The Golden Circle

In chapter 4 of my book I talk about a rectangle inscribed within a circle. Naturally there are an indefinite number of such figures. Take the diagram on the right, kindly produced by Michael Schneider. Look at the outermost circle, and the largest rectangle that lies inside it, touching its circumference at A, B and C. You could move points A and B nearer to the left-hand end of the horizontal diameter of the large circle, or else push them further apart towards the two ends of the vertical diameter, producing an ever-thinner oblong shape. Halfway between these  extremes the rectangle would become a square. But the shape Michael has drawn is a Golden Rectangle, so we can call the whole figure a Golden Circle ("Golden" because of the presence of the Rectangle). The G.R. is famous for being the "most beautiful" of rectangles, possessing the peculiar property that its sides are in the ratio of 1 to Phi (1.618...), so that if you cut off a square portion what remains is a smaller Golden Rectangle - and so forth, forming a logarithmic spiral, as in the following image.

When I wrote the book I was intending to use the Golden Circle as a way of exploring the relationship between Pi and Phi, but I never got around to it. My reason for being intrigued is simple. What we learn from Simone Weil - and what she learned from the Greeks - is that geometry is full of theological meaning. We have forgotten how to make those connections. It is not that we can prove the Trinity or the Incarnation with diagrams, but that the mathematical world is full of analogies that echo theological and spiritual truth. You might even say that mathematical necessities are a portrait of divine freedom, since in God freedom and necessity coincide. The beauties of geometry and arithmetic are a world of metaphors and help to raise our minds towards the contemplation of divine truth. My book only touches on this, but a much fuller and richer account is given by Vance G. Morgan of Providence College in his book Weaving the World: Simone Weil on Science, Mathematics and Love, reviewed here.

The Golden Circle

In chapter 4 of my book I talk about a rectangle inscribed within a circle. Naturally there are an indefinite number of such figures. Take the diagram on the right, kindly produced by Michael Schneider. Look at the outermost circle, and the largest rectangle that lies inside it, touching its circumference at A, B and C. You could move points A and B nearer to the left-hand end of the horizontal diameter of the large circle, or else push them further apart towards the two ends of the vertical diameter, producing an ever-thinner oblong shape. Halfway between these  extremes the rectangle would become a square. But the shape Michael has drawn is a Golden Rectangle, so we can call the whole figure a Golden Circle ("Golden" because of the presence of the Rectangle). The G.R. is famous for being the "most beautiful" of rectangles, possessing the peculiar property that its sides are in the ratio of 1 to Phi (1.618...), so that if you cut off a square portion what remains is a smaller Golden Rectangle - and so forth, forming a logarithmic spiral, as in the following image.

When I wrote the book I was intending to use the Golden Circle as a way of exploring the relationship between Pi and Phi, but I never got around to it. My reason for being intrigued is simple. What we learn from Simone Weil - and what she learned from the Greeks - is that geometry is full of theological meaning. We have forgotten how to make those connections. It is not that we can prove the Trinity or the Incarnation with diagrams, but that the mathematical world is full of analogies that echo theological and spiritual truth. You might even say that mathematical necessities are a portrait of divine freedom, since in God freedom and necessity coincide. The beauties of geometry and arithmetic are a world of metaphors and help to raise our minds towards the contemplation of divine truth. My book only touches on this, but a much fuller and richer account is given by Vance G. Morgan of Providence College in his book Weaving the World: Simone Weil on Science, Mathematics and Love, reviewed here.

January issue of Science & Education

The January issue of Science & Education has been published. One of the articles contained in the issue is of relevance to mathematics education: A Pilot Study of a Cultural-Historical Approach to Teaching Geometry. The article is written by Stuart Rowlands from the University of Plymouth. Here is the abstract of his article:

There appears to be a widespread assumption that deductive geometry is inappropriate for most learners and that they are incapable of engaging with the abstract and rule-governed intellectual processes that became the world’s first fully developed and comprehensive formalised system of thought. This article discusses a curriculum initiative that aims to ‘bring to life’ the major transformative (primary) events in the history of Greek geometry, aims to encourage a meta-discourse that can develop a reflective consciousness and aims to provide an opportunity for the induction into the formalities of proof and to engage with the abstract. The results of a pilot study to see whether 14–15 year old ‘mixed ability’ and 15–16 year old ‘gifted and talented’ students can be meaningfully engaged with two such transformative events are discussed.

