LAUSD’s Refusal to Adopt the
California Mathematics Standards
by David Klein
Professor of Mathematics
California State University, Northridge
Shortly after the adoption of the California Math Standards by the California Board of Education, LAUSD Superintendent of Schools, Ruben Zacarias, issued an Informative stating that the LAUSD Standards include and go beyond the State Board standards.
No adjustment of LAUSD’s math standards is necessary to accomodate the California math standards, according to the Informative. It explains that:
… the high expectations for student achievement set forth by the [LAUSD] school board and the Superintendent will be met by implementing the standards-based curriculum recommended by the Los Angeles Systemic Initiative.
The Los Angeles Systemic Initiative (LASI) is a project to impose an integrated math curriculum in LAUSD. Among other weak curricula, LASI promotes the highly discredited elementary school curriculum, MathLand.
The LAUSD Informative further claims that textbooks aligned with the new California State Standards would have to be supplemented to rise to the level of the LAUSD math standards.
In response to this statement, on March 16, 1998 a number of mathematicians and others familiar with the California Mathematics Standards released a document entitled A Comparison of the LAUSD Math Standards and the California Math Standards.
This study was based on the idea that such an analysis requires comparison of the California Mathematics Standards with the Los Angeles Unified School District Standards not by what might be meant, but by what is actually stated.
On this basis it was concluded that the LAUSD Standards are woefully lacking in critical mathematical content areas and lacking in precision, clarity and completeness.
In the conclusion, the comparison of the math standards notes:
The vacuousness of the LAUSD math standards facilitates poor achievement of LAUSD students in mathematics. Since these standards can mean whatever a particular reader wants them to mean, they are not standards at all. They serve only to protect poor achievement in mathematics — the status quo for LAUSD.
The comparison document also forewarned of wildly inflated interpretations of the LAUSD standards. For example, it might be claimed that standards #3 and #15 … subsume all California math standards related to linear equations, quadratics, simultaneous equations, perhaps even linear algebra, and all possible functions and all possible relations.
LAUSD STAFF RESPOND
At the behest of LASI, Ruben Zacharias, LAUSD Superintendent, released a second Informative to the LAUSD Board of Education on June 4, 1998. It state, There has recently been a great deal of confusion about the LAUSD mathematics standards, their rigor, and how they align with the California state standards. The LAUSD/LASI Informative continues, While at first glance, the LAUSD standards and the state standards appear to be fairly different, this is largely due to formatting choices.
The LAUSD/LASI Informative undermines its own purpose by reinforcing many of the public criticisms of the LAUSD Standards. For example:
- Aside from a brief statement that students should know basic numerical skills, the Informatives acknowledges that students really are allowed to use calculators from the earliest ages — third grade and below. In fact, the LAUSD math standards themselves explicity require the use of technology such as calculators or computers in its third grade benchmark. The LAUSD math standards are in conflicht with California state policy which does not allow calculator usage on the STAR exams in the elementary grades.
- The LAUSD/LASI Informative states that LAUSD has written broad statements at only the selected benchmark grades of third, seventh, ninth and twelfth. In other words, these grades are treated minimally, while other grades are missing altogether.
- The second Informative admits that the LAUSD Standards lack not only Trigonometry, they lack second year Algebra.
Thus, the LAUSD Standards have serious defects – defects that do not disappear even with this second attempt to deny them.
THE LAUSD COMPARISON ANALYZED
The document, A Comparison of the LAUSD Math Standards and the California Math Standards, dispels any notion that LA standards are even remotely comparable to the California Mathematics Standards. Even though it was intended to rebutt criticisms of the LAUSD math standards, a careful reading of the June 4 LAUSD/LASI Informative itself provides evidence that:
- The LAUSD standards present some required topics too late
- The LAUSD standards omit some required topics
- The LAUSD standards are unclear and subject to a wide range of interpretation
- The LAUSD standards omit grade-by-grade benchmarks and contradict the California standards
The LAUSD/LASI Informative includes detailed tables which juxtapose LAUSD Math Standards with California Math Standards. In the first table, LAUSD mathematics graduation requirements are equated with California algebra I standards and geometry standards. These California algebra I and geometry standards are designed to target 8th and 9th grade students, but LASI identifies these as graduation (i.e., 12th grade) standards. This is a clear (though perhaps unintended) admission by LASI and LAUSD staff that the LAUSD standards expect less of students than the California Math Standards. Here is a list of California algebra and geometry standards which, according to the LASI/LASUD Informative, LAUSD requires only AFTER the 9th grade:
1. Students identify and use the arithmetic properties of subsets of integers, rational, irrational and real numbers. This includes closure properties for the four basic arithmetic operations where applicable.
