APA Paper Template - cdyball.files.wordpress.com file · Web viewUnderstanding the way the brain works has fascinated great thinkers for thousands of years. Scientists have studied

Running head: PERKINS ANALYSIS 1

Perkins Analysis of Mental Representation

Christopher Dyball

The Cognitive Science of Teaching & Learning

Dr. Ruby Parker

Perkins Analysis of Mental Representation 2

Introduction

Understanding the way the brain works has fascinated great thinkers for thousands of

years. Scientists have studied the brain and mind in order to make sense of the way humans

think, remember and act which developed the field of science we know as cognitive science.

There are now many branches of science stemming from cognitive science: neuroscience,

anthropology, pedagogy, philosophy, artificial intelligence and psychology. (Thagard, 2012)

In the 1800s, scientists began to develop and perform experiments in order to try and test

theories on how the brain worked and so experimental psychology was developed. Cognitive

science continued to grow as a scientific field as more scientists developed and carried out

experiments. In the mid-1900s, as computers were being developed, scientists began to predict

that they could make machines work in the same way that the brain could. Cognitive scientists

were able to draw parallels between the way computers process information and the way the

brain processes information. It is widely accepted that the human mind has mental

representations similar to computer processing (Pavel, 2009). Understanding the way the brain

learns most effectively means understanding the processes involved in learning. Relating the

process of learning, to playing a sport is an analogy that many people in field of cognitive

science can understand. Almost everyone has had the experience of playing sports and games,

and has memorable moments when they felt emotion from winning, loosing or another event that

meant something in their life. David Perkins (2009) makes this connection, and then goes further

to develop seven principals of teaching and how they can transform education. He develops an

idea that there are seven principals of learning, and he personally relates them to playing

baseball, and, or other sports and games.


Analysis of Mental Representation and Application of Perkin’s Principles

There are several current theories that have proposed that the mind contains mental

representations and mental procedures to formulate concepts (Thagard, 2010, p5). Some of these

mental representations are listed below with an example, followed by the application of one or

more of Perkins seven principles of teaching. With the application of these principles, the

educational example can be more effectively taught to enhance the learning experience.

Formal Logic

Summary Logic is a method of reasoning in order to form hypothesis, conclusion

statements. These statements can be formed using either deductive reasoning or inductive

reasoning. Inductive reasoning is reasoning based on pattern that have been observed. It relies on

an element of inference. An example using geometry might be as follows:

Example From the following pattern in figure 1, use inductive reasoning to find the next

two terms in the sequence.

fig. 1 (Charles, 2010)

A student might say that the first shape is a large circle with a three-sided shape inside, the

second term is a large circle with a four-sided shape and the third is a large circle with a five-

sided shape. They use an observed pattern to predict what diagram might come next. All

diagrams have a circle as the outside shape and the inside shape increases by one additional side


each diagram. The next term might be a large circle with a six-sided shape inside and then a large

circle with a seven-sided shape inside (figure 2)

fig. 2 (Charles, 2010)

Application of Perkins Principle. Perkins (2009) suggests that when people give

examples of learning experiences they, more often than not, give examples that revolve around

families, deaths, war, jobs and hobbies, rather than giving academic examples (p. 68). These

examples are times in people’s lives when they are most likely operating at a high emotional

state, thus making the learning experience memorable. They hold value to the learner and

Perkins calls this principle: Make the game worth playing. These examples are important to

people as they form part of their lives. They form a context within their day-to-day existence.

Perkins (2009) suggests that pieces of the puzzle do not make sense unless the whole gets played

often enough to make it familiar and have context and meaning. Ensuring a student has an

emotional connection to a mathematical topic sounds like an impossible task, but it really

involves finding something that the students relate to or can connect with. In the example in

figure 1 and 2, there may not directly be a situation where this could be used in life, however, the

skill that is being taught is very useful. To make this lesson more relevant to motivate the

students, the instructor might explain how this concept would relate to the learner as a life skill.

Throughout life, adult looks for patterns in behavior and apply inductive reasoning to form a

conclusion. A parent buying groceries identifies patterns in family eating behavior, uses this


prediction to form a conclusion and buys food accordingly. A bank manager looks for patterns in

borrowing behavior and sets interest rates to suit. It is an important life skill and, as a way of

connecting with students, a teacher should explain why in order to make the game worth playing.

Rules

Summary Deductive reasoning is the process of reasoning logically from given

statements of facts to form a conclusion. Deductive reasoning can be split into two further

categories: Syllogism and detachment. Syllogism allows you to state a conclusion from two

conditional statements when the conclusion of one statement of the hypothesis of the other.

Detachment is the process of reasoning logically (forming a conclusion) from given statements

(hypothesis).

Example In order to form deductive and inductive statements rules need to be used.

Rules help to explain and justify logic and could be considered the vocabulary of reasoning. A

geometric example using detachment is as follows:

If a ray divides an angle into two congruent angles, then the ray is an angle bisector.

