Running head: EVERYDAY MATHEMATICS CURRICULUM – DOES IT ADD UP?
Everyday Mathematics Curriculum – Does It Add Up?
Sarah DeLeeuw
George Mason University
EVERYDAY MATHEMATICS CURRICULUM – DOES IT ADD UP?
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Everyday Mathematics Curriculum – Does it Add Up?
A curriculum is one of several important components in a successful and cohesive
mathematics program, along with standards, assessment, teaching, and professional
development. Although it is not the sole contributor to student learning outcomes, choosing a
curriculum is a critical decision, and transitioning to a new curriculum is equally significant.
Most would agree that teachers rely on textbooks and other supporting materials from their
respective adopted curricula to drive decisions of scope and sequence in the classroom.
Specifically, if particular content is not included in the textbook, teachers will not consider
teaching it, and consequently, students will not have a chance to learn it. On the other hand, this
also infers that the inclusion of material deemed less valuable will result in losing instruction
time that could have been used teaching and learning topics deemed more valuable.
Besides influencing the content that is taught and learned in a classroom, curriculum also
drives the decision of how that content is taught. Sure, all teachers have taken methods courses,
but I’m willing to bet that even those methods courses had a textbook that guided them how to
learn how to teach. The way content is presented in a curriculum, the pedagogy, will greatly
influence how the content is taught and learned in a classroom. Further, if a new curriculum is
introduced, even if the content is very similar to the previous, teachers may have an extremely
difficult time transitioning to the new methods of teaching and learning within. In this case, a
great deal of professional development may be necessary before student learning outcomes can
be expected to benefit.
It follows then that we need a way to evaluate a curriculum, in regards to both the content
and the pedagogy. As researchers, we are trained to first investigate whether or not there is an
existing evaluation instrument that is respected in the field, with high validity and reliability,
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which could be used to measure it. Even without a thorough investigation, it became clear that
evaluation criteria varied immensely; and so it is understandable that the results also varied
immensely. The discrepancies between ratings on the same curricula do not necessarily signify
that one evaluation tool is more effective than another; rather they reflect the values (and biases,
if you will) of those reviewing the curricula.
Again, curriculum can be broken down into three components: content, pedagogy, and
implementation. Often, content is measured by comparing topics in the curriculum with a set of
standards, pedagogy by comparing how it is presented to a philosophy of learning, and
implementation by comparing student outcomes on standardized tests over time.
Although choosing a curriculum is a high-stakes matter, high-stakes tests should not be
the only measure of effectiveness. Often high-stakes tests rely on procedural rather than
conceptual understanding. Especially as we strive for ideals outlined in the Common Core State
Standards (CCSS), we must give our undivided attention toward greater focus and coherence. As
outlined at www.corestandards.org, the CCSS:
“stress not only procedural skill but also conceptual understanding, to make sure
students are learning and absorbing the critical information they need to succeed at
higher levels - rather than the current practices by which many students learn
enough to get by on the next test, but forget it shortly thereafter, only to review
again the following year.”
Although procedural knowledge is still important, a greater focus on conceptual understanding
calls for greater attention to teaching and learning problem solving, reasoning and sense-making,
connections, and multiple representations of mathematical modeling.
Purpose
Within the scope of this essay, I will (1) address conclusions from past research on EM,
(2) evaluate one particular component of curriculum materials from EM – namely, the Online
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EM Games – according to the predefined analysis procedure, and (3) both identify weaknesses
and suggest improvements.
Today’s Learners in this Age of New Digital Media
Students are living and learning in an age of new media – where they give constant
attention to the latest scoop on TV, the hottest music for their iPods, newest games for their
game systems, instantaneous updates in their online communities and social networks, and they
have mobile apps that manage all of these interests simultaneously. Students are constantly (an
average of 7.5 hours a day!) interacting with media – more than ANY other activity besides
(maybe) sleeping – according to a popular report, compiled by the Kaiser Family Foundation
(Rideout, 2010).
Consequently, this age of new media implies an implication to teaching and learning.
Traditional methods of teaching may not be engaging today’s learners who are used to these
dynamic and interactive platforms. Since these new media forms have altered how youth
socialize and learn, how are we altering how we teach? Must we develop and implement an
entirely new curriculum to reach our learners?
To react to this age of new media, the commercial industry has capitalized by providing a
tremendous variety of technological approaches to teaching and learning. One attempt to pique
the interest of all types and ages of learners is through educational games.
