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This thesis is on the automation of diagrammatic proofs, a novel approach to mechanised mathematical reasoning. Theorems in automated theorem proving are usually proved by formal logical proofs. However, there are some conjectures which humans can prove by the use of geometric operations on diagrams that somehow represent these conjectures, so called diagrammatic proofs. Insight is often more clearly perceived in these diagrammatic proofs than in the algebraic proofs. We are investigating and automating such diagrammatic reasoning about mathematical theorems. Concrete rather than general diagrams are used to prove ground instances of a universally quantified theorem. The diagrammatic proof in constructed by applying geometric operations to the diagram. These operations are in the inference steps of the proof. A general schematic proof is extracted from the ground instances of a proof. it is represented as a recursive program that consists of a general number of applications of geometric operations. When gien a particular diagram, a schematic proof generates a proof for that diagram. To verify that the schematic proof produces a correct proof of the conjecture for each ground instance we check its correctness in a theory of diagrams. We use the constructive omega-rule and schematic proofs to make a translation from concrete instances to a general argument about the diagrammatic proof. The realisation of our ideas is a diagrammatic reasoning system DIAMOND. DIAMOND allows a user to interactively construct instances of a diagrammatic proof. It then automatically abstracts these into a general schematic proof and checks the correctness of this proof using an inductive theorem prover.

Student Number : 0008975N - M Ed research report - School of Education - Faculty of Humanities / This qualitative study was conducted in one school in an informal settlement, West of Johannesburg. The study explored how a grade 7 teacher promoted mathematical reasoning in multilingual mathematics class of English second language learners. The focus of the research was on how a Grade 7 mathematics teacher interacts with the learners to encourage mathematical reasoning during his teaching in a multilingual class. The study also looked at the kind of tasks the teacher used to promote mathematical reasoning and how he uses language to enable mathematical reasoning. The study was informed by a theory of learning which emphasises the importance of social interaction in the classroom where the teacher encourages learners to interact with each other to explain their thinking and to justify their answers. Data was collected through lesson and teacher interviews. The study shows the teacher focused more on developing the learners’ procedural fluency. This focus on procedural fluency was accompanied by the dominance of the use of English by the learners.

Abstrakt Syftet med denna studie är att undersöka faktorer som kan påverka elevers matematiska resonemang. Detta gjordes genom att undersöka vilken typ av resonemang elever i årskurs 9 använde när de löste olika uppgifter och även elevernas uppfattningar om matematik. Teoretiska perspektiv för studien är baserad på Lithners teoretiska ramverk som skiljer på två huvudtyper av matematiska resonemang: den första är imitativt resonemang, där eleven imiterar lösningsalgoritmer som hen känner till, och kreativt matematiskt resonemang, där eleven skapar en lösning för att lösa en uppgift. Metoderna som användes i studien var observation och ostrukturerade intervjuer. Resultaten visar att elever använde imitativa resonemang för att lösa uppgifter och sällan använde kreativa matematiska resonemang. Det är rimligt att anta att arbetssättet hade en roll för vilken typ av matematiska resonemang elever använde för att lösa olika övningsuppgifter. Resultatet tyder även på att elevens träning på att lösa uppgifter som krävde användning av kreativt matematiskt resonemang påverkade betyget eleven fick i det skriftliga provet. Eleverna indikerade uppfattningar om att uppgifter som krävde användning av kreativa matematiska resonemang för att lösas var svåra, särskilt svåra att förstå. De slutsatser som dras är att antalet uppgifter som kräver användning av kreativt matematiskt resonemang för att lösas bör utökas och eleverna behöver utveckla sitt matematiska språk.

Thesis (M.A.)--Brigham Young University. Dept. of Mathematics Education, 2007. / Includes bibliographical references (p. 54-56).

