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Contact Website Author
"Dr. Dave" Moursund

Miscellaneous Other Important Ideas

This Website is a "work in progress." This page contains miscellaneous materials that might be relevant to the Website, but to-date have not been developed sufficiently so that they fit into the general outline for the Website.

Self Assessment in Math for a CoE Research Course

George Washington University has a Website at http://www.gwu.edu/~assess1/index.html that contains a number of self assessment instruments that can be used by students who are planning to take a quantitative research methods course in the Graduate School of Education and Human Development. There are a number of different short multiple choice quizzes covering a wide variety of topics relevant to the quantitative methods course.
    1. Algebraic Notation and Operations
    2. Descriptive Statistics
    3. Tabular Displays of Data
    4. Graphical Displays of Data
    5. Research Topics, Theory, Constructs, Questions, and Hypothesis
    6. Research Designs
    7. Measurement
    8. Sampling
    9. Probability
    10. Basics of Inferential Statistics
    11. One-Group Inferential Statistics
    12. Two-Group Inferential Statistics
    13. Critiquing Education and Social Science Research

Eighth Grade Algebra

Algebra Poses a Problem of Timing Not Taking Class Early Could Close Gateway to Sciences. http://www.washingtonpost.com/ac2/wp-dyn?pagename=article&node=&contentId=A29022-2002Aug16&notFound=true

By Jay Mathews

Washington Post Staff Writer Sunday, August 18, 2002; Page A01

This article summarizes the pros and cons of Algebra becoming a common option or a requirement at the 8th grade. It indicates that the latest NCTM Standards addressed the issue in a preliminary edition, but did not address it in the final edition.

The Reston-based National Council of Teachers of Mathematics cautioned against eighth-grade algebra in an early draft of its latest standards but opted out of the debate in the final version. Its standards document says: "Students' understanding of foundational algebraic and geometric ideas should be developed through extended experience over all three years in the middle grades and across a broad range of mathematics content, including statistics, numbers and measurement. How these ideas are packaged into courses and what names are given to the resulting arrangement are far less important than ensuring that students have opportunities to see and understand the connections among related ideas."

A standard argument against such a requirement is based on Developmental Theory.

No Children Left Behind

TECHNOS QUARTERLY Summer 2002 Vol. 11 No. 2

Commentary: Leaving Children Behind

By William L. Bainbridge

http://www.technos.net/tq_11/2bainbridge.htm

Here are a few quotes. Note that for the Math Website, these need to be analyzed from a math education point of view.

In January 2002, the federal government's No Child Left Behind Act was signed into law. This legislation aims to "improve overall student performance and close the achievement gap between rich and poor students." No Child Left Behind focuses on school accountability, higher standards for students, and some of the very measurements educational evaluators advocate from coast to coast.

In addition to recognizing the positive aspects of this legislation, however, it also seems prudent to be concerned about what the national legislation lacks. The concern is that measurement alone will not bridge the learning gap that exists between children from homes of various socioeconomic levels.

The introduction to this legislation states that "In America, no child should be left behind. Every child should be educated to his or her full potential." Mandating standards and tests in and of itself cannot erase the fact that children from homes where parents have little education and minimal resources have many strikes against them.

Research in cognitive brain development shows that formation of synaptic contacts in the human cerebral cortex occurs between birth and age 10, and most of the brain gets built within a few years after birth. Environment matters greatly in brain development. The period of early childhood is critical to brain development, and those who have high-protein diets and lots of sensory stimulation tend to have more synaptic connections. Brains that do not receive enough protein and stimulation in their environments lose connections, and some potential neural pathways are shut down. These facts help to explain what educators have long observed: Children from impoverished environments, in which they do not receive good nutrition and stimulating experiences, generally achieve at lower levels than children from more enriching environments.