Everyday Mathematics Curriculum
Everyday Mathematics (EM) is one of many standards-based mathematics curricula
funded by the National Science Foundation (NSF) that stresses the use of games for learning. On
its website, http://everydaymath.uchicago.edu, EM is introduced as: “a comprehensive Pre-K
through 6th grade mathematics curriculum developed by the University of Chicago School
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Mathematics Project” that “is currently being used in over 185,000 classrooms by almost
3,000,000 students.” EM claims its distinguishing features are real-life problem solving,
balanced instruction, multiple methods for basic skills practice, emphasis on communication,
enhanced home/school partnerships, and appropriate use of technology.
The EM Teacher’s Reference Manual promotes games and defends their place in the
curriculum. EM asserts that many parents and educators make a sharp distinction between work
and play, and tend to ‘allow’ play only during prescribed times. EM also asserts that children
naturally carry out their playfulness into all of their activities. EM emphasizes that “games are an
integral part of the EM program, rather than an optional extra as they are traditionally used in
many classrooms.” EM purports that all children should have sufficient time to play games,
especially those that work at a slower pace or encounter more difficulty than classmates.
EM stresses that a major benefit of games is that, even when used over and over again, it
is unlikely the exercises will not repeat because often the numbers are generated randomly.
Games are also a different approach then the “monotonous, rote pencil pushing” that “has helped
produce generations of people who see mathematics as little else.”
Personal Connection to EM
I work closely with Terraset Elementary School in Reston, VA. I am the liaison for
NCTM’s collaboration with the school - both for a Professional Development series for teachers
and a Lunch Bunch tutoring partnership for the students, where NCTM employees meet with
designated students weekly to play math games over lunch. When this project (curriculum
research paper) was assigned, I spoke to one of Terraset’s math lead teachers that I know well. I
had intentions of investigating the curriculum that they used, if it were NSF funded. She pointed
out EM, and said that when I was ready to start writing, she would get a couple of the EM
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materials together that I could take home to have a look at. She told me that they do not use EM
as a comprehensive curriculum, but they do use select components. In the meantime, I collected
the research I could find on EM and considered several different evaluation frameworks. When I
was organized, I asked her to lend me some of the EM materials. She handed me the Everyday
Calendar Problems and the Everyday Partner Games. When I returned back to my office, I
realized that these materials were not from the EM curriculum, a Wright Group product. They
were actually from Great Source, which is a sector of Houghton Mifflin Harcourt publishers. I
returned the materials, but was still determined to investigate EM for my project.
In addition to my relationship with Terraset, I am personally interested in using games as
part of instruction. I have learned from math games myself, led several presentations at
professional conferences about facilitating learning with math games, and am working on
designing new math games and applets for my position at NCTM.
In addition, last semester, I was enrolled in two classes that informed my assumptions
about math games. First, I explored teacher’s perceptions of the use of math games as part of
instruction for my qualitative project. After analyzing the transcripts from these interviews, I was
able to add to my own understanding of teachers’ perceptions of using math games in their
classrooms. I concluded six major findings: (1) Games engage today’s learners. (2) Games teach
life skills. (3) Games inherently differentiate. (4) Stakeholders support the use of games. (5)
Classroom management is the role of the teacher. (6) ‘Good’ math games have a hook, simple
rules, non-threatening competition, instructional value, and an appropriate length of play. I
expected my findings, except I was surprised to uncover that all three teachers that I interviewed
raved about using math games to teach life skills.
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Second, I took an instructional technology class titled “Design Issues in Educational
Gaming and Media.” This class inspired me to situate my interest in math games in the context
of the digital divide, and explore games as a means to engage the current interaction-craving
generation. I created an affinity bundle of 55 free online math games for grades 3-7, in which I
evaluated according to a rubric that I developed. The rubric had four major sections: content –
evidence of the CCSS, students – characteristics identified and tested by students at Terraset,
teachers – characteristics identified in my qualitative study and tested by teachers at Terraset,
and design – informed by issues in educational gaming literature.
My own interest in games and the fact that EM purports that using games for learning is
an integral part of the program led me to choose this particular component of the curriculum for
this assignment.
Evaluation Framework
After reviewing the literature on frameworks to evaluate curricula, I was initially
determined to evaluate EM through the lens of a framework for evaluating curricular
effectiveness, as defined by the National Research Council (NRC) in the 2004 publication titled
On Evaluating Curricular Effectiveness: Judging the Quality of K-12 Mathematics Evaluations.
