mathematics.” (p. 479). As some of the papers already discussed, the author gives a brief rundown on the history of proof. “If the sequence for understanding mathematics [intuition, trial, error, speculation, conjecture, proof ] as espoused by MacLane also applies in mathematics learning as well as in research and if the theory of levels in proving is accepted then there are implications for proof in mathematics education” (p. 482). The author describes a study where the students manipulated descrined shapes in a computer package and had to come up with a conjecture based on their observations and then prove the conjecture. The results were interesting, with students coming up with conjectures and providing proofs across a range of formality. Interestingly, the students displayed positive proof perceptions and practices.

Rodd, M.M. (2000). On mathematical warrants: proof does not always warrant and a warrant may be other than a proof. Mathematical Thinking and Learning, 2(3), 221-244.

The author defines various terms, such as justification: “rationale for a belief” and warrant: “that which secures knowledge” (p. 222). He cites Plantinga in defining “warrant” as that quality which distinguishes knowledge from belief. He distinguishes between justification and warrant in that “warrants are for truth, and justifications are reasons” (p. 223). “I may justify a proposition by reproducing a proof of it, but this may not warrant my knowledge of the proposition because the proof is not my own reasoning. Moreover, it may be possible to know without being able to justify in terms of reasons (visualization, for example, may warrant).” The author cites several very interesting studies, notably one by Coe and Ruthven (1994) where they found that students understand the role of proof, but do not tend to use it themselves, preferring empirical observations. “The students were better able to explain the nature of proof than to use proofs in their mathematical investigations: They were aware that proof is required for knowledge, but proof was not the means by which they secured their beliefs in the truth values of mathematical propositions” (p. 225).

The author emphasises how so many studies show that proving and knowing are not the same thing for students. (Note to self: When teaching engineering students, are we really looking for “warrants” to help them understand, where warrants could be formal proofs but could equally be something else?) More definitions: “warranting (truth exhibiting) and justifying (giving reasons for the belief)” (p. 227). The author asks “What might warranting mean in classroom practice? For example, if there is a way in which students can come to know through visualizing, how can this epistemic visualization be taught?” (p. 230). There is much in this long paper which I cannot summarise here, for example philosophy, the nature of knowledge and embodiment.

“It is this autonomous needer of proof that I suggest is central to a developing mathematician. This need is more than an attitude adopted for math lessons, it is a part of the student’s body–mind. “I just do not believe the result is true unless I have proved it”: In more formal analytic-philosophy language, such a student’s knowledge is dependent on the student’s own proof production. Such students exist, but, in my school and college teaching experience, they are quite rare.” (p. 234). I find this quote interestingly provocative. There’s a nice quote from Polya about proof – mainly geometric. There’s a lovely section on “visual proofs” showing how visualisation can constitute a warrant. However, effective visualisation can be hard to teach and is hard to assess. While the author never mentions engineering maths in this article, I keep finding links. This is a very interesting paper and quite different in focus to the ones I’ve read up to this point.

Jones, K. (2000). The student experience of proof at university level. International Journal of Mathematical Education in Science and Technology, 31(1), 53-60.

Interesting: I am starting to see some themes running through these papers. There was the bit on empirical validation I mentioned in the paper just before this. Here I am seeing, again, the emphasis on how students don’t know what a proof is or what it is supposed to do. This is really important. It’s rearranging the deckchairs on the Titanic for us to quibble over misuse of symbols, for instance, in an algebraic proof, when the students don’t even know what constitutes a proof or why we bother to place value on them. Anyway, back to the paper.

The students involved in this study are pre-service mathematics teachers. If they don’t, as university students, have a good conception of proof, then they’re not going to be able to teach proof well to the learners in their classrooms. (Interesting segue into grading concept maps – something I’ve never thought about.) There was a correlation between the complexity of the proof concept map and the student’s passing grade on the course, however the author argues that neither passing grade nor complexity of map relates to being a better teacher. The author makes a good point that many of the studies on student’s conceptions of proof are based on small studies – not that he’s disagreeing with them. The author closes with a call to break the “vicious circle” of training maths teachers without attention to proof who then fail to pass on understanding of proof to their classes.