 

Out of Field Teaching

All Talk, No Action: Putting an End to Out-of-Field Teaching. http://www.edtrust.org/main/documents/AllTalk.pdf

--------------------

Math Archives: Topics in Mathematics [Online]. Accessed http://archives.math.utk.edu/topics/index.html

Math Forum @ Drexel [Online]. Accessed 5/3/02: http://mathforum.org/

 

Recently I have been Reading Jim Cassidy's doctoral dissertation on standards ion foreign language instruction. It occurs to me that in foreign language instruction, they have some well defined goals and they have ways to measure achievement of the goals. Of course, this is true in lots of disciplines. But, one aspect of the assessment is actual language proficiency performance under settings that are somewhat similar to what one might find as a person uses their foreign language knowledge and skills. It is not clear to me that we have the equivalent of this in math education. The testing is what might be called situated Testing, in line with the Situated learning. It does not appear to be testing for transfer of learning to situations that are a lot different than the math classroom.

There are two different mathematical learning theories in the list of 50 given at:

http://tip.psychology.org/theories.html

http://mathforum.org/~sarah/
Discussion.Sessions/Schoenfeld.html. Quoting from this Website:

Schoenfeld wrote this chapter in response to a challenge from mathematicians (among them Joe Crosswhite, Henry Pollak, Anna Henderson, and Steve Maurer) to explain "what metacognition is, why it's important, and what to do about it -- all in clear language that we can understand." Schoenfeld's explanation describes metacognition, or reflecting on how we think, through a discussion of how a problem was solved and what it was about, or where and why a difficulty occurred in the process of problem solving. He also proposes some ways metacognition could be used in the classroom.

Chapter:

Schoenfeld, A. H. (1987). What's all the fuss about metacognition? In A. H. Schoenfeld (Ed.), Cognitive science and mathematics education (pp. 189-215). Hillsdale, NJ: Lawrence Erlbaum Associates. [Online]. Accessed 10/27/01: http://mathforum.org/~sarah/
Discussion.Sessions/Schoenfeld.html

Overview:

Many students don't feel good about math, largely as a result of the way they have been taught. Because of the prevalent belief that classroom mathematics consists of mastering formulas, these students do not understand how mathematics can be meaningful.

Metacognition has the potential to increase the meaningfulness of students' classroom learning, and the creation of a "mathematics culture" in the class room best fosters metacognition. Schoenfeld believes that a "microcosm of mathematical culture" would encourage students to think of mathematics as an integral part of their everyday lives, promote the possibility of students making connections between mathematical concepts in different contexts, and build a sense of a community of learners working out the intricacies of mathematics together.

 

Date: Thu, 18 Oct 2001 06:05:39 -0400 (EDT) From: NSF Custom News Service <cns-admin@nsf.gov> To: CNS Subscribers <cns-subscribers@nsf.gov> Subject: [pr0180] - News Releases

The following document (pr0180) is now available from the NSF Online Document System

Title: NSF Initiates Massive Effort to Rebuild Teaching Leadership in Science and Mathematics Type: News Releases Subtype: Education

It may be found at:

http://www.nsf.gov/cgi-bin/getpub?pr0180

 ---------------------------

Two references that relate to the topic of virtual manipulatives:

 

http://mathforum.org/sum95/suzanne/whattess.html

http://www.geom.umn.edu/software/tilings/

 ---------------------

 

 

 

News Items

NSF Initiates Massive Effort to Rebuild Teaching Leadership in Science and

Mathematics [Online]. Accessed 1/3/02: http://www.nsf.gov/od/lpa/news/
press/01/pr0180.htm

Annotated Reference List

 

1. Assessment, and Self-Assessment

Rubrics and Self-Assessment Project from Project Zero [Online]. Accessed 1/25/02: http://pzweb.harvard.edu/Research/StuSA.htm. Quoting from the Website:

Scoring rubrics are among the most popular innovations in education (Goodrich, 1997a; Jensen, 1995; Ketter, 1997; Luft, 1997; Popham, 1997). However, little research on their design and their effectiveness has been undertaken. Moreover, few of the existing research and development efforts have focused on the ways in which rubrics can serve the purposes of learning and cognitive development as well as the demands of evaluation and accountability. The two studies that made up the Project Zero's research focused on the effect of instructional rubrics and rubric-referenced self-assessment on the development of 7th and 8th grade students' writing skills and their understandings of the qualities of good writing.

