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Self Assessment in Math for a CoE Research CourseGeorge 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.
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¬Found=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
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.
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:
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.
Date: Thu, 18 Oct 2001 06:05:39 -0400 (EDT) From: NSF Custom News Service <firstname.lastname@example.org> To: CNS Subscribers <email@example.com> 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:
Two references that relate to the topic of virtual manipulatives:
NSF Initiates Massive Effort to Rebuild Teaching Leadership in Science and
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:
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:
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:
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.
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....
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.
All contribute to a steadily increasing accumulation of data, information, and knowledge.
"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.
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.
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
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)
"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)
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
Stanford, California 94305-5479, USA
Stanley R. Jacob, Administrator
Tel: (650) 736-0728
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:
What Works Clearinghouse
2277 Research Boulevard, MS 6M
Rockville, MD 20850
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
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
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