History of Mathematics Version: 2011-2012

Scholastic Area that Course Is Active In: Mathematics

Strand: Number and Operations

Students will explore and use historical computational methods.
Element: MHMN1.a
Use Babylonian, Roman, Egyptian (hieratic and hieroglyphic), Chinese, and Greek number systems to represent quantities.
Element: MHMN1.b
b. Use historical multiplication and division algorithms (including the Egyptian method of duplation and mediation, the medieval method of gelosia, and Napier‚€™s rods).

Students will explore the implications of infinite sets of real numbers.
Element: MNHN2.a
Describe denumerable and nondenumerable sets and provide examples of each.
Element: NHMN2.b
Identify algebraic and transcendental numbers.

Strand: Algebra

Students will explore and use historical methods for expressing and solving equations.
Element: MHMA1.a
Solve linear equations using the method of false position.
Element: MHMA1.b
Express the geometrical algebra found in historical works (such as the Elements of Euclid) in modern algebraic notation.
Element: MHMA1.c
Solve systems of linear and nonlinear equations using Diophantus' method.
Element: MHMA1.d
Translate into modern notation problems appearing in ancient and medieval texts that involve linear, quadratic, or cubic equations and solve them.
Element: MHMA1.e
Use Cardano‚€™s cubic formula and Khayyam‚€™s gemoetric construction to find a solution to a cubic equation.

Students will explore abstract algebra and group-theoretic concepts.
Element: MHMA2.a
Add, subtract, and multiply two quaternions.
Element: MNMA2.b
Explore matrix products other than the Cayley product (including Lie and Jordan) by determining whether these products are associative or commutative.
Element: MNMA2.c
Identify whether a given set with a binary operation is a group.

Students will use and apply number theoretic concepts.
Element: MHMA3.a
Find the first four perfect numbers using Euclid's formula.
Element: NHMA3.b
Prove statements concerning figurate numbers using both graphical (as in the manner of the Greeks) and algebraic methods.
Element: MNHA3.c
Solve simple linear congruences of the form ax = b mod m.
Element: MHMA3.d
Use Fermat‚€™s Little Theorem and Euler‚€™s Theorem to simplify expressions of the form ak mod m.
Element: MHMA3.e
Use Gauss‚€™ Law of Quadratic Reciprocity to determine quadratic residues of two odd primes; i.e., solve quadratic congruences of the form x2 = p mod q.
Element: MNMA3.f
Discover that the real primes that can be expressed as the sum of two squares are no longer prime in the field of Gaussian integers.

Students will use the algebraic techniques of Fermat, Barrow, and Newton to determine tangents to quadratic curves.

Strand: Data Analysis and Probability

Students will trace the centers of development of mathematical ideas from the 5th century to the 18th century.
Element: MHMH3.a
Describe the transmission of ideas from the Greeks, through the Islamic peoples, to medieval Europe.
Element: MHMH3.b
Describe the influence of the Catholic Church and Charlemagne on the establishment of mathematics as one of the central pillars of education.
Element: MHMH3.c
Explain the cultural factors that encouraged the development of algebra in 15th century Italy, and how this development influenced mathematical thought throughout Europe.
Element: MHMH3.d
Identify the works of Galileo, Copernicus, and Kepler as a landmark in scientific thought, describe the conflict between their explanation of the workings of the solar system and then-current perspectives, and contrast their works to those of Aristotle.
Element: MHMH3.e
Describe the contributions of Fermat, Pascal, Descartes, Newton, and Gauss to mathematics.
Element: MHMH3.f
Identify Euler as the first modern mathematician and a motivating force behind all aspects of mathematics for the 18th century.
Element: MHMH3.g
Describe the influence the French Revolution had on education (establishment of the Ecole Normale and the Ecole Polytechnique, Monge, Lagrange, Legendre, Laplace).

Students will identify Hindu-Arabic numerals as a prime scientific advancement.
Element: MHMH1.a
Describe the limitations of the Babylonian, Roman, Egyptian (hieratic and hieroglyphic), Chinese, and Greek number systems as compared to Hindu-Arabic numerals.
Element: MHMH1.b
Describe the transition of Hindu-Arabic numerals from regional use in the 10th century to wide-spread use in the 15th (including the influence of Fibonacci for the use of the numerals and the Italian abascists against their use).
Element: MHMH1.c
Identify the number system and notation used by a society as an influence on the types of mathematics developed by that society.