<div> <div> <div> <p>Currently there is a need for research that explores students’ understanding of advanced topics in order to improve teaching and learning beyond the context of introductory-level courses. This work investigates students’ reasoning about graphs used in enzyme kinetics. Using semi-structured interviews and a think aloud-protocol, 14 second-year students enrolled in a biochemistry course were provided two graphs to prompt their reasoning, a typical Michaelis-Menten graph and a Michaelis-Menten graph involving enzyme inhibition. Student responses were coded using a combination of inductive and deductive analysis, influenced by the resource-based model of cognition. Results involve a discussion regarding how students utilized mathematical resources to reason about chemical kinetics and enzyme kinetics, such as engaging in the use of symbolic/graphical forms and focusing on surface-level features of the equations/graphs. This work also addresses student conceptions of the particulate-level mechanism associated with competitive, noncompetitive, and uncompetitive enzyme inhibition. Based on the findings of this study, suggestions are made regarding the teaching and learning of enzyme kinetics. </p> </div> </div> </div>

Avhandlingen består av två ganska olika delar som ändå har en del gemensamt. Del A är baserad på två artiklar i matematik och del B är baserad på två matematikdidaktiska artiklar. De matematiska artiklarna utgår från ett begrepp som heter polynomkonvexitet. Grundidén är att man skulle kunna se vissa ytor som en sorts ”tak” (tänk på taket till en carport). Alla punkter, eller positioner, ”under taket” (ungefär som de platser som skyddas från regn av carporttaket) ligger i något som kallas ”polynomkonvexa höljet.” Tidigare forskning har visat att för ett givet tak och en given punkt så finns det ett sätt att avgöra om punkten ligger ”under taket”. Det finns nämligen i så fall alltid en sorts matematisk funktion med vissa egenskaper. Finns det ingen sådan funktion så ligger inte punkten under taket och tvärt om; ligger punkten utanför taket så finns det heller ingen sådan funktion. Jag visar i min första artikel att det kan finnas flera olika sådana funktioner till en punkt som ligger under taket. I den andra artikeln visar jag några exempel på hur man kan konstruera sådana funktioner när man vet hur taket ser ut och var under taket punkten ligger. De matematikdidaktiska artiklarna i avhandlingen handlar om vad som krävs av studenterna när de gör universitetstentor i matematik. Vissa uppgifter kan gå att lösa genom att studenterna lär sig någonting utantill ur läroboken och sen skriver ner det på tentan. Andra går kanske att lösa med hjälp en algoritm, ett ”recept,” som studenterna har övat på att använda. Båda dessa sätt att resonera kallas imitativt resonemang. Om uppgiften kräver att studenterna ”tänker själva” och skapar en (för dem) ny lösning, så kallas det kreativt resonemang. Forskning visar att elever i stor utsträckning väljer att jobba med imitativt resonemang, även när uppgifterna inte går att lösa på det sättet. Mycket pekar också på att de svårigheter med att lära sig matematik som elever ofta har är nära kopplat till detta arbetssätt. Det är därför viktigt att undersöka i vilken utsträckning de möter olika typer av resonemang i undervisningen. Den första artikeln består av en genomgång av tentauppgifter där det noggrant avgörs vilken typ av resonemang som de kräver av studenterna. Resultatet visar att studenterna kunde bli godkända på nästan alla tentorna med hjälp av imitativt resonemang. Den andra artikeln baserades på intervjuer med sex av de lärare som konstruerat tentorna. Syftet var att ta reda på varför tentorna såg ut som de gjorde och varför det räckte med imitativt resonemang för att klara dem. Det visade sig att lärarna kopplade uppgifternas svårighetsgrad till resonemangstypen. De ansåg att om uppgiften krävde kreativt resonemang så var den svår och att de uppgifter som gick att lösa med imitativt resonemang var lättare. Lärarna menade att under rådande omständigheter, t.ex. studenternas försämrade förkunskaper, så är det inte rimligt att kräva mer kreativt resonemang vid tentamenstillfället. / This dissertation consists of two different but connected parts. Part A is based on two articles in mathematics and Part B on two articles in mathematics education. Part A mainly focus on properties of positive currents in connection to polynomial convexity. Earlier research has shown that a point z0 lies in the polynomial hull of a compact set K if and only if there is a positive current with compact support such that ddcT = μ−δz0. Here μ is a probability measure on K and δz0 denotes the Dirac mass at z0. The main result of Article I is that the current T does not have to be unique. The second paper, Article II, contains two examples of different constructions of this type of currents. The paper is concluded by the proof of a proposition that might be the first step towards generalising the method used in the first example. Part B consider the types of reasoning that are required by students taking introductory calculus courses at Swedish universities. Two main concepts are used to describe the students’ reasoning: imitative reasoning and creative reasoning. Imitative reasoning consists basically of remembering facts or recalling algorithms. Creative reasoning includes flexible thinking founded on the relevant mathematical properties of ob jects in the task. Earlier research results show that students often choose imitative reasoning to solve mathematical tasks, even when it is not a successful method. In this context the word choose does not necessarily mean that the students make a conscious and well considered selection between methods, but just as well that they have a subconscious preference for certain types of procedures. The research also show examples of how students that work with algorithms seem to focus solely on remembering the steps, and researchers argue that this weakens the students’ understanding of the underlying mathematics. Article III examine to what extent students at Swedish universities can solve exam tasks in introductory calculus courses using only imitative reasoning. The results show that about 70 % of the tasks were solvable by imitative reasoning and that the students were required to use creative reasoning in only one of 16 exams in order to pass. In Article IV, six of the teachers that constructed the analysed exams in Article III were interviewed. The purpose was to examine their views and opinions on the reasoning required in the exams. The analysis showed that the teachers are quite content with the present situation. The teachers expressed the opinion that tasks demanding creative reasoning are usually more difficult than tasks solvable with imitative reasoning. They therefore use the required reasoning as a tool to regulate the tasks’ degree of difficulty, rather than as a task dimension of its own. The exams demand mostly imitative reasoning since the teachers believe that they otherwise would, under the current circumstances, be too difficult and lead to too low passing rates.

