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Running head: PERKINS ANALYSIS WITH CONNECTIONS & REFLECTIONS 1

Perkins Analysis with Connections & Reflections

Cheryl Fox

EDU 510.90

Dr. Ruby Parker

PERKINS ANALYSIS WITH CONNECTIONS & REFLECTIONS 2

Perkins Analysis with Connections & Reflections

Author Paul Thagard (1996) asserts that knowledge in our minds consist of what are

called mental representations (p. 4). These are presentations to the mind in the form of ideas or

images. He mentions the following five mental representations: logic, rules, concepts,

analogies, and images. When working in a classroom, teachers should try and provide engaging

examples in the form of problem- or inquiry-based learning such that these mental

representations are highlighted and reinforced. Using author David Perkins’ (2009) seven

Principles of Teaching can allow teachers to do just that. These are: Play the Whole Game,

Make the Game Worth Playing, Work on the Hard Parts, Play Out of Town, Uncover the Hidden

Game, Learn from the Team, and Learn the Game of Learning. By incorporating these teaching

strategies into the classroom while focusing on mental representations, students will become

active, social learners who will want to take control of their own learning.

Logic

Logic is a mental process where one can infer or assume some sort of conclusion when

provided with certain information. When certain pieces of information are presented and

purported to be true, one can deduce a conclusion from that information (this is also called

deductive reasoning). Or, sometimes the conclusion or outcome to a situation or series of

situations is already made, and one can infer the basic principles or ideas upon which the

conclusion or outcome was made (this is also called inductive reasoning). Logic is a higher

order thinking skill.

A perfect example of using logic in the mathematics classroom would be to solve a

system of equations and determine whether there are solutions (intersection points) to the system

or not. Any system of equations just represents more than one equation. Visually, 99% of the


time it is where two lines cross on a plane, which represents a point. A point always has two

coordinates: an x-coordinate, and a y-coordinate (written as: (x,y)). However, the other 1% of

the time, two lines will either be parallel in space and never cross (no intersection point), or be

parallel in space and sit right on top of each other (intersect at all points). See the visual aid

below:

(Stewart, Redlin, & Watson, 2008)

In addition to having students algebraically solve a system of equations and determine

how many solutions exist, they would need to logically infer how the lines would look on a plane

in space. One solution means the two lines cross at a point. No solutions means the lines are

parallel but never touch. Infinitely many solutions means the lines sit atop each other in the

plane. There are two of Perkins’ Principles that apply here: Work on the Hard Parts and Learn

From the Team. Students reach this part in the course and can solve the system of equations


using algebra. This does not tend to cause them problems. What they do have issues with is

logically inferring what the lines look like in space using the data they have. What is difficult for

them is moving from given data ---> visualization. This is where Working on the Hard Parts

comes in. As an instructor, one could provide them with a theory of difficulty when the lesson

has begun. Perkins (2009) says these are theories that “warn teachers and learners about

potholes on the learning road and tell us where we need a special spring in our educational feet”

(p. 101). They should try and be specific to the “content of what’s being taught, explaining what

makes it hard” (Perkins, p. 103). A possible theory of difficulty for systems of equations would

be to advise students to algebraically solve the system of equations first, and

then graphically solve the system (as opposed to what many books suggest, which is the

opposite). They can then check their answers when they have drawn a graph by using

substitution or elimination (i.e. algebraic methods). This works out much better for them. They

learn about what it means to solve a system of equations and then their algebraic answer is that

intersection point. Then, they graphically show that, and check it using algebraic methods. Just

starting out drawing pictures is not advantageous to them, since they do not really understand the

meaning behind what intersecting lines on a plane mean. Students can then incorporate any new

learned skills they have acquired into future assignments, while teachers can make sure they

revisit this material in later lessons to reinforce what students have learned.

