Areas of Increased Emphasis in the Learning of Elementary Mathematics
The belief that "learning is limitless for learners who can solve problems", supports the initiative to have problem solving as a central focus for the curriculum. It is through problem solving that Mathematics can be presented in a meaningful,
applicable, constructive manner for students.
Problem solving, by its very nature, does not necessarily follow a sequential pattern. Students must analyze the problem and combine prior knowledge and experiences into a procedure that will yield success in arriving at a solution.
To establish a problem-solving environment in which students feel motivated and confident, a number of practices:
- collecting a good supply of a variety of quality problems;
- presenting problems orally, visually, and in written form
- using a variety of instructional approaches and student groupings;
- making manipulatives available for students to use to assist in solving problems;
- integrating problem solving with other topics and subject areas;
- discussing various ways to solve a problem and accepting alternate solutions where applicable;
- reinforcing positive attitudes and perseverance; and,
- providing sufficient time - it is better to solve fewer problems and fully explore other possible processes, solutions, and related problems.
Students need to recognize the applicability of mathematics. Problem solving is the link between mathematics in the classroom and the real world.
There are many strategies, processes, concepts, and skills that students should learn and be able to apply in order to become successful, life-long mathematical problem solvers. The following areas of increased emphasis in the teaching and learning of
Mathematics are critical to the overall development of problem solvers who are confident and competent.
Activity-Based Classroom
It is crucial that students develop a positive attitude towards mathematics. Without this positive attitude they will not realize their full potential. They must view mathematics as more than paper-and-pencil assignments with the primary goal of producing correct answers. They must see mathematics as a means to solving a variety of challenging problems. This involves manipulating concrete materials, using pictures and diagrams, collecting and analyzing data, and sharing their mathematical experiences.
Activity-based classrooms usually have:
- an assortment of manipulatives, stored so that they are readily available to both students and teacher;
- a mathematics centre or area where problems, activity cards, manipulatives, displays, and games are kept;
- areas (floor and tables) where students can work cooperatively; and,
- adaptability, to accommodate the various tasks in which students actively participate.
Manipulative Materials
Students at the elementary level best learn mathematical concepts through the manipulation of concrete materials because it assists them in building a mental representation of the concept. Manipulatives provide concrete introductions to abstract ideas.
Each student should have an opportunity to have adequate "hands on" experiences with appropriate manipulatives before engaging in paper-and-pencil activities. All print resources, including textbooks and workbooks, offer only the pictorial and symbolic representation of mathematical concepts. Therefore, it is highly recommended that every classroom have an assortment of manipulatives (purchased, constructed, collected) that are accessible to students at all times.
Students need time for free exploration when each type of manipulative is first introduced. They must have the opportunity to play, experiment, and observe characteristics of the concrete materials. Students should talk to classmates and their teacher (
using appropriate mathematical terminology) about their experiences.
As objects are manipulated and new concepts introduced, teachers must help students make the connections between their actions and the concepts. A gradual transition to pictorial representations and when appropriate to symbolic representations is made.
During this process the appropriate verbal representation is incorporated. Students must eventually understand the relationship between the physical manipulation of materials and the concepts. The way students record their results of manipulative activities often affects their bridging of the gap between the concrete and the abstract.
Although not all objectives need to be introduced using manipulatives, most students will benefit enormously if given the opportunity to proceed from concrete to pictorial to abstract when learning the majority of new concepts.
It is extremely beneficial for students to use a variety of manipulatives when learning a major mathematical concept. This will help to assure that students do not develop a narrow view of the concept.
Many manipulatives can be collected (odds and ends, counters) and/or constructed (10-frames, geoboards) by teachers, students, and parents. Others, (linking cubes, Mira) are best purchased. The storage and distribution of manipulatives is an important consideration for the classroom teacher and many excellent suggestions can be found in various teacher resources.
Mental Calculation
Mental calculation is a life skill that assists in solving many mathematically related problems. For elementary students, developing the ability to calculate mentally helps with understanding basic number concepts and relationships. It improves students
' paper and pencil calculations and eliminates many common errors produced on electronic calculators. Mental calculation is also the cornerstone to all estimation. This mental ability develops the confidence that assures students that they have the skills to quickly solve basic mathematical problems.