The hypothesis, a ray divides an angle into two congruent angles, is true so the

conclusion, the ray is an angle bisector, is also true.

An example of syllogism is: If a figure is a square, then the figure is a rectangle. If a

figure is a rectangle, then it has four sides. Conclusion: If a figure is a square, then it has four

sides.

Application of Perkins Principle– In this seventh principle: Learn the Game of

Learning, Perkins (2009) suggests that teachers should help students to understand that process

of learning and let them use that process to help them learn in the future. Students should have an


active part in the learning process to understand what skills they used to learn something. He

uses a driving analogy to describe the different parts of this principle and states “We want to

teach them to drive, and we can’t do this without letting them drive!” (p201). This example

teaches the student to use the vocabulary and logic of hypothesis, conclusion statements. As a

way of engaging students, the instructor could add some more relevant, real-life example to use

rules. It might also be motivate the students more if it could be turned into a game where teams

try to identify whether a set of statements shows detachment, syllogism or neither. For example:

if it is Saturday, then you walk to school. If you walk to school, then you wear sneakers. (Law of

Syllogism – If ti is Saturday, you wear sneaker).

Concept

Summary A concept is the conclusion formed from logic. Concept formation is the

process by which humans learn to categorize things. Using logic and rules, students can start to

classify shapes into different categories to form concepts.

Example Concept formation can be described as follows:

If a triangle has two congruent legs (figure 3), then it has two congruent angles (figure 4)

Fig. 3 (Charles, 2010) Fig. 4 (Charles, 2010)

Concept: A triangle is isosceles when it has two congruent lengths and two congruent

angles.

Application of Perkins Principle Learn from the team – This is a principle that suggests


the one of the best ways to learn is by watching someone else. It may be present in the form of

watching someone succeed at a particular task or watching someone who is proficient in a task.

After all, apprentices have been learning trades for centuries from watching a ‘master’ at the

trade. The principle may also show itself in the form of watching someone else fail in a task.

Perkins (2009) states: “Not only might they learn from their mistakes, others can learn from their

mistakes.” (p 176).

Analogies

Summary An analogy is a mental representation that links previous experience to help

with a new problem.

Example A student could use their experience of isosceles triangles to solve the problem

in figure 5. Solve for x and y.

Fig. 5 (Charles, 2010)

Using previous experience, a student could use their knowledge of isosceles triangles to

say that both interior base angles are yo. Coupled with prior knowledge that supplementary

angles add up to 180o , they could solve for y.

Application of Perkins Principle. Perkins refers to the idea of having to take your game

up a notch when playing at another team’s field as “Play out of town”. In learning terms, this


principle looks at how knowledge is transferred from one example to another. This principle

draws a natural parallel with analogies as the definitions are very similar. Perkins states: “The

whole point of education is to prepare people with skills and knowledge and understanding for

use elsewhere, often very elsewhere” (p123). This is probably the most important real-life skills

that school aged children learn. Much of the fact-based information they learn throughout school

is irrelevant in real-life. In answer to the frequently heard question “When are we going to need

to know this in real life?” a teacher could (although, may never) say “Never”. It is true. A ninth

grader will most likely never have to accurately use the quadratic formula in life, ever, but they

may have to find all solutions to a problem (even a complex one) which may not be visible.

Getting a student to understand that they are learning the processes of learning by applying one

to another may be a way of motivating them to succeed at a certain skill.

Images

Summary The ability to represent information visually is much more useful than a

lengthy verbal description. In geometry, assumptions can be made in diagrams from the marking

on a diagram and can be less confusing than a written description of a shape.

Example Triangle ABC has a segment DE at midpoints E and D on segment CB and CA

respectively (Figure 6).

Fig 6 (Charles, 2010)


Application of Perkins Principle – Perkins (2009) suggests, in his Uncover the Hidden

Game principle, that identifying underlying factors related to learning, is an important part in

understanding the learning process. Perkins uses ‘Uncovering the hidden game’ as an analogy for

this identification. He states “Almost everything that people learning in school and out of school

has its hidden aspects, dimensions and layers and perspectives not apparent on the surface” (p

142). It may not be clear to a student why the written form of this construction suggests that

segment DE is parallel to segment AB. They might not even recognize that it is parallel,

therefore rendering it a hidden fact. By drawing a diagram, a student could see that when a

segment is drawn between two midpoints of a triangle, they form a segment parallel to the base.

Simply using an image uncovers hidden aspect to a problem which may become invisible. As a

way to engage learners using this principle, an instructor could encourage students to draw silly

pictures of a word problem to help understand the question and motivate them. An example

might be: A lighthouse is 200 feet tall and stands on a cliff 455 feet above sea level. If the angle

of depression from the top of the lighthouse to a boat on the water is 22 degrees, how far away

from the base of the cliff is the boat? Students could draw an image of a lighthouse (Figure 7)


Fig. 6

Students would be able to label the height of the cliff and the lighthouse to understand

that the heights need to be added to form the opposite side.

