While there are ways districts can avoid the use of these Framework Texts, it is common for them to go along with the state-directed designs. It must be stressed that there are alternatives to this situation. There are three programs that were added to the approved list that were not judged to be highly consistent with the Framework. Districts should consider evaluating these texts. Also, districts may apply for a waiver to select materials not found on the list of adopted materials.
Despite available options, districts typically take the default course of selecting from the approved Framework Texts. The following material was produced by Robert Herriot, a parent in the Palo Alto Unified School District, in an attempt to influence such a process. This report therefore makes a comparison among three specific Framework Texts. No comparison to other programs, either traditional or reform, was provided.
This parent finds one program superior to the other two, and has made an extensive effort to elaborate the reasons for his conclusion.
The views expressed below are those of the author, Robert Herriot, and do not necessarily represent the views of the Palo Alto Unified School District or of Mathematically Correct. In fact, Mathematically Correct has taken the position that it is best to include Non-Framework texts when evaluating curriculum materials. Permission to use Non-Framework materials can be received from the State Board of Education.
copyright © 1996 by Robert Herriot, 650-326-8279
permission granted to copy this document for noncommercial purposes only.
Any copy must include this copyright statement.
This packet compares the following math series
In comparing math books, there are two essential components that must be considered:
To evaluate mathematical content, I have spent the past 8 months doing extensive studying of the above 3 textbook series. I have read through the materials, grade by grade, and have followed a few strands through all the grades. I have used my extensive experience in the mathematical field
To evaluate suitability for teachers and children, I have consulted with teachers in about a dozen school districts, most near Boston and Chicago to get their opinions. I have included teachers' quotes in this packet.
These three books do not represent the two sides of the traditional versus constructivist curriculum (i.e., kids construct and discover math rather than being taught it). All three use the constructivist approach, make use of calculators, and avoid "drill and kill" worksheets. Rather the difference is degree. Everyday Math is balanced. It is partly traditional (teachers instruct and kids must know "the basic facts as reflexes") and pragmatically constructivist (the discovery is well channeled). It gives kids a sound mathematical foundation both in understanding concepts and in computational skills fluency. Investigations and MathLand are dogmatically constructivist. Activities frequently have unclear objectives. The books omit many important concepts. They leave kids with an incomplete understanding of math and low computational skills.
From my extensive analysis, I have concluded that Everyday Math is the best of the 3 series by far. It is not perfect. It is not as traditional as some would like, and the student textbook does not contain the sort of explanations that will help students and parents at home. But the mathematics is sound, and the lessons are intricately designed to lead kids through important mathematical concepts so that kids achieve an excellent understanding of mathematics. Also, teachers like it, and kids are learning more from it than from any other curriculum.
Both Investigations and MathLand are severely flawed. Investigations is very uneven. Some of its activities are excellent, and others are on the mathematically unimportant. But even the good activities are designed so that neither the teacher nor the kids know whether the kids' solutions are great, mediocre or wrong, and some creative and correct solutions are indicated as wrong. Some parts of the curriculum develop concepts well, and other parts jump around skipping important parts of math. The book also dogmatically discourages the learning of algorithms (e.g. long division), and does a poor job of developing computational skills. There is no student textbook and no student workbook. Some of the books activities would be a good supplement, but is too incomplete to be a full curriculum without a very large amount of work.
MathLand is similar to Investigation, only far worse. The curriculum is about 1 year behind the other two by 5th grade, and its activities have none of the imagination of Investigations. It also contains many mathematical inaccuracies. Some activities keep kids busy, but don't teach anything. Teachers are strongly discouraged from saying whether a kid's solution is right or wrong. There is no student textbook, and the student workbook problems are based on a repetitive gimmick which doesn't have enough variation or the right sort of skills development.
The packet contains the following documents:
The following is a matrix of the strengths and weaknesses of the 3 math series. Each row discusses some characteristic of the series.
