The goal of incorporating Numeracy into Saskatchewan's K-12 curricula is to develop individuals who can cope confidently and competently with everyday situations demanding the use of mathematical concepts, as well as developing their ability to learn new concepts when necessary. What is desired are students who know how to compute, measure, estimate and interpret mathematical data, know when to apply these same skills and techniques, and understand why these particular processes apply. Further, Numeracy is intended to strengthen students' learning in all school subjects through providing them with grounded understanding of the quantitative aspects of each subject.
Numeracy can be described as the knowledge, skills and appreciations needed for students to understand and utilize mathematical ideas, techniques and applications. Numerate people demonstrate an "at-homeness" with numbers and are able to understand information that is presented quantitatively: in graphs, charts and tables; through reference to percentage increase or decrease; and in timelines. Numeracy involves students in integrating such skills as interpreting quantitative information, performing straightforward calculations mentally, estimating values that are known and unknown, and developing an intuitive knowledge of measurement units. See Figure 3.1 for an overview of some of the skills, abilities and processes which underly Numeracy. It should be noted that this visual representation is not meant to be comprehensive or prescriptive, but rather to be descriptive of this area. This framework can be- helpful in determining those aspects of Numeracy which can be incorporated into a particular subject area. Educators are encouraged to evaluate the usefulness of this framework in designing appropriate learning activities as they become more familiar with incorporating Numeracy into their teaching.
Students need not master all of the skills, abilities and processes represented in Figure 3.1 in order to be numerate. This depends on their individual ability and level of development.
Alternative processes and basic skills are needed for students with severe disabilities, as some students may not proceed beyond simple ordering, one-to-one correspondence or an elementary knowledge of numbers. Conversely, capable students should not be limited by the skills, abilities and processes represented in this figure.
A focus upon Numeracy in Saskatchewan's core curricula is justified by the demands of modern life - both within and outside the workplace. It is also justified by the demands of acquiring an education.
Numeracy can be considered as a way of knowing - one which helps students better understand all of their school subjects. Strengthening students' mathematical abilities by exercising them in other subject areas may offer students another means of comprehending these subjects. Increasing students' opportunities to apply their numeracy skills should result in more students integrating them into their everyday understandings.
An examination of the demands of modern life (such as that undertaken by Hope, 1987) reveals that numeracy skills are drawn upon daily. Numeracy demands encompass a number of routine tasks such as counting objects, reading clocks, counting money and finding the page location of a book. Non-routine tasks such as completing an annual tax return or estimating the amount and cost of new carpeting are often accomplished with outside help. Adults do, however, exhibit flexibility of thinking while performing everyday tasks and tend to select techniques according to a number of criteria, including: the purpose of calculation, the degree of precision and accuracy needed, the size and types of numbers, and (most importantly) helpful contextual cues in the environment. For example, shoppers use a variety of ways to compare prices and take into account contextual cues when they compare the prices of two products of equal size or quantity.
A great variety of mathematical demands are placed on people in contemporary Canadian society (both inside and outside educational institutions); a review of these demands illustrates the general types of abilities and competencies required. These can be discussed under the general headings of: data analysis and interpretation, calculation, and measurement.
Figure 3.1
appreciation and knowledge of numbers and their
interrelationships
C - 3
e.g. pencil and paper methods mental calculation - calculator
<=~> .~
e.g. doorways are about 2 m high milk cartons are usually 1 or 2 litres in capacity the thickness of a dime is about 1 mm
interpreting quantitative information
analysis of
e.g.
—graphs
—charts
—tables
—timelines
—surveys
Computers are forcing more and more members of society to become familiar with numbers and statistics, and in particular the interpretation of quantitatively expressed information. Most government agencies and business groups produce statistics. Business managers, administrators, public servants and consumers need to be sufficiently numerate to interpret the data that are now routinely used for decision making. Consumer reports, cost of living indexes and surveys are examples of data-oriented messages that people receive and need to process. Adults see graphs, tables, charts, scales and dials almost every day and are expected to synthesize information from them. In order to function in an information society, it is essential for students to learn how to read and interpret quantitative information which may be presented in tables of figures or visually displayed in graphs.