The teacher could also provide numerous examples of this type of problem for homework

or group work and provide communicative feedback. This is feedback “structured to ensure good

communication” (Perkins, 2009, p. 86) and is clear, identifies positive features, and shows the

recipient what to do better next time. This is constructive criticism, not being mean and not just

putting X’s here and there.


Additionally, students can really benefit from working together in groups or teams. The

teacher can assign extra word problems that are applications of systems of equations for groups

to work on and present such as the following:

A checking account pays an annual interest rate of 2.3% and a savings account pays an

annual interest rate of 5.4%. A total investment of $700 paid $34.50 in interest for the year.

How much was deposited in each account?

This is a perfect example of a system of equations problem, and the students would need

to use problem-based learning and could work together to come up with the equations and figure

out the answer. The teacher could have them do pair problem solving. This allows for students

to pair up and one learner takes the role of problem solver while the other takes the role of

listener. The problem solver tackles the problem out loud and the listener listens. If the problem

solver stumbles a little bit with the explanantion, he/she looks for feedback and/or clarification

from the listener. When done, they change roles. This really allows both people to focus on their

thought processes and how to solve the problem. Another thing the instructor can do is to have

students break up into larger groups and try and work together to tackle the problem. This could

be considered a community of practice (COP). These students all share a common mission – to

solve this system of equations and present their answer. By working with others, they can

bounce ideas off of each other and they can answer each other’s questions. They can all

contribute and each has a part in the problem-solving process. One can visualize and draw a

picture, while one can solve the equations. While another can decide to present the material to

the class. Learning From the Team is worthwhile, since “social interactions develop language –

which supports thinking – and they provide feedback and assistance that support ongoing

learning” (Stanford University School of Education, n.d., p. 126).


Rules

When one encounters a certain varying condition in his/her environment, the brain

decides how s/he should react to that condition. One takes an action based on that condition to

help suit his/her needs to satisfy a situation. The condition paired with the action is a procedure

or rule.

A good example of a rule in the mathematics classroom to illustrate would be the

distributive property which reads:

a (b+c )=a∗b+a∗c∨a (b−c )=a∗b−a∗c

In layman’s terms, it simply says in order to multiply a number (“a”) by the sum of two

numbers (“b” and “c”), one would multiply each term of the sum (“b” and “c”) by the first

number (“a”), then add the result. The same holds for the difference of two numbers being

multiplied by a number. Students studying mathematics need to understand rules or theorems,

since these rules apply to different situations that will arise. Once they deduce what rule to use

and when to use it, they have to know what the rule means (“says”) in order to take an action

based on the conditions the rule sets forth. This property is widely used in lower- and higher-

level algebra, so knowing how and when to use this rule is exceedingly beneficial to students.

Perkins’ Play the Whole Game is applicable in this situation. Students not only have to be able

to define this rule, but have to know when to use it, and where. They have to be able to use it in

different situations, and apply it in the real world. In other words, teachers really need to stress

that students need to see the whole picture when it comes to using the distributive property, and

not just have them memorize it for the sake of memorizing it. If instructors assign examples in

class, ask students, “Why are we using this property here?” “Could we use a different property?

Why/Why not?” A great project would be to first highlight the property in class. Talk about the


definition and explain what it means to students. Next, show some basic examples on the board.

For example, an instructor could write:

Expand the following, then combine like terms: 2(−3 x−5 y+7 x−15 y )

and show the following steps, written out for students, on the board:

{original example} 2 (−3 x−5 y+7x−15 y )

{apply distribute property} 2∗−3 x+2∗−5 y+2∗7 x+2∗−15 y

{multiply} −6 x−10 y+14 x−30 y

{rearrange like terms} −6 x+14 x−10 y−30 y

{combine like terms to simplify} 8 x−40 y

Next, teachers could ask students if they have seen examples of this property being used

anywhere in real life. If so, where and when? Why was this particular property used? Next,

split the class up into groups (COPs), and assign a problem-based learning project. Have them

come up with a real-world example of how they could use the distributive property. The group

must write up a report (being explicit and using diagrams) summarizing their findings, and at

least one person must present the findings to the class. They must all hand in a paper and the

presentation must include a PowerPoint or Prezi. An example might be as follows:

A farmer has three plots of land that he has divided up as follows. How much area does

the farmers’ land encompass?