Hoyles, C. (1997). The curricular shaping of students’ approaches to proof. For the Learning of Mathematics, 17(1), 7-16.

The author criticises studies which focus on students’ relationship with and understanding of proof independent of borader educational and social context. She argues that such an approach is locked into producing results which suggest a hierarchy of interaction, such as a proof heiracrchy “with empirical "proof" or procedural validation by action at the bottom, and rigorous deductive argument or relational validation based on premises and properties at the pinnacle” (p. 7). The author suggests that we need to look for more wide ranging influences on student responses as well as how the goals for incorporating proof into the curriculum are operationalised. A social dimension can be advantageously used in explain proofs by getting students to explain to a peer. (To link this idea to something currently trendy, whiteboard tutorials could incorporate validation or explanation of proofs.)

The author discusses the role of proof in the current (1997) UK national curriculum, which has an unfortunate implication that “it is now official that proof is very hard and only for the most able” (p. 9), due to the layered and hierarchical nature of the curriculum. Just a side note: all the papers I’ve read so far emphasis the intent student focus on “empirical validation” and the complexities around how that differs from formal symbolic proofs. Some take it as a big problem, some see it as a difference which is unavoidable and both sides have worth, some see it as a step on a developmental path. The author goes on to discuss a study on justification and proof, which looked at students conceptions of proofs and how they constructed proofs, rather than taking a deficit view. The author goes on to discuss interesting data which I don’t have space for here. Finally the author argues that, if we wish students to develop a robust conception of proof and skills in proof construction and validation, such skills development needs to be explicitly included in a scaffolded curriculum. We must also avoid thinking that development of skills related to proof are developed linearly.

Do not treat this blog entry as a replacement for reading the paper. This blog post represents the understandings and opinions of Torquetum only and could contain errors, misunderstandings and subjective views.

Selden, J. & Selden, A. (1995). Unpacking the logic of mathematical statements. Educational Studies in Mathematics, 29(2), 123-151.

From the abstract: “This study focuses on undergraduate students' ability to unpack informally written mathematical statements into the language of predicate calculus... We discuss this data from a perspective that extends the notion of concept image to that of statement image and introduces the notion of proof framework to indicate that part of a theorem's image which corresponds to the top-level logical structure of a proof... We infer that these students would be unable to reliably relate informally stated theorems with the top-level logical structure of their proofs and hence could not be expected to construct proofs or validate them, i.e., determine their correctness” (p. 123).

Students can be confused between proof and evidence, also, some accept proofs as valid, but still like to see empirical evidence. There are some nice references to different ways to teach proofs to students. The authors look at students’ abilities to clarify the logical structure of mathematical statements in the context of constructing and validating proofs themselves. They found that this ability was not successfully represented by most of the students in the study. When judging the correctness of a proof, there are questions which need to be asked and answered by the person reading the proof. Their study showed that students could neither ask those questions nor answer them. The paper goes on to define and discuss terms like “proof”, “unpack” and “formal and informal statements”. Also a proof framework is defined: “By a proof framework we mean a representation of the "top-level" logical structure of a proof, which does not depend on detailed knowledge of the relevant mathematical concepts, but which is rich enough to allow the reconstruction of the statement being proved or one equivalent to it. A written representation of a proof framework might be a sequence of statements, interspersed with blank spaces, with the potential for being expanded into a proof by additional arguments” (p. 129).

The authors argue that students “who cannot reliably unpack the logical structure of informally stated theorems, also cannot reliably validate their proofs” (p. 130) and support their argument with discussion of proof frameworks. There is a lovely discussion of “statement images” which is a collection and network of all knowledge items associated with a particular statement and which can be very rich and detailed. There is reference to a Radford 1990 paper which I’d like to dig out. The data is presented in detail and finally three suggestions for teachers are made (summarised from p. 142):

First, since many undergraduate students have difficulty unpacking informally written statements, it might be useful to present theorems and definitions both in a more informal way and in a more formal way.