2. Tip of the (Month, Week, Day): A general theme or activity that can be laid over and/or integrated into the general curriculum, and that reflects important aspects of ICT in mathematics education. An example is Self-assessment. At the beginning of each major new topic, ask students to: A) Share with the whole class in a whole class brainstorming and sharing; B) Share and discuss in small groups; and/or C) Write about in their journals:

  • What they already know about the topic.
  • How they know that they know what they say they know. That is, how can they demonstrate this knowledge?
  • Their level of expertise on a scale that goes from ... This is an interesting question. Does the scale use a comparison with competence levels of others in the class? Or, perhaps it uses a scale based on the students insights on what a student might know, or could know, or should know, or needs to know?
  • Where/when did they acquire this knowledge?
  • Roles of calculators in knowing and doing (knowing and using?) this topic.
  • Roles of computers in knowing and using this topic.
  • Roles of telecommunications in knowing and using this topic.

3. The case against ICT in education, or against ICT in mathematics education. This could be one of the main topics (see the Home Page) that would be addressed in the workshop. It seems to me that a person needs to know the pros and cons of the issue. This is a good area in which to illustrate constructivism. Each person has their opinion, feelings, thoughts, intuition on appropriate roles of ICT in math education. They study the materials in this workshop with that as part of their background, and it colors or flavors all of their learning. They also have knowledge, skills, opinions, and so on about change in the curriculum, and their role in implementing change. Hmm. This is an interesting topic. Suppose that the overall goal of the workshop is to improve math education through having the participants improve their capacity to make "appropriate" use of ICT in math education. Presumably a participant might well learn to make less use, or to stop doing some of the things he/she is doing.

4. For each topic, think in terms of the Expertise Scale. We want to help a person find out where they are on this scale. We want to help them move up on the scale. We do this by three things:

  1. Helping them to understand that their current knowledge and skill is sufficient to carry out some useful ICT/Math related activities with their students--activities designed to help both them and their students to learn.
  2. Helping them to directly gain some increased ICT knowledge and skills, along with a possible sense of direction on this particular topic.
  3. Helping them to see how their current knowledge and skills on this ICT/Math topic fits in with their other knowledge and skills both for the purposes of facilitating transfer of learning and also to support their use of constructivism. This could be done by a brief summary/overview of the topic and a metacognitive activity. (And, if they are in a position to do small group discussion, a small group discussion. And, if they are in a position to talk with students, a discussion with a small group of students or the whole class.)

5. A cumulative science: A body of knowledge where one result leads to another. I think of this as a vertical structure. Thus, a result builds on a pyramid or chain of previous results, adding to the height and breadth of the pyramid and/or the length of the (vertical) chain.

6. Metacognition. What roles does it play in math education?

7. We tend to think that a math problem is solved or it is not solved. But of course, this does not take into consideration partial credit or reasonable progress toward achieving a solution. Note, however, there is a weak parallel between this and Process Writing, or writing in general. In writing, one can produce a product that can then be improved. A piece of writing is never "perfect." What can we do in math education to create more situations in which a student can make visible progress and still have room for more progress? One answer lies in PBL.

8. What is the meaning of the term "learning?" What is the meaning of the term "knowledge?"

Some people think it is useful to look at a "continuum" consisting of Data, Information, Knowledge, and Wisdom. Others think this is a wrong approach to (whatever). I, personally, find the continuum useful in promoting discussion of whether a computer system can have (has) knowledge, and the educational implications of this.

Certainly one of the goals in formal education is to help students learn. My spur of the moment definition of learning is something like the following:

Gaining increased knowledge, skills, understanding, insights, etc. that can be used in decision making, problem solving, and task accomplishing.

