Understanding Mathematics

Mathematicians distinguish between proofs that explain their results and those that merely prove. This paper explores the nature of explanatory proofs, their role in mathematical practice, and some of the reasons why philosophers should care about them. Among the questions addressed are the following: what kinds of proofs are generally explanatory (or not)? What makes a proof explanatory? Do all mathematical explanations involve proof in an essential way? Are there really such things as explanatory proofs, and if so, how do they relate to the sorts of explanation encountered in philosophy of science and metaphysics?

Gauss's quadratic reciprocity theorem is among the most important results in the history of number theory. It's also among the most mysterious: since its discovery in the late 18th century, mathematicians have regarded reciprocity as a deeply surprising fact in need of explanation. Intriguingly, though, there's little agreement on how the theorem is best explained. Two quite different kinds of proof are most often praised as explanatory: an elementary argument that gives the theorem an intuitive geometric interpretation, due to Gauss and Eisenstein, and a sophisticated proof using algebraic number theory, due to Hilbert. Philosophers have yet to look carefully at such explanatory disagreements in mathematics. I do so here. According to the view I defend, there are two important explanatory virtues-depth and transparency-which different proofs (and other potential explanations) possess to different degrees. Although not mutually exclusive in principle, the packages of features associated with the two stand in some tension with one another, so that very deep explanations are rarely transparent , and vice versa. After developing the theory of depth and transparency and applying it to the case of quadratic reciprocity, I draw some morals about the nature of mathematical explanation.

Full version of my doctoral thesis, Concept Attributivism.


We often encounter situations where we must decide which concepts someone possesses. Teachers must decide which of their students has grasped the material, coworkers must figure out whether their colleagues have the theoretical skills necessary to perform some task, and scientists must judge whether they’ve finally really figured out something new.  In all these situations, it’s necessary to attribute certain concepts to certain people.  The question to the philosopher is: what does that mean? When is it appropriate to say that someone possesses some concept?

This thesis provides an answer to these questions, and it does so by taking into account the confounding factor of time. For example, concepts tend to change over time, which makes it hard to say which concepts people in the past grasped, or partially grasped. In a case study about Leibniz, Newton and Cauchy, I show that the answer to the question “which concept did this person grasp?” depends on when and where the question is asked, against the classical view on the matter. Moreover, I show that almost all concepts will keep changing, due to a phenomenon called “open texture”, the tendency for concepts to keep changing in response to difficult cases. I also show that concepts can retain their identity throughout such changes.

In Concept Attributivism, I develop and defend an alternative view of concepts. According to attributivism, we must take seriously the process of attributing concepts to one another. Concepts are tools for understanding one another. We project a concept on some individual when we find that the other acts according to some relevant theory. For example, a teacher may judge that high school students grasp the concept of gravity if they say and write things according to the norms of Newtonian physics. Questions about concept grasp are then really questions about theory and an individual’s actions.

This article aims to describe the transition process from procedural understanding to conceptual understanding in solving mathematical problems. Subjects in this study were three students from 20 fifth grade students of SDN 01 Sumberberas Banyuwangi selected based on the results of the students' answers. The transition process from procedural to conceptually based on three aspects: (1) identify problems in the use of an algorithm, (2) the process algorithm, (3) connect multiple concepts to transform into another shape through the symbolic/picture representations. The results showed that the majority of students (18 students out of 20 students) only meets two (2) aspects of 10 students (50%) can identify the algorithms and the use of algorithms, 8 students (40%) able to use algorithms and connect with other forms. While other students (two students from 20 students) of 10% that meet only three aspects of the transitions. Thus, understanding the procedural has an important role in developing a conceptual understanding. Because the component/aspect of procedural understanding exist on components/aspects of conceptual understanding. Thus, the association acquired several components/aspects that support the process of transition from procedural understanding to a conceptual understanding.

The objective of the current study was to investigate the mechanisms of xylanase action in a maize-soya diet and its effect on starch digestion. A total of 60 broilers were divided into 6 treatment groups; a control group without xylanase, and five other groups supplemented with xylanase (Econase XT 25; 100 g/t) from 1, 2, 3, 4 or 5 weeks before slaughter. At the end of the experiment, digesta was collected from the gizzard, upper and lower small intestine, and both caeca. Digesta pH ranged from pH 2.2-4.4, 5.9-6.6, 6.7-7.8 and 5.7-7.3 in the gizzard, upper small intestine, lower small intestine, and both caeca, respectively, with no effect of xylanase (P > 0.05). Scanning Electron Microscope (SEM) images along with total starch measurements showed the progression of starch digestion through the tract. The SEM did not show any greater disruption to cell wall material with xylanase supplementation. This suggests that xylanase was not working directly on the cell wall and provides evidence for the hypothesis that xylanase works through an indirect mechanism. Peptide YY (PYY) concentration in the blood was higher during the first few weeks of supplementation, with longer periods of supplementation nulling this effect, implying that xylanase may be acting through a prebiotic mechanism. The RT-q PCR results revealed a numerical increase in glucose transporter (GLUT2 and SGLT1) expression at 2 and 3 weeks of xylanase supplementation, respectively, which might suggest a greater absorption capacity of birds. From these results, a potential mechanism of xylanase action in maize-based diets has been proposed.

One of the most common challenges expressed by classroom teachers in mathematics teaching and in the literature is the concretisation of abstract concepts in teaching basic subjects. Lego MoreToMath is a training set focusing on the mathematical lesson gains of elementary school first and second grade students. The aim of this study is to enable teachers to experience Lego MoreToMath, which is a combination of building toys and educational software, by using the summer camp as a TÜBİTAK 4005 project. The study was conducted with 25 classroom teachers and their opinions were taken during the 1-week training process. Data were collected through observation and interview techniques by keeping field notes. In the study, a descriptive analysis approach was used as the data analysis technique. The teachers stated that using a different approach in mathematics teaching developed them professionally and they wanted to use such approaches in their lessons.Keywords: Educational software, flue...

We present in this paper an alternative geometric interpretation of the Newton-Raphson method. The method is used widely for finding successively better approximations for the roots of real-valued functions which are at least once differentiable. It is shown that an incredibly crude approximation for the area bounded by a function, in the neighbourhood of its roots, leads to the recovery of the algorithm.

The pictorial of representation in solving word problems in this study is the interpretation of using illustrations of images in solving problems in the form of stories through polya steps. The representation shown by students when the process of understands is done by identifying the problem to find important information on the problem, the process of preparing a plan is shown through translations and conversions of information into the form of picture illustrations, the process of Carry out the plan is done through calculations using reduction operations, the look back process is carried out to check or clarify the results obtained according to the question information. The results of this study found that subject 1 did not complete the polya steps, only identified and devise a plan by picture illustration the accompanied by symbolic descriptions, subject 2 was only inaccurate caused by ignoring the last step which was to look back the results obtained, so that the impact of the image illustration with result of calculations inconsistent or not the same.

... Investigating Forms of Understandings in the Context of Trigonometry. - 발행기관, 한국수학교육학회. - 발행정보, D-수학교육연구 , 13 권 , 2 호 , Startpage 151 , Endpage 170 , Totalpage 20. - 저자, ( Ali Delice ) , ( Berna Adatoz Sidi ) , ( Emin Aydin ). - 가격, 4,000 원. - ...

Abstract. Here we report some results of a two-year study for grades 5-6 and 7-8 (dur-ing the academic years 2001–03). The study included a quantitative survey for ap-proximately 150 Finnish mathematics classes out of which 10 classes were selected to a longitudinal part of the ...