Everyday Math Investigations MathLand
contains accurate math contains accurate math except contains too many errors in
for a major conceptual error math; authors frequently
about silhouettes in 4th display a lack of
grade, and some misleading mathematical understanding of
solutions. what they are teaching.
developed 1 grade per year is complete and well field is complete in K-5, but put
starting in K; is complete tested in 3-4, 2nd and 5th together too quickly.
and well field tested in K-4, just finished this year; K-1
5th finished this year; still in preparation;
mature program immature program
gives excellent understanding gives excellent understanding gives some understanding of
of basic concepts; is of basic concepts; omits many basic concepts, although at a
intricate and carefully basic concepts that are very slow pace compared to
planned to cover basic needed for math in later the other 2 series.
concepts. grades
leads from particular results fails to lead from particular same comment as for
in an activity to an results in an activity to an Investigations
understanding of general understanding of general (on the left)
principles, which allows principles, and keeps
children to solve unexpected children from solving
problems. unexpected problems.
spends time mostly on spends too much time on same comment as for
concepts that are building concepts that are cute but Investigations
blocks to future learning. not mathematically useful. (on the left)
makes math fun and lessons makes math fun but lessons same comment as for
are well connected to are often Investigations
important math concepts. unrelated to any important (on the left)
math concepts.
provides excellent connection provides little connection to same comment as for
to real world applications real world applications and Investigations
and children's everyday life. children's everyday life. (on the left)
Activities are mostly based
on pure math
relates problems to a has occasional brilliant has nothing outstanding.
superlative student book on activities, though it is hard
the world (4th), and US to know what a child has
(5th). learned
contains activities that are contains activities that are same comment as for
usually fairly short and that usually fairly long but Investigations
have a clear objective. without clear objectives. (on the left)
contains a good balance is unbalanced and oriented is unbalanced and oriented
between teacher based towards student discovery towards student discovery
instruction and student activities with no direct activities; teachers are
discovery activities. teacher instruction. strongly discouraged from
instructing
gives teachers an excellent gives teachers an excellent gives teachers virtually no
explanation of the lesson's explanation of the lesson's explanation of the lesson's
concepts and purpose. concepts and purpose. concepts and purpose.
explains directly what are explains likely answers same comment as for
good answers to a problem or through simulated class Investigations
activity. discussion without saying (on the left)
which answers are great,
mediocre or wrong.1
is nearly a complete needs a lot of supplementing needs a great deal of
curriculum and needs no because of the lack of supplementing; the
supplementing except for more student workbooks, skill Arithmetwists seem boring and
worksheets on skills. sheets, and missing topics. not designed to teach
necessary skills.
needs no pruning needs pruning of digressions. needs pruning of digressions.
has Journals that are has no text book for student same comment as for
somewhat like text books for and parent reference. Investigations
student and parent reference. (on the left)
has Journals, which are has no student workbooks, and has student workbooks
combination textbooks and there are only occasional (Arithmetwists), but they are
workbooks; there are too few black line masters with a few very narrowly focused with
problems for each lesson problems on them. only a few types of problems
sometimes. .
has daily Home Links which has Family Letters (one page has weekly Family Letters
contain a few short well summaries of a unit) sent which contain one paragraph
defined skill problems and home every 3 weeks; there are summaries and a simple
sometimes a one-page summary. no problems sheets to send problem, often not well
home. directed.
has written assessment has no written assessment at has an open-ended oral
problems at the end of each end of unit; depends on assessment for each unit.2
unit ongoing teacher observations.
has improved test scores. has severely reduced no information
computation scores.
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1. For examples, see p. 64 Turtle Paths (3rd) or p. 48 of 3-D geometry (5th). See also footnote 2 of Geometry page in this packet,.
2. Examples for scoring will cause some teachers to give low scores for correct but unexpected answers
I have talked to teachers in about a dozen districts, mostly near Chicago and Boston. Some of the schools have classes around 20-22, but many range from 25 to 29. Most of the schools are demographically similar to Palo Alto. Some are more average; others have a wider range.
Most of the districts are using Everyday Math, and they have been using it for 2 to 4 years. A few are using Investigations. A common theme with Everyday Math users is an incredible excitement for it. Below is a summary of quotations on Everyday Math. On page 2, there are quotations for Investigations.