9. An Extended Epistemology for Transformative Learning Theory and Its Application Through Collaborative Inquiry [Online]. Accessed 2/4/02: http://www.tcrecord.org/Content.asp?ContentID=10878. Elizabeth Kasl California Institute of Integral Studies. Lyle Yorks Teachers College, Columbia University. Quoting from the article:

Before proceeding with our discussion, we make a short comment about the development of Transformation Theory (Mezirow 1978, 1981, 1991, 1995, 2000). The original purpose of Mezirow's project was to introduce a theory of adult learning into the discourse about adult education. Writing at a time when the literature in adult education was largely focused on describing a set of educational practices that assumed beliefs about adults as learners, Mezirow called attention to the need for a formal theory of adult learning and offered his own vision. He has been successful in sparking an extensive discussion about how adults learn (Taylor, 1998, 2000). Mezirow rests his work on the assumption that learning transformatively is rooted in learning from experience.
Learning is understood as the process of using a prior interpretation to construe a new or revised interpretation of the meaning of one's experience as a guide to future action....

Transformative learning refers to the process by which we transform our taken-for-granted frames of reference (meaning perspectives, habits of mind, mind-sets) to make them more inclusive, discriminating, open, emotionally capable of change, and reflective so that they may generate beliefs and opinions that will prove more true or justified to guide action (Mezirow, 2000, pp. 5-8).

 

 

11. Human-developed "languages."

We can explore four major categories of mind tools have affected overall human capabilities to represent and solve problems, define and accomplish tasks.

1. Writing

2. Mathematics

3. Science

4. ICT

All contribute to a steadily increasing accumulation of data, information, and knowledge.

Each:

  • Is an aid to communication face to face as well as over time and distance).
  • Is an aid to representing and processing certain types of information & knowledge.
  • Is a cognitive aid (a mind tool).
  • Has helped facilitate development of a global library.

"Chomsky's new view of [spoken] language as a biologically based universal feature of our brain has taken hold. Steven Pinker, a colleague of Chomsky at MIT, has extended it by successfully arguing that [spoken] language is an instinct -- just like any other adaptation. Syntax is not learned by Skinnerian associative systems; rather, we can all communicate through [spoken] language because all members of our species have an innate capacity to manipulate symbols in a temporal code that maps sounds onto meaning." (Gazzaniga 1998, p7)

12. Interaction among curriculum, instruction, assessment, and teacher.

This Website is designed to support the preservice and inservice education of K-12 math teachers. This Website is a multimedia book design to be used either for self study or in conjunction with a workshop or other type of formal instruction.

The Website explores current knowledge in the fields of Brain Science and ICT, and applies this knowledge to K-12 Mathematics Education.

The Venn diagram indicates that there is a substantial interaction among curriculum, instruction, assessment, and the teacher. The overall diagram suggests that Math Education is carried out in a complex environment (for example, contains stakeholders such as parents, business people, politicians, and professional societies). The Website specifically focuses on how our overall Math Education system can be improved by appropriate use of ICT facilities and our knowledge of Brain Science.

The Website is designed for all K-12 educators who have some involvement in Math Education, and it assumes only minimal knowledge of ICT and of Brain Science. The Website has a focus on the future -- where Math Education is going as ICT facilities become more powerful and readily available, and as Brain Science continues its rapid pace of gaining new knowledge.

This Website is designed to support a face-to-face, a distance learning, or a self-study workshop or course. The Website is not designed to be read or used in a "cover to cover" mode. It is a nonlinear hypertext. Especially in its current, very rough draft mode, effective use of the materials is highly dependent upon the individual users.

 

13. Collaboration

There is good research to support collaborative learning and collaborative problem solving. Both can be facilitated by ICT. Applications to math education ...?

