[This page was last updated on April 9, 2006.]
There are several links below that are related to K-12 mathematics education. Some are closely related to my service on the Academic Standards Committee formed by the Commissioner of Education in February 2003. The Mathematics Subcommittee, consisting of about 40 people wrote mathematics standards and benchmarks which were approved by the Legislature and Governor in Spring 2003, and to which, according to my understanding, the state mathematics tests in Spring 2006 are to be aligned.
Problems Based on the Minnesota Mathematics Standards Document, without Extra Commentary
There following seven links correspond to the fact that Minnesota tests, at the state level, mathematics at seven grade levels in approximately late April ---grades 3-8 and 11. The seventh link below is also relevant for mathematics beyond what is relevant for the state test in grade 11.
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Problems Based on the Minnesota Mathematics Standards Document, with Comments
There are 7 links in this category are similar to the above 7, but contain some extra comments which I hope will prove useful.
Problems Based on the Spring 2003 Mathematics Standards and Benchmarks FOR GRADE 3, WITH COMMENTS
Problems Based on the Spring 2003 Mathematics Standards and Benchmarks FOR GRADE 4, WITH COMMENTS
Problems Based on the Spring 2003 Mathematics Standards and Benchmarks FOR GRADE 5, WITH COMMENTS
Problems Based on the Spring 2003 Mathematics Standards and Benchmarks FOR GRADE 6, WITH COMMENTS
Problems Based on the Spring 2003 Mathematics Standards and Benchmarks FOR GRADE 7, WITH COMMENTS
Problems Based on the Spring 2003 Mathematics Standards and Benchmarks FOR GRADE 8, WITH COMMENTS
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Solutions of problems based on the mathematics standards document
There might eventually be 7 links in this section, but for now there is only one; it is for Grade 7.
Solutions of problems based on the mathematics standards document FOR GRADE 7
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Standards-Related Material for Possible K-12 Usage
The Minnesota Standards for K-12 mathematics underwent a major change
in Spring 2003. The most far-reaching changes concerned content and
the nature of mathematical reasoning, not overall level of difficulty.
As an introduction to the problem links (only part of one at present)
below, I want to first give a brief over-simplified description of those changes.
The 2003 standards place much less emphasis than
the previous standards on sorting through
non-mathematical terminology to find the mathematics problem underneath
the camouflage, but they place more emphasis on logical reasoning
and the words that occur naturally in logical reasoning such as `and',
`or', `not', `if', and `then'.
In the early grades the 2003 standards emphasize calculational skill
with whole numbers of moderate size without the aid of calculator.
In the middle school and high school level the 2003 standards are much
more focused than the preceding standards. Thus, for example,
vertex-edge graphs only appear very narrowly in the 2003
standards---at one place in connection with probability, whereas these
constituted a significant topic in the pre-2003 standards. Calculational
skill with fractions, mixed numbers, negative numbers, percentages,
and decimals play a much more prominent role in the 2003 middle school
standards than they did previously. Algebraic skill and step-by-step
justification of conclusions in geometry play a central role in the high school
2003 standards whereas the pre-2203 standards were somewhat vague
about these aspects of mathematics.
The various lead-paragraphs in the standards for Grades 3, 4, 5, 6, 7, 8
and also 9-11 make two points (although not as strongly as I, with hindsight,
would have preferred): (i) that the standards focus on the time at which
appropriate questions could with fairness be asked on state MCA's,
not on the time when the topic should be first treated in the schools;
(ii) that the standards for a particular grade do not constitute a
stand-alone document; the standards for preceding and subsequent
grades are also
relevant for any particular grade. [The 2003 standards for K-2 constitute
a guide for teachers (rather than topics for a state test)
so that third grade teachers and students are not swamped
in advance of the Grade-3 state MCA. The 11-12 standards are not
to be reflected in any state test but do provide guidelines for students
and their teachers in order to prepare for a math-rich path in college.]
Given the fact that schools sometime change materials no more than
once every seven years, the change in standards that occurred in 2003
can create a dilemma. Generally speaking, materials chosen to align
with the previous standards do contain the mathematics relevant for
the 2003 standards, but often the appropriate emphasis is lacking.
The links below are for lists of problems that teachers (or parents)
might want to download in order to give students more practice on
things emphasized in the 2003 standards, especially if the teacher feels that the
materials he or she is using do not give sufficient practice. In view
of the preceding paragraph, particular grade levels are not identified for
the problems but grade spans will be indicated, and the problems will
generally progress from lower-grade level to higher-grade level.
I want to emphasize that the problem lists in the links below
are not designed to
represent a balanced representation of the 2003 standards. Rather
I am focusing on what I view as shortfalls in emphasis in some
materials that were chosen on the basis of the pre-2003 standards.
Arithmetic with fractions, mixed numbers, and prime numbers
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Additional Comments on Minnesota Benchmarks
The following links (only one link at present) concern certain individual benchmarks in the Minnesota K-12 mathematics standards approved in Spring 2003. My hope is that the comments I make for these benchmarks will be of some use.
Angular Measure in Radians in the 9-10-11 Benchmarks
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Mathematics Not in the Standards
The following links (only one link at present) concern topics suitable for math instruction in K-12 but which are not mentioned in the Minnesota K-12 mathematics standards approved in Spring 2003. Three reasons: (1) to highlight that it was not the intent of the standards to discourage teachers from teaching such topics; (2) some were advocated by some members of the mathematics subcommittee of the Academic Standards Committee; (3) to indicate ways in which such topics can be used to reinforce mathematics that is explicitly mentioned in the standards and benchmarks. These three reasons are intertwined; if time is spent on topics not mentioned in the standards and benchmarks and if in doing so, no connections are made with the standards and benchmarks, then it is likely that time will be lost for the mathematics mentioned in the standards and benchmarks. Although there are many places on this page and the links therein where it is my opinion that you are seeing, it seems particularly important mention this presence of personal opinion in connection with the following links.
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Mathematics Related to Other Subjects
One concern of the Profiles of Learning, which Minnesota has now repealed, is the interrelation among various K-12 subject areas. I view this as a legitimate concern. The personal opinion links (only one link at present) below address this concern.
Mathematics Related to Other Subjects
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I think there are several issues connected with the Minnesota state tests in mathematics, called Minnesota Comprehensive Assessments (abbreviated MCA's), which deserve serious consideration. The issue of alignment with the state standards was one motivation for my constructing the problem lists described above. Another motivation for construction of the problem lists is to help communicate the content of the mathematics standards and benchmarks by illustrating them (faithfully I hope). Some related opinionated short essays in this general area are given in the following links.
Inappropriate Contexts for Testing Which, However, Might Be Great for Teaching
Test Formats: Minimizing the Effect on Test Scores
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Problems Relevant for Preparedness for Calculus
The first of the following two links consists of problems that might be regarded as typical prerequisite problems for a student in order to study any calculus sequence that includes trigonometric functions; they have been written without any direct reference to the Minnesota standards in mathematics. The second link contains the solutions of the problems, hopefully with few, if any, errors.
Desirable Outcomes from a Full K-12 Mathematics Program
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Contests for Students in Grades 5-12
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The views and opinions expressed in this page are strictly those of the page author. The contents of this page have not been reviewed or approved by the University of Minnesota.