"Teachers who looked at it, initially dismissed it. Now 90% of K-5 teachers love it, and would never go back." HS
"The 1st year, I had 6 volunteer pilots. The 2nd year, I asked for 30 more volunteer pilots. I accepted 30 and turned away many more teachers, 30 of whom did it on their own as 'pirates'." CY
"Before starting with Everyday Math, primary teachers said kids couldn't do Everyday Math, and they would never figure out xyz. During the 1st year, teachers were saying things like 'I'm glad I didn't bet my salary because kids are thinking in ways they have never thought, and drawing conclusions they have never had in the past.'" FH
"Teachers are thrilled. Teachers love it!" KA
"Teachers love it! They have learned so much math. I never would have believed children could do what they do." CY
"Kids are doing remarkable things. Teachers are very impressed with what they learn." MW
"Teachers love it. A teacher, who had never taught math, wanted to start teaching math after seeing how much fun it was." WM
"The first year was a struggle. Now they like it." RM
"Absolutely wrong! Kids come through, and then some." JS
"It's very hard and challenging, but kids rise to it beautifully." KA
"Laughs. We haven't found that. It provides multiple entry points. It offers a challenge. Teachers find that kids can do more than previously. It stretches students. Teachers are impressed that kids can do it." MW
"Laughs. No, it's not too hard! Kids, including grade 1 thrive on it. First grade builds on information kids already have." WM
"It's challenging, but 1st graders can do things they used not to be able to do." RM
"Reading is not a problem; it is at the right level for kids." CG
"It does not require reading level above kids' ability." WM
"No. It is at the right level, not too fast." BW
"Kids love it, even those struggling." HC
"Good balance of teaching and activities." HS
"Most actively teach, and they guide too." MW
"Reasonable balance, almost every teacher instructs." RM
"Amazed at what slow kids, including special ed. kids, are understanding." HS
"Healthy component for gifted. Have to be really gifted to need more. Some at the bottom struggle." WM
"Provides a lot of entry points to activities. Gifted find additional ways to extend activities." CY
"It has something for gifted kids. Learning disabled kids have trouble." BW
"Builds number sense extremely well before computation starts." FH
"Real focus on problem solving and thinking." KA
"Way ahead in teaching important building blocks." FH
"It's systematized; ensures that kids learn good math and don't miss things." FH
"Parents who are professors at MIT say it builds the mathematical structure that they would like to see in their students." BW
"Curriculum is intricate and so carefully planned." FH
"Kids can do more than with traditional curriculum." RM
"Delivers accurate math." FH
"Games are critical to learning." HS
"There was a basic skills concern, but it's happening." MW
"We spent only $700 per class for books and manipulatives by getting generic manipulatives, where appropriate; we saved hundreds of dollars per class, and got more for our money." HS
"Every concept is taught 5 times in 5 different ways over two years before mastery is expected. It works well." FH
"Every concept is taught 5 times in 2 years, so teachers have to learn to let go." BW
"The jumping around threw teachers off at first; now they realize it is a solid approach." HS
"Like spiraling aspect." WM
"Love the spiral and connection to other areas of the curriculum. No longer reviewing a lot and not challenging students." KA
"Spiraling threw people off. After training, some are very comfortable; others are still learning not to require mastery initially." MW
"Biggest criticism is spiraling approach. Those who used it 3 years or more, like it. Absolutely, the book is the correct choice." DN
"Have to supplement computation." BW
"Have not needed to supplement computation." WM
"Added test twice a year for accountability." MW
"Added traditional problems to Home Links." MW
"Kids did incredibly well on SAT. It was very easy for them." HS
"There was a statistically significant rise in CTBS scores." CG
"Scores went up in understanding and in computation." HC
"Test scores have gone up." MW
"Tests in two nearby districts using Everyday Math found 6th graders were algebra ready." FH
"Not every parent likes it; there isn't enough drill in computation, and it isn't what they had as children." RM
"Had to do parent education." BW
"Some parents objected. But after parent nights, they liked it."
"No parent complaints about it." CY
"Had to do parent workshops." HS
"Had parent nights to avoid discontent; it's non traditional for parents." MW
"Need a parent reference. Home Links and student journals are better than nothing, but not sufficient." MW
The following are comments on Investigations [TERC] from people who are using Everyday Math and know or use TERC.
"TERC is not a complete program. It's missing the scope and sequence of Everyday Math. It would be a lot of work to scaffold material. Some use TERC to supplement Everyday Math." CY
"TERC has its merits, but we're happy with Everyday Math." DN
"TERC went too far to avoid algorithms. Everyday Math is more balanced. Yes, TERC is dogmatic, and Everyday Math is pragmatic." FH
"TERC doesn't go far enough in covering subjects and misses areas." FH
"TERC lessons take much longer than the 1 hour allocated in the book." FH
"Very displeased with CTBS results last April for 3rd, 4th, and 5th grades using TERC. Conceptual thinking went up; percentile was in the 80's. Computation was way, way down-the lowest we had ever seen. After adding computation this year, we are still not pleased with kids' thinking. We are changing to Everyday Math for the entire K-5 program." FH
All books cover the same basic topics. But only Everyday Math teaches kids the general way of creating equivalent fractions with different denominators and addition with different denominators using common denominator rule.