Students will describe factors involved in the rise and fall of ancient Greek society.
Element: MHMH2.a
Describe the theories for the rise of intellectual thought in ancient Greece and the factors involved in its collapse.
Element: MHMH2.b
Describe the cultural aspects of Greek society that influenced the way mathematics developed in ancient Greece.
Element: MHMH2.c
Explain the distinction made between number and magnitude, commensurable and incommensurable, and arithmetic and logistic, the cultural factors inherent in this distinction, and the logical crisis that occurred concerning incommensurable (irrational) magnitudes.

Students will trace the centers of development of mathematical ideas from the 5th century to the 18th century. a. Describe the transmission of

Students will identify the 19th and 20th centuries as the time when mathematics became more specialized and more rigorous.
Element: MHMH4.a
Describe the societal factors that inhibited the developement of non-Euclidean geometry.
Element: NHMH4.b
Explain how the ancient Greek pattern of material axiomatics evolved into abstract axiomatics (non-Euclidean geometry, non-commutative algebra)
Element: MHMH4.c
Identify Cantor as the most original mathematician since the ancient Greeks.
Element: MHMH4.d
Describe the implications of Klein‚€™s Erlanger Programme and Godel's Incompleteness Theorem on the nature of mathematical discovery and proof.

Strand: Geometry

Students will prove geometry theorems.
Element: MHMG1.a
Students will understand and recognize the use of definitions, postulates, and axioms in defining a deductive system such as Euclidean geometry.
Element: MHMG1.b
Prove the first five propositions in Book I of Euclid‚€™s Elements.
Element: MHMG1.c
Construct a regular pentagon with a straight-edge and compass.

Students will compute lengths, areas, and volumes according to historical formulas.
Element: MHMG2.a
Find the volume of a truncated pyramid using the Babylonian, Chinese, and Egyptian formulas.
Element: MHMG2.b
Compute the areas of regular polygons by Heron‚€™s formulas.
Element: MHMG2.c
Identify cyclic quadrilaterals and find associated lengths by Ptolemy‚€™s Theorem.

Students will explore and prove statements in non-Euclidean geometry.
Element: MHMG3.a
Prove that the summit angles of an isosceles birectangle are congruent, but that it is impossible to prove they are right without referring to the parallel postulate or one of its consequences.
Element: MHMG3.b
Describe the hypothesis of the acute angle (Hyperbolic), the hypothesis of the right angle (Euclidean), and the hypothesis of the obtuse angle (Spherical).
Element: MHMG3.c
Prove that under the hypothesis of the acute angle, similarity implies congruence.

Strand: Process Standards

Students will solve problems (using appropriate technology).
Element: MHMP1.a
Build new mathematical knowledge through problem solving.
Element: MHMP1.b
Solve problems that arise in mathematics and in other contexts.
Element: MHMP1.c
Apply and adapt a variety of appropriate strategies to solve problems.
Element: MHMP1.d
Monitor and reflect on the process of mathematical problem solving.

Students will reason and evaluate mathematical arguments.
Element: MHMP2.a
Recognize reasoning and proof as fundamental aspects of mathematics.
Element: MHMP2.b
Make and investigate mathematical conjectures.
Element: MHMP2.c
Develop and evaluate mathematical arguments and proofs.
Element: MHMP2.d
Select and use various types of reasoning and methods of proof.

Students will communicate mathematically.
Element: MHMP3.a
Organize and consolidate their mathematical thinking through communication.
Element: MHMP3.b
Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.
Element: MHMP3.c
Analyze and evaluate the mathematical thinking and strategies of others.
Element: MHMP3.d
Use the language of mathematics to express mathematical ideas precisely.

Students will make connections among mathematical ideas and to other disciplines.
Element: MHMP4.a
Recognize and use connections among mathematical ideas.
Element: MHMP4.b
Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
Element: MHMP4.c
Recognize and apply mathematics in contexts outside of mathematics.

Students will represent mathematics in multiple ways.
Element: MHMP5.a
Create and use representations to organize, record, and communicate mathematical ideas.
Element: MHMP5.b
Select, apply, and translate among mathematical representations to solve problems.
Element: MHMP5.c
Use representations to model and interpret physical, social, and mathematical phenomena.

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