(figure not to scale)

190 ft.

190 ft. 220 ft. 300 ft.


Students can draw the picture above, and then show how to use the distributive property to find

the total area. Although their work may be shown in the papers they hand in, students must show

all work on the board as well:

190 (190+220+300 )=¿

190∗190+190∗220+190∗300=¿

36100+41800+57000=¿

134900 ft2

Is this the same thing as adding up all the sides of the length (190 + 220 + 300) = 710, and

multiplying by 190? Yes! Not only can students find the smaller individuals areas of the plots

and add them up to find the total area, but they should remember to let their classmates know

that they could also find the total area of the large rectangle by adding up all the lengths of the

plots and multiplying by the width:

Areatriangle=l∗w

A=710∗190

A=134900 ft2

What are the advantages of using the distributive property in this case? Perhaps, it is easier to

measure 190 feet, 220 feet, and 300 feet, instead of 710 feet lengthwise. Students should be able

to come up with some reasons why using the distributive property might be beneficial (or not!) in

their particular examples. In this way, students are using inquiry, strategy, skills, and actually

710 ft.

190 ft.


doing something to illustrate the property, instead of just sitting in a classroom and memorizing a

property.

Concepts

Concepts are basic units of thought or knowledge that are representations of

typical/constant/permanent entities or situations. They are usually organized into a hierarchy,

where the higher level elements depend upon the lower level (more basic) elements. According

to Pavel (2009), a concept should be defined in terms of a context as this provides a meaningful

interpretation of the concept itself (p. 2). Concepts can be combined with other existing

concepts.

Many students are familiar with the concept of slope in a mathematics classroom. Slope

is discussed when talking about points and lines. In order to find the slope of a line, a student

should know some basic concepts first – rise and run -- between two points on a line, which

leads to the definition:

slope of aline= rise(change∈horizontal direction between2 points)run (change∈vertical direction between2 points)

Therefore, when discussing lines, the instructor first needs to not only explain how to plot points

in a plane and connect them to make lines, but should discuss how to move horizontally or

vertically from one point to the other. This allows students to comprehend the definition of slope

more easily. In addition, the teacher should discuss and demonstrate how to determine the direct

distance between two points (using the distance formula below), and then provide students a

context within which the slope can be found and why they are finding it.

distance formula=√(rise)2+(run)2

For example, the following problem could be discussed in class: Given the following two points:

(4, -4) and (-7, 5), let us say you move from (4, -4) to (-7, 5). What is the rise? What is the run?


What is the direct distance between the two points?

(plane is opensourced)

The instructor would begin by showing students that the rise is the change in the vertical

direction. Rise is positive if you move upwards, and negative if you move downwards. Since

students need to move upwards 9 units, the rise would be +9. Next, instructors can show

students that run is a change in the horizontal direction. Run is negative if you move left and

positive if you move right. In this case students need to more left 11 units to get to (-7, 5).

Therefore, the run is -11. To determine the direct distance between the two points (which is

always a straight line) students need to use the distance formula:

distance formula=√(rise)2+(run)2

d=√(+9)2+(−11)2

d=√81+121

d=√202 or d ≈ 14.21units

Instructors ask: Can d ever be negative? Why or why not? What about the fact that the run is

RISE

Start at this

point

RUNFinish at

this point

Direct Distance Between 2 Points


negative – should not that mean that our distance could be negative? These questions are

important to ask, since it gets students thinking about the order of operations and the fact that the

distance of a line can never be negative. This speaks to deep, not surface, learning.

Once this main discussion has occurred, instructors should split the class up into groups.

The following project could be assigned:

Go outside and measure the height (rise) and length (run) of a handicap-access ramp.

