APA Format 6th Edition Template - George Mason …mason.gmu.edu/~sdeleeuw/portfolio/docs/MATH ED...

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? 1

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,


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

http://www.corestandards.org/


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

http://everydaymath.uchicago.edu/


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


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.


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.


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


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.


. 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


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 upper-

middle 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


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

https://www.everydaymathonline.com/


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.


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 ChildhoodStandard: Kindergarten / Counting and Cardinality

Name of Game: Angle Race / Grade 3Standard: Grade 4 / Measurement and Data

Name of Game: Factor Captor / Grades 5 and 6Standard: Grade 4 / Operations and Algebraic Thinking

Content Analysis

SUBSTANCE: Does the activity address the specific substance of the standard?

Note: If only a “topic” match to the standard is present, the activity is not included in the list.

Yes, see the specific standards listed below.

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.

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 OR WHOLE: Which part or parts of the standard are addressed?

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:

An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the


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.

An angle that turns through n one-degree angles is said to have an angle measure of n degrees.

(2) Recognize angle measure as additive. When an angle is decomposed into non-overlapping 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 purposePURPOSE: 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.

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.

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

No, does not convey the relationships to other activities.


the 360 degree mark.

Object: to get a higher total than the computer.

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?

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 ideasPREREQUISITE KNOWLEDGE AND SKILLS: Does the material specify prerequisite knowledge and skills that are necessary for learning the standard?

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.

ALERTING THE TEACHER TO COMMONLY HELD IDEAS: Does the material alert teachers to commonly held student ideas, both troublesome and helpful?

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.

ADDRESSING COMMONLY HELD IDEAS: No. See above.


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 StudentsVARIETY OF CONTEXTS: Does the material provide experiences with ideas in multiple, different contexts to support the formation of generalizations?

No, students are limited to the context in the game.

QUALITY OF EXPERIENCES: Does the material include activities that promote first-hand experiences or applications when practical or present students with other meaningful experiences?

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 ideasBUILDING A CASE: Does the material develop justifications or arguments for the importance or significance of mathematical generalizations or procedures?

No, although the games are classified as Skill Builder or Challenge games, there is no evidence of ‘building a case.’

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?

Yes, unnecessary terms are not evident in the games.

The game uses “smaller than” and “bigger than” instead of “less than” and “greater than.” Although young children may be used to small/big and not yet less/greater, it


seems it would be manageable.

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.

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 reflectionPROVIDING 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?

No, the only feedback provided is “Great job,” “Try again,” or an explanation of when a wrong move is made.

GUIDING STUDENT INTERPRETATION AND REASONING: Does the material include tasks and question sequences to guide student interpretation and reasoning about activities and readings?

No. Again, even in the teachers’ manuals, there are nothing more than descriptions and directions for the games.

PROMOTING REFLECTION AND SELF-MONITORING: 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, there are not suggestions of how students can extend their ideas or reflect


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.

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: AssessmentALIGNMENT TO GOALS: Are assessment items and tasks aligned with the identified standard or benchmark?

Yes, there is feedback on the particular skill.

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?

Students that develop a strategy of guessing somewhere near the middle of the remaining numbers would theoretically do better than those randomly guessing.

Students that develop a strategy and apply their knowledge of factors to determine optimal selections will theoretically win more often.

No, there are not assessments to evaluate conceptual understanding.

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


of steps it took before the student was able to guess the ‘mystery number,’ but this may not show conceptual understanding.

MULTIPLE FORMATS: Do assessments include multiple formats and response modes that afford students diverse opportunities to exhibit their skills and understanding?

No, there is not even a suggestion for a more summative assessment of the concept.

Cluster 7: Other featuresTEACHER CONTENT KNOWLEDGE: Would the material help teachers improve their understanding of mathematics and its applications?

Yes, it may be helpful for teachers to become comfortable with the representations in the games, although they are not diverse.

No, there are not applications of the mathematics beyond the mathematics of the game itself.

CLASSROOM ENVIRONMENT: Does the material help teachers create a classroom environment that welcomes student curiosity, rewards creativity, encourages inquiry, and avoids dogmatism?