One of the most common mathematical tasks carried out by adults is that of calculation, using pencil and paper methods, calculators, mental calculations and estimations. Adding, subtracting, multiplying and dividing with whole numbers are done frequently, but rarely with large numbers. Multiplying and dividing usually involve only one-digit factors and divisors. Adults use a wide variety of methods for such tasks; these methods are usually either self-invented or taught by other adults.
Calculations involving decimal fractions to two places are common in relation to money - especially addition and subtraction. The use of percentages is widespread in laboratories and offices; again, most uses involve money. Since the changeover to metric few common fractions are seen, but recognizing fractional parts such as halves and quarters is needed by most people. Common fractions almost always have denominators that are powers of 2, such as 2, 4, 8 and 16. The ability to calculate with whole numbers and fractions increases students' understanding in subject areas such as social studies (e.g. calculating the population density or GNP of a country).
In order to increase speed and accuracy, and ultimately productivity, most employers encourage the use of calculators for the tasks described above. Calculators are faster, and in many instances, more accessible than computing with pencil and paper. Opposition to the use of calculators in classrooms or workplaces may be based on the fear that students will not gain
knowledge of basic calculation skills or that workers will fail to notice errors produced by the inappropriate use of calculators. These difficulties can be largely overcome by instruction in the optimal use of calculators, supported by instruction in estimating approximate answers in order to check the accuracy of calculations.
In a large number of situations, there is no choice but to estimate because exact values are not obtainable. Predictions about the future, guesses about the past, and estimates of military strength would fall into this category. Sometimes estimates are easier to use and are clearer than the more precise actual figures. For example, a newspaper headline may state that the population of a city is 30 000 people, instead of using the actual figure of 29 876. Further, the use of estimates usually gives consistency. Government reports on the percentage of people unemployed give that percent to the nearest tenth and sports columns customarily round off a winning "percentage" to the nearest thousandth.
The ability to estimate is a highly valued work skill and can take several forms. Numerical estimation is used to detect calculation errors produced by calculators or pencil-and-paper methods. Another form of estimation involves measurement, as some decisions often have to be made based on visual inspection. Although skills of measurement estimation develop further on the job, it is essential that workers demonstrate an intuitive knowledge of or "feel" for measurement units. Students' understanding in the Required Areas of Study and other subjects is increased when they are able to estimate the accuracy of calculations or the length of time a particular assignment, such as an essay, will require.
A knowledge of measurement is vital. Even adults who do not take an active part in the measurement process need to be aware of the meaning of measurement, as they are likely to be consumers of measurement information - ordering supplies, calculating profit margins, or evaluating the success of a program.
Understanding in school subjects such as science also require that students have a feel for measurement, understand the options presented by specific measurement situations and take into account their margin of error.
In the future, the numeracy demands of adult life will be less concerned with complex pencil-and-paper calculations, since technology will continue to replace such skills. The focus of Numeracy will shift to other abilities, such as:
| — | interpreting quantitative information; |
| — | performing straightforward calculations; |
| — | estimating values that are known and unknown; and |
| — | developing an intuitive knowledge of measurement and measurement units. |
Regardless of the future advances in technology, all members of our society should be sufficiently numerate to be able to do the following tasks independently and with a minimum of intellectual effort:
| — | wise consuming (e.g. reading prices, calculating costs); |
| — | homemaking (e.g. measuring, reading gauges); |
| — | reading, interpreting and evaluating quantitative information found in newspapers, popular magazines and other publications; |
| — | using and working with common technological devices (e.g. cars, calculators); |
| — | participating in and enjoying leisure activities, games and sports; and |
| — | adapting to new mathematical demands as needed (e.g. learning new employment skills, using instruction manuals). |
School subjects will continue to pose quantitative as well as qualitative demands upon students' learning. Thus the knowledge, skills and values of Numeracy are necessary to support present learning and future life. How teachers can support the achievement of Numeracy is the focus of the discussion which follows. See Table 3.1 at the end of this chapter for an overview of some of the student and teacher behaviors that might be seen when developing Numeracy.
Mathematics teachers will continue to be primarily responsible for developing an understanding of mathematical concepts, rules, principles, relationships and procedures, but Numeracy must also be supported in other subject areas. Almost all subjects can include some meaningful opportunities for students to apply mathematical knowledge. Most students in the elementary and middle grades learn best if they are involved in meaningful activities that require the use of manipulative materials. Secondary students also benefit from hands-on experiences with instruments and tools. A great deal of discussion and work with concrete materials should precede any introduction of symbolic abstractions. Opportunities to support the achievement of Numeracy can be discussed under the general headings of: problem solving, calculation, measurement, space and shape, and data analysis and interpretation.
