What Makes the SAIL program different?
Most of the 700 Alberta classroom teachers who have taken inservice courses introducing the mathematics learning activities found in Richard Skemp's SAIL program have become thoroughly convinced of the soundness of the materials by trying them out with elementary school children (after discussing them with peers and while becoming familiar with the theory which the activities embody). Short of viewing some of the 31 video taped vignettes that have been produced at the University of Calgary (including activity introduction demonstrations by Richard Skemp or Marilyn Harrison with children of appropriate ages, sample wholeclassroom activity management sequences, and an hourlong round table discussion of the theorybased activities by Skemp and five Canadian mathematics educators), it is difficult to appreciate just how the activities, and the mathematics for which they are the vehicles, come to life in the hands of children and their teachers. What follows is a description of some of the unique features of the SAIL program.
SAIL Design Strengths
An undeniable strength of the SAIL program is that it has been designed and field tested by an internationally renowned learning theorist and mathematics educator, Professor Richard Skemp, in cooperation with outstanding classroom practitioners in several countries. Embodying Richard Skemp's pioneering theory of intelligent learning in practical, readytouse classroom activities, it is the product of thorough development, field testing, and refinement with schoolchildren and their teachers in British and Canadian schools. What other elementary school mathematics program could lay claim to the following?
 completely integrated and wellstructured, spanning Kindergarten through Grade 6
 exquisitely detailed and consistent, having been written entirely by an eminent scholar, yet thoroughly fieldtested over many years in schools with children and refined in collaboration with knowledgeable teachers
 developed from basic principles by an author whose books on the psychology of learning mathematics are internationally respected
 winsome empathy with children and teachers so that the learning activities are not only sound mathematically but also universally appealing to children while successfully immersing them in real mathematics
 generous investment of time and talent by the author and his Canadian collaborators in inservice courses designed for and delivered to more than 700 teachers over a period of 7 years
 unprecedented collaboration between an author of international stature and North American educators to ensure thorough implementation by the SAIL program of the NCTM Standards for elementary school mathematics.
Somewhat paradoxically, some of the strengths of the SAIL program are virtually invisible but nevertheless extremely powerful. For example, three important invisible components in the SAIL activities are the mathematical content, learning via schema construction, and structure.
In a sense, the mathematical content is the least invisible component. Nevertheless, unlike typical elementary school mathematics programs, the SAIL program frequently encourages teachers to reflect on the large amount of mathematics which the children need to do while they are engaged in a given activity. This is often very illuminating but not likely to occur fortuitously. Because adults may well be able to do the mathematics very easily, perhaps even intuitively, it is easy to overlook the mathematical content component (and even easier when only reading a description of the activity procedures).
At the heart of Skemp's theory of intelligent learning and reflected in the design of the activities are three modes of schema construction. Schemas, connected groups of ideas, are constructed at the Mode 1 level by building from physical experience and testing to see whether predictions are confirmed. Mode 2 schema building proceeds via communication and testing through discussion. Mode 3 schema construction involves mental creativity and testing of new ideas for consistency with what is already known. The SAIL activities are specifically designed to foster all three modes of learning.
The third invisible component, structure, is particularly important for learning mathematics with understanding. To understand something is to relate it to what is already known (an existing schema). With the right prerequisite knowledge, intelligent learning is possible. The new knowledge becomes part of the schema to which it has been assimilated, the schema expands, and new things can be learned with understanding which could not have been understood before. Without the prerequisite knowledge, intelligent learning cannot be applied and the result is rote learning, or no learning at all.
