Transformations in U.S. Commercial High School Geometry Textbooks since 1960 – A Brief Report

1. Transformations in U.S. Commercial High School Geometry Textbooks since 1960 – A Brief Report International Conference on Mathematics Textbooks University of Southampton 31 July 2014 by Zalman Usiskin The University of Chicago z-usiskin@uchicago.edu

2. BACKGROUND

3. Mathematics courses taken by U.S. high school graduates (2009) Red indicates that the course is required for high school graduation in most states. In 2009, 18% of high school graduates had completed calculus, 35% had completed precalculus, and 76% had completed a second year of algebra. About 85% take a one-year geometry course; perhaps 5% study an integrated curriculum.

4. Content in almost all U.S. geometry texts since 1960 (the scope of this study) • properties of points, lines, and planes • angles, parallel and perpendicular lines • triangle congruence, including proof • properties of special quadrilaterals (parallelograms, etc.) • similar triangles and similar polygons, including regular polygons • properties of segments and angles in circles • figures in 3-dimensional space (prisms, pyramids, cones, cylinders, spheres) • area, surface area, and volume • coordinate geometry (figures on the coordinate plane) • indirect proof.

5. Transformations and Euclidean geometry Transformation = 1-1 function that maps point of a set F onto a set G. Euclidean geometry = study of those properties of figures preserved by transformations that preserve ratios of distances (Felix Klein, 1872) Similarity transformation = transformation that preserves ratio of distances = distance-multiplying transformation Isometry = distance-preserving transformation = similarity transformation with multiplier 1.

6. Definitions of congruence and similarity using transformations • Two figures α and β are congruent if and only if there is a distance-preserving transformation (isometry) that maps α onto β. β α • Two figures α and β are similar if and only if there is a distance-multiplying transformation (similarity transformation) that maps α onto β. α β

7. Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989) (all students) “…deduce properties of figures using transformations; identify congruent and similar figures using transformations; analyzeproperties of Euclidean transformations and relate translations to vectors.” (college-intending students) “…apply transformations in problem solving.”

8. Principles and Standards for School Mathematics (NCTM, 2000) (all students) “understand and represent translations, reflections, rotations, and dilations of objects in the plane by using sketches, coordinates, vectors, function notation, and matrices; use various representations to help understand the effects of simple transformations and their compositions.”

9. Common Core State Standards for Mathematics (National Governors’ Association and Council of Chief State School Officers, 2010)http://www.corestandards.org/the-standards/mathematics Three clusters of standards:* • Experiment with transformations in the plane. (4 standards) • Understand congruence in terms of rigid motions. (3 standards) • Understand similarity in terms of similarity transformations. (3 standards) *Detail is given on slides 28 and 29 of this ppt.

10. THE RAW DATA 64 geometry texts published commercially in the U.S. from 1961 to 2015

11. Mean length* of 62 geometry textbooks 1961-2015 *Length includes all features but does not include introductory material (title page, table of contents, etc.). Two experimental texts not included.

12. RESULTS AND DISCUSSION

13. Four analyses of the data • Extent of opportunities to encounter geometric transformations • Scope and language of transformations in the texts that discussed them • Coverage of six basic reasons for studying transformations • Attention to specific activities mentioned in the U.S. Common Core Standards of 2010

14. Treatment of transformations in 62 geometry textbooks 1961-2015

15. Treatment of transformations in 62 geometry textbooks 1961-2015

16. Treatment of transformations in 62 geometry textbooks 1961-2015

17. Culling down the sample 64 texts in collection -2 texts deemed purely experimental = 62 texts in sample -22 texts that do not discuss transformations in more than a few pages = 40 texts discuss transformations in a full chapter or more

18. Four analyses of the data • Extent of opportunities to encounter geometric transformations • Scope and language of transformations in the texts that discussed them • Coverage of six basic reasons for studying transformations • Attention to specific activities mentioned in the U.S. Common Core Standards of 2010

19. Definition of “transformation” in 35 geometry textbooks 1961-2015

20. Scope and language of transformations • Euler’s f( ) function notation in 12 of 35 texts that define transformations. • 37 of the 40 texts discuss composition of transformations, calling the answer: composition (17 texts), composite (9), product (6), combination (2), resultant (1), with 1 text using two of these names and three texts giving no special name.

21. Scope and language of transformations (more) • Of the 40 texts, 22 mention all four types of isometries (reflections, translations, rotations, and glide reflections); 18 mention all but glide reflections. • The transformation T that fixes a single point C, and for which the image of P is the point P’ on ray CP so that CP’ = k•CP is almost universally called a dilation (other names: dilatation, size change, size transformation, expansion). If k > 1, an expansion, enlargement, or stretch; if k < 1, a contraction, a reduction, or a shrink.

22. Four analyses of the data • Extent of opportunities to encounter geometric transformations • Scope and language of transformations in the texts that discussed them • Coverage of six basic reasons for studying transformations • Attention to specific activities mentioned in the U.S. Common Core Standards of 2010

23. Basic reasons for studying transformations (1) They enable single definitions of congruence and similarity to apply to all figures. • They are functions, so can provide experience with function notation and composition. • They provide a mathematical explanation of symmetry. • They provide mathematical models of real-world actions. • They solve problems that might be more difficult to solve without them.

24. The most common shortest distance problem solved using transformations Find the path of shortest total length from A to B that intersects line m. .B A. m

25. Basic reasons for studying transformations (1) They enable single definitions of congruence and similarity to apply to all figures. • They are functions, so can provide experience with function notation and composition. • They provide a mathematical explanation of symmetry. • They provide mathematical models of real-world actions. • They solve problems that might be more difficult to solve without them. • There are interesting and important algebraic relations involving them.

26. Numbers of texts discussing various aspects of transformations

27. Four analyses of the data • Extent of opportunities to encounter geometric transformations • Scope and language of transformations in the texts that discussed them • Coverage of six basic reasons for studying transformations • Attention to specific activities mentioned in the U.S. Common Core Standards of 2010

28. Degree of coverage of transformation activities now in Common Core

29. Degree of coverage of transformation activities now in Common Core

30. Summary • High school mathematics teachers in the U.S. today are unlikely to have studied geometric transformations as high school students. • The language and approaches of the material in books that cover transformations vary significantly. • Only a few of the activities in the Common Core are likely to be familiar to teachers. • Professional development will be necessary for most teachers if this content is to be taught to students in the manner dictated in the Common Core.

31. Comments and questions are welcome! z-usiskin@uchicago.edu