If students encounter only the simplest of problems, or use formal procedures they don't. understand when solving problems, their approach to problem solving can become mechanical. For example, an 11 year-old student explained this approach to solving problems: "Problem solving is easy. If there are more than two numbers, I always add; otherwise, I subtract. If I'm not sure if it's multiplication or division, I divide and, if there is a remainder, I multiply instead" (Hope, 1987, p. 57).
When students are given problems to solve, instruction should focus on the understanding of the problem statement, the analytic tools or general problem-solving strategies that might be profitably used, and the assessment of the reasonableness of a solution. Even after a solution has been determined and discussed, analysis of the problem should continue. Students could discuss other methods of solution, make any generalizations, and show how other problems could be created by changing some of the features of the original problem. A few problems studied intensively contribute far more to developing the ability to solve problems than many problems treated superficially.
It is important to include examples of real-life problem solving in all school subjects, especially those problems that require continual reformulation, involve the collecting of information, and allow students to try a number of methods of solution - including ones that may not appear to be mathematically elegant.3 To discuss and solve real-life experiences, a teacher of second-language and other special needs students must be well aware of the level of language competency of students and then direct discussion through appropriate questioning in order to make use of what students know.
By participating in practical activities, students develop their abilities to carry out routine tasks and approach non-routine problems with some confidence that solutions can be found. Experiences with real-world problem solving can begin as early as Grade 1, using simple examples such as deciding how many sheets of paper are needed by the class for an art project. Older students can estimate the amount of food or money needed for a 3-day class trip. A school subject such as social studies can provide students with the opportunity to participate in examining solutions to real-life problems, such as inner-city poverty or where to locate a community recreation center.
Opportunities for students to practice and apply calculation skills can be found in all school subjects. In the Required Areas of Study, students can be encouraged to select the most appropriate means of performing any necessary calculations from pencil-and-paper methods, calculators, mental calculations, and estimations. Although not a replacement for thinking and reasoning, the hand-held calculator can provide teachers with additional time to help students become numerate. It is critical, as well, that students learn to distinguish between appropriate and inappropriate uses of calculators. "Just as literate people do not rely entirely on a dictionary to carry out their daily communication tasks, numerate people do not rely entirely on a calculator to carry out their computational tasks" (Hope, 1987, p. 70).
Despite the obvious advantages of using calculators, the ability to calculate mentally and to provide quick estimates of the accuracy of a calculation, whether produced electronically or manually, continues to be extremely useful. Since the great majority of everyday problems are solved mentally, it is important that students are taught a variety of ways to calculate in their heads. Once the students understand how and when to apply a mental calculation technique, practice is needed to increase speed and accuracy.4 Students also require considerable help and practice if they are to learn to distinguish between situations calling for rough estimates or exact answers.
Teachers can develop students' intuitive notions of mathematical concepts through practical activities with varied materials. Students can perceive the attributes of mass (weight) by comparing the differing weights of objects balanced in their hands. By pouring liquids from one container into another, students can observe the properties of volume. Direct experiences related to measurement ideas can lead students to realize that area requires a "covering" of surfaces and perimeter forms a "boundary" around those surfaces. If students have an intuitive notion about what they are measuring, they will be able to distinguish between--and be less likely to confuse - measurement processes which appear to be similar. Unconventional instruments and non-standard measurement units, such as feet or hands, can be used initially to help students develop an understanding of the measurement process.
Students of all ages benefit from experiences in carrying out measurements using a variety of conventional measurement tools. Providing students with extensive practical work ensures that they will have less difficulty in learning to use and select tools and instruments encountered outside of school. Whenever measuring takes place in any subject area, it should be done primarily for the purpose of gathering data needed for the solution of a problem, with some discussion about precision and error and their effects on computation.
Perhaps the most significant contribution schools can make to a student's understanding about the nature of measurement is the establishment of a wide variety of everyday equivalencies that can be used whenever measurement units are being considered. For example, if students know that doorways are usually 2 m high, then students can be asked to compare the length or height of people or objects in relation to the standard height of a doorway.
