Analysis of Mathematics Standards from APEC: Core Content Expectations
From APEC HRDWG Wiki
Contents |
Introduction
APEC used this particular portion of the analysis to describe what topics economies address in their standards, the extent to which those topics are common across most or all of the economies, and the proportion of economies' standards that is made up of the common topics.
APEC set out to examine the core content included in the standards across the different economies in order to determine the extent to which there is commonality. To do this, the topics were analyzed by grade level or span. Although there is variation across economies, a number of topics were found to be common across economies. The data was divided into three grade level spans: primary (grades 1-6); lower secondary (grades 7-9); and upper secondary (grades 10-12). Although the standards vary in detail and are introduced and emphasized at different grade levels, there an identifiable set of common topics exists across most or all of the economies participating in this study at each grade span.
Common Topics across Economies by Grade Span
It was found that there is a set of topics at each grade span that are common across the economies. The topics included at each grade span are listed below alongside the percentage of economies addressing that topic in their standards.
Primary School: Grades 1-6
The primary school standards introduce essential basic concepts and skills.
- Numbers: Most economies expect students in grades 1-6 to learn number sense and operations with whole numbers, fractions, decimals and percents, as well as how to order and compare such numbers. In addition to the basic operations, standards at this level address basic number theory concepts.
- Measurement: Economies expect primary students to learn common measures, units, conversion between untits and estimation of measurements. The standards also tend to cover perimeter, area, volume and calculation of these measurements for a variety of shapes and figures.
- Geometry: The set of common topics includes the properties and classification of two- and three-dimentional shapes, facilitated by study of angles, parallelism and perpedicularity.
- Data: Basic data collection and the representation and interpretation of that data in a variety of formats.
Lower Secondary School: Grades 7-9
Economies share more topics in common at this level than in the primary and upper secondary school levels.
- Numbers: Expand beyond whole numbers, fractions, decimals and percents to include treatment of integers. Economies commonly call for students to conduct multi-step operational problems with rational numbers, requiring the application of order of operations and absolute value. They are expected to round numbers and work with significant digits.
- Geometry: At this level, many economies include line and coordinate geometry, the Pythagorean Theorem and its applications, transformations and congruence. Proportionality is futher developed.
- Algebra: The set of common topics expands to include functional relationships and their graphs, with a focus on linear functions. Students are expected to be able to solve linear equations and their systems, and are also expected to master simplification and factorization skills.
- Data: These overlapping topics reflect a greater degree of sophistication in the types of plots and graphs students must construct and interpret. Economies tend to cover the uses and misuses of statistics, as well as basic concepts in probability.
Upper Secondary School: Grades 10-12
- Geometry: Although the number of Geometry topics in common decrease at this level, the few topics carried over are joined by the expectation that students be able to understand equations of lines in a plane. Right triangle trigonometry and slope in line graphs also appear in this grade span, linked with addition of trigonometric functions in algebra.
- Algebra: Economies tend to place emphasis on the relationship between functions and equations. The content moves beyond linear functions and the solution of trigonometric functions and the solution of quadratic and polynomial equations.
- Data: The topics economies emphasize in this strand suggest a tendency to maintain a focus on data representation and interpretation, but to apply it to more sophisticated types of plots and graphs. Data topics here include counting principles, such as permutations and combinations.
Summary of Common Topics
This analysis indicates that at the Primary School level, there is a robust set of common topics that includes and emphasis on Number Sense, Number Operations and Measurement, which provide students with foundational knowledge and skills they need to be successful in other domains of mathematics, such as algebra. Measurement and geometry concepts provide foundational knowledge and skills that students can then apply in more sophisticated and abstract contexts later in their schooling. The few algebra and data concepts in the common topics at the primary level serve as foundations upon which greater sophistication is built at the lower and upper secondary levels. By the upper secondary level, the set of common topics has decreased, likely as a result of a greater number of curricular choices for students.
Featured Economy: New Zealand’s focus on Data Representation
Unlike most economies, New Zealand devotes about a third of its standards to statistics at every grade span. Each pass through statistical content emphasizes the statistical enquiry cycle, placing individual tasks and skills in the context of a larger process of research and discovery. By 5th grade, the standards indicate that students are "gathering, sorting, and displaying multivariate category data, discrete numeric data and simple time-series data to answer questions." At 10th grade, students are planning and conducting their own surveys and experiments. By the end of secondary school, they have critiqued and refined the process of statistical enquiry using margins of error, experimental randomization schemes, data modeling and more. These expectations are considerably different from other economies' expectations of their students, not only in the level of mastery expected, but in the consistent focus across all grade levels on data.
Topics that persist across grade spans
It was found that some of the common topics are covered across economies in more than one grade span. The table below shows the common topics, as well as the percent of economies that address those topics in each grade span.
The following table illustrates how two topics from Table 8-- "representing data" and "circles and their properties"-- increase in complexity in the standards of four economies.
As Table 9 shows, each economy emphasizes collecting data on issues that pertain to daily life in the standards for the early grades. All four economies expect their students to master a collection of graph types including line, bar, pie and histogram by the end of the lower secondary grades and to continue toward more challengin explorations of data in the upper secondary grades.
Similarly, the standards covering the topic of circles and their properties show a pattern of both diverse content and increased depth across the grade spans. Each approach offers a slightly different template for guiding students from the fundamentals through the finer points of an important topic as they move from childhod toward the adult world. In each case, however, topics covered deepen in complexity over the grades.
Common Topics as Proportion of Economy Standards at Different Grade Spans
Having identified a set of topics that most economies address in common, it was then possible to determine what proportion of the content addressed in each economy's standards is comprised of that set of topics. Table 10 below indicates overall overlap between standards and common sets of topics.
These data reflect the fact that on average the proportion of the content addressed by the economies in their standards that are from the set of common topics decreases as th egrade levels progress. Looking more specifically at the upper secondary level, as Table 11 shows, this trend continues.
Conclusion
The primary goals of this analysis was to identify and describe expectations for material students should know and practice that are common across the participating economies as well as to determine the extent to which these common expectations reflect a substantial portion of the standards developed by each economy. Common expectations provide a benchmark that economies can use to review their own standards. Where an economy’s standards depart from this international benchmark, for example, that economy may wish to consider whether changes are indicated. At the same time, economies may also want to pay attention to instances in which only one or two economies value a particular set of knowledge or skills. In some cases, these outliers may be signaling skills that will take on increased importance in the global economy, such as New Zealand’s relatively greater emphasis on probability and statistics in its mathematics standards.