Chapter 3 of this book outlines primary and secondary components as well as evaluation design,
measurement, and evidence within the framework, all contributing to an overall rating of a
curriculum (Confrey). Although the framework was extremely thorough, I didn’t find it easy to
use. I struggled applying it to the curricular materials that I selected, and I attribute this to the
fact that I chose just two small components of EM, a mere slice of the curriculum as a whole, to
analyze and evaluate.
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I resisted creating my own evaluation tool, and I am glad that I did. It allowed me to dig
deeper until I found an existing analysis procedure that I was satisfied with. I was looking for a
framework that I could use flexibly in conjunction with a set of standards to evaluate the content,
pedagogy, and implementation, as outlined above. It seemed that often analyses failed because
they relied on charts full of check marks (or the absence thereof) to determine if topics in the
given curriculum aligned with a set of standards. These surface-level analyses lacked an in-depth
evaluation of what and how the mathematics topics were explored. It was quickly clear that these
charts were merely selling-points to decision-makers.
The mathematics-curriculum-analysis procedure from the American Association for the
Advancement of Science (AAAS) seemed to offer an in-depth analysis that could be applied to
curriculum (or components of a curriculum) and also was flexible in that any set of standards
could be used as evaluation criteria. I especially liked this component of the AAAS framework
because it allowed me to view a subset of curricular materials from EM through the lens of the
CCSS.
The AAAS framework consisted of three phases: preliminary analysis, content analysis,
and instructional analysis. It seemed instructional analysis covered both pedagogy and
implementation. The goal of the preliminary analysis was to identify the standards to be used as
a focus for the content and instructional analysis. I had already chosen the CCSS, and hoped to
use both the grade-level content standards and the standards for mathematical practice. The
content analysis was just that – to identify examples within the curriculum that address each of
the chosen standards. For the content analysis, the AAAS framework offered specific criteria that
had to be met in order to be considered ‘content-matched to the standard’. Last, the instructional
analysis used those examples that met the criteria to develop strengths and deficiencies of the
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material. This final phase was broken into seven specific clusters, each with questions to guide
the analysis.
Each step in the analysis depends on the previous. Both the content analysis and
instructional analysis objectively and clearly outline the criteria for both selecting and evaluating
activities that are content-matched to the standards. The clusters in the instructional analysis
demand analysis of criteria from a variety of venues, yet each is followed by several prompts to
ensure that the analysis is in-depth rather than on-the-surface. Clusters included in the analysis
are: identifying and maintaining a sense of purpose, taking account of student ideas, engaging
students, developing and using mathematical ideas, promoting student reflection, assessment,
and other features. Overall, the AAAS framework seemed to provide an evaluation that
effectively assessed a curriculum on all three fronts: extent, depth, and quality.
Past Research on EM
The research on the effectiveness of EM is diverse. For example, in the Department of
Education review (1999), EM was commended for its depth of understanding. A reviewer wrote,
“Mathematics concepts are visited several times before they are formally taught… This
procedure gives students a better understanding of concepts being learned and takes into
consideration that students possess different learning styles and abilities” (Confrey, p. 83). In
contrast, Braams (2003) reviewed the same materials and wrote that “The EM philosophical
statement quoted earlier describes the rapid spiraling as a way to avoid student anxiety, in effect
because it does not matter if students don’t understand things the first time around. It strikes me
as a very strange philosophy, and seeing it in practice does not make it any more attractive or
convincing. (Confrey, p. 83)” As I discussed earlier, discrepancies between ratings on the same
curricula seem to be a reflection of the values of the reviewers.
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. Most studies focused on a particular aspect of the curriculum. I did not find any studies
that analyzed the online games component separately as I sought out to do. I did, however, find
several studies on EM. For example, Yackel and Cobb wrote about a study by Fuson that
illustrated the evaluator’s view of the importance of classroom discourse that draws on students’
ideas. In that study, Fuson analyzed first grade EM materials to discern the social and
sociomathematical norm assumed by the curriculum designers. Findings were that EM: (1)
extended students’ thinking, (2) used errors as opportunities for learning, and (3) fostered student
to student discussion of mathematical thinking (Confrey). A follow-up study showed that the
norms were rarely implemented by teachers in classrooms, even though they used the EM
materials.