Major Minor Topic Everyday Investigations MathLand
Topic Math
dividing ½ 1 2 3 3 4 2 3
whole
¼ 1 2 3 3 4 2 3
1/3 1 2 3 3 4 2 3
1/6 1 2 3 3 4 2 3
1/8 1 2 3 3 4 2 3
1/10 2 3
1/12 2 3
1/20 2
collection member is fraction of 2 3 5
whole
several 2/3 1 2 3 4 3 4 4
parts
2/4 1 2 3 4 3 4 4
3/4 1 2 3 4 3 4 4
m/n for 1 digits 3 4 4 4
numbers
3/3 4/4 5/5 = 1 3 3
0/1 0/2 0/3 0/4 = 0 3 5
greater with ½ 3 4
than 1
general mixed number 5 3 4 5
inches and fractions 3 5
on ruler
measurement fractions of larger 3 5
units
terms denominator defined 1 2 5
numerator defined 1 2 5
comparison different numerators 1 2 3 4
different denominators 1 2 3 3 4
¼, ½ of different 2 4 4
wholes
equivalents 1, fractions tiles 4 3 4 3
that sum to 1
of ½ 1 2 3 4 3 4 5
of 1/3 2 4 3 4 5
of 1/4 , 1/3 , ½, 3 4 4 3* 4 5
2/3
how to produce 3 4 5 -#
decimals 4 5 3 5
percentage 5 5
adding of m/n , specific 4
values with tiles
of m/n , general 5
case
subtracting of m/n , specific 4
values with tiles
multiplying m/n of (m out of n 3 4 5 3 4 5 4 5
of ) a number
number position on 4 5 4
line
position for 4
comparison
miscellaneo things we can split 3$
us
halves of 4%
non-symmetric shapes
shapes on geoboard 4@
that are ¼
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* Children work with tiles that represent different fractions. The work with specific and poorly sequenced fractions is unlikely to give a good understanding that will last past the loss of the tiles. Example of what's wrong: 1/5 + 1/5 + 1/10 = 1/2 . Building with such tiles does not give a general understanding.
# In Fractions and Area book, p55, the teacher's text says that some students may notice that 1/3 =2/6 because 1x2=2 and 3x2=6. But Investigations does not suggest that all students learn about this.
$ Children investigate what they can split, e.g. apples, beverages, and what they cannot split, e.g. balloons, books, shirt (even though it is possible cut a book or shirt in half). It is not clear how this investigation helps understand fractions or anything else. It's more a philosophical issue.
% This exercise is interesting, but it isn't clear how much this helps with understanding of fractions, especially because it is easier to understand a fractions of a whole when it has a simple regular shape.
@ Children divide a 4*4 geoboard into 4 or 8 equal areas experimenting with different shapes. These may help a child understand area (which is a part of this unit), but they do not add to intuition about fractions.