Yes, it seems that games in general welcome student curiosity, and these also avoid dogmatism.

No, there are not rewards in these particular games that reward creativity or encourage inquiry.

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, all students can take part in the games.

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


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


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


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


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


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.


References

Brehe Pixler, P. Mild disability students and Everyday Mathematics: Serving the needs of this

population. Ed.D. dissertation, The University of Toledo, United States -- Ohio.

Carroll, W. M. (1998). Geometric knowledge of middle school students in a reform-based

mathematics curriculum. School Science and Mathematics, 98, 188-197.

Confrey, J. & Stohl, V., eds. (2004). On Evaluating Curricular Effectiveness: Judging the

Quality of K-12 Mathematics Evaluations. Washington, DC: The National Academies

Press.

Department of Education Sources Cite Effectiveness and Popularity of McGraw-Hill Everyday

Math Program. (2011, January 13). PR Newswire.

Dillon, S. (2003, January 22). Curriculum with Roots in Global Competition. New York

Times [Late edition (east coast)], B6.

Esposito, L. M. (2005). The implementation of an elementary mathematics program: A series of

Everyday Mathematics case studies. Ed.D. dissertation, Columbia University Teachers

College, United States -- New York.

Everyday Mathematics Adds Online Demonstrations of Different Solution Methods for Teachers

and Parents. (2009, June 1). PR Newswire.

Fraivillig, J. L., Murphy, L. A., & Fuson, K. C. (1999). Advancing children's mathematical

thinking in everyday mathematics classrooms. Journal for Research in Mathematics

Education, 30, 148-170.

Fuson, K. C., Carroll, W. M., & Drueck, J. V. (2000). Achievement results for second and third

graders using the standards-based curriculum everyday mathematics. Journal for

Research in Mathematics Education, 31, 277-295.


Goodnough, A. (2003, May 3). Teachers in the dark on curriculum. New York Times [Late

edition (east coast)], B1.

Hurst, D. The impact of a reform-based elementary mathematics textbook on students' fractional

number sense. Ed.D. dissertation, Widener University, United States -- Pennsylvania.

Letters to the Editor: Our `New Math' Adds Up to Success. (2000, January 13). Wall Street

Journal [Eastern Edition], A23.

McGraw-Hill Education; Everyday Mathematics; McGraw-Hill and University of Chicago

School Mathematics Project Announce First Continuous Pre-K-12 Math

Curriculum. (2010, May). Journal of Mathematics, 199.

McGraw-Hill Education; Everyday Mathematics; McGraw-Hill's Everyday Mathematics Drives

Student Achievement in Washington, D.C., School System. (2010, March). Education

Letter, 57.

Minnich, S. How does the implementation of a reform-based mathematics program, Everyday

Mathematics, change teachers' beliefs and perceived pedagogy of mathematics

instruction? Ed.D. dissertation, Temple University, United States -- Pennsylvania.

Salvo, L. C. (2006). Effects of an experimental curriculum on third graders' knowledge of

multiplication facts. Ph.D. dissertation, George Mason University, United States --

Virginia.

Schoenfeld, Ed Researcher article

Smiddy, J. Everyday Mathematics and its use as curricular reform to stimulate stronger school

peformance. M.A.E. dissertation, Pacific Lutheran University, United States --

Washington.


Sood, S., & Jitendra, A. (2007). A comparative analysis of number sense instruction in reform-

based and traditional mathematics textbooks. The Journal of Special Education, 41, 145-

157.

Wright Group/McGraw-Hill; Everyday Mathematics; New York City Students Narrow the

Achievement Gap With McGraw-Hill Education's Everyday Mathematics and IMPACT

Mathematics. (2009, June). Journal of Mathematics, 43.

Wright Group/McGraw-Hill; Everyday Mathematics' Online Assessment Management System

Tracks Student Progress Effortlessly. (2009, January). Journal of Mathematics, 86.

APA Format 6th Edition Template - George Mason …mason.gmu.edu/~sdeleeuw/portfolio/docs/MATH ED...

Documents

Transcript of APA Format 6th Edition Template - George Mason …mason.gmu.edu/~sdeleeuw/portfolio/docs/MATH ED...