Carroll and Isaacs conducted a synthesis study that summarized six different quantitative
studies that all measured student outcomes by comparing achievements of students using the EM
curriculum with those who used other curricula. They also concluded three main findings: (1) On
traditional topics, such as fact knowledge and paper-and-pencil computation, EM students
perform as well as students in other programs. However, EM students used a greater variety of
computation solution methods. They are especially strong on mental computation. (2) On topics
that the authors claim have been underrepresented in elementary curriculum - such as geometry,
measurement, data, etc. – EM students score substantially higher than students in traditional
programs. They also perform better on questions that assess problem solving, reasoning, and
communication. (3) Although some districts report a decline in computation, especially during
the first year or two of implementation, this is usually offset by gains in other areas. (Confrey)
Fuson, Carroll, and Drueck conducted a study that compared achievement results,
specifically of second and third graders, from a heterogeneous sample from the US using EM, a
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sample of upper-middle-class US students using traditional curriculum, and a sample of Japanese
children. Results showed that the EM second graders scored equal or higher than the uppermiddle class traditional students in number sense and scored the same as the middle-class
Japanese students. The EM students outperformed the upper-class traditional students on
computation, but were outperformed by the Japanese students. EM third grade students also
outperformed the traditional students in place value, numeration, reasoning, geometry, data and
number-story items.
Sood compared number sense instruction in EM to three different traditional mathematics
textbooks. Results showed that the traditional textbooks included more opportunities number
relationship tasks, but EM emphasized more real-world connections. EM also did better at
promoting relational understanding and integrating spatial relationship tasks with other skills.
EM excelled in scaffolding instruction by devoting more lessons to particular activities. EM
offered a variety of models to develop number sense concepts, a sequence of representations
(concrete to semiconcrete to symbolic), and hands-on activities using real-world objects to
enhance engagement. Traditional textbooks provided more opportunities for practice within the
lessons, but EM provided more opportunities for ongoing review.
Carroll studied geometric knowledge of middle school students in EM compared to
students in traditional curricula. He also concluded that EM students outperformed the
comparison students. Further, the fifth-graders in EM showed a mean gain of nearly 2.5 times
that of the comparison students between the pre and post tests.
Online Games in EM
Online EM Games have the slogan “Everyone Wins … When Everyone Plays!” On their
website, EM advertises that games are an integral part of the EM program. They offer computer
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games that aim to make basic skills practice fun. The games are available both online and on
CD-Rom. EM games are organized by grade-level. The directions are available to be read on the
screen, but there are also in-game audio instructions. EM markets that the games “add a little
friendly competition to help motivate learning.”
On the website https://www.everydaymathonline.com/, several of the games from EM are
available as demos. I selected a demo game from the lowest level, early childhood, and a game
from the highest level, Grade 6. I decided to select a third game the middle level, Grade 3. The
games I selected were: Monster Squeeze, Angle Race, and Factor Captor.
Monster Squeeze is played on a number line with a monster at each end, one at zero and
one at ten. There is a mystery number that the player is prompted to guess. If the guess is too
high, the monster on the right moves toward the left to cover up the student’s guess and all of the
other numbers that are also too high. If the guess is too small, the monster on the left will move
toward the right to cover up all of the numbers that are equal to or smaller than the guessed
number. The player continues to guess until he or she is able to correctly guess identify the
mystery number. The computer keeps track of how many guesses it takes each time. There are
two levels of the game: one that includes the numbers zero to ten, and another that goes from ten
to twenty. There are also two player versions.
Angle Race is a race to 360 degrees and played on a circular gameboard. The player and
the computer take turns drawing cards from a deck. Each card has an angle measure on it, and
the player is expected to move the ray to the resulting angle on the board after adding the
measure on the chosen card. The board is designed in increments of 15 degrees. The player to get
to exactly 360 degrees first wins the round. This may mean that near the end of the round, the
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players find themselves passing on their turns often. The player who wins the best of five rounds
wins the game.
Factor Captor is played on a rectangular number chart, made up of counting numbers
ranging from 1 to 60. Some of the lower digits are repeated several times. The game begins with
the computer selecting a number on the number chart, and the student is prompted to click on all
of the factors in that number. The student receives points equal to the sum of the correct
identified factors. The game continues as the computer and player switch roles until all of the
numbers on the number chart have been selected. The player with the most points wins.