Connections and Reflections of Mental Representation

The cognitive processing involved in solving geometry problems naturally lends itself to

mental representation. As a teacher new to geometry, I was surprised how different the reasoning

and problem-solving processes are to those in algebra. The five aforementioned mental

representations allow a student to deconstruct a problem and reconstruct it with an appropriate

solution.

Different brain structures are engaged depending on the structure of the problem (Barbey,

2009 p.2). Barbey suggests that when a problem is not complex, the associative system is used.

This is a system that uses basic cognitive operations to produce a result quickly. However, more

complex task may implement a more advanced mechanism such as the rule-based system,

whereby the brain carries out reasoning procedures deliberately and consciously.

Understanding and taking in to account the different factors affecting these systems being

engaged means more effective teaching. Perkins’ Principles (2009) give a way of unlocking the

barriers to let the mental representation through.

To understand the way mental representation is constructed gives an insight into the way

the brain functions. On reflection, some problem solving activities I had given to my class, and

subsequently worked through a solution, used mental representation to find a solution. This was

not deliberate, and often missed steps in the cognitive process. In the future, I can identify which


steps might be missing in a students work and address that gap in understanding. An example of

a geometry question where deconstructing the steps might have helped students to more

effectively solve a problem is as follows (figure 8):

Problem: Triangle GAB is isosceles with vertex angle A and triangle BCD is isosceles

with vertex angle C. Is triangle BGH congruent to triangle BDH? Justify you reasoning.

5

Fig. 8 (Charles, 2010)

The way I would have previously solved the problem may not have different, but the way

I would explain the solution or set the assignment up would use more mental representation

language and employ Perkins’ principles.

Firstly, I would allow the task to be completed in partners. I would not have done this in

the past for fear of students becoming distracted. Partner work would allow students to discuss

the problem and learn from each other. One might have a better way to begin the problem, one


might see a way to solve it but not know the postulates involved. In this manor they are ‘learning

from the team’. I would ask the students to redraw the diagram with all the relevant information.

I would suggest drawing all triangles separately. This would allow the students to see the whole

problem split into part that could be worked no separately. This part of the problem solving

process uses the visualization representation and could ‘uncover the hidden game’: parts of the

triangle might look different when separated from the whole. I would then ask if there were any

parts of the triangle that looked familiar and ask if they could apply any prior knowledge of

triangles to any of these triangles. This transfer of knowledge employs Perkins ‘play out of town’

principle and the analogies mental representation. Once the students had come up with a relevant

proof, I would ask them to pick out any logic-based vocabulary (we had previously looked at the

language of hypothesis-conclusion statements and reasoning statements).

Sample of a verbal solution: “If2 triangle GAB is isosceles, then2 it will have two

congruent legs1. If it has two congruent legs then it has two congruent base angles3. Triangle

GAB is isosceles, and then the two base angles are congruent4. I can use the same process with

the given information on triangle BCD to show that triangle GAB and BCD are congruent”4.

This statement uses logic1, rules2, concept3, analogy4 and images5. As a plenary activity, I

would ask the class what aspect of the task helped the most in finding a solution. This gives

students the ability to recognize what parts of the learning process they identify with most and

begin to learn the game of learning.

Conclusion

“One never knows when one is going to be ambushed into learning something.”

(Perkins, 2009, p.220). This statement recognizes that a learning experience can take place at any


time whether in the classroom or on the athletic fields. Regardless of the context, the cognitive

process is the same and ultimately ends in the same result: an increase in experience.

Understanding how and why these cognitive processes take place give instructors valuable keys

to unlock student brain function.

References

Barbey, A. K., & Barsalou, L. W. (2009). Reasoning and problem solving: Models. Retrieved

from http://psychology.emory.edu/cognition/barsalou/papers/Barbey_Barsalou_

Enclopedia_Neuroscience_2009_reasoning.pdf

Charles, R., Hall, B., Kennedy, D., Bass, L. E., Johnson, A., Murphy, S. J. and Wiggins, G.

(2010). Geometry: Common Core. Prentice Hall.

Pavel, G. (2009). Concept learning: Investigating the possibilities for a human-machine dialogue.

Knowledge Media Institute. Retrieved from http://kmi.open.ac.uk/publications/pdf/kmi-

09-01.pdf

Thagard, P. (2010). Retrieved October 30, 2013 from http://plato.stanford.edu/entries/ cognitive-

science/

Perkins, D. (2009). Making learning whole. [E-reader version] San Francisco, CA: Jossey-Bass.

Victor Valley College, (n.d.). Bronfenbrenner’s Microsystems and Mesosystems. Retrieved from

http://www.vvc.edu/academic/child_development/droege/ht/course2/

faculty/lecture/cd6lectmicro.html


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APA Paper Template - cdyball.files.wordpress.com file · Web viewUnderstanding the way the brain works has fascinated great thinkers for thousands of years. Scientists have studied