The following instructional ideas are suggested to promote development of student mental calculation abilities:
- use mental calculation whenever possible in all areas of study;
- teach a variety of mental calculation strategies as students do not always learn these on their own;
- allow student choice, as a strategy that is preferred by one student may not be the best for another student;
- encourage students to develop, use, and explain their own strategies; and,
- practice in a meaningful setting - present real-life problems, activities, and games in which mental calculations must be used.
Students involved in repetitive paper and pencil exercises often become accustomed to following with minimum thought, pre-determined steps. Mental calculation forces students to think about numbers and number relationships.
Estimation
Mathematics is a discipline that we often characterize by its precision in common usage. However, we do not always need, nor are we sometimes able, to attain a high degree of accuracy in our calculations. Approximate numbers are often easier to comprehend and they can also help to develop consistency. When counting, measuring, or calculating it is often advantageous to estimate prior to finding a more exact solution. Development of the concept and skills of estimation helps students to adapt mathematics in a variety of situations.
The increased emphasis of estimation in the curriculum corresponds with the important role estimation assumes in daily life.
Integration
Mathematics should be applicable and meaningful for all students. Students must understand the need for them to learn the concepts and skills of mathematics. This is best accomplished by teaching a topic or theme that is relevant to students' lives and that offers many and varied opportunities for mathematical applications. When choosing and planning appropriate topics/themes (e.g., bicycles), teachers should attempt to meet objectives from all five strands. This gives them the opportunity to create problems that require students to collect and analyze data, learn geometrical concepts, incorporate measurement understandings and skills, apply concepts and skills of numbers and operations, as well as develop problem-solving abilities.
Teachers may also wish to integrate across subject areas by developing a single theme (e.g., animals) within their classroom. This allows for a more focused planning of units and further demonstrates to students the use of mathematics in daily life.
Calculators
Calculators should be an important contributing factor in students' number development. Quality designed calculator activities and problems enhance the growth and formation of students' understanding of mathematical patterns and relationships. The concepts of computation are usually understood in advance of the mystery of the algorithms. Therefore, with a calculator, more complex calculations can be performed and problems solved. This can eliminate the drudgery and frustration that may hinder rather t
han enhance mathematical thinking and creativity when students are expected to use and master algorithms they are not developmentally ready to comprehend.
All elementary students should have regular access to calculators. The calculator should be used as a tool to:
- formulate generalizations from patterns of displayed numbers e.g., counting, place value;
- assist in the learning of basic number concepts and mental calculation strategies;
- enhance understanding of number order and magnitude;
- assist with solving problems and therefore promote independence; and,
- encourage experimentation with mathematical ideas.
Calculators do not replace the need for the acquisition of basic number facts. Since there is an increasing need to be able to calculate mentally, estimate, and determine the reasonableness of answers to computed problems, students need to learn strategies that assist them with number fact recall.
Students must use calculators regularly if they are to learn how, when, and why they are effective.
Microcomputers
The microcomputer has many beneficial applications in mathematics. It has been identified as a tool for students to use to explore and discover concepts by assisting with the transition from concrete representation to the more abstract mathematical stage
s of learning. Appropriate computer software can be the link between concrete manipulatives and the symbolic representation. Using a combination of manipulatives and corresponding computer software supports the constructivist perspective that learning is a process of constructing, building and fitting ideas, and making connections. By giving students the opportunity to experiment with various representations, we are offering more models with which to make these connections.
Computer hardware and software is continuing to become increasingly sophisticated in design in areas such as problem solving, manipulating, creative programming, games, tutorials, drill and practise, and managing. Within these applications it is crucial that software correlates with the curriculum and the ability levels of the students.
The microcomputer also has tremendous potential to help students with special needs. Students with learning difficulties, those who require enrichment activities, and students in multi-graded classrooms can all benefit from the individual assistance a computer can offer.
Assignments
Assignments, intended to be completed in class or at home, enhance students' understanding, skills, and proficiency in mathematics. But, care must be taken to assure that assignments are meaningful extensions of the concepts taught in class. Repetitive
computation or other similar homework assignments can often inhibit a student's creativity, love of mathematics, and desire to independently extend their learning. Assignments should develop students' higher levels of thinking by being structured in a problem solving mode so that students have the opportunity to apply the mathematical ideas learned.
Parents/caregivers can be significant contributors of this learning process. Opportunities for parents/caregivers to be involved in the data collection and problem solving processes allows them to display their interest in the child's work. It also offers them the opportunity to become familiar with the student's program.