## Classroom-Ready Resources and Teaching Suggestions

Browse index of informative background articles for courses for prospective K–12 teachers.

For additional activities of interest to prospective K–12 teachers, see the resources listed in the courses for K–12 students index.

Return to master index.

**General**

Pathways from the Past: Classroom-Ready Materials for Using History to Teach Mathematics, by Bill Berlinghoff and Fernando Gouvêa

Reproducible student activity sheets developed by the authors of the well-regarded textbook, *Math through the Ages*, and especially suitable for practicing and pre-service teachers of secondary mathematics and those involved in teacher training.

**Problem Solving**

Euler Squares, by Elaine Young

An elementary introduction to Euler squares and how they can be used in teacher training.

**Numeration and Number Operations**

Babylonian Numeration: A Mini-Primary Source Project for Pre-service Teachers and Other Students__,__ by Dominic Klyve

One of a collection of student-ready modules based on primary historical sources presented in the article A Series of Mini-projects from TRIUMPHS: TRansforming Instruction in Undergraduate Mathematics via Primary Historical Source.

Converting the Old Babylonian Tablet ‘Plimpton 322’ into the Decimal System as a Classroom Exercise, by Antonella Perucca and Deborah Stranen

A student-ready activity, ideal for pre-service elementary mathematics teachers.

“In these numbers we use no fractions”: A Classroom Module on Stevin’s Decimal Numbers, by Kathleen M. Clark

A classroom assignment on Simon Stevin's treatment of decimal numbers in his 1585 *De Thiende* that helps pre-service mathematics teachers understand why our usual procedure for multiplying such numbers works.

**Number Theory**

Cuisenaire Art: Modeling Figurate Number Sequences and Gnomonic Structures in a History of Mathematics Classroom, by Günhan Caglayan

Students construct Cuisenaire-rod models per instructions from Theon and Nicomachus.

Generating Pythagorean Triples: A Mini-Primary Source Project in Number Theory for Mathematics Majors, Elementary Teachers and Others__,__ by Janet Heine Barnett

One of a collection of student-ready modules based on primary historical sources presented in the article A Series of Mini-projects from TRIUMPHS: TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources.

**Algebra **(including logarithms)

Algebra Tiles Explorations of al-Khwārizmī’s Equation Types, by Günhan Caglayan

Activities for visualizing al-Khwārizmī's algebraic solution methods using algebra-tile manipulatives.

Completing the Square: From the Roots of Algebra, A Mini-Primary Source Project for Students of Algebra and Their Teachers__,__ by Daniel E. Otero

One of a collection of student-ready modules based on primary historical sources presented in the article A Series of Mini-projects from TRIUMPHS: TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources.

Logarithms: The Early History of a Familiar Function, by Kathleen M. Clark and Clemency Montelle

A recounting of Napier's and Burgi's parallel development of logarithms told as a ‘great tale’ for use in the classroom. Includes student exercises.

**Geometry**

Aiding the Teaching of Geometry and Affording Mathematical Recreation: Paper Folding in the Spirit of Sundara Rao of Madras by Peggy Aldrich Kidwell

A history of paper folding in mathematics education that focuses on the background, publication, and reception of Sundara Rao’s 1893 *Geometrical Exercises in Paper Folding*. The article also describes several potential classroom activities for secondary and undergraduate students of geometry and preservice teachers.

Episodes in the History of Geometry through Models in Dynamic Geometry, by Eduardo Veloso and Rita Bastos

Four episodes in the history of geometry where dynamic geometry (e.g., *The Geometer’s Sketchpad*) helps in understanding the ideas.

Exploring Liu Hui’s Cube Puzzle: From Paper Folding to 3-D Design, by Lingguo Bu

How your students can explore an ancient cube dissection using paper models, computer animations, and/or 3D printing!

Need the Area of a Triangle? The Pope Can Help! by Betty Mayfield

Gerbert d’Aurillac on finding the area of an equilateral triangle, with exploration activities for students.