The Geometry sections cover traditional topics for Everyday Math and MathLand. Investigations omits many standard topics and covers other topics instead; some are mathematically important and some aren't.*
Major Minor Topic Everyday Investigations MathLand
Topic Math
2-D shapes triangles 1 2 3 4 3# 1 3
relation of sides, 3 3 4 5
angles
sum of angles = 180º 5
equilateral triangle 5 3 5
isosceles triangle 5 5
scalene triangle 5 5
congruent triangles 5
quadrilaterals 1 2 3 4 1 4
relation of sides, 3 4 5
angles
squares 1 2 3 4 5 3 3
rectangles 1 2 3 4 3
parallelogram 1 2 3 4 2 4
trapezoid 1 2 3 4 2
rhombus 1 2 3 4
pentagon 1 2 3 4 4
hexagon 1 2 3 4 2 4
heptagon 1 2 3 4 4
octagon 1 2 3 4 4
nonagon 3 4
decagon 3
tetrominoes 3
pentomino 3
tangrams 3 4
tessellations 5 5
polygons 1 2 3 4 5 3 4
relation of sides, 3 5
angles
sides 1 2 3
vertices 1 2 4 3
angles 1 2 4
convex, concave 4
circles 4
diameter 5
radius 4
concentric circles 4
tangent to circle 4
patterns with circles 4
compass and 4 5
straightedge
Shapes congruence 3
Angles general 3 4
right angle 3 5
acute angle 5 5
obtuse angle 5 5
measuring angles in º 4 5 3 5
latitude & longitude 4
Lines line segments 1 2 3 4 3
named line segments 1 2 3 4
rays 3 4
parallel lines 3 4
intersecting lines 3
paths, closed 3
paths, open 3
Points general 1 2 3 4
named points 1 2 3 4
Transformat rotation of lines 180º 3
ion
rotation of lines 90º 3 3
rotation of shapes 2 4 3 2 3
slide (translation) 4 3
symmetry 2 4
reflections 4
line symmetry 4
quilts 4 3
real world designs 4
Perimeter triangles 4
quadrilaterals 4 3 4
polygons
circle 5
5
distance along curves 5
Area with grids 4 5 4 3
triangles 4 5 3
squares 4 5 3 4
rectangles 4 5 4
parallelogram 4 5
polygon 5
circle
scaling 5
circles 5
shapes 3 4
X y locate point 5 4$
coordinates
coordinate pairs 5 4
negative coordinates 5 4
scale pictures 5
translate, flip, 5
rotate
Turtle rectangles 4%
3-D shapes faces 1 2 4 1 2 3 4
edges 2 4 4
corners (vertices) 1 2 4 4
square prisms 3
rectangular prisms 1 2 3 4 3 1 2 3
triangular prism 1 2 3 4 3 1 2 3
cubes 1 2 3 4 3 2 3 4
pyramids 1 2 3 4 3 2 3 4
spheres 1 2 3 4 3
cylinders 1 2 3 4 3
cones 1 2 3 4 3
regular polyhedra 2 4 3 4
pattern into 2 4
tetrahedron
3-D shapes pattern into 4
dodecahedron
pattern into polyhedra 3
build 3-D from 2 2-D 4
views
build 3-D from 2-D 4@
picture
silhouettes of solids 4&
similarities of solids 3 3
parallel sides 3
3-D tetranominoes 3
Volume rectangular prism 4 5
cylinders 5
cones 5
comparisons 5
standard units 5
packing problems 5+
---------------------------------------------------------------------------------
* Investigations is difficult to analyze because it covers topics in a nonstandard way, sometimes with activities that are hard to categorize. As a result, it is hard to find where a particular topic is covered.
# The following quote is an example of what is really wrong with the discussion of solutions to activities in Investigations. The following is a quote (p 64, Turtle Paths, 3rd grade). "Most students agreed that figure N was not a triangle. Ryan, however, thought it was indeed OK because it was two triangles. The class disagreed, convincing Ryan by emphasizing that "it crossed itself" and it "really has four sides."" Here is an example wrong mathematical conclusions by democracy. Ryan is correct. It really is two triangles that happen to share a point. Ryan is the sort of creative kid who will be stifled by this curriculum.
$ Investigations devotes 6 sessions (hours) to x-y coordinates (15 including the Turtle part described below); Everyday Math devotes 3. Investigations covers more ground, but not 5 times more.
% Investigations devotes 9 more hours using the Turtle program to draw rectangles in the x-y coordinate space, to rotate rectangles, repeat them, and design rectangle patterns. All of this is done in the Turtle programming language. Much of the time is spent on learning something that has no future value.
@ The book devotes 9 hours to various projects of constructing cube buildings. This seems much too long.
& This section contains a gross error. The activities on silhouettes are actually about projections. For example a cylinder has a silhouette that is a rectangle, but when seen from a nearby point, its projection contains at least one curved edge. The book devotes 5 hours to silhouettes which is far too much time for its value. This topic and the one on cubes devote 14 hours, nearly 3 weeks to these parts of 3-D geometry.
+ This is a great example of Investigations at its best. The problem presents 5 packages: A: 2x2x2, B: 1x1x3, C: 1x2x2, D: 2x2x3, E: 1x1x5. The children design a box for packing (with no empty space) packages A, B, C or D. And then as a bonus, package E too. This problem brings together a lot of skills. Unfortunately, the teacher explanation relies on a simulated class discussion (p 48 of 3-D Geometry, Volume) and could leave many teachers confused about the best answer. Because of the required pedagogy of "no instruction," the children won't learn the best solutions, and understand why they are the best.