Using the distance formula, compute the length of the actual ramp (the distance). Now, measure

the actual distance of the ramp to see how close your answer came. Does your answer make

sense? Next, find a space where this is no ramp, but see if you can “construct” one using

formulas you know. What would the height, length, and hypotenuse (ramp) length be? Share

your answers with your classmates.

This is using Perkins’ Make the Game Worth Playing and Learn From the Team. Again,

this is applying Inquiry- or Problem-Based Learning, since it asks students to apply base

knowledge they have to a world-world example. Students are first given a basic overview of the

definitions of slope and the distance formula in class. Then, by having students go outside with a

clear set of goals and formulas to find the lengths of the ramps, instructors are indirectly

intrinsically motivating students to be engaged in this topic. This is not learning for rewards or

punishments, but learning for the sake of learning. Intrinsic motivation is what teachers and

students should strive for, since it predicts greater achievement (Perkins, 2009, p. 55).

Memorizing for the sake of the test to get an A will only go so far. If you link the definitions

with an activity, now the definitions have meaning, and are encoded into memory for later

retrieval. Students are learning by doing, and they can ask questions while they are doing it.

They can see why the distance can never be negative. Secondly, the teacher has worked to set up


the basic foundation for the concept. Perhaps the student is not that confident regarding slope or

the distance formula. He/she thinks they understand the ideas, but they are not really sure.

These topics are just out of reach for them; outside their zone of proximal development (ZPD).

Here is where working within a social network can come in handy, since students learn in a

social context. The people in a person’s immediate environment that they have the most direct

interaction with would be called their microsystem. This would be parents, friends, and

teachers, for example. Students can all learn from these people because they work and

communicate so closely together (The Sociocultural Approach, 2012). The Zone of Proximal

Development (ZPD) was posited by Lev Vygotsky and it has to do with his theory of social

development. “In order to learn, we must be presented with tasks that are right out of reach of

our present abilities” (Lev Vygotsky, 2013). Using the basic knowledge that students have, those

people in a student’s microsystem can build on that and help the student to move a little bit closer

to the ZPD. Teachers can help students move their ZPD toward “higher levels of competence

and complexity … [by providing] clear and effective communication…” (Stanford University

School of Education, n.d., p. 129). In essence, they can scaffold, and provide students with just

enough support to get them to the next level of thinking to help them improve their skills and

understanding within the ZPD. By doing this, learners become active participants in

constructing their own learning and will build their own knowledge base to help them reach

higher levels of understanding to use in different situations.

Analogies

Analogies are relational patterns whereby they describe the relationship between a source

analog (an old situation) and a target analog (a new situation).

Source Analog ———–Analogy————> Target Analog


The Source Analog can be retrieved in memory and applied to the new situation (Target

Analog) using higher-order relations.

A good example of using an analogy in the math classroom is with explaining the

concept of combining positive and negative numbers and using money to help do this. Often,

when students add and subtract signed numbers, they get very flustered and frustrated as to how

to go about tackling the problem. Positive and negative numbers are used in the real world every

day, so discussing them in class is very beneficial, since they apply to many situations that they

currently have and will encounter outside of the classroom setting. Using money as an analogy

can work wonders with students since it is something they are so familiar with and they

understand what it is like to have money (positive numbers) and not have money or owe money

(negative numbers).

Give an example in class such as: add “negative twenty-two and positive eight”.

If one student presently had eight dollars in their pocket, but then owed their classmate

twenty-two dollars, how much would they have left to pay their friend after paying them the

eight dollars? They work the math out in their heads by “taking away” eight from twenty-two

and getting 14. But they know they are in the “whole” $14, so they associate that with a negative

value. Hence, since they owe money, combining -22 and +8 comes to a negative number, or -14.

