Mathematical Writing. Franco Vivaldi Queen Mary, University of London

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1 Mathematical Writing Franco Vivaldi Queen Mary, University of London

2 Reading Mathematical Writing Translating Franco Vivaldi Queen Mary, University of London

3 Why mathematical writing?

4 Why mathematical writing? 3. Writing is an essential skill, and many students are not good at it.

5 Why mathematical writing? 3. Writing is an essential skill, and many students are not good at it. 2. A preparation for --or an alternative to-- a final year project.

6 Why mathematical writing? 3. Writing is an essential skill, and many students are not good at it. 2. A preparation for --or an alternative to-- a final year project. 1. Writing forces one to think about structures.

7 Why mathematical writing? 3. Writing is an essential skill, and many students are not good at it. 2. A preparation for --or an alternative to-- a final year project. 1. Writing forces one to think about structures. 0. Teaching MW gives you X-rays into the studentsʼ thoughts.

8 Why mathematical writing? 3. Writing is an essential skill, and many students are not good at it. 2. A preparation for --or an alternative to-- a final year project. 1. Writing forces one to think about structures. 0. Teaching MW gives you X-rays into the studentsʼ thoughts. L. Alcock & A. Simpson, Ideas from mathematics education, MSOR (2009). We have found it illuminating to ask students to think aloud... as they work on mathematical tasks. This often uncovers fragility in the knowledge of even the strongest students.

9 Why mathematical writing? 3. Writing is an essential skill, and many students are not good at it. 2. A preparation for --or an alternative to-- a final year project. 1. Writing forces one to think about structures. 0. Teaching MW gives you X-rays into the studentsʼ thoughts. L. Alcock & A. Simpson, Ideas from mathematics education, MSOR (2009). We have found it illuminating to ask students to think aloud... as they work on mathematical tasks. This often uncovers fragility in the knowledge of even the strongest students. MW students commented on the unexpected depth required of their thinking, when asked to offer verbal explanations.

10 An exercise

11 An exercise On the plane, I have a circle and a point outside it, and I must find the lines through this point which are tangent to the circle. What shall I do?

12 An exercise On the plane, I have a circle and a point outside it, and I must find the lines through this point which are tangent to the circle. What shall I do? Answer this question over the phone.

13 An exercise On the plane, I have a circle and a point outside it, and I must find the lines through this point which are tangent to the circle. What shall I do? Answer this question over the phone. This task requires grasp of structure and organisation;

14 An exercise On the plane, I have a circle and a point outside it, and I must find the lines through this point which are tangent to the circle. What shall I do? Answer this question over the phone. This task requires grasp of structure and organisation; it gives the students an opportunity to express their knowledge, intelligence, and individuality;

15 An exercise On the plane, I have a circle and a point outside it, and I must find the lines through this point which are tangent to the circle. What shall I do? Answer this question over the phone. This task requires grasp of structure and organisation; it gives the students an opportunity to express their knowledge, intelligence, and individuality; but it also exposes logical faults, immaturity, incompetence.

16 ~100 words

17 ~100 words Write down the equation of a line passing trough the point. This equation will depend on one parameter, the line s slope, which is the quantity to be determined.

18 ~100 words Write down the equation of a line passing trough the point. This equation will depend on one parameter, the line s slope, which is the quantity to be determined.

19 ~100 words Write down the equation of a line passing trough the point. This equation will depend on one parameter, the line s slope, which is the quantity to be determined. Adjoin the equation of the line to that of the circle and eliminate one of the unknowns. After a substitution, you ll end up with a quadratic equation in one unknown, whose coefficients depend on the parameter.

20 ~100 words Write down the equation of a line passing trough the point. This equation will depend on one parameter, the line s slope, which is the quantity to be determined. Adjoin the equation of the line to that of the circle and eliminate one of the unknowns. After a substitution, you ll end up with a quadratic equation in one unknown, whose coefficients depend on the parameter.

21 ~100 words Write down the equation of a line passing trough the point. This equation will depend on one parameter, the line s slope, which is the quantity to be determined. Adjoin the equation of the line to that of the circle and eliminate one of the unknowns. After a substitution, you ll end up with a quadratic equation in one unknown, whose coefficients depend on the parameter. Equate the discriminant of the quadratic equation to zero, to obtain an equation -also quadratic- for the slope. Its two solutions are the desired slopes of the tangent lines. Any configuration involving vertical lines (infinite slope) will require a variant of the above procedure.