According to everydaymathonline.com, EM Games are easy to manage, fun to play, and
designed for success. ‘Easy to manage’ may be appealing to teachers. EM flaunts that no
additional planning is needed to incorporate the games into instruction, that there are menus for
different levels (ie. they inherently differentiate instruction), and that there is a built-in
management system to monitor student progress (ie. one type of evaluation tool). They also note
that there are both single player and two player versions for students, such that they can practice
on their own or compete against a friend. With respect to being ‘designed for success,’ EM
claims that the games both promote practice of basic skills as well as build critical thinking,
develop mathematical strands, develop computation and fact fluency, and offer student feedback
on all mathematical concepts.
Evaluation Tools
Although the publisher claimed all of the above, I wanted to investigate further how
credible the publishers’ promises really were. As described above, I decided to use the AAAS
mathematics-curriculum procedure using the CCSS to take a closer look at just the online games
component of EM.
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Below you will see the chart that I filled in for each of the three games. I used a different
color for each, but where all three had the same response, it remains black. In the instructional
analysis, I arranged the positive feedback in the left column and negative feedback on the right.
___________________________________________________________________
Preliminary Analysis – CCSS
Name of Game: Monster Squeeze for Early Childhood
Standard: Kindergarten / Counting and Cardinality
Name of Game: Angle Race / Grade 3
Standard: Grade 4 / Measurement and Data
Name of Game: Factor Captor / Grades 5 and 6
Standard: Grade 4 / Operations and Algebraic Thinking
Content Analysis
SUBSTANCE: Does the activity address the
specific substance of the standard?
Yes, see the specific standards listed below.
Note: If only a “topic” match to the standard
is present, the activity is not included in the
list.
SOPHISTICATION: Does the activity reflect
the level of sophistication of the standard?
Note: If research and best practice indicate
that the activity is below or above the
intended grade level, the activity is not
included in the list.
PART OR WHOLE: Which part or parts of
the standard are addressed?
Yes, exactly matches up.
Yes, although intended for Grade 3 in EM,
satisfies Grade 4 of CCSS.
Yes, although intended for Grades 5 and 6 in
EM, satisfies Grade 4 CCSS.
Part: Compare two numbers between 1 and
10 as written numerals.

Part (s): (1) Recognize angles as geometric
shapes that are formed wherever two rays
share a common endpoint, and understand
concepts of angle measurement:
o
An angle is measured with reference to a
circle with its center at the common
endpoint of the rays, by considering the
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fraction of the circular arc between the
points where the two rays intersect the
circle. An angle that turns through 1/360 of
a circle is called a “one-degree angle,” and
can be used to measure angles.
o
An angle that turns through n one-degree
angles is said to have an angle measure of n
degrees.
o
(2) Recognize angle measure as additive.
When an angle is decomposed into nonoverlapping parts, the angle measure of the
whole is the sum of the angle measures of
the parts.
Part: Gain familiarity with factors and
multiples.
Instructional Analysis
Cluster 1 : Identifying and maintaining a sense of purpose
PURPOSE: Does the material convey an
overall sense of purpose and direction that
is understandable and motivating to
students?
Yes, students
understand the
purpose and are
motivated to at least
try the game.
ACTIVITY PURPOSE: Does the material
convey the purpose of each activity and its
relationship to others?
Yes, all convey the
purpose of the
activity:
Object: To identify a
“mystery number.”
Object: To complete
an angle at exactly
The directions are
unclear. The
directions in the
tutorial say nothing
about being able to
‘steal’ the points
from the opponent
when factors are
missed. They also do
not include that the
value of the target
number contribute
to points for the
player than selects
the target number.
No, does not convey
the relationships to
other activities.
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the 360 degree
mark.
ACTIVITY SEQUENCE: Does the material
involve students in a logical or strategic
sequence of activities, versus a collection of
activities, that build toward an
understanding of a standard?
Object: to get a
higher total than the
computer.
There are two levels:
one on a number line
from zero to ten, and
another from ten to
twenty. (In my
opinion, it would
make more sense if
the higher level went
from zero to
twenty.)
No, there are not
‘levels’ of complexity
to cater to diverse
needs in a
classroom.
Cluster 2: Taking account of student ideas
PREREQUISITE KNOWLEDGE AND
SKILLS: Does the material specify
prerequisite knowledge and skills that are
necessary for learning the standard?