Establish year long routines
Numbers and how they are used in the world: coins, telephone numbers, time, counting
Number Patterns: even and odd, other pattern in the number grid 1 to 100.
Measurement: temperature, linear measure, time
Relations: =, < and > with numbers, ordinals
Addition and Subtraction: many explorations
Geometry: common 3-D and 2-D shapes, lines, points, attributes of shapes
Mental Arithmetic: games, e.g. with money
Fractions: dividing into 2 or 4 equal parts, things that come in equal parts, e.g. file drawers.
Numeration: strategies for adding and subtracting
Place Value
Probability (incidental to a lesson, toss 2 dice 30 time to see distribution)
Counting and sorting
Patterns: colored blocks arrangements
Counting and comparing numbers to 20
Sorting items by attribute
Combinations of things (but not necessarily all combinations)
Number pair relations: with tiles, e.g. 1+4, 2+3, 3+2, 4+1
Measurement: linear measurement, time, and balance
Adding and Subtracting
Numbers on the grid 1 to 100: explorations
Geometry: explore a few 3-D and 2-D shapes.
Probability: (does a very poor job on probability).
This series has a number of very attractive qualities that put it far ahead of its competition.
The following is a summary of topics in the book.
This section shows the topics covered by the 10 chapters of the teacher's guidebook. Overall, many of the activities seem interesting, but many of them also seem to be below first grade level. The book is especially weak on topics that have mathematical depth, such as combinatorics and probability. The probability section explains almost nothing to the teacher and seems to teach almost nothing to the kids.
Another big concern is that the kids will not master addition and subtraction. There are many activities involving these subjects, but the curriculum does not include any methodical way to assure that every kid visits all combinations of number facts enough times until they are mastered. Too much is left to what each kid encounters in random activities. For some, these encounters will be sufficient, but for others it will not be enough.
How many ?
The answer book says 6, but 8 is correct when the two large triangles are counted too.
How many ?
The answer book says 4, but 5 is correct when the one large square is counted too.
Similar question with triangle and diamonds. The answer misses the case of two triangle that form a diamond.
How many ?
The answer book says 6, but 7 is correct when the one large square is counted too.
How many ?
How many ?
How many ?
The answer book says 5, 2, 4 for the 3 shapes respectively, but there are 3 more triangles and 2 more trapezoids of the same proportions, one the same size as the others and one that is the whole shape.
There are not very many problems, and many don't seem very interesting.
Arithmetwists offer a child some variety from the standard drill, but they tend to become rather repetitive with the same theme on page after page. In order for a child to solve most of the Arithmetwists, the child must perform some addition or subtraction to get the correct answer. For children who need assurance that they understand the basic concepts, there are no exercises that build to these higher level ones.
Although some exercises direct the child on what to do, others are so open that the child may not have enough direction. For example there is an exercise that show 8 rows, each with 5 adjacent empty squares. The child is to "Find different ways to color 5. Use 2 colors. Keep the colors together." The answer book shows 1&4, 5&0, 3&2, 4&1, 2&3, which are all combinations except 0&5. The answer book does not say how many combinations should be shown. For similar problems which sum to 6 and 7, the answer book shows all combinations. For similar problems which sum to 8, 9, 10, 11, and 12, the problem doesn't have enough rows for the child to enumerate all combinations.
The assessment process is very subjective and open-ended. Although most assessments are simple and straightforward, there are some where the authors have missed the ambiguity of their question or don't understand the depth of their question. Below are some examples of these issues. A child have a level 1 (lowest), 2, or 3 (highest) answer.
The teacher gives the kids a pattern, such as green, green, red, green, green, red. The teacher then asks the kids to "draw a design that is different in shape or color, but uses the same pattern." There are no examples using a different shape, but there is an example of a solution that is only level 2 because only one of the colors is changed. The problem does not state that it should be different in both colors.
The child has created a 20 page book about things the child has counted, one page for each number 1 through 20. The teacher ask the kids "What numbers (pages) go between these two? How do you know?" A level 3 answer is "Here's the page for 5, so 6, 7, 8, and 9 come next." A level 2 answer is "6 and 8 are between those pages." The teacher's guide downgrades this response because the kid "Names some of the numbers, but not all." But the question does not state that the kid must name all numbers between the two.
