http://www.ed.gov/about/bdscomm/list/mathpanel/report/conceptual-knowledge.doc
The exec summary says this:
What is usually called Algebra I would, in most cases, cover the topics in Symbols and Expressions, and Linear Equations, and at least the first two topics in Quadratic Equations. The typical Algebra II course would cover the other topics, although the last topic in Functions (Fitting Simple Mathematical Models to Data), the last two topics in Algebra of Polynomials (Binomial Coefficients and the Binomial Theorem), and Combinatorics and Finite Probability are sometimes left out and then included in a pre-calculus course. It should be stressed that this list of topics reflects professional judgment as well as a review of other sources.Symbols and Expressions
• Polynomial expressions
• Rational expressions
• Arithmetic and finite geometric series
Linear Equations
• Real numbers as points on the number line
• Linear equations and their graphs
• Solving problems with linear equations
• Linear inequalities and their graphs
• Graphing and solving systems of simultaneous linear equations
Quadratic Equations
• Factors and factoring of quadratic polynomials with integer coefficients
• Completing the square in quadratic expressions
• Quadratic formula and factoring of general quadratic polynomials
• Using the quadratic formula to solve equations
Functions
• Linear functions
• Quadratic functions – word problems involving quadratic functions
• Graphs of quadratic functions and completing the square
• Polynomial functions (including graphs of basic functions)
• Simple nonlinear functions (e.g., square and cube root functions; absolute value; rational functions; step functions)
• Rational exponents, radical expressions, and exponential functions
• Logarithmic functions
• Trigonometric functions
• Fitting simple mathematical models to data
Algebra of Polynomials
• Roots and factorization of polynomials
• Complex numbers and operations
• Fundamental theorem of algebra
• Binomial coefficients (and Pascal’s Triangle)
• Mathematical induction and the binomial theorem
Combinatorics and Finite Probability
• Combinations and permutations, as applications of the binomial theorem and Pascal’s Triangle
The doc itself is much more specific, as you'd guess. The tables and figures are quite insightful.
APPENDIX A: Comparison of the Major Algebra Topics in Five Sets of Algebra I and Algebra II Textbooks with the List of Major Topics of School Algebra
APPENDIX B: Errors in Algebra Textbooks
List of Figures
Figure 1: Percent of Students At or Above Proficient in Mathematics Achievement on the Main NAEP Test: 1990, 2003, and 2007
Figure 2: States with Standards for Algebra I and II Courses
Figure 3: Topics in a 1913 High School Algebra Textbook
Figure 4: Singapore 2007 Algebra Standards for Grades 7-10
Figure 5: Algebra Objectives for NAEP’s Grade 12 Mathematics Assessment
Figure 6: Topics to Be Assessed in the ADP Algebra II End-of-Course Test
Figure 7: Mathematics Topics Intended from Grade 1 to Grade 8 by a Majority of TIMSS 1995 Top-Achieving Countries
Figure 8: Mathematics Topics Intended from Grade 1 to Grade 8 in the 2006 NCTM Focal Points Compared with the Topics Intended by a Majority of TIMSS 1995 Top-Achieving Countries
List of Tables
Table 1: Frequency Counts for Broad Topics in 22 States’ Standards for Algebra I and II Courses
Table 2: Algebra Elements Covered by State Algebra or Integrated Mathematics Frameworks, by State and Two-Thirds Composite
Table 3: Comparison of the Major Topics of School Algebra with Singapore’s Secondary Curriculum
Table 4: Comparison of the Major Topics of School Algebra with the 2005 NAEP Grade 12 Algebra Topics
Table 5: Comparison of the Major Topics of School Algebra with ADP’s High School Algebra Benchmarks, Core Topics in its Algebra II Test, and the Topics in the Optional Modules for its Algebra II Test
Table 6: K-8 Grade Level Expectations in the Six Highest-Rated State Curriculum Frameworks in Mathematics Compared with the Topics Intended by a Majority of TIMSS 1995 Top-Achieving Countries
Thanks for the link. I'm interested in comparing it with the textbook our school uses.
ReplyDeleteWhile the inclusion of a topic into a textbook allows for someone reviewing the book to check it off on a list, one gets no insight into if the topic is treated in such a way as to do any good.
ReplyDeleteSince mathematical induction is on the list and of personal interest to me I will use it as an example and discuss its inclusion and treatment in several texts that I have:
1. One textbook I have tells the student to prove a particular result by mathematical induction. A sketch is given of the proof as a footnote at the bottom of the page. No example is given before the student attempts his proof.
2. Another textbook includes it as the last chapter in a doorstop tome. Fat chance anyone will get to it. It's technically there though.
3. Another "teaches" induction by giving such a generalized example of how it works that only a mathematician would get anything out of it. Student is provided with only three difficult practice problems.
4. The textbook that I'm using gives 8 complete examples of mathematical induction, each with not a detail left out so that each proof takes up an entire page in small print (the much decried mindless formalism, but that's magically what it took for me to finally understand) Forty practice problems are given beginning with the easist and most concrete and ending with very general formulas. Many of the problems provide a result which will be used in a later problem so that increasingly sophisticated results are obtained.
5. Thomas and Finney teaches mathematical induction by use of an informal sketch in a side bar. I haven't gone through the entire calculus book to see which problems require this, but let's hope students got a good dose of it in high school before getting to Calculus, because I don't think the token gesture of including it as a topic would be enough for me. No doubt, if I had to rely on side bars for learning important topics I'd be one of the 50% who drops that class.