Pythagorean Cuts, by Martin Bonsangue and Harris Shultz

Discussion of how Euclid’s proof of the Pythagorean Theorem can be adapted to shapes other than squares.

**Probability and Statistics**

How to Calculate \(\pi\): Buffon's Needle – A Mini-Primary Source Project on Geometric Probability for Calculus 2 Students, Pre-service Teachers and Others, by Dominic Klyve

One of a collection of student-ready modules based on primary historical sources presented in the article A Series of Mini-projects from TRIUMPHS: TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources. Suitable for use in courses for pre-service elementary teachers.

Seeing and Understanding Data: A Mini-Primary Source Project for Students of Statistics, by Charlotte Bolch and Beverly Wood

One of a collection of student-ready modules based on primary historical sources presented in the article A Series of Mini-projects from TRIUMPHS: TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources. Suitable for use in courses for pre-service elementary teachers.

## Informative Background Articles

Browse index of classroom-ready resources and teaching suggestions for courses for prospective K–12 teachers.

For additional activities of interest to prospective K–12 teachers, see the resources listed in the courses for K–12 students index.

Return to master index.

**General**

Introducing the History of Mathematics: An Italian Experience Using Original Documents, by Adriano Dematte

A discussion of a collaborative effort in Italy to produce materials enabling secondary school teachers to use the history of mathematics in the classroom.

The High School Mathematics Curriculum—What Can We Learn from History?, by Robert Reys and Barbara ReysA reminder that, while change is hard and pleasing everyone about anything in education is difficult, awareness and knowledge of the history of mathematics education may be helpful to those confronted with current and future challenges related to mathematics curriculum and teaching at the secondary level.

**Numeration and Number Operations**

An Investigation of Subtraction Algorithms from the 18th and 19th Centuries, by Nicole M. Wessman-Enzinger

This survey of four subtraction algorithms used in North America includes as sources both handwritten “cyphering books” and printed arithmetic texts.

Descriptions of the Integer Number Line in United States School Mathematics in the 19th Century, by Nicole M. Wessman-Enzinger

Demonstrates the gradual development of the now-ubiquitous number line, traced through textbooks of the time.

Elementary Soroban Arithmetic Techniques in Edo Period Japan, by Rosalie Joan Hosking, Tsukane Ogawa, and Mitsuo Morimoto

Common techniques for doing basic arithmetic on the Japanese abacus from the *Taisei Sankei* (ca 1700) that show modern students, mathematicians, and historians alike how this device was used for over 250 years in Japan.

“He Advanced Him 200 Lambs of Gold”: The Pamiers Manuscript, by Randy Schwartz

A discussion of the context and content of a 15th-century *abbaco *textbook, the Pamiers manuscript, with translations of its problems, including one for which a negative number was accepted as the answer to a problem.

“Large” Roman Numerals, by Phillip S. Jones with commentary by Victor J. Katz

A discussion from 1954 of the challenges of writing large numbers with Roman symbols and a 2024 essay on Roman achievements in approximating areas, with examples.

Moses ibn Tibbon’s Hebrew Translation of al-Hassar's *Kitāb al Bayān**, *by Jeremy I. Pfeffer

An exploration of Abu Bakr al-Hassar's influential work about arithmetic of fractions,* Kitāb al Bayān wa-l-tadhkār *(*Book of Proof and Recall*).

Reflections on Chinese Numeration Systems, by Frank J. Swetz

Recommends ancient Chinese rod numerals to the instructors of pre-service elementary teachers as an alternative place-value numeration system for helping students understand the structures and operations of arithmetic. Includes historical descriptions and classroom suggestions.

Russian Multiplication, Microprocessors, and Leibniz, by Sid Kolpas

A traditional method of multiplication via binary arithmetic finds a modern use.

The Great Calculation According to the Indians of Maximus Planudes, by Peter G. Brown

A translation of part of a 13th-century work by the Byzantine monk Maximus Planudes on the Hindu-Arabic numerals and the algorithms for calculation.

**Algebra**

Completing the Square, by Barnabas Hughes

Explains the geometric basis for “completing the square,” the original method of solving quadratic equations, to your students.