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
BACKGROUND
Mathematics courses taken by U.S.
high school graduates (2009)
Grade
8
9
10
11
12
Better students
Average students
Algebra (1st year)
Prealgebra
Geometry
Algebra (1st year)
Algebra (2nd year)
Geometry
Precalculus
Algebra (2nd year)
Calculus
Other students
8th grade math
Prealgebra
Algebra (1st year)
Geometry
Precalculus
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.
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.
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.
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
Curriculum and Evaluation Standards
for School Mathematics (NCTM, 1989)
(all students)
“…deduce properties of figures using transformations;
identify congruent and similar figures using transformations;
analyze properties of Euclidean transformations and relate
translations to vectors.”
(college-intending students)
“…apply transformations in problem solving.”
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.”
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:*
A. Experiment with transformations in the plane.
(4 standards)
B. Understand congruence in terms of rigid motions.
(3 standards)
C. Understand similarity in terms of similarity
transformations. (3 standards)
*Detail is given on slides 28 and 29 of this ppt.
THE RAW DATA
64 geometry texts published commercially
in the U.S. from 1961 to 2015
Mean length* of 62 geometry
textbooks 1961-2015
Number of textbooks
Mean length (pages)
1960s
1970s
1980s
1990s 2000s 2010s
14
18
9
9
6
6
612
636
697
765
851
956
*Length includes all features but does not include introductory material
(title page, table of contents, etc.). Two experimental texts not included.
RESULTS AND DISCUSSION
Four analyses of the data
1. Extent of opportunities to encounter
geometric transformations
2. Scope and language of transformations in the
texts that discussed them
3. Coverage of six basic reasons for studying
transformations
4. Attention to specific activities mentioned in
the U.S. Common Core Standards of 2010
Treatment of transformations in 62
geometry textbooks 1961-2015
1960s
1970s
1980s
1990s 2000s 2010s Totals
No treatment
11
5
1
1
0
0
18
One lesson only
One chapter at
end of book
One chapter in middle,
unconnected
In various places,
unconnected
0
2
1
1
0
0
4
1
5
5
3
2
0
16
0
4
0
2
1
1
8
1
1
2
0
1
4
9
Treatment throughout
1
1
0
2
2
1
7
Totals
14
18
9
9
6
6
62
Treatment of transformations in 62
geometry textbooks 1961-2015
1960s
1970s
1980s
1990s 2000s 2010s Totals
No treatment
11
5
1
1
0
0
18
One lesson only
One chapter at
end of book
One chapter in middle,
unconnected
In various places,
unconnected
0
2
1
1
0
0
4
1
5
5
3
2
0
16
0
4
0
2
1
1
8
1
1
2
0
1
4
9
Treatment throughout
1
1
0
2
2
1
7
Totals
14
18
9
9
6
6
62
Treatment of transformations in 62
geometry textbooks 1961-2015
1960s
1970s
1980s
1990s 2000s 2010s Totals
No treatment
11
5
1
1
0
0
18
One lesson only
One chapter at
end of book
One chapter in middle,
unconnected
In various places,
unconnected
0
2
1
1
0
0
4
1
5
5
3
2
0
16
0
4
0
2
1
1
8
1
1
2
0
1
4
9
Treatment throughout
1
1
0
2
2
1
7
Totals
14
18
9
9
6
6
62
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
Four analyses of the data
1. Extent of opportunities to encounter
geometric transformations
2. Scope and language of transformations in
the texts that discussed them
3. Coverage of six basic reasons for studying
transformations
4. Attention to specific activities mentioned in
the U.S. Common Core Standards of 2010
Definition of “transformation” in 35
geometry textbooks 1961-2015
Transformation identified as
F
U
N
C
T
I
O
N
S
Y
N
O
N
Y
M
S
Correspondence
Textbooks numbered (in
date of publ. order)
10, 17, 23, 37, 44, 46, 48,
58
11, 22, 25, 28, 32, 38, 41,
49, 52, 53, 59
Function
27, 34, 42, 47, 64, 66
6
Operation
21, 61, 65
3
Set of ordered pairs
30
1
Matching
45
1
Process
63
1
Movement or change
54, 55, 56, 62
4
Mapping
Number of
textbooks
8
11
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.
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.
Four analyses of the data
1. Extent of opportunities to encounter
geometric transformations
2. Scope and language of transformations in the
texts that discussed them
3. Coverage of six basic reasons for studying
transformations
4. Attention to specific activities mentioned in
the U.S. Common Core Standards of 2010
Basic reasons for studying
transformations
(1) They enable single definitions of congruence and
(2)
(3)
(4)
(5)
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.
The most common shortest distance
problem solved using transformations
Find the path of shortest total length
from A to B that intersects line m.
.
A
.B
m
Basic reasons for studying
transformations
(1) They enable single definitions of congruence and
similarity to apply to all figures.
(2) They are functions, so can provide experience with
function notation and composition.
(3) They provide a mathematical explanation of symmetry.
(4) They provide mathematical models of real-world
actions.
(5) They solve problems that might be more difficult to
solve without them.
(6) There are interesting and important algebraic relations
involving them.
Numbers of texts discussing various
aspects of transformations
One chapter at
end of book
(n = 16)
One chapter in
middle
(n = 8)
Various places,
unconnected
(n = 9)
Treatment
throughout
(n = 7)
Symmetry is explained using transformations.
11
6
8
7
Transformations model the real world.
7
5
3
7
Shortest distance problem solved using reflections.
5
0
4
5
Algebra of transformations is discussed.
1
1
0
5
Four analyses of the data
1. Extent of opportunities to encounter
geometric transformations
2. Scope and language of transformations in the
texts that discussed them
3. Coverage of six basic reasons for studying
transformations
4. Attention to specific activities mentioned in
the U.S. Common Core Standards of 2010
Degree of coverage of transformation
activities now in Common Core
Activity
Degree of attention
in 40 texts
Cluster: Experiment with transformations in the plane.
1. Give a definition of “transformation” as a function from points
of the plane into points of the plane.
2. Compare transformations that preserve distance and angle
to those that do not (e.g., translation versus horizontal stretch).
3. Describe the rotations and reflections that map rectangles,
parallelograms, trapezoids, and regular polygons onto
themselves.
4. Give definitions of rotations, reflections, and translations in
terms of angles, circles, perpendicular lines, parallel lines, and
line segments.
5. Given a geometric figure and a rotation, reflection, or
translation, draw the transformed figure using, e.g., graph
paper, tracing paper, or geometry software.
6. Given two congruent figures, determine a transformation or
sequence of transformations that will map one figure onto
another.
Few specifically mention
functions.
Coverage rare.
8 texts mention all the
symmetries.
In almost all texts.
In almost all texts.
In 18 texts.
Degree of coverage of transformation
activities now in Common Core
Activity
Degree of attention in
40 texts
Cluster: Understand congruence in terms of rigid motions.
In almost all texts.
7. Use geometric descriptions of rigid motions to transform
figures.
8. Deduce ASA, SAS, and SSS congruence through isometries. In 9 texts.
Cluster: Understand similarity in terms of similarity transformations.
9. Draw to show that a line not containing the center and its
dilation image are parallel and to show that a line containing the
center and its dilation image coincide.
10. Draw to show that the ratio of the length of a dilation image
to the length of its preimage equals the scale factor of the
dilation.
11. Deduce AA similarity through similarity transformations.
Difficult to determine;
drawings in almost all texts
that discuss dilations.
In 8 texts.
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.
Comments and questions are welcome!
z-usiskin@uchicago.edu