1b. Learning models

Learning models that underpin understanding of maths/numeracy

1. Jean Piaget

Piaget identified four primary stages of development for children: 

1a. Sensorimotor:

• • • mental and cognitive attributes develop from birth until the appearance of language

    • progressive acquisition of object permanence in which the child becomes able to find objects after they have been displaced, even if the objects have been taken out of his field of vision

    • characteristic of children at this stage is their ability to link numbers to objects (Piaget, 1977) (e.g., one dog, two cats, three pigs, four hippos)

  • Developing maths skills in this stage should include:

    • opportunity to act on the environment in unrestricted (but safe) ways in order to start building concepts (Martin, 2000)

    • children at the sensorimotor stage have some understanding of the concepts of numbers and counting

    • solid mathematical foundation by providing activities that incorporate counting and thus enhance children’s conceptual development of number

    • teachers and parents can help children count their fingers, toys, and candies

    • questions such as “Who has more?” or “Are there enough?” could be a part of the daily lives of children as young as two or three years of age

    • enhance the mathematical development of children at this stage with literature - use children’s books that embed mathematical content; books should include pictorial illustrations

    • children at this stage can link numbers to objects, learners can benefit from seeing pictures of objects and their respective numbers simultaneously

    • children’s books will also contribute to the development of their reading skills and comprehension

1b. Preoperational:

• • • an increase in language ability (with over-generalizations), 

    • symbolic thought

    • egocentric perspective

    • limited logic

    • lack of logic associated with this stage of development; rational thought makes little appearance

    • link together unrelated events, sees objects as possessing life, does not understand point-of-view, and cannot reverse operations

    • perceptions in this stage of development are generally restricted to one aspect or dimension of an object at the expense of the other aspect

    • Piaget tested the concept of conservation by pouring the same amount of liquid into two similar containers - when the liquid from one container is poured into a third, wider container, the level is lower and the child thinks there is less liquid in the third container. The child is using one dimension, height, as the basis for his judgment of another dimension, volume, although they saw that there was an equal amount initially.

  • Developing maths skills in this stage should include:

    • while the child is working with a problem talk to them about what they are doing 

    • engage with problem-solving tasks that incorporate available materials such as blocks, sand, and water

    • what you hear and what they do with the materials, will tell you the child’s thought processes

    • child at this stage can understand that four added to five gives nine cannot yet perform the reverse operation of taking four from nine

    • teaching students in this stage of development should employ effective questioning about characterizing objects - students investigate geometric shapes, you could ask students to group the shapes according to similar characteristics - questions following the investigation could include, “How did you decide where each object belonged?  Are there other ways to group these together?”

    • engaging in discussion or interactions with students can develop their discovery of the variety of ways to group objects, helping them think about the quantities in novel ways 

1c. Concrete operational:

• • • remarkable cognitive growth, student's development of language and acquisition of basic skills accelerates dramatically

    • utilize their senses in order to know; they can now consider two or three dimensions simultaneously instead of successively. For example, in the liquids experiment, if the child notices the lowered level of the liquid, he also notices the dish is wider, seeing both dimensions at the same time

    • seriation (ability to order objects according to increasing or decreasing length, weight, or volume) and classification (involves grouping objects on the basis of a common characteristic) are the two logical operations that develop during this stage (Piaget, 1977) and both are essential for understanding number concepts

  • Developing maths skills in this stage should include:

    • importance of hands-on activities cannot be overemphasized at this stage. These activities provide students an avenue to make abstract ideas concrete.

    • “hands-on experiences and multiple ways of representing a mathematical solution can be ways of fostering the development of this cognitive stage”

    • concrete experiences are needed, use manipulatives with students to explore concepts such as place value and arithmetical operations. Manipulative materials include: pattern blocks, Diene's blocks, Cuisenaire rods, algebra tiles, algebra cubes, geoboards, tangrams, counters, dice, and spinners, paper folding and cutting. As students use the materials, they acquire experiences that help lay the foundation for more advanced mathematical thinking. Students’ use of materials helps to build their mathematical confidence by giving them a way to test and confirm their reasoning.

    • allow them to get their hands on mathematical ideas and concepts as useful tools for solving problems

    • important challenge in mathematics teaching is to help students make connections between the mathematics concepts and the activity. Children may not automatically make connections between the work they do with manipulative materials and the corresponding abstract mathematics: “children tend to think that the manipulations they do with models are one method for finding a solution and pencil-and-paper math is entirely separate”

    • For example, it may be difficult for children to conceptualize how a four by six inch rectangle built with wooden tiles relates to four multiplied by six, or four groups of six. Help students make connections by showing how the rectangles can be separated into four rows of six tiles each and by demonstrating how the rectangle is another representation of four groups of six. Providing various mathematical representations allows for the uniqueness of students and provides multiple paths for making ideas meaningful. 

    • opportunities for students to present mathematical solutions in multiple ways (e.g., symbols, graphs, tables, and words) is one tool for cognitive development in this stage

    • while a specific way of representing an idea is meaningful to some students, a different representation might be more meaningful to others

1d. Formal operational

• • • child at this stage is capable of forming hypotheses and deducing possible consequences, allowing the child to construct his own mathematics

    • typically begins to develop abstract thought patterns where reasoning is executed using pure symbols without the necessity of data. For example, the formal operational learner can solve x + 2x = 9 without having to refer to a concrete situation presented by the teacher, such as, “Tony ate a certain number of candies. His sister ate twice as many. Together they ate nine. How many did Tony eat?”

