The close correlation between the NCTM Standards and Richard Skemp’s SAIL through Mathematics program is undoubtedly the result of the convergence of two independent, thorough, and insightful explorations to the heart of what is needed for the intelligent learning of mathematics.

Accepting that the NCTM Standards have established a broad framework for guiding needed reform in school mathematics, an examination of some of the ways in which SAIL through Mathematics fits that framework follows.

Mathematics as Problem Solving (Standard 1). The Skemp learning activities are, themselves, problem-solving tasks. The students are led to construct mathematical concepts and relationships from physical experiences designed to appeal to their imagination and to build on their real-world experiences. Students work cooperatively on well-designed, goal-directed tasks, making predictions, testing hypotheses and building relational understandings that facilitate routine and non-routine problem solving.

Mathematics as Communication (Standard 2). The Skemp learning activities are designed to foster communication about mathematical concepts between students and between students and adults. A typical cooperative-group activity, Number targets (Num 2.8/1), engages the students in communicating about place value concepts with physical embodiments, spoken/heard symbols, and written/read symbols, all the while exploring the underlying mathematical meanings and using problem solving strategies to predict their best move.

Mathematics as Reasoning (Standard 3). Using patterns and relationships to make sense out of situations is an integral component of the Skemp learning activities. The activities frequently lead the students to explore, conjecture (make predictions), and test their conjectures. The program builds on relational understanding. Useful instrumental (habit) learning is promoted when appropriate.

Mathematical Connections (Standard 4). Skemp's detailed conceptual analysis of the elementary school mathematics curriculum has produced in SAIL through Mathematics a set of well-defined concept maps or networks. The networks arrange the activities in optimal learning sequences and provide teachers with the framework to make relational connections within and across networks. Many of the activities require assembling previously learned concepts and processes to deal with the task at hand. An entire network (NuSp 1, The number track and the number line) is devoted to number tracks and number lines, which are of importance throughout mathematics, from kindergarten through university-level mathematics, and beyond. They provide valuable support for our thinking about numbers in the form of a pictorial representation. Skemp's unerring notions about contexts that appeal to student imaginations have produced interesting lifelike settings in which the students learn. They compare possible outcomes of the moves they might make in One tonne van drivers (Num 3.10/3), and they are introduced to a budgeting activity in Catalogue shopping (Num 3.10/4). In adult life, planning the use of money and other resources (e.g., time, labour) is one of the major uses of arithmetic. Because of interesting real life situations, connections to other curriculum areas are easily integrated. Feeding the animals (Num 7.2/1) and Setting the table (Num 4.5/2, SAIL Volume 1) are examples of activities which have spin-offs to art and health.

Estimation, Number Sense & Numeration, Numbers & Operations, Computation, Number Systems (Standards 5-7). One entire network of Skemp learning activities, Num 1, treats Numbers and their properties from 'sorting dots' to 'square numbers' and 'relations between numbers.' Another complete network, Num 2, The naming of numbers, carefully builds everything one needs to know about place value for numerals of any number of digits. Networks are devoted to each of the basic arithmetic operations, Addition, Subtraction, Multiplication, and Division. In all of these networks there is emphasis on number sense, operations sense, reasonable estimates, making and testing predictions, mental computation, thinking strategies, and relationships between concepts and between operations . . . all recommended in Standards 5 through 8. Calculators are used when appropriate, as they would be in real-life situations. Activities which use calculators include:

One tonne van drivers (Num 3.10/3), Cargo boats (Num 5.7/3), Treasure chest (Num 5.10/2), Cargo airships (Num 6.8/4), Number targets: division by calculator (Num 6.9/1), Predict, then press (Num 7.8/2), "Are calculators clever?" (Num 7.8/3), and Number targets by calculator (Num 7.8/4).

Geometry, Measurement (Standards 9, 10 [Grades K-4]; 12, 13 [Grades 5-8]). The Space 1 and Space 2 networks cover properties and components of two- and three-dimensional shapes, symmetry, and motion geometry. NuSp 1, The number track and the number line network, develops basic linear measurement skills while providing useful embodiments for the activities developing number concepts and arithmetic operations. The SAIL measurement networks, Meas 1, 2, 3, 4, 5, and 6 treat Length, Area, Volume & capacity, Mass & weight, Time, and Temperature, covering the concepts and relationships of Standards 9 & 10 for Grades K-4 and Standards 12 & 13 for Grades 5-8.

Statistics, Probability (Standards 11 [Grades K-4]; 10, 11 [Grades 5-8]). The prerequisite skills for statistics and probability have been well developed in the networks. Students are introduced to probability from real-life situations. For example, in Crossing (Num 3.2/4, SAIL 1), students roll a die to move their counters up a 10-square number track. A student whose counter is on the ninth square, just needs to roll a '1' to finish. The other child is at square five. Each child rolls the die; the one at square five gets a '5' and wins the game. The first child says, "How did that happen? I was closer to the end." The teacher discusses how probability works when you throw a die. One child may be working on addition: 5 + 5 = 10 whereas the other child is thinking about probability.

Fractions and Decimals (Standard 12 [Grades K-4]; Standards 5, 6 [Grades 5-8]). The Num 7 network, Fractions, begins with the development of real-object concepts of 'equal parts,' 'denominators,' and 'numerators,' building to fully-symbolic treatments of fractions, decimal-fractions, and operations with decimals. The whole development of fraction 'number sense' is firmly grounded in the use of concrete and pictorial models. The relationships between fractions and decimal fractions are carefully developed.

The Standards and Professional Development

In-service courses, based on Skemp's learning theory, have been developed to give teachers opportunities to learn about the theory, to do a selection of activities together, to make them, and to discuss them. Follow-up professional development support has been made available as teachers have incorporated the activities in their classrooms. Thirty-nine VHS video clips of the SAIL activities and of the theory which they embody have been produced as part of that support. The SAIL program is well suited for addressing the NCTM Standards in a thorough, consistent, and well-organized manner.

Each of the SAIL activities embodies both a mathematical concept, and also one or more aspects of Skemp's theory of learning. So by doing SAIL activities with a group of children, both the children and their teacher benefit. By this approach to learning mathematics the children gain well-structured mathematical concepts and processes to support intelligent problem solving; and the teacher has an opportunity to learn about the theory of intelligent learning by seeing it in action. "Theoretical knowledge acquired in this way relates closely to classroom experience and to the needs of the classroom. It brings with it a bonus, since not only do the children benefit from this approach to mathematics, but it provides a good learning situation for teachers also. In this way we get 'two for the price of one,' time-wise." [3]

Notes

1 Skemp, R. R. (1989). Mathematics in the Primary School. London: Routledge. p. 166.

2 National Council of Teachers of Mathematics. (1989). Curriculum and Evaluation Standards for School Mathematics. Reston, Va.: The Council.

3 Skemp, 1989, op. cit., p. 111.