14. Dyslexia and Mathematics

Dyslexia and Mathematics (November 2000) [Online]. Accessed 10/2/01: http://www.bda-dyslexia.org.uk/d07xtra/x10maths.htm. Quoting from the Website:

The British Dyslexia Association welcomes the government's initiative of the National Year of Numeracy, with its National Numeracy Strategy. This leaflet is in response to interest of parents and teachers who are concerned about problems that some dyslexic children experience when learning mathematics.

Traditionally, dyslexia has focused very much on literacy - the learning of the reading and writing processes. For some dyslexic children and adults difficulties also transfer into the learning of mathematics. It is well known that dyslexic people are as able as many others but that they need to learn in ways which suit them best.

Too many dyslexic children are put in low sets for mathematics whether they receive "more of the same". Such methodology is of little value to them. As a result, frustration and tension grow. Many highly able dyslexic children are in these sets through misdiagnosis. It is hoped that many of them will be helped directly through the National Numeracy Strategy but, for those who are not, alternative methods must be found.

We look forward to seeing both teachers and student teachers trained in the recognition of dyslexia within the area of mathematics as well as literacy.

At the end of this leaflet there is a list of helpful books where both teachers and parents may get more information.

 

15. Computer Programming

Under this topic we may well cover related topics such as spreadsheet. A spreadsheet can be thought of as a limited purpose programming environment.

16. Computer Graphics

This is a mathematically intense field. Morphing, curve fitting, etc. types of tools used in graphics software are all mathematical. A small example comes from having the knowledge to understand the following news item:

GRAPHICS THAT RESIZE TO FIT ANY SCREEN

The World Wide Web Consortium has recently recommended general adoption of a new graphics technology designed to make Web pages fit any kind of display, including those on wireless phones. Developed with contributions from companies such as Adobe, Corel, Apple, Microsoft and others, the technology is called Scalable Vector Graphics (SVG). Whereas most graphics on the Web are created as bitmaps (collections of pixels plotted on a grid), vector graphics draw images based on mathematical algorithms that specify the length, position, and color of a graphic element. The chairman of the SVG working group of the WWW Consortium says: "People are accessing the Web with a wider range of devices that all want a different-size display. But users don't want to have to go a special version of the site for handhelds. They want to go to the same site as everyone else." (New York Times 4 Oct 2001) http://partners.nytimes.com/2001/10/
04/technology/circuits/
04NEXT.html (NewsScan Daily, 4 October 2001)

 

http://www.cec.sped.org/osep/news14.html

 

Development funded by the U.S. Office of Special Education Programs NEWS BRIEF

 

Math Curricula Don't Match Learning Needs of Students with Mild Disabilities

 

Neither students with mild disabilities nor their counterparts without disabilities learn math the way commercial or district math curricula are organized, according to a study funded by the Office of Special Education Programs of the US Department of Education.

 

While math standards, such as those published by the National Council of Teachers of Mathematics (NCTM) may group mathematics topics across grades, commercial materials provided to the schools and the curriculum guides of the states and districts themselves continue to specify grade-by-grade level content, which the data reveal to be an ineffective measure of student progress.

 

The study, by Cawley, Parmar, Foley, Salmon, and Roy, offers some suggestions on improving the mathematical achievement of students with and without mild disabilities and discusses the implications for math standards. Students with mild disabilities also have problems with vocabulary-laden math texts, say the researchers. The students' comprehension of text may be below grade level at the same time that their computational skill levels are higher.

 

The solution does not lie in simplifying the math. The authors believe that a major limitation in the mathematics performance of students with disabilities is caused by the fact that the math presented to students in special education intervention is of a much lower quality than the students are capable of mastering. "Both mathematics educators and the special educators," the authors state, "must ... identify an alternative that will be acceptable at the state levels where adoptions are made and at the classroom level where instruction is conducted." A possible alternative encouraged by the researchers is to focus on "big ideas," which are the central concepts within a learning domain, and which form the basis for generalization and expansion.