22 ~100 words Write down the equation of a line passing trough the point. This equation will depend on one parameter, the line s slope, which is the quantity to be determined. Adjoin the equation of the line to that of the circle and eliminate one of the unknowns. After a substitution, you ll end up with a quadratic equation in one unknown, whose coefficients depend on the parameter. Equate the discriminant of the quadratic equation to zero, to obtain an equation -also quadratic- for the slope. Its two solutions are the desired slopes of the tangent lines. Any configuration involving vertical lines (infinite slope) will require a variant of the above procedure.

23 ~100 words Write down the equation of a line passing trough the point. This equation will depend on one parameter, the line s slope, which is the quantity to be determined. Adjoin the equation of the line to that of the circle and eliminate one of the unknowns. After a substitution, you ll end up with a quadratic equation in one unknown, whose coefficients depend on the parameter. Equate the discriminant of the quadratic equation to zero, to obtain an equation -also quadratic- for the slope. Its two solutions are the desired slopes of the tangent lines. Any configuration involving vertical lines (infinite slope) will require a variant of the above procedure.

24 Classroom schizophrenia

25 Classroom schizophrenia Teacher Student

26 Classroom schizophrenia Teacher Student definitions theorems

27 Classroom schizophrenia Teacher Student definitions theorems examples

28 Classroom schizophrenia Teacher Student definitions theorems examples assessment

29 Classroom schizophrenia Teacher Student definitions definitions theorems theorems examples assessment

30 Classroom schizophrenia Teacher Student definitions theorems examples definitions theorems EXAMPLES assessment

31 Classroom schizophrenia Teacher Student definitions theorems examples definitions theorems EXAMPLES assessment assessment

32 Many mathematics students are not prepared for the mathematical language.

33 Many mathematics students are not prepared for the mathematical language. Why?

34 Many mathematics students are not prepared for the mathematical language. Why? Insufficient exposure to formal logical systems (grammar and syntax, Latin, Euclidean geometry, programming, philosophy, foreign languages).

35 Many mathematics students are not prepared for the mathematical language. Why? Insufficient exposure to formal logical systems (grammar and syntax, Latin, Euclidean geometry, programming, philosophy, foreign languages). Words and symbols donʼt have exact meaning.

36 Many mathematics students are not prepared for the mathematical language. Why? Insufficient exposure to formal logical systems (grammar and syntax, Latin, Euclidean geometry, programming, philosophy, foreign languages). Words and symbols donʼt have exact meaning. League tables.

37 Many mathematics students are not prepared for the mathematical language. Why? Insufficient exposure to formal logical systems (grammar and syntax, Latin, Euclidean geometry, programming, philosophy, foreign languages). Words and symbols donʼt have exact meaning. League tables. Concepts are replaced by processes.

38 Many mathematics students are not prepared for the mathematical language. Why? Insufficient exposure to formal logical systems (grammar and syntax, Latin, Euclidean geometry, programming, philosophy, foreign languages). Words and symbols donʼt have exact meaning. League tables. Concepts are replaced by processes. lack of conceptual accuracy

39 Many mathematics students are not prepared for the mathematical language. Why? Insufficient exposure to formal logical systems (grammar and syntax, Latin, Euclidean geometry, programming, philosophy, foreign languages). Words and symbols donʼt have exact meaning. League tables. Concepts are replaced by processes. lack of conceptual accuracy inadequate reading ineffective learning difficulties with abstraction difficulties with reasoning poor writing

40 The language of processes

41 The language of processes Compute the value of the following expression:

42 The language of processes Compute the value of the following expression: Great complexity, limited conceptual content.

43 The language of processes Compute the value of the following expression: Great complexity, limited conceptual content. Compute the derivative of the following function:

44 The language of processes Compute the value of the following expression: Great complexity, limited conceptual content. Compute the derivative of the following function:

45 The language of processes Compute the value of the following expression: Great complexity, limited conceptual content. Compute the derivative of the following function: To complete the task, knowledge of the exact meaning of words and symbols is irrelevant.

