Bruner's Theory

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Learning Theories

An effective teacher needs not only a high level of competence in the content area but also considerable knowledge of how students learn. For example, use of hands-on teaching strategies, technology, manipulatives, cooperative group work, individual tutoring, and other techniques can contribute to helping every student learn the mathematics
Richard Skemp, who wrote a popular book entitled The Psychology of Learning Mathematics (1971), stated, “[P]roblems of learning and teaching are psychological problems, and before we can make much improvement in the teaching of mathematics, we need to know more about how it is learned” (p. 14).

एक कुशल शिक्षक हुनको लागी बिषयबस्तुमा उच्च स्तरको दक्षता मात्र नभई विद्यार्थीहरूले कसरी सिक्छन भन्ने कुरामा पनि पर्याप्त कौशलता चाहिन्छ। उदाहरणका लागि, use of hands-on teaching strategies, technology, manipulatives, cooperative work, group work, individual tutoring जस्ता शिक्षण सिकाई विद्यार्थीलाई गणित सिक्नमा मद्दत गर्न सक्छ ।
यसै सम्बन्धमा Richard Skemp ले “The Psychology of Learning Mathematics (1971)” मा उल्लेख गरेका छन की “[P]roblems of learning and teaching are psychological problems, and before we can make much improvement in the teaching of mathematics, we need to know more about how it is learned” (p. 14).

Bruner (1915−2016)
Bruner was born blind on October 1, 1915, in New York City, to Herman and Rose Bruner. An operation at age 2 restored his vision. He received a bachelor's of arts degree in psychology, in 1937 from Duke University, and master's degree in psychology in 1939 and then a doctorate in psychology in 1941 from Harvard University.
Bruner theorized that learning passes through three stages of representation—inactive, iconic, and symbolic. His theory has led to the extensive use of hands-on materials and manipulative in mathematics classrooms.
अक्टोबर १, सन १९१५ मा न्यूयोर्क शहरमा हर्मन र रोज ब्रुनरको घरमा जन्मिएका ब्रुनर, जन्मदै अन्धा थिए। २ वर्षको उमेरमा शल्यक्रियाले उनको दृष्टि पुनर्स्थापित गर्यो। उनले सन १९३७ मा Duke University बाट मनोविज्ञानमा स्नातक डिग्री, सन १९३९ मा मनोविज्ञानमा स्नातकोत्तर र त्यसपछि Harvard University बाट सन १९४१ मा मनोविज्ञानमा डक्टरेट प्राप्त गरे। उनले सिकाईको प्रकृया तीन चरणहरू : inactive, iconic, and symbolic बाट हुन्छ भनी सिद्धान्त प्रतिपादन गरे। उनको यस सिद्धान्तले गर्दा आजभोली गणित कक्षाकोठामा hands-on materials and manipulative को व्यापक प्रयोग हुन थालेको छ ।
  1. inactive
    Bruner’s theory postulates that learning begins with an action—touching, feeling, and manipulating. It is difficult, if not impossible, to have a meaningful conversation about apples if one has never actually held an apple. This first stage is the inactive or concrete stage.
    ब्रुनरको सिद्धान्त अनुसार सिकाइको सुरुवात action बाट हुन्छ- जस्तै touching, feeling, and manipulating । जस्तै यदि सिकारुले वास्तवमा कहिल्यै स्याउ देखेको छैन भने स्याउको बारेमा अर्थपूर्ण सिकाई गर्न गाह्रो छ, लगभग असम्भव जस्तै छ। त्यसैले यो प्रारम्भिक चरण हो जसलाई inactive or concrete stage भनिन्छ ।
  2. iconic
    The second stage of learning, according to Bruner, is the iconic or pictorial phase. This stage depends on visuals, such as pictures, to summarize and represent concrete situations so that learners use drawings, infographics, flowcharts, pictures, charts, or graphs to understand and organize information, bridging the gap between concrete experiences and abstract reasoning, which help in grasping concepts that cannot yet be fully verbalized.
    ब्रुनरका अनुसार सिकाइको दोस्रो चरण iconic or pictorial चरण हो। यो चरणमा धारणा बुझ्न र व्यवस्थित गर्नको लागी तथा ठोस परिस्थितिहरूको प्रतिनिधित्व गर्नका लागि चित्रहरू वा दृश्यहरूको प्रयोग गरिन्छ जस्तै सिकारुले drawings, pictures, graphs, infographics, charts, flowchart आदी को गरेर ठोस अनुभवहरू र अमूर्त तर्कहरू बीचको सिकाईलाई पुरा गर्छन । जसबाट सांकेतिक सिकाईको आधार तय हुन्छ।
  3. symbolic
    Bruner’s third stage of learning is the symbolic or abstract phase. It is called symbolic because the words and symbols are representing information so that learners represent ideas through words, numbers, or other abstract symbols, enabling complex thinking and problem-solving, abstract reasoning, language comprehension, mathematical operations, and logical analysis.
    ब्रुनरको तेस्रो चरणको सिकाइ symbolic or abstract phase हो। यसलाई symbolic भनिन्छ किनभने शब्दहरू र symbol हरूले धारणाको प्रतिनिधित्व गरिरहेका हुन्छन् ताकि सिकारुले शब्दहरू, संख्याहरू, वा अन्य अमूर्त प्रतीकहरू मार्फत विचारहरू ब्यक्त गर्छन्, जटिल सोच, समस्या-समाधान, अमूर्त तर्क, भाषा, गणितीय क्रियााहरु, र तार्किक विश्लेषण सक्षम गर्छन्।

According to Bruner, abstract symbol actually does not have any inherent connection with information. For example, the numeral 3 has no meaning in and of itself and takes on meaning only if we have first held three of something in our hands and worked with pictures of things arranged in groups of three. Similarly, asking students to visualize an angle measuring 45° assumes that they have had experience with drawing angles and measuring the angles with a protractor.

