This sub-organizer contains the following sections:
Prescribed Learning Outcomes
Suggested Instructional Strategies
Suggested Assessment Strategies
Recommended Learning Resources

PRESCRIBED LEARNING OUTCOMES

It is expected that students will understand the historical background and problems that lead to the development of calculus.

It is expected that students will:

• describe the contributions made by various mathematicians and philosophers to the development of calculus, including:
  • Archimedes
  • Fermat
  • Descartes
  • Barrow
  • Newton
  • Leibniz
  • Jakob and Johann Bernoulli
  • Euler
  • L’Hopital

SUGGESTED INSTRUCTIONAL STRATEGIES

Students gain a better understanding and appreciation for this field of mathematics by studying the lives of principal mathematicians credited for the invention of calculus, including the period in which they lived and the significant mathematical problems they were attempting to solve.

• Conduct a brief initial overview of the historical development of calculus, then deal with specific historical developments when addressing related topics (e.g., the contributions of Fermat and Descares to solving the tangent line problem can be covered when dealing with functions, graphs, and limits). For additional ideas, see the suggested instructional strategies for other organizers.
• Encourage students to access the Internet for information on the history of mathematics.
• Ask students to investigate various mathematicians and the associated periods of calculus development and to present their findings to the class.
• Point out the connection between integral and differential calculus when students are working on the "area under the curve" problem.
• Conduct a class discussion about the contributions of Leibniz and Newton when the derivative is being introduced (e.g., the notation for the derivative and for the integral are due to Leibniz; the notation is due to Lagrange).
• Have students research the historical context of the applications of antidifferentiation. For example:
  • The area above the x axis, under to . Archimedes was able to show this without calculus, using a difficult argument.
  • The area problem for was still unsolved in the early 17th century. Using antiderivatives, the area under the curve to .
  • The area under , was calculated by Roberval, without calculus, using a difficult argument. It can be solved easily using antiderivatives.

SUGGESTED ASSESSMENT STRATEGIES

When students are aware of the outcomes they are responsible for and the criteria by which their work will be assessed, they can represent their comprehension of the historical development of calculus more creatively and effectively.

Observe

• Have students construct and present a calculus timeline, showing important discoveries and prominent mathematicians. Check the extent to which students’ work is:
  • accurate
  • complete
  • clear

Collect

• Ask students to write a short article about one mathematician’s contributions to calculus. The article should describe the mathematical context within which the person lived and worked and the contributions as seen in calculus today.

Research

• Use a research assignment to assess students’ abilities to synthesize information from more than one source. Have each student select a topic of personal interest, develop a list of three to five key questions, and locate relevant information from at least three different sources. Ask students to summarize what they learn by responding to each of the questions in note form, including diagrams if needed. Look for evidence that they are able to:
  • combine the information, avoiding duplications or contradictions
  • make decisions about which points are most important

Peer Assessment

• To check on students’ knowledge of historical figures, form small groups and ask each group to prepare a series of three to five questions about the contribution of a particular mathematician. Have groups exchange questions, then discuss, summarize, and present their answers. For each presentation, the group that designed the questions offers feedback on the extent to which the answers are thorough, logical, relevant, and supported by specific explanation of the mathematics involved.

RECOMMENDED LEARNING RESOURCES

Print Materials

• Calculus: Graphical, Numerical, Algebraic Integrated Throughout
• Calculus of a Single Variable Early Transcendental Functions, Second Edition Integrated Throughout
• Single Variable Calculus Early Transcendentals, Fourth Edition Integrated Throughout

Multimedia

• Calculus: A New Horizon, Sixth Edition
  pp. 10-11, 15, 19, 99, 225, 237, 278, 352, 378, 638, A3
• Calculus of a Single Variable, Sixth Edition Integrated Throughout

© 2000 Copyright. All Rights Reserved. Curriculum Branch.
Maintained by: Mathematics Coordinator
Revised: December 5, 2000

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