ALERTING THE TEACHER TO COMMONLY
HELD IDEAS: Does the material alert
teachers to commonly held student ideas,
both troublesome and helpful?
ADDRESSING COMMONLY HELD IDEAS:
No. A correlation
chart shows what
lesson the game
supplements. It
could be assumed
that the previous
lessons were
covered, although it
would be extremely
helpful if the specific
skills necessary for
the game were listed.
No. There are NO
supplemental
support materials
for the games. In the
student reference
book, the directions
are listed, just as
they are on the
screen and the audio.
These pages are
repeated in the
teachers’ manuals,
but there is nothing
additional for
teachers.
No. See above.
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Does the material include suggestions for
teachers to find out what their students
think before mathematical ideas are
introduced? Does the material explicitly
address commonly held student ideas?
Cluster 3: Engaging Students
VARIETY OF CONTEXTS: Does the material
provide experiences with ideas in multiple,
different contexts to support the formation
of generalizations?
QUALITY OF EXPERIENCES: Does the
material include activities that promote
first-hand experiences or applications when
practical or present students with other
meaningful experiences?
No, students are
limited to the
context in the game.
Yes, the experiences
are first-hand.
Although the games
do not demonstrate
practical uses of
math, it could be
argued that playing
the game itself is a
practical use. To me,
other meaningful
experiences would
entail transferring
the applications to
another context, or
at least developing a
strategy to the game.
It is not evident from
the materials that
strategy-building is a
goal.
Cluster 4: Developing and using mathematical ideas
BUILDING A CASE: Does the material
develop justifications or arguments for the
importance or significance of mathematical
generalizations or procedures?
INTRODUCING TERMS AND
PROCEDURES: Does the material introduce
terms or algorithms only in conjunction
with experience with the idea or process
and only as needed to facilitate thinking and
promote effective communication?
No, although the
games are classified
as Skill Builder or
Challenge games,
there is no evidence
of ‘building a case.’
Yes, unnecessary
The game uses
terms are not
“smaller than” and
evident in the games. “bigger than” instead
of “less than” and
“greater than.”
Although young
children may be
used to small/big
and not yet
less/greater, it
EVERYDAY MATHEMATICS CURRICULUM – DOES IT ADD UP?
RESPRESENTING IDEAS: Does the material
include appropriate and accurate
representations of mathematical ideas?
Yes. Although the
ideas seem to be
mostly accurate,
they are lacking.
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seems it would be
manageable.
No, it would be more
appropriate to make
the mathematics
explicit to the
students. For
example:
Show the
inequalities on the
screen using the
symbols correctly.
Show the addition of
the angles and the
resulting sum.
PRACTICE: Does the material present tasks
and questions for students to practice skills
or use knowledge in various situations?
Yes, it seems that the
main focus is to
practice a particular
math skill.
No, there are not
accompanying
activity sheets or
even suggestions
questions for
students in the
teachers’ manuals.
Cluster 5: Promoting student reflection
PROVIDING OPPORTUNITIES FOR
STUDENTS TO EXPRESS IDEAS: Does the
material routinely include suggestions for
students to express, clarify, justify, and
represent their ideas? Are suggestions made
for when and how students will get
feedback from peers and the teacher?
GUIDING STUDENT INTERPRETATION
AND REASONING: Does the material
include tasks and question sequences to
guide student interpretation and reasoning
about activities and readings?
PROMOTING REFLECTION AND SELFMONITORING: Does the material help, or
include suggestions on how to help,
students know when to use knowledge and
Yes, students that
use the online games
are able to monitor
their progress by
No, the only
feedback provided is
“Great job,” “Try
again,” or an
explanation of when
a wrong move is
made.
No. Again, even in
the teachers’
manuals, there are
nothing more than
descriptions and
directions for the
games.
No, there are not
suggestions of how
students can extend
their ideas or reflect
EVERYDAY MATHEMATICS CURRICULUM – DOES IT ADD UP?
skills in new situations? Does the material
suggest ways to have students check their
own progress and consider how their ideas
have changed and why?
accessing their past
scores.
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on how their ideas
have changed. The
hints in the games
are not really ‘hints.’
Rather, they repeat
the directions for the
particular task.
Cluster 6: Assessment
ALIGNMENT TO GOALS: Are assessment
items and tasks aligned with the identified
standard or benchmark?