The child is asked to "find some things that will balance this [scale]." The child who puts too many items on the initially empty side and then removes them one by one until balanced receives a level 3. The child who puts too many items on the initially empty side and then removes a bunch that is too many and then adds a bunch that is too many receives a level 2. But what the mathematically unsophisticated authors don't realize is that the level 2 student may be finding the rudiments of the binary search algorithm which is far more efficient for large numbers than the sequential search algorithm that the level 3 student uses. A good example of this is the telephone book search. People find names by jumping around the book using knowledge of alphabetic order.
I am writing this letter to complete the talk that was cut short at the last meeting. From comments made later at the meeting, it seemed that some board members may not have understood the intent of my talk.
I tried to present the highlights from three activities in Investigations. I did not pick these examples to show minor mistakes with Investigations. Rather I picked them to illustrate how its philosophy reduces the effectiveness of many of its activities, and makes it fall short of its potential. The authors start with good ideas, but they often don't execute them very well. If the authors could moderate their philosophy and improve their mathematics, Investigations could become an excellent supplement.
The following is a summary and analysis of the 3 activities.
(Grade 5, Statistics, pages 91-95)
This activity is the last of 3 in an investigation to determine the fraction of a newspaper that is ads. In the first activity students practice finding the fraction of a newspaper page that is made up of ads. In the second activity, students define a sampling strategy and collect proportions of ads from a sample of 10 to 15 pages. In the third activity they combine the data to get a single fraction for the entire newspaper.
In activity two, 10 groups of students record the fraction of each page that is ads. For each of 10 to 15 pages in a newspaper, they use a 3" strip of paper with various common fractions marked on them and color the fraction of each page that is ads. (The blackline master for the strips is at the end of the book.) In activity three, the students add the fractions on each of the 10 to 15 3" strips by cutting off the colored part. They then tape the colored parts together and the blank parts together. Finally they tape these two parts together to form one long strip. The students look at the colored part of this long strip to estimate the fraction of ads for the entire newspaper.
This investigation is a great idea, but there are several difficulties with it.
(Grade 5, 3-D Geometry, Volume, pages 45-48)
All 3 activities in this investigation involve packing rectangular packages of different sizes into rectangular boxes. In the second activity, students design a single box that can be completely filled with each of four different-shaped rectangular packages: 2 2 1, 2 2 2, 2 2 3, 1 1 3. They pack the box with only one type of package at a time, and it must fill the box to the top with no gaps. A problem extension adds a package that is 1 1 5.
This problem is quite interesting and has great potential, but the authors miss several important points when they discuss the problem in the Teacher Note on page 48:
(Grade 5, Introduction, pages 87-91)
This investigation involves working with factors of 100 and 1000. In the second activity, students work with factor pairs of 1000, such as 25 40 and 20 50. In the third activity, they are asked to find factor pairs of 1100. The book tells the teacher to expect kids to start with 22 50 which kids may reach in a variety of ways. They might skip count from 20 50, known from the previous activity. Or they might get it by first noticing that 1100 is 11 100 and that the factor 2 in 100 can be combined with the 11 (yes, the reasoning for this case is murky).
Once the child has 22 50, the child can divide one factor by 2 and multiply the other by 2 to produce 44 25 and 11 100. The kid derives other factor pairs using similar techniques. Another example suggests a child might notice that 1100 is even and start with 550 2. Then the child works from there.
The problem of finding factor pairs can be an interesting problem. But the children would have gotten a much better understanding of factor pairs if they had first broken 1100 into its prime factors 22 52 11. Then they could have methodically taken various combinations of these five numbers. The method suggested by the authors is very haphazard and does not help with understanding the fundamentals of factors. If the number had been 1125, these techniques would have failed.
Both Mathematics Unlimited (the current text) and Everyday Math teach about breaking numbers into their prime factors.
When I started working with the committee last June, I had an uneasy feeling that something was wrong with the way math was being taught, but I couldn't articulate it. I had heard the arguments where words like "traditional", "balanced", and "fuzzy" were used. But I didn't know what they really meant. I decided when I joined the committee for the math textbook selection that I would try to figure out what was really going on. I wanted to do an independent study of the concepts, free of the bias of any math advocacy groups.
I spent a great deal of time reading the various math books and talking with teachers in numerous districts around the country to get a sense of what the issues are. I have already reported some of the important things I learned in the handouts that most of you have already seen. In this letter, I want to make the following additional points. These ideas are my best current understanding of the math issues, and they continue to evolve as I improve my understanding.