January issue of Science & Education

The January issue of Science & Education has been published. One of the articles contained in the issue is of relevance to mathematics education: A Pilot Study of a Cultural-Historical Approach to Teaching Geometry. The article is written by Stuart Rowlands from the University of Plymouth. Here is the abstract of his article:

There appears to be a widespread assumption that deductive geometry is inappropriate for most learners and that they are incapable of engaging with the abstract and rule-governed intellectual processes that became the world’s first fully developed and comprehensive formalised system of thought. This article discusses a curriculum initiative that aims to ‘bring to life’ the major transformative (primary) events in the history of Greek geometry, aims to encourage a meta-discourse that can develop a reflective consciousness and aims to provide an opportunity for the induction into the formalities of proof and to engage with the abstract. The results of a pilot study to see whether 14–15 year old ‘mixed ability’ and 15–16 year old ‘gifted and talented’ students can be meaningfully engaged with two such transformative events are discussed.

Students' use of technological tools

Ioannis Papadopoulosa and Vassilios Dagdilelis have written an article that was published online in the Journal of Mathematical Behavior yesterday. The article is entitled Students’ use of technological tools for verification purposes in geometry problem solving. Here is a copy of the article abstract:

Despite its importance in mathematical problem solving, verification receives rather little attention by the students in classrooms, especially at the primary school level. Under the hypotheses that (a) non-standard tasks create a feeling of uncertainty that stimulates the students to proceed to verification processes and (b) computational environments – by providing more available tools compared to the traditional environment – might offer opportunities for more frequent usage of verification techniques, we posed to 5th and 6th graders non-routine problems dealing with area of plane irregular figures. The data collected gave us evidence that computational environments allow the development of verification processes in a wider variety compared to the traditional paper-and-pencil environment and at the same time we had the chance to propose a preliminary categorization of the students’ verification processes under certain conditions.

Students' use of technological tools

Ioannis Papadopoulosa and Vassilios Dagdilelis have written an article that was published online in the Journal of Mathematical Behavior yesterday. The article is entitled Students’ use of technological tools for verification purposes in geometry problem solving. Here is a copy of the article abstract:

Despite its importance in mathematical problem solving, verification receives rather little attention by the students in classrooms, especially at the primary school level. Under the hypotheses that (a) non-standard tasks create a feeling of uncertainty that stimulates the students to proceed to verification processes and (b) computational environments – by providing more available tools compared to the traditional environment – might offer opportunities for more frequent usage of verification techniques, we posed to 5th and 6th graders non-routine problems dealing with area of plane irregular figures. The data collected gave us evidence that computational environments allow the development of verification processes in a wider variety compared to the traditional paper-and-pencil environment and at the same time we had the chance to propose a preliminary categorization of the students’ verification processes under certain conditions.

A cultural-historical approach to teaching geometry

Stuart Rowlands has recently written an article called A Pilot Study of a Cultural-Historical Approach to Teaching Geometry, which was published in Science & Education on Wednesday. Here is the abstract of the article:

There appears to be a widespread assumption that deductive geometry is inappropriate for most learners and that they are incapable of engaging with the abstract and rule-governed intellectual processes that became the world’s first fully developed and comprehensive formalised system of thought. This article discusses a curriculum initiative that aims to ‘bring to life’ the major transformative (primary) events in the history of Greek geometry, aims to encourage a meta-discourse that can develop a reflective consciousness and aims to provide an opportunity for the induction into the formalities of proof and to engage with the abstract. The results of a pilot study to see whether 14–15 year old ‘mixed ability’ and 15–16 year old ‘gifted and talented’ students can be meaningfully engaged with two such transformative events are discussed.