 

Teachers who want to understand the progress of students in solving word problems should understand the complications created by extraneous information. For instance, some students attempt to include all the numbers in a problem regardless of their relevance. Many word problems pose difficulties for students because the students lack the contextual information to make their computations. Instruction in problem-solving skills should put less stress on understanding of cue words and more on situated language comprehension and information processing, because cue words sometimes misguide students. Student difficulties with word problems can result from a failure to comprehend language or to process information rather than any inability to do the math involved.

 

The pressure is on for teachers to have ALL students in their classes show progress in the general education curriculum and meet statewide educational standards. This OSEP-funded study sheds light on some issues that curriculum developers and teachers need to address in order to make that happen.

 

More complete information on this study can be found in Cawley, John, Parmar, Rene, Foley, Theresa E., Salmon, Susan, and Roy, Sharmila, "Arithmetic Performance of Students: Implications for Standards and Programming," Exceptional Children, 67, no. 3 (Spring 2001): 1-18. Funding for the original study was provided by a grant from the Division of Personnel Preparation, Office of Special Education Programs, US Department of Education (Grant #H029K890068; John Cawley, Project Director).

 

Note that the article is available in PDF format. I went to the page http://journals.cec.sped.org/index.cfm?fuseaction=ec_toc

 

and used the search engine there to access the document.

 

====================

Notice the answer to the question:

Better math education requires higher expectations, too

 

By Agnes Blum, Globe Correspondent, 12/15/2002

 

http://www.boston.com/dailyglobe2/349/learning/Better_math_education_requires_higher_expectations_too+.shtml ahesh Sharma, provost and executive vice president of Cambridge College, has been a professor of math education for 28 years. Sharma grew up in Rajasthan, a state in northwest India, and first came to the United States in 1965. He has written and spoken extensively about the state of math education. He works with schools to improve math education and collaborates with textbook publishers and designers of educational curricula. Sharma is married, with two grown children, and lives in Framingham. His two grandchildren attend Boston public schools.

Q. Why do you think so many American students are failing math?

A. There are three reasons. In American society, literacy is more important than numeracy. No one will admit they can't read, but they will readily admit they can't do math. It's almost like a badge of honor to not be able to do math. Second, the expectations of schoolchildren are low. In Asian and European countries, it's quite common for students to know their facts by age 7. Here we have students in the ninth grade who are still counting on their fingers. And, third, the focus here is procedural. But there are three parts to learning math: linguistic, the language of the problem; conceptual, the model of it; and procedural, how to get the answer. If a student doesn't have the conceptual model, they are not prepared for higher-order math education.

Q. How is learning math different from learning other subjects?

A. Look at the reading skill. Once you acquire it, you read more complex sentences and longer passages. You have acquired that skill. But with math, once you can add and subtract, then they add fractions, then real numbers, then negative numbers. It is not automatic. More and new concepts are constantly added. The larger concept subsumes the previous concept.

 ========

Two new math things from FREE 12/20/02

"Create a Graph"

helps students create their own graphs & charts. This online

tool can be used to make 4 kinds of charts & graphs: bar

graphs, line graphs, area graphs, & pie charts. (ED)

http://nces.ed.gov/nceskids/graphing/

 

"Explore Your Knowledge"

challenges students to try their hand at 8th grade math &

science questions taken from the Third International

Mathematics & Science Study (TIMSS). (ED)

http://nces.ed.gov/nceskids/eyk/index.asp?flash=true

 

----

 

http://www.edtrust.org/main/documents/k16_spring01.pdf

 

This gives info on math preparation of certain groups of students. Fits in well with developmental theory.

 

 

International Society for Computational Biology (ISCB)

Stanford Medical Informatics

Stanford University School of Medicine

251 Campus Drive

MSOB X215

Stanford, California 94305-5479, USA

Stanley R. Jacob, Administrator

E-mail: jacobs@smi.stanford.edu

Tel: (650) 736-0728

Fax: 650-725-7944

http://www.iscb.org

 

The International Society for Computational Biology is dedicated to advancing the scientific understanding of living systems through computation; our emphasis is on the role of computing and informatics in advancing molecular biology. The Society aims to serve its membership by facilitating scientific communication through meetings, tutorials, publications and by electronic means; by collecting and distributing information about training, education and employment in the field; and by increasing the understanding of the significance of our endeavor in the larger scientific community and in the public at large.