It's a far easier thing to come up with a list of topics that ought to be taught than it is to formally characterize how these topics need to be covered. Side bar coverage, foot notes, sketches, authors' digressing into monolgues on the topic in the text but without giving the student any problems as well as inadequate problem sets may permit a box checker to check a box saying that "we teach that too!" but they are very nearly tantamount to not including the topic at all.
"While the inclusion of a topic into a textbook allows for someone reviewing the book to check it off on a list, one gets no insight into if the topic is treated in such a way as to do any good."
ReplyDeleteGood point. I think Everyday Math fits this category. If you open it up to any page, you see some proper exercises. Unfortunately, it jumps all over the place and is almost impossible to finish.
Myrtle makes a good point.
ReplyDeleteI wonder if it’s possible to learn if my school is teaching NMAP Algebra I to any 8th graders without doing an in-depth review of the course syllabus. These two questions would be a start, I guess.
1. Does the class teach the NMAP Algebra I topics to mastery?
2. What textbook is used?
In most schools, it would actually be a start for them to even KNOW about the NMAP. The stark reality is that the majority of districts haven't been following this at all. They are slaves to their state standards which in almost every case look nothing like what NMAP has set forth.
ReplyDeleteThis is going to take awhile.
"These two questions would be a start, I guess.
ReplyDelete1. Does the class teach the NMAP Algebra I topics to mastery?
2. What textbook is used?"
Many textbooks wouldn't pass this first cut. You could use these simple questions to educate your school about the NMAP. Can one say that it has been approved by NCTM? They call it a first step, but they can't turn around and step in the opposite direction when it comes to specifically-defined content.
"Does the class teach the NMAP Algebra I topics to mastery?"
ReplyDeleteTo whom will this question be put?
What qualifies them as an authority as to what mastery of a particular topic consists of?
I thought I had mastery of inequalities until this problem mastered me: Prove that 4^n > n^4
for all integers greater than or equal to 5.
If it's true that teachers have not taken many math classes in the math department, much less courses in abstract algebra or real analysis how could they possibly be in any position to know what it means to master inequalities? As the joke goes, in the above problem my graphing calculator can't help me now.
"I thought I had mastery of inequalities until this problem mastered me: Prove that 4^n > n^4
ReplyDeletefor all integers greater than or equal to 5."
The point is that parents now have something tangible (and approved by NCTM?) to use with their schools. Perhaps questions of mastery are hard to answer or prove, but the panel's report gives parents a basis for expecting some kind of answers to these questions. Schools will have to deal with the contents of the report.
But I'm not overly optimistic because, as we say, "They do what they do". They'll figure out some kind of response.
I get the impression the proficiency levels recommended by the panel lag behind by one grade level. For example: "By the end of Grade 6, students should be proficient with multiplication and division of
ReplyDeletefractions and decimals."
I would expect proficiency with these types of operations in 5th grade.
Table 2: Benchmarks for the Critical Foundations
Fluency With Whole Numbers
1) By the end of Grade 3, students should be proficient with the addition and subtraction of
whole numbers.
2) By the end of Grade 5, students should be proficient with multiplication and division of
whole numbers.
Fluency With Fractions
1) By the end of Grade 4, students should be able to identify and represent fractions and
decimals, and compare them on a number line or with other common representations of
fractions and decimals.
2) By the end of Grade 5, students should be proficient with comparing fractions and decimals
and common percents, and with the addition and subtraction of fractions and decimals.
3) By the end of Grade 6, students should be proficient with multiplication and division of
fractions and decimals.
4) By the end of Grade 6, students should be proficient with all operations involving positive
and negative integers.
5) By the end of Grade 7, students should be proficient with all operations involving positive
and negative fractions.
6) By the end of Grade 7, students should be able to solve problems involving percent, ratio,
and rate and extend this work to proportionality.
Geometry and Measurement
1) By the end of Grade 5, students should be able to solve problems involving perimeter and
area of triangles and all quadrilaterals having at least one pair of parallel sides (i.e.,
trapezoids).
2) By the end of Grade 6, students should be able to analyze the properties of two-dimensional
shapes and solve problems involving perimeter and area, and analyze the properties of threedimensional
shapes and solve problems involving surface area and volume.
3) By the end of Grade 7, students should be familiar with the relationship between similar
triangles and the concept of the slope of a line.
Source: National Mathematics Advisory Panel, 2008.
I believe your impression is correct. It may be two full grade levels behind the Singapore Math curriculum. But at least they push for acceleration for capable students. This opens the door for following a timeline such as the one Singapore Math establishes.
ReplyDeleteThe Grade 7 topics -- rate, percent, ratio, and propotionality are covered extensively at the end of grade 5 and all of grade 6 in Singapore Math. So yes, having just struggled through those chapters with my 6th grader, the NMAP sequence is at least one grade level behind.
ReplyDeleteOTOH, at least they are covering it. Many a 7th or 8th grader has never seen a ratio or percent problem here in the US.
Perhaps the NMAP believe that they'll get better buy in if they don't start with a huge jump in expectations.
I gave C. the Singapore Math placement test at the end of 4th grade. He tested into 2nd semester 3rd grade in Primary Mathematics.
ReplyDeleteThat's a 1 1/2-year gap at the end of 4th grade, and it only grows larger as the next years go by.
He scored a "4" on the state math test that year.