A cultural-historical approach to teaching geometry

Stuart Rowlands has recently written an article called A Pilot Study of a Cultural-Historical Approach to Teaching Geometry, which was published in Science & Education on Wednesday. Here is the abstract of the article:

There appears to be a widespread assumption that deductive geometry is inappropriate for most learners and that they are incapable of engaging with the abstract and rule-governed intellectual processes that became the world’s first fully developed and comprehensive formalised system of thought. This article discusses a curriculum initiative that aims to ‘bring to life’ the major transformative (primary) events in the history of Greek geometry, aims to encourage a meta-discourse that can develop a reflective consciousness and aims to provide an opportunity for the induction into the formalities of proof and to engage with the abstract. The results of a pilot study to see whether 14–15 year old ‘mixed ability’ and 15–16 year old ‘gifted and talented’ students can be meaningfully engaged with two such transformative events are discussed.

Pearson's correlation between three variables

Pauline Vos has written an article called Pearson's correlation between three variables; using students' basic knowledge of geometry for an exercise in mathematical statistics. The article was recently published in International Journal of Mathematical Education in Science and Technology. Here is a copy of the article abstract:

When studying correlations, how do the three bivariate correlation coefficients between three variables relate? After transforming Pearson's correlation coefficient r into a Euclidean distance, undergraduate students can tackle this problem using their secondary school knowledge of geometry (Pythagoras' theorem and similarity of triangles). Through a geometric interpretation, we start from two correlation coefficients rAB and rBC and then estimate a range for the third correlation rAC. In the case of three records (n = 3), the third correlation rAC can only attain two possible values. Crossing borders between mathematical disciplines, such as statistics and geometry, can assist students in deepening their conceptual knowledge.

Pearson's correlation between three variables

Pauline Vos has written an article called Pearson's correlation between three variables; using students' basic knowledge of geometry for an exercise in mathematical statistics. The article was recently published in International Journal of Mathematical Education in Science and Technology. Here is a copy of the article abstract:

When studying correlations, how do the three bivariate correlation coefficients between three variables relate? After transforming Pearson's correlation coefficient r into a Euclidean distance, undergraduate students can tackle this problem using their secondary school knowledge of geometry (Pythagoras' theorem and similarity of triangles). Through a geometric interpretation, we start from two correlation coefficients rAB and rBC and then estimate a range for the third correlation rAC. In the case of three records (n = 3), the third correlation rAC can only attain two possible values. Crossing borders between mathematical disciplines, such as statistics and geometry, can assist students in deepening their conceptual knowledge.

The decorative impulse

Swapna Mukhopadhyay has written an article entitled The decorative impulse: ethnomathematics and Tlingit basketry. The article was published online in ZDM earlier this week. Here is the article abstract:

Pattern is a key element in both the esthetics of design and mathematics, one definition of which is “the study of all possible patterns”. Thus, the geometric patterns that adorn cultural artifacts manifest mathematical thinking in the artisans who create them, albeit their lack of “formal” mathematics learning. In describing human constructions, Franz Boas affirmed that people, regardless of their economic conditions, always have been engaged in activities that reveal their deeply held esthetic sense. The Tlingit Indians from Sitka, Alaska, are known for their artistic endeavors. Art aficionados and museum collectors revere their baskets and other artifacts. Taking the approach of ethnomathematics, I report my analysis of the complex geometrical patterns in Tlingit basketry.

The decorative impulse

Swapna Mukhopadhyay has written an article entitled The decorative impulse: ethnomathematics and Tlingit basketry. The article was published online in ZDM earlier this week. Here is the article abstract:

Pattern is a key element in both the esthetics of design and mathematics, one definition of which is “the study of all possible patterns”. Thus, the geometric patterns that adorn cultural artifacts manifest mathematical thinking in the artisans who create them, albeit their lack of “formal” mathematics learning. In describing human constructions, Franz Boas affirmed that people, regardless of their economic conditions, always have been engaged in activities that reveal their deeply held esthetic sense. The Tlingit Indians from Sitka, Alaska, are known for their artistic endeavors. Art aficionados and museum collectors revere their baskets and other artifacts. Taking the approach of ethnomathematics, I report my analysis of the complex geometrical patterns in Tlingit basketry.