A very important part of teaching for intelligent learning is a conceptual analysis of the subject matter to ensure that intelligent learning is continually facilitated. Just such an analysis has been carried out by Richard Skemp for teachers using the SAIL program, the results of which can be seen, for example, in the concept maps in SAIL Volumes 1 and 2. In one sense the concept maps illustrate an arrangement of activities that embody just one new concept at a time, building upon the schemas which the learners already have. In another sense, the concept maps illustrate just how flexible the SAIL program is, while preserving the overall mathematical structure. Many different routes can be followed through any of the concept maps, depending on prerequisite understandings demonstrated by the learners. Accordingly, a mathematical program can be tailored to the needs of any student or group of students. On the other hand, experience with teachers has made it clear that there is a real concern that the curriculum be covered (and be seen to be covered) and that any assistance that could be given regarding daytoday sequencing of activities is much appreciated by classroom teachers. Marilyn Harrison's response to these concerns can be seen in the suggested Sequencing Guides for Kindergarten (Pregrade Level 1) and Grade Levels 1 through 6 provided on foldout flaps on the covers of SAIL, Volume 1 and on the final pages of SAIL, Volume 2. Skemp's mathematical/psychological sequencing and the topic placement in exemplary North American programs of study have been seamlessly blended in the SAIL Sequencing Guides and in the SAIL Progress Records.
SAIL Evaluation
Interviewbased cognitive response assessments of Grade 1 and 2 students exposed to 6 weeks of Skemp subtraction activities have been compared with those of students following a conventional textbookbased program. Before the experimental treatment began, the cognitive response distribution of the Skemp Grade 1 students was not significantly different from that of the Textbookbased Grade 1 students. At the end of the subtraction unit, the cognitive response distribution of the Skemp Grade 1 students was not significantly different from that of the Textbookbased Grade 2 students. The Grade 1 and Grade 2 data showed marked shifts from Early Concrete responses to Late Concrete during the brief experimental treatment. For those students who were ready, exposure to the subtraction learning activities was associated with marked overall cognitive response level gains.
A master's thesis completed by Helen Gardner, a British primary school advisory teacher, reported and classified seven distinct levels of strategy developed and exhibited by children engaged in a single Skemp activity, Slippery Slope . . . a promising alternative assessment of mathematical ability that uncovered a wealth of information not available from the standardized tests administered.
A study completed in a Canadian high needs school in November 1993 on "The Effects of Skemp Cooperative Learning Activities on the Ability of Grade 2 and 3 Students to Count by 2's and 5's" showed a significant difference favouring students using Skemp activities when their countingtask performance was compared with that of students using a traditional textbook approach.
Sample SAIL Feedback from Classrooms
Teachers regularly share interesting and revealing anecdotes about their students' insightful experiences as they explore Skemp's mathematics learning activities. Here is a brief sampling.
Connections: Having learned previously via Skemp activities that addition is commutative and subtraction is not, a Grade 3 student doing a beginning multiplication activity exclaimed: “Oh, so multiplication is commutative, too!”
Appreciation of understanding: Having experienced Skemp’s approach to long division, a class of Grade 5 students asked their teacher if she wanted them to do a succeeding division assignment the ‘old way’ or the ‘intelligent way.’
Accommodating individual differences: Playing the SAIL activity, Crossing (Num 3.2/4), some Grade 1 students were rolling a die to move their counters up a 10square number track. One student whose counter was on the ninth square, just needed to roll a ‘1’ to finish. The other child was at square five. Each child rolled the die; the one at square five got a ‘5’ and won the game. The first child said, “How did that happen? I was closer to the end.” The teacher discussed how probability works when you throw dice. In an activity like this, one child may be working on addition (e.g., 5 + 5 = 10) while another other child is thinking about more advanced concepts and applications or related ideas.
Evaluation: After using some Skemp activities with her students, a veteran Calgary Kindergarten teacher/consultant remarked, “It’s as though the children's thinking was out there on the table.” She knew just where they were and what they needed next.
Numeration: An urban Administrator/Resource Teacher has reported that a Grade 3 teacher's regular class has demonstrated remarkable gains in understanding and applying numeration concepts when using a combination of Skemp numeration activities and a currently authorized textbook. When they wrote a systemwide mathematics achievement test, 80% of the class scored at or above the 70th percentile in Numeration (60% at or above the 80th). The other Grade 3 students in the school, who represented a comparable ability range but had used only the authorized textbook, scored between the 30th and 50th percentiles in Numeration