Besides using the analogy of money, however, teachers need to think about Perkins’

teaching technique of Uncover the Hidden Game. Even though this money analogy may help,

many students may still simply add and subtract numbers using their calculators, and will never

truly understand why they are getting the answers they get, or how to add or subtract signed

numbers on their own. The Hidden Games (strategies) underneath adding and subtracting

signed numbers are using the concepts of absolute value & the number line, and applying the


rule for subtracting signed numbers. When in doubt, students can use these rules and visualize a

number line to help them combine signed numbers together. The absolute value of a number can

be defined as that number’s distance on a number line from zero. Distance can never be

negative, so the absolute value can never be negative (there is an exception, but it will not be

addressed in this paper). In simplest terms, the absolute value of a number x can be written as:

|x|. When one takes the absolute value of a number it will always be a positive value, therefore,

|x|= x.

So, |2|=2

(number line is opensourced)

and, |−6|=6

In order to add two signed numbers:

If the numbers have the same sign, add their absolute values and keep the

sign they share

If the numbers have opposite signs, subtract their absolute values

(subtract the smaller number from the larger number) and take the sign of

the higher number

The following are two examples that could be shown in class to first illustrate the two

“2” is 2 units away from 0 on the number line

“-6” is 6 units away from 0 on the number line


rules:

For example: Combine −1+−8.

{original example} −1+−8=?

{set up absolute values} |−1|+|−8|=?

{take the absolute values and add} 1+8=9

{take the sign they share} −9

Using the number line, students start at -1 and “add” eight more negatives, or move to the left

eight units on the number line and end up at -9.

Another example: Combine−15+33.

{original example} −15+33=?

{set up absolute values} |33|−|−15|=?

{take the absolute values and subtract} 33−15=18

{take the sign of the higher #} +18

In addition to adding two signed numbers, students should really know the following rule

for subtracting two signed numbers:

a−b=a+−b

This definition says if one subtracts two signed numbers, one can change the sign of the second

Start at -1 and move left -8


number as well as change the operation from subtraction to addition. In layman’s terms,

subtracting b from a is the same as adding a to the opposite of b.

So, 6−10

becomes 6+−10

Once students change the operation from subtraction to addition, they can apply the rules of

addition using absolute values as illustrated above.

Students can then be asked to explain adding and subtracting signed numbers to other

students in front of the class, and each give examples to their classmates using absolute value

and the number line, or money for an analogy. For example, a student could offer the following:

The temperature at noon was 30º F. By 3:00 pm, it fell to -10º F. What was the change in

temperature?

Do their fellow classmates understand? If not, the student can make sure that they

answer the questions, getting help from other classmates (or the instructor). Peer feedback is

very important, since questions from fellow classmates really touch home, since chances are if

one student has a question, others do, too. Allow students to ask questions, and provide a

comfortable learning and inquisitive environment in the classroom. Are they using methods of

inquiry to come up with the answer? If not, guide them along to do so. The teacher can give a

quiz after the student has taught a lesson on the underlying principles and has asked the question

above to see if their fellow classmates really understood the lesson and the “hidden” games

underneath. Have them show how they came up with the answer, not just give an answer.

Images

Images are pictoral representations in our mind of actual objects. Images are especially

helpful in problem solving, since they aid students in figuring out the path to get from A —>B,


when words are not enough. And problem solving is not just important in mathematics courses.

According to the Partnership for 21st Century Skills website (2009), “Within the context of core

knowledge instruction, students must also learn the essential skills for success in today’s world,

such as critical thinking, problem solving (my emphasis), communication and collaboration”

(p.1). When a school builds on this foundation, students become more engaged and are better

prepared to “thrive in today’s global economy” (Partnership for 21st Century Skills, 2009, p. 1).

Problem solving is not just about memorizing steps (but then again, nothing in math

should be!) It is an essential skill to have in the real world. People solve problems all the time

and do not realize they are following steps along the way to arrive at an answer. Drawing a

picture in mathematics is one of the essential problem-solving steps, and teachers should stress

this step. Problem-solving is one of the most difficult concepts for algebra students to grasp.