46 The language of concepts: reading symbols

47 The language of concepts: reading symbols Chinese:

48 The language of concepts: reading symbols Chinese: Mathematics:

49 The language of concepts: reading symbols Chinese: Mathematics: One symbol, three meanings:

50 The language of concepts: reading symbols Chinese: Mathematics: the value of an inverse function at a point the inverse image of a set under a function One symbol, three meanings: the reciprocal of the value of a function at a point

51 The language of concepts: reading symbols Chinese: Mathematics: the value of an inverse function at a point the inverse image of a set under a function One symbol, three meanings: the reciprocal of the value of a function at a point... within a single expression!

52 The language of concepts: reading symbols Chinese: Mathematics: the value of an inverse function at a point the inverse image of a set under a function One symbol, three meanings: the reciprocal of the value of a function at a point... within a single expression!

53 The language of concepts: reading symbols Chinese: Mathematics: the value of an inverse function at a point the inverse image of a set under a function One symbol, three meanings: the reciprocal of the value of a function at a point... within a single expression! set sentence nonsense

54 The language of concepts: reading symbols Chinese: Mathematics: the value of an inverse function at a point the inverse image of a set under a function One symbol, three meanings: the reciprocal of the value of a function at a point... within a single expression! set sentence nonsense The dog is black. The black is dog.

55 The language of concepts: reading symbols Chinese: Mathematics: the value of an inverse function at a point the inverse image of a set under a function One symbol, three meanings: the reciprocal of the value of a function at a point... within a single expression! relational operator set operator set sentence nonsense The dog is black. The black is dog.

56 The language of concepts: reading words

57 The language of concepts: reading words Every detail matters:

58 The language of concepts: reading words Every detail matters: The set of even integers. A set of even integers. The set of the divisors of a large integer. A set of divisors of a multiple of 24.

59 The language of concepts: reading words Every detail matters: The set of even integers. A set of even integers. The set of the divisors of a large integer. A set of divisors of a multiple of 24. definite indefinite

60 The language of concepts: reading words Every detail matters: The set of even integers. A set of even integers. The set of the divisors of a large integer. A set of divisors of a multiple of 24. definite indefinite The dictionary is constantly needed:

61 The language of concepts: reading words Every detail matters: The set of even integers. A set of even integers. The set of the divisors of a large integer. A set of divisors of a multiple of 24. definite indefinite The dictionary is constantly needed: Let the restriction of f to the open unit interval be surjective.

62 The language of concepts: reading words Every detail matters: The set of even integers. A set of even integers. The set of the divisors of a large integer. A set of divisors of a multiple of 24. definite indefinite The dictionary is constantly needed: Let the restriction of f to the open unit interval be surjective.

63 The language of concepts: reading words Every detail matters: The set of even integers. A set of even integers. The set of the divisors of a large integer. A set of divisors of a multiple of 24. definite indefinite The dictionary is constantly needed: Let the restriction of f to the open unit interval be surjective. Verba volant, scripta manent. Heb je geen paard, gebruik dan een ezel.

definite indefinite The dictionary is constantly needed: EXAMPLES Let the restriction of f EXAMPLES to

64 The language of concepts: reading words Every detail matters: The set of even integers. A set of even integers. The set of the divisors of a large integer. A set of divisors of a multiple of 24. definite indefinite The dictionary is constantly needed: EXAMPLES Let the restriction of f EXAMPLES to the open unit interval be surjective. EXAMPLES Verba volant, scripta manent. Heb je geen paard, EXAMPLES gebruik dan een ezel.

65 The mathematical language is full of idiomatic expressions:

66 The mathematical language is full of idiomatic expressions:

67 The mathematical language is full of idiomatic expressions: There is pathological reluctance to standardise basic symbols:

68 The mathematical language is full of idiomatic expressions: There is pathological reluctance to standardise basic symbols:

69 The mathematical language is full of idiomatic expressions: There is pathological reluctance to standardise basic symbols:...and words:

70 The mathematical language is full of idiomatic expressions: There is pathological reluctance to standardise basic symbols:...and words: By a triangle we mean a metric space of cardinality three. By a segment we mean a maximal subpath of P that contains only light or only heavy edges. By a circle we mean an affinoid isomorphic to max Cp(T,T-1).