This thesis explores two aspects of mathematical reasoning: affect and gender. I started by looking at the reasoning of upper secondary students when solving tasks. This work revealed that when not guided by an interviewer, algorithmic reasoning, based on memorising algorithms which may or may not be appropriate for the task, was predominant in the students reasoning. Given this lack of mathematical grounding in students reasoning I looked in a second study at what grounds they had for different strategy choices and conclusions. This qualitative study suggested that beliefs about safety, expectation and motivation were important in the central decisions made during task solving.  But are reasoning and beliefs gendered? The third study explored upper secondary school teachers conceptions about gender and students mathematical reasoning. In this study I found that upper secondary school teachers attributed gender symbols including insecurity, use of standard methods and imitative reasoning to girls and symbols such as multiple strategies especially on the calculator, guessing and chance-taking were assigned to boys. In the fourth and final study I found that students, both male and female, shared their teachers view of rather traditional feminities and masculinities. Remarkably however, this result did not repeat itself when students were asked to reflect on their own behaviour: there were some discrepancies between the traits the students ascribed as gender different and the traits they ascribed to themselves. Taken together the thesis suggests that, contrary to conceptions, girls and boys share many of the same core beliefs about mathematics, but much work is still needed if we should create learning environments that provide better opportunities for students to develop beliefs that guide them towards well-grounded mathematical reasoning.

This dissertation consists of four articles and a summary. The main focus of the studies is students' explorations in upper secondary school mathematics. In the first study the central research question was to find out if the students could learn something difficult by using the graphing calculator. The students were working with questions connected to factorisation of quadratic polynomials, and the factor theorem. The results indicate that the students got a better understanding for the factor theorem, and for the connection between graphical and algebraical representations. The second study focused on a the last part of an investigation, the verification of an idea or a conjecture. Students were given three conjectures and asked to decide if they were true or false, and also to explain why the conjectures were true or false. In this study I found that the students wanted to use rather abstract mathematics in order to verify the conjectures. Since the results from the second study disagreed with other research in similar situations, I wanted to see what Swedish teachers had to say of the students' ways to verify the conjectures. The third study is an interview study where some teachers were asked what expectations they had on students who were supposed to verify the three conjectures from the second study. The teachers were also confronted with examples from my second study, and asked to comment on how the students performed. The results indicate that teachers tend to underestimate students' mathematical reasoning. A central focus to all my three studies is explorations in mathematics. My fourth study, a revised version of a pilot study performed 1998, concerns exactly that: how students in upper secondary school explore a mathematical concept. The results indicate that the students are able to perform explorations in mathematics, and that the graphing calculator has a potential as a pedagogical aid, it can be a support for the students' mathematical reasoning.