1.1 Students use properties of numbers to demonstrate that assertions are true or false.
3. Students solve equations and inequalities involving absolute values.
4. Students simplify expressions prior to solving linear equations and inequalities in one variable such as 3(2x-5) + 4(x-2) = 12.
19. Students know the quadratic formula and are familiar with its proof by completing the square.
23. Students apply quadratic equations to physical problems such as the motion of an object under the force of gravity.
25. Students use properties of the number system to judge the validity of results, to justify each step of a procedure and to prove or disprove statements.
25.1 Students use properties of numbers to construct simple valid arguments (direct and indirect) for, or formulate counterexamples to, claimed assertions.
25.2 Students judge the validity of an argument based on whether the properties of the real number system and order of operations have been applied correctly at each step.
25.3 Given a specific algebraic statement involving linear, quadratic or absolute value expressions, equations or inequalities, students determine if the statement is true sometimes, always, or never.
7. Students prove and use theorems involving the properties of parallel lines cut by a transversal, the properties of quadrilaterals, and the properties of circles.
11. Students determine how changes in dimensions affect the perimeter, area, and volume of common geometric figures and solids.
13. Students prove relationships between angles in polygons using properties of complementary, supplementary, vertical, and exterior angles.
14. Students prove the Pythagorean Theorem.
17. Students prove theorems using coordinate geometry, including the midpoint of a line segment, distance formula, and various forms of equations of lines and circles.
21. Students graph quadratic functions and know that their roots are the x-intercepts.
The LAUSD/LASI Informative acknowledges that topics of Algebra II and above are not covered in the LAUSD standards. But this admission does not go far enough. Nowhere in any of the tables of the LAUSD/LASI Informative are the following California algebra I and geometry standards assigned counterparts in the LAUSD math standards. These topics are entirely omitted by the LAUSD Math Standards:
9. Students solve a system of two linear equations in two variables algebraically, and are able to interpret the answer graphically. Students are able to use this to solve a system of two linear inequalities in two variables, and to sketch the solution sets.
10. Students add, subtract, multiply and divide monomials and polynomials. Students solve multistep problems, including word problems, using these techniques.
11. Students apply basic factoring techniques to second and simple third degree polynomials. These techniques include finding a common factor to all of the terms in a polynomial and recognizing the difference of two squares, and recognizing perfect squares of binomials.
14. Students solve a quadratic equation by factoring or completing the square.
1. Students demonstrate understanding by identifying and giving examples of undefined terms, axioms, theorems, and inductive and deductive reasoning.
2. Students write geometric proofs, including proofs by contradiction.
3. Students construct and judge the validity of a logical argument. This includes giving counter examples to disprove a statement.
18. Students know the definitions of the basic trigonometric functions defined by the angles of a right triangle. They also know and are able to use elementary relationships between them, (e.g., tan(x) = sin(x)/cos(x), (sin(x))2 + (cos(x))2 = 1).
19. Students use trigonometric functions to solve for an unknown length of a side of a right triangle, given an angle and a length of a side.
20. Students know and are able to use angle and side relationships in problems with special right triangles such as 30-60-90 triangles and 45-45-90 triangles.
The LAUSD/LASI Informative arbitrarily juxtaposes state standards with LAUSD standards in an effort to demonstrate consistency. However, it is evident that the Informative takes great liberties in interpreting the meaning of the LAUSD standards. For example, the LAUSD standard number 28 states:
Identify, describe, compare, and classify geometric figures; apply geometric properties and relationships to solve problems; and use geometric concepts as a means to describe the physical world.
This could mean:
2. Students describe and compare the attributes of plane and solid geometric figures and use their understanding to show relationships and solve problems.
2.1 identify and describe and classify polygons (including pentagons, hexagons and octagons)
2.2 identify attributes of triangles, (e.g., two equal sides for the isosceles triangle, three equal sides for the equilateral triangle, right angle for the right triangle)
2.3 identify attributes of quadrilaterals (e.g., parallel sides for the parallelogram, right angles for the rectangle, equal sides and right angles for the square)
2.4 identify right angles in geometric figures or in appropriate objects and determine whether other angles are greater or less than a right angle
2.5 identify, describe and classify common three-dimensional geometric objects (cube, rectangular solid, sphere, prism, pyramid, cone, cylinder)
2.6 identify the common solid objects that are the component parts needed to make a more complex solid object
However, LASI AND LAUSD STAFF claim that it means:
2. Students compute the perimeter, area and volume of common geometric objects and use these to find measures of less common objects; they know how perimeter, area, and volume are affected under changes of scale.