71 Oblivious teaching

72 Oblivious teaching Conditional probability:

73 Oblivious teaching Conditional probability:

74 Oblivious teaching Conditional probability: The probability of A intersection B

75 Oblivious teaching Conditional probability: The probability of A intersection B The probability of A given B

76 Oblivious teaching Conditional probability: The probability of A intersection B The probability of A given B,

77 Oblivious teaching Conditional probability: The probability of A intersection B The probability of A given B,

78 Oblivious teaching Conditional probability: The probability of A intersection B The probability of A given B,

79 Oblivious teaching Conditional probability: The probability of A intersection B The probability of A given B,

80 Oblivious teaching Conditional probability: The probability of A intersection B The probability of A given B, Grammar and syntax should take precedence over semantics.

81 Injectivity: A function is injective if distinct elements of the domain have distinct images.

82 Injectivity: icons A function is injective if distinct elements of the domain have distinct images. metaphors

83 Injectivity: icons A function is injective if distinct elements of the domain have distinct images. metaphors

84 Injectivity: icons A function is injective if distinct elements of the domain have distinct images. metaphors We have a group of archers (the elements of the domain), each with one arrow (the function). If all enemies get killed, the function is surjective, if nobody is hit twice, the function is injective.

85 Injectivity: icons A function is injective if distinct elements of the domain have distinct images. metaphors We have a group of archers (the elements of the domain), each with one arrow (the function). If all enemies get killed, the function is surjective, if nobody is hit twice, the function is injective. In absence of a definition, icons and metaphors may only illustrate themselves.

86 Injectivity: icons A function is injective if distinct elements of the domain have distinct images. metaphors We have a group of archers (the elements of the domain), each with one arrow (the function). If all enemies get killed, the function is surjective, if nobody is hit twice, the function is injective. In absence of a definition, icons and metaphors may only illustrate themselves. Attention should shift to the defining sentence.

87 Injectivity: icons A function is injective if distinct elements of the domain have distinct images. metaphors We have a group of archers (the elements of the domain), each with one arrow (the function). If all enemies get killed, the function is surjective, if nobody is hit twice, the function is injective. In absence of a definition, icons and metaphors may only illustrate themselves. Change words: Attention should shift to the defining sentence. A diet is varied if distinct days of the week have distinct menus.

88 Injectivity: icons A function is injective if distinct elements of the domain have distinct images. metaphors We have a group of archers (the elements of the domain), each with one arrow (the function). If all enemies get killed, the function is surjective, if nobody is hit twice, the function is injective. In absence of a definition, icons and metaphors may only illustrate themselves. Change words: Attention should shift to the defining sentence. A diet is varied if distinct days of the week have distinct menus. Introduce symbols: Let D be a diet and let x and y be two days of the week...

89 Encourage logical analysis:

90 Encourage logical analysis: A function f is injective if distinct elements of the domain of f have distinct images.

91 Encourage logical analysis: A function f is injective if distinct elements of the domain of f have distinct images. ( f is a dummy variable)

92 Encourage logical analysis: A function f is injective if distinct elements of the domain of f have distinct images. ( f is a dummy variable) The function f is injective because distinct elements of the domain of f have distinct images.

93 Encourage logical analysis: A function f is injective if distinct elements of the domain of f have distinct images. ( f is a dummy variable) The function f is injective because distinct elements of the domain of f have distinct images. ( f is not a dummy variable)

94 Encourage logical analysis: A function f is injective if distinct elements of the domain of f have distinct images. ( f is a dummy variable) The function f is injective because distinct elements of the domain of f have distinct images. ( f is not a dummy variable) A function f is injective if any two elements of the domain of f have distinct images.

95 Encourage logical analysis: A function f is injective if distinct elements of the domain of f have distinct images. ( f is a dummy variable) The function f is injective because distinct elements of the domain of f have distinct images. ( f is not a dummy variable) A function f is injective if any two elements of the domain of f have distinct images. (empty definition)

96 The Mathematical Writing course: syllabus

97 The Mathematical Writing course: syllabus Essential dictionary: translating words into symbols, and vice-versa, with short phrases and sentences.