 

===========================

 

Teaching and Learning Mathematics Using Research to Shift from the "Yesterday" Mind to the "Tomorrow" Mind. March 2000. Accessed 4/3/03: http://www.k12.wa.us/publications/docs/MathBook.pdf Quoting from page1-2:

RESEARCH IN MATHEMATICS EDUCATION: WHAT IT CAN AND CANNOT DO

Think of the many things that can be investigated in mathematics education; it is easy to be overwhelmed. Four key ingredients can be identified:

  • The students trying to learn mathematics&emdash;their maturity, their intellectual ability, their past experiences and performances in mathematics, their preferred learning styles, their attitude toward mathematics, and their social adjustment.
  • The teachers trying to teach mathematics&emdash;their own understanding of mathematics, their beliefs relative to both mathematics itself and how it is learned, their preferred styles of instruction and interaction with students, their views on the role of assessment, their professionalism, and their effectiveness as a teacher of mathematics
  • The content of mathematics and its organization into a curriculum&emdash;its difficulty level, its scope and position in possible sequences, its required prerequisite knowledge, and its separation into skills, concepts, and contextual applications.
  • The pedagogical models for presenting and experiencing this mathematical content&emdash;the use of optimal instructional techniques, the design of instructional materials, the use of multimedia and computing technologies, the use of manipulatives, the use of classroom grouping schemes, the influences of learning psychology, teacher requirements, the role of parents and significant others, and the integration of alternative assessment techniques. All of these ingredients, and their interactions, need to be investigated by careful research. Again, it is easy to be overwhelmed (Begle and Gibb, 1980). Our position is that educational research cannot take into account all of these variables. The result we must live with is acceptance that educational research cannot answer definitively all of the questions we might ask about mathematics education. At best, we can expect research in mathematics education to be helpful in these ways:

 

 ================

4/20/03

 

What Works Clearinghouse

2277 Research Boulevard, MS 6M

Rockville, MD 20850

Email: wwcinfo@w-w-c.org

Phone: 1-866-WWC-9799

Fax: 301-519-6760

Systematic reviews of evidence in this topic area will address the following questions: * Which curriculum-based interventions are effective in increasing the learning of mathematics content and skills (that is, what students should know and be able to do) among elementary, middle, and high school students?

 

* Are some interventions more effective than others for learning certain types of math content and skills?

 

* Are some interventions more effective for certain types of students, particularly students who lag behind in mathematics achievement? Topic Area Focus

 

Sample Problems

In attempting to explain what math is and what mathematicians do, it is helpful to have exmaples of prolbems that are easy to understand but challenging. Here is an example:

The "3n + 1" Problem

The "3n + 1" problem involves starting with a particular integern, and repeatedly performing the following operation:

If (n is even) divide n by 2;
Else multiply n by 3, and add 1.

For example, starting with the number 6, we get the following series:

6, 3, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1...

The 4,2,1 loop repeats over and over, so it's usually convenient to terminate the process once it is entered. All numbers tested so far eventually hit this loop, although it has not been proven that all numbers do.