Second, it might be useful to offer university students some explicit instruction or advice on validation, an area currently more or less neglected. There is no reason for such instruction to be restricted to proofs.

Finally, because university students who still have weak validation skills may not be able to distinguish proofs from supplementary or explanatory comments, it might be good to present material in a way that makes this distinction clear.

Moore, R.C. (1994). Making the transition to formal proof. Educational Studies in Mathematics, 27(3), 249-266.

From the abstract: “This study examined the cognitive difficulties that university students experience in learning to do formal mathematical proofs... An inductive analysis of the data revealed three major sources of the students' difficulties: (a) conceptual understanding, (b) mathematical language and notation, and (c) getting started on a proof. Also, the students' perceptions of mathematics and proof influenced their proof writing” (p. 249). The author did a grounded study of students’ difficulties with proof, by observing classroom activities and interviewing instructors and students. The author found seven major difficulties (p. 251-252):

The students did not know the definitions.

The students had little intuitive understanding of the concepts.

The students' concept images were inadequate for doing the proofs.

The students were unable, or unwilling, to generate and use their own examples.

The students did not know how to use definitions to obtain the overall structure of proofs.

The students were unable to understand and use mathematical language and notation.

The students did not know how to begin proofs.

The author explores these in greater detail, particularly concept image and concept definition, using definitions and generating examples. He finds that students need to develop concept images through diagrams, examples and graphs before being able to move to symbolic definitions and manipulations. Also, the students’ understanding of a concept seemed to be cognitively separated into different schemata, whereas the instructor had the entirely of the knowledge needed for the proof incorporated into a single schema. This makes working through a proof have a greater cognitive load for students than for the instructor. All the various difficulties seem to combine to make the starting of a proof really difficult. The author recommends a less abrupt transition to proof in the education system and perhaps a transition course on mathematical language and proof before doing courses which heavily depend on a knowledge of proof.

Hanna, G. & Jahnke, H.N. (1993). Proof and application. Educational Studies in Mathematics, 24(4), 421-438.

Rigorous proof in geometry has existed since the time of the ancient Greeks. Rigorous proof in analysis and algebra was slower to develop, only becoming ubiquitous in the 19th century. “Curiously enough, the traces of these two different traditions can be seen to this very day in school mathematics, where “proving” has remained for the most part peculiar to geometry” (p. 421). During the 1960s and 1970s there was a worldwide attempt, the authors say, to include formal proof at the school level, however there was poor teaching and much complaint and the movement did not succeed. There has been a “shift to a pragmatic view of proof” (p. 422) in which application is key - both “extra- and intramathematical” applications (p. 426). The authors say, “in the classroom proof has a status in algebra quite different from and inferior to its status in geometry” (p. 423) which definitely then has implications for university teaching, especially if one’s students have done very little geometry. The authors draw distinctions between the role of proof amongst mathematical scholars and in the classroom. For the former, the deductive nature of proof and syntactic correctness are of greatest importance. In the classroom, the relation of mathematics to reality is of greater importance and “the complex relationship between deductive reasoning or symbolic manipulation on the one hand and its application on the other must be developed afresh for each individual case” (p. 433), making teaching proof multi-dimensional and hard to do successfully.

“Our first thesis … is as follows: communication in scholarly mathematics serves mainly to cope with mathematical complexity, while communication at school serves more to cope with epistemological complexity” (p. 433)

“Our second thesis can … be stated as follows: in order to understand the meaning of a theorem and the value of its proof, students must have extensive and coherent experience in the appropriate application area. This pragmatic foundation can and should be taught in conscious separation from the formal derivation. Only then will students be able to see the real point of a proof” (p. 434).

This is a lovely informative paper which I enjoyed reading a lot. There is a lot of detail on the history of proof which I have omitted in my little summary here, but this paper is definitely worth a reread should I ever need to do something on proof again.