Most subject areas can contribute significantly to the development of a measurement sense. Whenever a country is being studied in social studies, its size can be compared with the size of other geographical areas familiar to the students. For example, a chart portraying a map of Saskatchewan can be overlaid with an outline of the country of France to demonstrate comparative size. In the physical sciences, where measurement is commonplace but the units are not, teachers can facilitate students' understanding of units such as light years, pascals, joules, newtons, milliamperes and nanometres by providing them with a concrete reference. For example, a dollar bill lying flat on a table exerts a pressure of about 1 Pa.
It is the responsibility of all subject areas to help students to think and estimate with metric units. All measurements and measuring tools seen in school should continue to involve metric units, especially in the elementary and middle grades. As students need considerable practice in converting from one metric measurement unit to another, teachers can provide opportunities for students to convert from cm to mm, and back again.
Although it is important for students to learn the terminology used to describe geometric shapes (i.e. diagonal, perimeter, square and radius), the focus of instruction should be on the properties of plane and solid objects and how such knowledge can be applied to real-world situations. Spatial concepts are developed best through kinesthetic experiences. Plane and solid shapes should be available for pupils to handle, measure, and use in building. Students can learn a variety of ways that plane and solid objects can be constructed from simple materials or drawn with conventional instruments. All students can learn to use such drawing instruments as a ruler or compass.
Students can also learn how to read a variety of scale drawings and plans, make reasonably accurate sketches of simple or complex structures, and occasionally construct a full or scale model using a plan or drawing. Such activities can help students learn spatial concepts and relationships, use appropriate terminology, and improve their ability to visualize.
Knowing how to read and interpret quantitative information presented in any number of forms is essential for functioning in an information society. At a minimum, students need a facility with the base ten number system and an understanding of how symbols can be used to represent quantities. Emphasis in all subject areas should be on the practical handling of numbers, such as interpreting graphs, using an instructional manual or reading prices. Structured concrete materials - such as base 10 arithmetic blocks, bundling sticks, place value charts and containers, abacuses, money and metric measuring instruments - should be used to develop an understanding of decimal place value relationships.
An important aspect of data analysis is the reading and interpretation of information organized in tables and charts or displayed as graphs. To develop a facility with reading these types of data presentations, it is important that almost all school subjects contain some instruction on reading and using tables, charts and graphs commonly seen in a particular area of study.
One of the most important areas of Numeracy utilized in our technological society is statistics. Statistics refers broadly to the collection, analysis and interpretation of numerical data. Because of its importance, students should become familiar with rudimentary statistical notions; in all school subjects students should be encouraged to critically examine any statistics cited as evidence for an argument or used to express a point of view. It is important that students learn to withhold judgments about any argument based on statistical information until they find out how this information was obtained.
The study of elementary probability should emphasize its use in estimating risks, chances, odds and proportions. Students should see how probability is used, as in forecasting weather conditions, in estimating risks to the environment from chemical pollutants, or in determining the chances of winning a lottery. In all subject areas an attempt should be made to help students understand common expressions and uses of probability.5
To summarize the theory and practice of promoting Numeracy:
| — | learning in the content areas should promote the ability of students to function effectively in quantitative situations; |
| — | students require opportunities to encounter, solve and discuss real-life experiences utilizing quantitative information in all of the Required Areas of Study and other school subjects; |
| — | it is important to develop students' intuitive knowledge of mathematical concepts and techniques through practical experiences involving a variety of concrete materials. |
Table 3.1
Numeracy
An overview of some of the student and teacher behaviors that might be seen when developing Numeracy.
| Teacher Behaviors | Student Behaviors |
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1. The discussion of Numeracy in this chapter makes extensive use of the work of Jack Hope (Numeracy, Saskatchewan Education, 1987).
2. Through questioning and discussion, students' thinking can be explored with regard to calculating, measuring and problem solving. By talking about their thought processes, students can share how they "see" a problem and discuss alternate strategies.
3. See Lesh (1985) for an example of such a problem.
4. See the Practical Resources at the end of this handbook (Appendix A) for a list of publications for mental calculation and estimation.
5. See the Practical Resources at the end of this handbook (Appendix A) for a list of publications that suggest real world uses of statistics and probability.
6. See Schoen and Sweng (1986), NCTM Yearbook, for further suggestions related to estimation.
7. See Schulte and Smart (1981), NCTM Yearbook, for further suggestions related to statistics and probability.