APPLICATION: Does the material include
assessment tasks that require applying
ideas and avoid allowing students a trivial
way out, such as using a formula or
repeating a memorized term without
understanding?
Yes, there is
feedback on the
particular skill.
Students that
develop a strategy of
guessing somewhere
near the middle of
the remaining
numbers would
theoretically do
better than those
randomly guessing.
No, there are not
assessments to
evaluate conceptual
understanding.
Students that
develop a strategy
and apply their
knowledge of factors
to determine optimal
selections will
theoretically win
more often.
EMBEDDED: Are some assessments
embedded in the material along the way,
with advice to teachers about how they
might use the results to choose or modify
activities?
No. Students receive
feedback
(correct/incorrect)
on each turn during
a game. If incorrect,
the computer
explains what the
correct move would
have been.
However, there is
not advice for
teachers on how to
modify the activities.
The computer keeps
track of the number
EVERYDAY MATHEMATICS CURRICULUM – DOES IT ADD UP?
19
of steps it took
before the student
was able to guess the
‘mystery number,’
but this may not
show conceptual
understanding.
No, there is not even
a suggestion for a
more summative
assessment of the
concept.
MULTIPLE FORMATS: Do assessments
include multiple formats and response
modes that afford students diverse
opportunities to exhibit their skills and
understanding?
Cluster 7: Other features
TEACHER CONTENT KNOWLEDGE: Would
the material help teachers improve their
understanding of mathematics and its
applications?
CLASSROOM ENVIRONMENT: Does the
material help teachers create a classroom
environment that welcomes student
curiosity, rewards creativity, encourages
inquiry, and avoids dogmatism?
WELCOMING ALL STUDENTS: Does the
material help teachers create a classroom
that encourages high expectations for all
students and enables all students to
experience success?
Yes, it may be helpful
for teachers to
become comfortable
with the
representations in
the games, although
they are not diverse.
Yes, it seems that
games in general
welcome student
curiosity, and these
also avoid
dogmatism.
Yes, all students can
take part in the
games.
No, there are not
applications of the
mathematics beyond
the mathematics of
the game itself.
No, there are not
rewards in these
particular games
that reward
creativity or
encourage inquiry.
No, the games would
be better in diverse
classrooms if there
were more levels.
There is no way to
possibly ‘lose’ the
game. The best score
is 1 (luckily guess it
correctly on the first
try) and the worst is
10 (literally guess
every number until
correct).
The game is
frustrating at the
end. It is possible for
the weaker student
EVERYDAY MATHEMATICS CURRICULUM – DOES IT ADD UP?
20
to get lucky and win
the game because
there score is
determined by who
gets to the end (360
degrees) first. The
game often ends
with luck – whoever
gets the card to
complete the circle.
Areas of Weakness and Suggested Improvements
After filling out the above grid for each of the three games, I looked for areas of
weakness that were common to all of the games. They are arranged by cluster here:
In identifying and maintaining a sense of purpose, more levels would have been
beneficial in order to help inherently scaffold students in a logical sequence as they develop
deeper understanding of the concepts.
In taking account of student ideas, there was no guidance for teachers. The games seemed
to be ‘add-ons’ to lessons on particular topics and not be integrated into the curriculum itself. A
teacher’s guide for the games would have been particularly helpful. The items from cluster two
of the evaluation framework are just one section of items that would be helpful to include in such
a guide - prerequisite knowledge, commonly held student ideas, and how to address
misconceptions.
In engaging students, the games could have offered multiple representations of concepts.
Although they did offer students first-hand experiences, the experiences weren’t necessarily
meaningful to the students. There was often only one correct answer on a turn, and the games
seemed to focus on procedural knowledge. A better balance of chance versus choice would have
encouraged students to practice critical thinking and problem solving skills in addition to skill
EVERYDAY MATHEMATICS CURRICULUM – DOES IT ADD UP?
21
and fluency. Math games that provide a balance of chance and choice cater both to weaker
students who need to practice procedural skills and also challenge those at higher levels of
conceptual understanding to seek out an optimal strategy. In this way, math games could be
designed to differentiate instruction, which also address maintaining purpose (cluster one).
In developing and using mathematical ideas, building a case for the games was lacking.
Although the games were correlated with particular lessons within a grade level, there was not an
explanation about what the games added to the curriculum. Also missing was the opportunity to
connect the math explicitly to the math implicit in the game. For example, the computer should
have shown the inequalities symbolically on the screen in Monster Squeeze, the sums of the
angles in Angle Race, and the difference of sum of the factors and the target number in Factor
Captor.