Therefore, it is necessary, according to Bruner’s recommendations for sequencing teaching episodes, that learners progress through all three stages: inactive, iconic, and symbolic.
According to Bruner, if students, in general or in regular basis, make the mistake of thinking, for example: \(3x + 6x\) is \(9x^2\), then it is logical for the teacher to take the class back to the concrete or pictorial stage of representation for intervention.
Example 1: number addition: Bruner's theory
Bruner's theory of cognitive development emphasizes that learning is an active process in which learners construct new ideas based on their current and past knowledge, in which, Concrete, Pictorial, Abstract (CPA) approach lies at the heart of instruction design (ID). Bruner highlights the importance of scaffolding, where educators provide support to help students progress through these stages, ultimately fostering deeper understanding and critical thinking. According to Bruner number addition can be represented with his three as in figure.
  1. Inactive Representation (Concrete): learning through direct experience and action, learners interact with the physical world, manipulating objects and observe outcomes. For example, a student use physical objects to understand number addition through touch and movement.
  2. Iconic Representation (Semi-concert): knowledge is represented through images or diagrams, learners use visual aids to grasp concepts, such as drawing number line, and executing number addition.
  3. Symbolic Representation (Abstract): learners use abstract symbols and language to represent ideas. This includes mathematical notation or algebraic expressions, allowing students to articulate complex concepts symbolically.

    \( 4+2=?\) or \( 4+?=6\) or \( ?+2=6\)

Example 2: reflection: Bruner's theory
Bruner's theory, known as constructivist learning theory, highlights the importance of active learning and (re)discovery, where learners construct knowledge through engagement and exploration rather than passive reception. According to Bruner reflection can be represented with his three stages as in figure.
  1. Concrete (Inactive): Activity using mirrors to explore reflection
    Students can engage in a hands-on learning activity that involves using a grid drawn on the ground to explore the concept of reflection. This activity combines physical movement with visual learning, making abstract mathematical concepts tangible and fun. One student stands at a specific point on the grid (e.g., at (1, 1)). Another student stands at the reflected point (e.g., at (-1, 1) if the y-axis is the line of reflection). After the activity, students can discuss what they observed and learned.
  2. Semi-concrete (Iconic): Activity using drawing of reflected images on graph paper, to determine the line of reflection (perpendicular bisector of object and image), encouraging students to color the original shape and the reflected shape to verify it in understanding of the relationship between shapes and their reflections.
    Also, utilize software or apps that allow students to manipulate shapes on a screen. They can reflect shapes across designated lines and observe changes in coordinates. This method provides a visual representation that bridges concrete experiences with abstract understanding.
  3. Abstract (symbolic): Activity using algebraic representation to express reflection transformations analytically, for example reflection over the x-axis and y-axis, and other lines presenting in the formulas.
Let \(l\) is a line and \(A\) be a point, then, reflection about \(l\) is denoted by \(T_l\), is a transformation on a plane, where
  1. if \(A \notin l\), then \(T_l (A) = A'\) s.t. \(l\) is perpendicular bisector of \(AA'\).
  2. if \(A \in l\), then \(T_l (A)=A\)
In this process, teacher can present real-world scenarios involving reflections, asking students to write equations or expressions. This encourages critical thinking and application of mathematical concepts to everyday life.
Example 3: multiplication: Bruner's theory
Bruner's theory, often referred to as constructivist learning theory, emphasizes the significance of active learning and rediscovery. In this approach, learners actively build their knowledge through three stage of knowledge representation. Bruner's model of multiplication can be illustrated, as shown in figure.
  1. Concrete (Inactive): Let student think of what it means to multiply, for example \(12 \times 13\). Geometrically, this multiplication problem can be expressed as the process of finding the area of a rectangle measuring 12 by 13. The tens and units pieces of a set of base ten blocks have been separated for emphasis in figure below. Let them think of the traditional algorithm. Four multiplications take place: 10 × 10, 10 × 2, 3 × 10, and 3 × 2. The diagram shows 1 flat that represents 100, 5 longs that represent 50, and 6 units, modeling the product of 156. Using base ten blocks, a student can visualize what it means to multiply.
  2. Semi-Concrete (Iconic): \((x+2)(x+3)\), can be discussed as area of the rectangle, which is made up of one \(x^2\) tile, five \(x\) tiles, and six unit tiles, so the product would be \(x^2 + 5x + 6\), in figure below.
  3. Abstract(language): lest student explain about “how to FOIL” realizing that FOIL (First, Outside, Inside, Last) so that this algorithm applies only to multiplication to binomials.

    \((x+2)(x+3)\)
    \(=\)
    \(x(x+3)+2(x+3)\)
    \(x^2+3x+2x+6\)
    \(x^2+5x+6\)

Conclusion
Historically, the lack of conceptual understanding has been a substantial problem in mathematics education. For many students, mathematics is nothing more than a set of rules and procedures because they have been taught almost entirely on the symbolic level or in a classroom in which getting the answer was valued over making sense of the mathematics.
This theory helps students progresses through the phases which allows learner to reason out why terms are multiplied through FOIL, to communicate the process through the use of models and pictures as representations, and to generalize the idea. This goes beyond procedural knowledge to an understanding of the process. And although some students can be taught at the symbolic level and be successful in school, many cannot and are in need of other representations of problems.
Please note that, students with well-developed symbolic systems might be able to bypass the first two stages when studying some concepts, but teachers can NOT take risk because students will not possess the visual images on which to fall back if the symbolic approach does not work (Bruner, 1966).

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