Visualizing a situation can be helpful, since students can benefit from the verbal messages and

the visual images. Visual students prefer learning using pictures and images since they have

spatial understanding (Overview of learning styles, 2014).

As an example of problem solving, instructors can start off discussing the following example:

A triangle has a height of 10 m. The base of the triangle is three less than four times the height.

Find the area of the triangle.

This word problem can be figured out using algebra alone, but visualizing the problem

makes finding the answer so much easier for students, since they are combining verbal and

visual skills together (hence using more than one part of their brain to tackle a problem). Have

them try and figure out the problem first without drawing a picture. Can they do it? Are they

having trouble? Next, have them draw any triangle and label the triangle with the pertinent

information:


b=4 h−3

(base or “b” is 3 feet less than 4 times the height or “h”)

Using the figure, they can substitute the known information into the given formulas to

come up with an answer. They would substitute 10 for h into “b=4h-3” to find “b”. They would

find “b” to be 37. Then, they would need to remember that the area of a triangle is equal to the

triangle’s base times its height. (A=b*h). Substituting this information into the formula:

A=b∗h

A=37∗10

A=370 m2

Take a poll. Was it easier for students to find the answer with the picture? Most likely,

students will agree that it was.

This not only works well for visual students, but all students, since they now have a

picture in their minds of how images are important in the problem-solving process. They can

refer back to this process when they encounter a similar problem in the future

Perkins’ Principle Play Out of Town is applicable with this mental representation and

example, since one of the main ideas to take away from problem solving is to be able to solve a

problem and then transfer the gained skills of problem solving to a totally different type of

problem that students have not yet seen. Drawing a picture can definitely help with this skill.

Students can truly learn something by practicing it over and over, and explaining it to other

students. In addition, they can move out of their comfort zones and try something a little bit

novel and difficult to challenge themselves, all the while using their base knowledge and skills

10 m.


they have acquired in other contexts. Students can then acquire new skills to help them when

they encounter similar examples in the future. They can ask themselves questions such as, What

skills worked in this new situation? Did working with others help me to understand the material

more deeply? Did using a different learning or study strategy help me retain the concepts? What

positive things came out of this new experience? Only by moving out of familiar territory, can

students learn other strategies that can better serve them in different situations. As Perkins

(2009) states, “the whole point of formal education is to prepare for other times and other places,

not just to get better in the classroom. What we learn today is not for today but for the day after

tomorrow” (p. 12). In addition, when moving students out of their comfort zone, it is important

to make the learning relevant and engaging. Also, since students may be uncomfortable,

presenting them with smaller bits of significant information instead of large amounts works well

since the brain is limited in its capacity to multitask. This lets students focus their attention on

one or a few topics and allows information to be encoded into memory for later retrieval. Miller

(2011), declares, “little information is encoded in the absence of focused attention”. Simply

stated, without attention, there is no memory.

A perfect example of incorporating images in the problem-solving process with the

transfer of learning could be:

At 12:30 pm, the Sebathia steamed toward the Luthania at a rate of 15.5 miles per hour.

Suppose the Luthania was also drifting toward the Sebathia at a rate of 1.5 miles per hour. If the

two boats started 85 miles apart, at what time would they meet?

The example shown in class had to do with area of triangles, not rates of boats. Would a

picture help solve this problem? Teachers could assign groups of two students to work together

to solve this problem and then share their answers out loud. They must draw a picture as one of


the problem-solving steps. The following could be an image (obviously, the image need not be

this detailed):

Next, the following table (another visual aid!) can help students come up with an equation to

solve this problem:

Distance (d=r∗t ) Rate (r) Time (t)

Sebathia 15.5*t 15.5 t

Luthania 1.5*t 1.5 t

dSebathia+d Luthania=d total

15.5 t+1.5 t=85

17 t=85

t=8517

=5hours

12 :30 pm+5 hours=5 :30 pm

Lastly, instructors can ask students how images have helped them solve problems in real life.