Pedagogy is one of the most important issues. Many of the new textbooks are designed around a particular pedagogy that is used to teach every concept regardless of whether it works. I am a pragmatist; I have found over the years that whenever people try to design a complete working system with a single pure theoretical basis, it fails. (When your only tool is a hammer, everything looks like a nail.) Systems succeed when they contain a pragmatic blend of theoretical bases. Teaching, like many other activities, works best when teachers are free to blend many different teaching methods.
Dale Seymour's Investigations (written by TERC) and Creative Publication's MathLand are two examples of textbooks that are based on a single pure pedagogy and that lack the necessary blending of pedagogies. There are several important beliefs that make up this pedagogy.
This pedagogy is certainly a component of Everyday Learning's Everyday Math (University of Chicago). But the authors of Everyday Math are much more pragmatic and have blended this new pedagogy with the older instructional style. The result is a very strong curriculum that takes the best of both.
Investigations is popular with many of the teachers because it implements the latest research. Teachers, like engineers, scientists and many others, like to be at the forefront of research and work with the latest technology.
But customers frequently have a very different point of view. Most customers prefer a technology that is modern, but proven. Today's latest research may be tomorrow's established knowledge or tomorrow's forgotten dead-end. Ideas that have survived a few years are more likely to still be valid in the future.
Parents, likewise, prefer a curriculum that is modern, but proven. Parents' and teachers' primary desire is for children to get the best possible education in order to achieve their potential.
Investigations is at the forefront of research and very few classrooms are using it yet. In five years it might be a well established success or it might be a forgotten failure. Everyday Math is about 5 years older and has an established track record. About 500,000 children in 20,000 classrooms are currently using Everyday Math.
For children to succeed in middle school or high school math, they must understand mathematical abstractions. Those who never get past concrete concepts will hit a glass ceiling that hinders their progress in math.
Both Everyday Math and Investigations give children a good understanding of concrete math concepts, but only Everyday Math leads children consistently to understand the abstraction, which allows them to solve any problem. Investigations' single pedagogy seems not to be capable of consistently leading to an abstraction. For example, with fractions, children develop a good understanding of the relationships of common fractions that consist of eighths, sixths, fourths, thirds and halves. Children gain this understanding primarily by using manipulatives and by folding paper into sequential segments. In fifth grade, Everyday Math teaches the rule for achieving a common denominator; Investigations does not. Even in fifth grade, students of Investigations are still comparing 11/25 with ½ by folding paper.
Abstractions are frequently easy to learn when directly taught, but very hard to learn by discovery. Because Everyday Math teaches abstractions directly, children are likely to learn them. Because Investigations teaches abstractions by discovery, children are less likely to learn them.
There are two important components to math education: knowledge and process. Knowledge includes the information that a child can instantaneously retrieve, such as sums and products of single digit numbers, what an even number is, and what a triangle is. Process includes mathematical reasoning, problem solving and explanation of mathematical understanding. To be successful in math, children must master both components. Without the process component, knowledge is of little use; without the knowledge component, the process component cannot yield valid results (the garbage-in, garbage-out syndrome). Everyday Math balances the teaching of process and knowledge. Investigations seems to emphasize the teaching of process and de-emphasize the learning of knowledge. Because learning by discovery, by its nature, takes longer than learning by direct instruction, Investigations cannot cover the amount of mathematical material that Everyday Math can As a result Investigations appears to need considerably more supplementation to correct these deficiencies.
The two quotes below show the difference in philosophies between Everyday Math and MathLand.
In the Everyday Math teacher's manual, Third Grade, Volume A, Page 33, there is the following statement:
"Everyday Mathematics believes that automaticity-knowing the basic facts as reflexes, without having to figure them out-is an essential prerequisite for mental arithmetic, estimation, and paper-and-pencil computation."
This quote is in considerable contrast to one in Creative Publications: 2nd grade, page 25 of the teacher's guide. This unit, which teaches addition and subtraction, starts with a quote attributed to Constance Kamii Young Children Reinvent Arithmetic.
"There is no such thing as a "number fact". There are only relationships and these relationships are created inside the child's head (mind)."
These two quotes illustrate the essence of the philosophical difference of the two series and how important they believe it is for kids to learn their number facts.