  • Developing maths skills in this stage should include:

    • Reasoning skills within this stage include clarification, inference, evaluation, and application.

    • Clarification requires students to identify and analyze elements of a problem, allowing them to decipher the information needed in solving a problem. By encouraging students to extract relevant information from a problem statement, teachers can help students enhance their mathematical understanding.

    • Inference requires students to make inductive and deductive inferences in mathematics. Deductive inferences involve reasoning from general concepts to specific instances. On the other hand, inductive inferences are based on extracting similarities and differences among specific objects and events and arriving at generalizations.

    • Evaluation involves using criteria to judge the adequacy of a problem solution. For example, the student can follow a predetermined rubric to judge the correctness of his solution to a problem. Evaluation leads to formulating hypotheses about future events, assuming one’s problem solving is correct thus far.

    • Application involves students connecting mathematical concepts to real-life situations. For example, the student could apply his knowledge of rational equations to the following situation: “You can clean your house in 4 hours. Your sister can clean it in 6 hours. How long will it take you to clean the house, working together?”

1e. To note:

• Piaget is criticized for underestimating the abilities of young children. Abstract directions and requirements may cause young children to fail at tasks they can do under simpler conditions (Gelman, Meck, & Merkin, 1986). 

• He has also been criticized for overestimating the abilities of older learners, having implications for learners, SLSOs and teachers. For example, middle school teachers interpreting Piaget’s work may assume that their students can always think logically in the abstract, yet this is often not the case. Although it is not possible to teach cognitive development explicitly, research has demonstrated that it can be accelerated. 

• Piaget believed that the amount of time each child spends in each stage varies by environment. All students in a class are not necessarily operating at the same level. Teachers and SLSOs can benefit from understanding the levels at which their students are functioning and should try to adjust their teaching accordingly. 

• By emphasizing methods of reasoning, the teacher provides critical direction so that the child can discover concepts through investigation. The child should be encouraged to self-check, approximate, reflect and reason while the teacher studies the child’s work to better understand his thinking (Piaget, 1970). 

• The numbers and quantities used to teach the children number should be meaningful to them. Various situations can be set up that encourage mathematical reasoning. For example, a child may be asked to bring enough cups for everybody in the class, without being explicitly told to count. This will require them to compare the number of people to the number of cups needed. 

• Other examples include dividing objects among a group fairly, keeping classroom records like attendance, and voting to make class decisions. Games are also a good way to acquire understanding of mathematical principles (Kamii, 1982). For example, the game of musical chairs requires coordination between the set of children and the set of chairs.

• Scorekeeping in marbles and bowling requires comparison of quantities and simple arithmetical operations. Comparisons of quantities are required in a guessing game where one child chooses a number between one and ten and another attempts to determine it, being told if his guesses are too high or too low. 

2b. Discovery learning 

Discovery learning is learning that occurs when students are required to find out something by themselves. For example, rather than telling students how big a shape is, the teacher asks them to measure the objects to find out the measurements themselves. 

Teachers use discovery learning to achieve three educational goals. 

• they would like learners to recognize how to find out things and think on their own. They would like them to be less dependent on getting knowledge from teachers and acknowledge the conclusions of others in the class. 

• to get the learners to see how knowledge is achieved or gained. Students are enabled to learn by gathering, organizing, and analyzing information to achieve their own conclusions. 

• learners employ their higher order thinking skills. Students analyze, synthesize, and evaluate. 

Developing maths skills in this learning model should include the following:

• The role of the teacher/SLSO is not imparting knowledge but rather creating and guiding classroom experiences in which learners are engaged to discover knowledge

• As learners are making their discoveries, the teacher/SLSO motivates them to think profoundly. 

• Learners are required to acknowledge the challenge of realizing something for themselves rather than requiring the teacher to provide for them answers.

Activity: use discovery learning to undertake a measurement task

All student in pairs or threes have a piece of A4 paper. Without using a ruler, find out what the dimensions of the paper are (length, breadth, perimeter, area). 

Make notes about what they asked, what they considered, how they assisted each other, how they came to their answer.

Discuss terms: informal and formal measurement

5. What does the research say?

Problem-solving and discovery-learning skills not only contribute to better mathematics learning but also enhance students’ creativity to cope with life challenges. Since constructivist approaches give students the opportunity to think creatively, there should be more emphasis on teaching methods which include less lecture, more student-directed classes and more discussions. 

In classes that used the problem solving method, students are more active, they think better, and they have less anxiety for exams. 

In summary, research indicates that students who learn mathematics by problem-solving and discovery-learning methods are more active in comparison with the students under the traditional teacher-centred method. These approaches encourage students to think rationally in their daily life, and enhance their thinking, and reasoning power.  Students are more successful and encouraged when systematic problem solving based on Polya’s approach is incorporated in lessons. These methods prepare students better in solving problems and facing discovery learning.