2.1 routinely use formulas for finding the perimeter and area of basic two-dimensional figures and for the surface area and volume of basic three-dimensional figures, including rectangles, parallelograms, trapezoids, squares, triangles, circles, prisms, cones and circular cylinders
2.2 estimate and compute the area of more complex or irregular two- and three-dimensional figures by breaking them up into more basic geometric objects
2.3 compute the length of the perimeter, the surface areas of the faces, and the volume of a 3-d object built from rectangular solids. They understand that when the length of all dimensions are doubled or tripled, the unit measures are increased by the same factor
2.4 relate the changes in measurement under change of scale to the units used (e.g., square inches, cubic feet) and to conversions between units (1 square foot = 122 square inches, 1 cubic inch = 2.63 cubic centimeters)
3. Students know the Pythagorean Theorem and deepen their understanding of plane and solid geometric shapes by constructing figures that meet given conditions and by identifying attributes of figures.
3.1 identify and construct basic elements of geometric figures, (e.g., altitudes, midpoints, diagonals, angle bisectors and perpendicular bisectors; and central angles, radii, diameters and chords of circles) using compass and straight-edge
3.2 understand and use coordinate graphs to plot simple figures, determine lengths and areas related to them, and determine their image under translations and reflections
3.3 know and understand the Pythagorean Theorem and use it to find the length of the missing side of a right triangle and lengths of other line segments, and, in some situations, empirically verify the Pythagorean Theorem by direct measurement
3.4 demonstrate an understanding of when two geometrical figures are congruent and what congruence means about the relationships between the sides and angles of the two figures
3.5 construct two-dimensional patterns for three-dimensional models such as cylinders, prisms and cones
3.6 identify elements of three-dimensional geometric objects (e.g., diagonals of rectangular solids) and how two or more objects are related in space (e.g., skew lines, the possible ways three planes could intersect)
The LAUSD standard quoted above is a seventh grade standard. The first list is from the state’s third grade standards, while the second is from the state’s seventh grade standards. The LAUSD standard is not particularly better aligned with the state seventh grade standards than with the state third grade standards. LASI has demonstrated in the Informative that the LAUSD standards are not adequately specific.
The LAUSD/LASI Informative admits that, LAUSD has written broad statements at only the selected benchmark grades of third, seventh, ninth, and twelfth. This obvious difference leaves the students of Los Angeles without a road map for their progress – most of the grades are omitted. In addition, the California state mandated STAR exams will not allow calculator usage in the first six grades. This contradicts the LAUSD third grade benchmark which prematurely requires children to use calculators in the early grades, Kindergarten, first, second, and third grades.
The LAUSD/LASI Informative even contradicts itself.
On page 2, the LAUSD/LASI Informative indicates, in a chart, that the LAUSD math standards include topics which the California math standards do not. According to the chart, discrete mathematics is included in the LAUSD standards, but not in the California State standards. But the LAUSD/LAUSD Informative then lists topics in discrete mathematics from the California math standards on the very next page, contradicting its own assertion. The treatment of discrete mathematics in the California math standards is superior to the treatment in the LAUSD standards.
The only remaining standards included in the LAUSD document, but missing in the California math standards, correspond to those noted in A Comparison of the LAUSD Math Standards and the California Math Standards under the heading, “Topics Which Either do not Exist or Have Nothing to do with Mathematics.”
LASI’s attempt to present the LAUSD math standards in a positive light cannot mask the deficiencies. The district’s own comparison actually confirms criticisms found in A Comparison of the LAUSD Math Standards and the California Math Standards.
Why would LASI go to such lengths to defend its obviously inferior math standards? Perhaps it is merely pride of authorship. Perhaps LASI’s hollow claims are dictated by LAUSD’S need to meet or surpass the state standards in order to secure future grant funding. Perhaps the district is trying to justify itself knowing that there will eventually be a review by the state. In any case, however, the district would be well-advised simply to adopt the state standards and be done with it. Afterall, … you can’t fool all of the people all of the time. (Abraham Lincoln, 1864)
LAUSD’s current resistance to the state standards, on the advice of LASI, is likely to cause delays in revising LAUSD curriculum to meet the new state standards. This will undoubtedly contribute to an even worse showing in 1999 on the STAR exams relative to other school districts in California than occurred this year. Other districts have already either aligned their math standards with the state standards or adopted the state standards outright.
LAUSD cannot succeed merely by claiming to have high expectations. In the strongest possible terms, the Los Angeles Unified School District must make it official and clear to all involved – students, parents, teachers, principals, administrators and the public – that district students are expected to meet the California Mathematics Standards. The continuing resistance to adopting the California mathematics standards serves only to save face for LASI and district staff, but it does so at the cost of a proper mathematics education for the children of Los Angeles.