98 The Mathematical Writing course: syllabus Essential dictionary: translating words into symbols, and vice-versa, with short phrases and sentences. Formal mathematical sentences: predicate calculus, quantifiers.

99 The Mathematical Writing course: syllabus Essential dictionary: translating words into symbols, and vice-versa, with short phrases and sentences. Formal mathematical sentences: predicate calculus, quantifiers. Describing real functions: the language of analysis.

100 The Mathematical Writing course: syllabus Essential dictionary: translating words into symbols, and vice-versa, with short phrases and sentences. Formal mathematical sentences: predicate calculus, quantifiers. Describing real functions: the language of analysis. Writing effectively: choosing notation; some techniques; writing a short summary (150 words).

101 The Mathematical Writing course: syllabus Essential dictionary: translating words into symbols, and vice-versa, with short phrases and sentences. Formal mathematical sentences: predicate calculus, quantifiers. Describing real functions: the language of analysis. Writing effectively: Forms of arguments: choosing notation; some techniques; writing a short summary (150 words). methods of proof; spotting and correcting bad arguments.

102 The Mathematical Writing course: syllabus Essential dictionary: translating words into symbols, and vice-versa, with short phrases and sentences. Formal mathematical sentences: predicate calculus, quantifiers. Describing real functions: the language of analysis. Writing effectively: Forms of arguments: choosing notation; some techniques; writing a short summary (150 words). methods of proof; spotting and correcting bad arguments. Existence and definitions: existence proofs, unique existence.

103 Most mathematics students require explicit instructions on how to read and analyse mathematical expressions.

104 Most mathematics students require explicit instructions on how to read and analyse mathematical expressions.

105 Essential dictionary: from symbols to words

106 Essential dictionary: from symbols to words A sequence.

107 Essential dictionary: from symbols to words A sequence. An infinite sequence.

108 Essential dictionary: from symbols to words A sequence. An infinite sequence. An infinite sequence of polynomials.

109 Essential dictionary: from symbols to words A sequence. An infinite sequence. An infinite sequence of polynomials. An infinite sequence of polynomials in one indeterminate.

110 Essential dictionary: from symbols to words A sequence. An infinite sequence. An infinite sequence of polynomials. An infinite sequence of polynomials in one indeterminate. put symbols in a context: with integer coefficients. with increasing degree. with bounded coefficients....

111 Structure of expressions

112 Structure of expressions

113 Structure of expressions

114 Structure of expressions

115 Structure of expressions

116

117 polynomial equation identity inequality sentence set sequence function

118 From symbols to words: synthesis

119 From symbols to words: synthesis

120 From symbols to words: synthesis The image of the set of integers under the function f is the set consisting of the integer 0. [Robotic, no understanding]

121 From symbols to words: synthesis The image of the set of integers under the function f is the set consisting of the integer 0. The function f vanishes at all integers. [Robotic, no understanding] [Good]

122 From words to symbols

123 From words to symbols

124 From words to symbols

125 Describing functions

126 Describing functions This is a smooth function, which is bounded and non-negative. It features an infinite sequence of evenly spaced local maxima, whose height decreases monotonically to zero. The function has a zero between any two consecutive maxima.

127 Describing functions This is a smooth function, which is bounded and non-negative. It features an infinite sequence of evenly spaced local maxima, whose height decreases monotonically to zero. The function has a zero between any two consecutive maxima.

128 The bigger picture

129 The bigger picture Writing is essential for learning: students should write more.

130 The bigger picture Writing is essential for learning: students should write more. The development of conceptual accuracy requires small-scale writing exercises (words, symbols, phrases, short sentences).

131 The bigger picture Writing is essential for learning: students should write more. The development of conceptual accuracy requires small-scale writing exercises (words, symbols, phrases, short sentences). One specialised course is insufficient: elements of writing should be embedded in most courses (as in the Writing in the Disciplines programme at American universities).

132 The bigger picture Writing is essential for learning: students should write more. The development of conceptual accuracy requires small-scale writing exercises (words, symbols, phrases, short sentences). One specialised course is insufficient: elements of writing should be embedded in most courses (as in the Writing in the Disciplines programme at American universities). Universities should develop centrally run schemes to raise the profile of writing and to support departments.

133 Thank you for your attention

134 Only 22.49

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