Part 1
Fouts, Jeffrey (April 2003). A decade of reform: A summary of research findings on classroom, school, and district effectiveness in Washington State. Accessed 1/20/04: http://www.spu.edu/wsrc/currentresearch.html.
The overall document summarizes nearly 10 years of work in the State of Washington, as they have moved towards a “standards based” educational system. In essence, the idea is to carefully define standards, have the teachers teach to the standards, and have assessment aligned with the standards. As might be expected this approach leads to higher test scores. Quoting from page 24 of the report:
My Thoughts
In the quote, the term “successful schools” refers to schools that succeeded in raising their test scores in reading, writing, and math. I found this quote to be quite disturbing. Surely education is much more than scoring well on reading, writing, and math tests. Surely the tests do not adequately reflect our goals in reading, writing, and math education.
For example, here is a sample question from Washington Assessment of Student Learning (WASL) 2002 Math-Grade 4 (Fouts, 2003, page 8):
As I thought about this math assessment question, I was first struck by the first word, “Ming’s.” I thought about whether Ming might be the name of a city, University, soccer team owner, Dynasty, or type of vase. Since I don’t know anybody named Ming, I did not easily connect with the intended idea that Ming is a person connected in some way with a soccer team, that the team has some money, and that Ming has some responsibility for spending the money to provide uniforms for the team.
I then thought about the statement “buy a new uniform.” My first thought was that this might well mean buying new uniforms for the whole team. After much more thought it occurred to me that perhaps a new player has joined the team, or a uniform has been lost or stolen, or a uniform is in such ill repair that it needs to be replaced. I suppose that this is the situation, since the intent seems to be to buy one new uniform.
I then thought about what I know about soccer uniforms. The answer is “not much.” Perhaps we are talking about a team of fourth grade students. (The test question is aimed at the fourth grade level.) Do fourth graders who play on a soccer team buy their own soccer shoes, or does someone else pay for them? Do they have quite expensive uniforms, or does a uniform consist mainly of a jersey and a pair of pants or shorts? The statement, “figure out how much money his team must raise,” suggests to me that $100 is not adequate to purchase the needed uniform. But, this certainly goes against my guess that $100 is more than enough to buy one uniform, especially if the purchase does not need to include a pair of soccer shoes.
One additional thought. It seems like the intent is to create an “authentic, real world” problem for the fourth graders. The problem is made authentic and real world by casting it in a authentic, real world environment. However, this environment may be far removed from the typical student taking the test. This adds some layers of complexity that might not have been intended by the people who made up this problem.
Here is a grade 7 math question from Fouts, page 20.
This is another attempt to create an authentic, real-world problem. I think it an even worse attempt than the previous example. Have you ever looked at workers at a construction site? Presumably you noticed that they worn clothing and, typically, had a tool belt with tools. If a worker’s clothing and tools weigh more than 2 1/2 pounds, then the problem has no solution.
Of course, I presume that the workers could walk up the stairs. But, I was puzzled by why they needed to be doing “repairs” at a construction site. My image of a construction site is one of building a new building—hence “repairs” is a puzzling term. In summation, this seems to me to be a poorly stated problem.
The problem itself is a “puzzle” type of problem. It brought to my mind the puzzle:
Since I was experienced in solving this puzzle, I was able to use the same ideas in solving the construction site problem. However, I am left wondering what there is about math up through grade 7 that is related to solving such puzzle problems. And, here is another observation. In all of my life, I don’t recall ever being faced by this type of problem except in puzzle-solving environments.

Some leading math educators are given here. The list came from:

The Mathematics Education Portfolio Brief Document Number: nsf0503 http://www.nsf.gov/pubsys/ods/getpub.cfm?nsf0503

- Joan Ferrini-Mundy, Michigan State
University, Senior Advisor
- Ronald Graham, University of California, San
Diego, Chair
- Deborah Ball, University of Michigan
- Hyman Bass, University of Michigan
- Sylvia Bozeman, Spelman College
- Robert Floden, Michigan State University
- Kathleen House, Frederick County, MD Public
Schools
- Roger Howe, Yale University
- Jeremy Kilpatrick, University of Georgia
- William Lewis, University of Nebraska, Lincoln
- Kenneth Millett, University of California, Santa
Barbara
- Paul Sally, University of Chicago
- Richard Schaeffer, University of Florida
- Alan Schoenfeld, University of California,
Berkeley
- William Tate, Washington University
- Jeffrey Witmer, Oberlin College
- Patricia Wright, Virginia Department of
Education 

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