In promoting student reflection, the games do not offer opportunities for students to
interact with peers or opponents, or even on their own strategies. It may have been productive for
the computer to display questions for students about the ‘big ideas’ in the game. That way, they
could monitor their own development of strategies they more they play the games. It may also be
productive to design games such that students work together toward a common goal, instead of
competing against each other. In this case, they may be encouraged to work together to advance
their strategy. They might communicate more and negotiate meaning that leads to deeper
understanding of concepts.
Assessing learning, in my opinion, is as equally important as integrating the games into
instruction and providing the supports for teachers to do so effectively. I considered assessment
as a current critical issue for Dr. Suh’s last assignment. If we are to accept that students in this
new age of digital media are learning differently, we also accept that we are responsible for
EVERYDAY MATHEMATICS CURRICULUM – DOES IT ADD UP?
22
teaching differently as well. It follows then, that we will also have to accept that students’
understanding may have to be assessed differently.
Research has shown that students perform best on exams that that test skills in the same
manner as they were learned (Fennell, 2006). If students practice by doing worksheets, they will
perform better on a test that models those worksheets. Likewise, if students practice by engaging
in an online math game, it follows that they will perform best on a test that models their learning
in that mode of learning.
This is a call to action. Teachers much look to formative assessments, not in place of, but
in addition to, summative assessments. Furthermore, we must use alternate forms of both
assessments. Alternate forms of assessment include promoting student discussion, taking time to
observe, including presentations, involving students in developing rubrics, interviewing students,
making writing a routine, and using the web to keep students talking outside of class. Although
this list is not comprehensive, it offers a variety of ways to assess understanding.
Assessing learning ties back into the preceding clusters of the evaluation framework.
Earlier we considered student reflection as a component of curriculum that is necessary for
productive learning, and suggested games that promote communication. We also stated that we
sought conceptual understanding and flexible use of multiple representations over procedural
knowledge and mastery of skills. Principles and Standards notes that communication deepens
understanding. Students need opportunities to discuss their reasoning and negotiate meaning with
their peers. When solving problems, it is important that teachers allow time for students to
discuss, explain, and justify their solutions, even if they are not completely correct. And, even
when students are struggling, it is critical that teachers allow some time for students to grapple
with the mathematical ideas (Principles and Standards, 2000). Often, for teachers, it is harder to
EVERYDAY MATHEMATICS CURRICULUM – DOES IT ADD UP?
23
watch students struggle than it is to intervene, but there are real benefits to allowing them the
opportunity to make sense of the mathematics on their own. Talking with and presenting to
others helps students become comfortable talking about mathematics. It is crucial that all
students are taking active roles in presentation, and can be motivating to allow them to create the
guidelines and score their peers. Besides observing students’ communication orally, critiquing
their writing can give insight to misconceptions and help understand the extent of their
understanding. Procedural knowledge is meaningless until its significance to an application can
be communicated effectively and inform decisions.
Creating a blog or other safe place for students to exchange ideas outside of class can
allow teachers to monitor progress. Assigning open-ended problems that lend themselves to
multiple paths to a solution will often get students talking on their own. It may be worthwhile not
to grade such a platform, unless grading is based solely on contributing.
Observations from discussions, blogs, and blogs are all effective alternate forms of
assessment, but nothing can replace an interview with a student. Conversing one on one with a
student allows teachers to modify their follow-up questions and responses specifically for each
student, and this will generate the clearest assessment of misconceptions and mastery.
In summary, suggestions for the University of Chicago School Mathematics Project,
would be to both modify the existing games and develop additional supporting materials both for
teachers and students. Modifying the games would entail creating games that had a balance of
chance and choice, more levels and contexts, and more variety in paths for solutions/winning the
games. Supporting materials for teachers would include a guide for each game that described
how the game added value to a particular content standard, while also offering extensions.
Teacher materials would also include questions for students, assessment options, and reflection
EVERYDAY MATHEMATICS CURRICULUM – DOES IT ADD UP?
24
questions for teachers themselves. Materials for students would encourage them to debrief about
their experiences playing the games. These materials would be diverse, and would range all of
the above-mentioned assessment options.
EVERYDAY MATHEMATICS CURRICULUM – DOES IT ADD UP?
25
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