What examples can they come up with to illustrate when a picture or image aided them in

formulating an answer to a problem?

Sebathia – traveling at 15.5 Luthania – traveling at 1.5 mph

Meeting Point

85 miles

distanceSebathia distance Luthania


Connections & Reflections

I have learned that our brain uses mental representations to help us sort knowledge in our

minds. We defined them in the beginning of the course, and I thought, “This is SO hard!” It

paid off in the end of the course, though, because I came to see just how important these are in

relation to everything else we studied in the course. They were really the basis for the rest of the

material we studied. It helped me to see how my students organize knowledge and where they

may have issues in their learning. Many of them have problems coming up with analogies and

drawing pictures (images) and I see that many of them need to work on their visual learning

skills. Administering a learning styles inventory to all of my students might not be a bad idea!

For example, they could be visual, aural, kinesthetic, or reflective. They may not know what

their strengths and weaknesses are when it comes to learning, and this may really help them hone

in on a particular set of skills.

I am still not convinced that Artificial Intelligence (AI) will or should try to delve into the

world of emotions and cognition. I think some things should be left alone. Just because we

might have the scientific capability someday, what about the ethical concerns? In addition to the

ethical concerns, I am finding my students are becoming more and more reliant on technology

for everything, and they cannot think for themselves. I have students that cannot add 2 + 7

without using their calculators. This truly scares me. This shows me they are learning on the

surface, but not truly understanding the concepts beneath. I fear that we will end up having

computers do everything for us, and that we will lose the power to think and reason for

ourselves. As educators, we must know when to introduce technology and know that it should

not take over our classrooms, but be an aid for our teaching purposes. Technology will never

replace an instructor (I hope!) and we need to be aware of how we can use technology and AI in


our classrooms at all times. I am certainly willing to take a closer look and consider the positives

of how to integrate AI and technology into my classroom, but right now I am still very leery

about introducing too much of it into my classroom.

Social contexts are extremely important in knowledge acquisition and learning. Ideally,

you want students interacting with each other and with the teacher to get the most out of the

classroom experience. This way, they grow intellectually and cognitively and take control of

their own learning. I always loved working alone as a student, and throughout this course I have

truly come to see the value in working in a social context. I have realized the value of PLE’s and

COP’s because you can meet with other people that share the same interests as you do and

bounce ideas off of other people for input. You can make positive changes when you have other

people working with you to get something accomplished. What we learn cannot be separated

from how we learn it and this becomes invaluable, since we can take this with us when we move

to other environments. I have also learned that as an instructor, I, along with others in a student’s

microsystem, can help students to move into their ZPD. I truly did not understand how valuable

other people can be in helping each other learn. Our environments truly can shape our learning.

Not only can I have an impact, but as an instructor, I should try and encourage students to be

interactive with each other. Peer feedback is very important!

I have learned valuable teaching strategies based on Perkins’ (2009) seven Principles of

Teaching:

a. deconstruct and reconstruct the hard parts so they can be exe-cuted in new and better ways (p. 80)

b. teachers can provide feedback that touches not just on matters of correctness but strengths and shortfalls of understanding (p. 84)

c. students can evaluate one another’s work or even self-evaluate with the help of rubrics (p. 84)


d. incorporate improved understanding of the hard parts into the whole game (p. 88)

e. try and anticipate the hard parts with learning by wholes and playing “out of town” (p. 89)

f. as educators, we can ask, “what makes this hard?” and when we can answer this question (i.e. come up with a theory of diffi-culty), we can try and prevent those hard parts from doing their worse damage (p. 101)

g. try not to just focus on the surface characteristics, but also look at the underlying principles. This will allow for deeper learning and enable students to gain the skills to tackle the hard parts (p. 112)

h. have students learn by DOING! This allows for effective transfer of material and aids in students making connections (p. 123)

One of the most important ones on this list really struck home for me throughout the

course, and that was item “h.”. I really am walking away with a better understanding of why I

should be incorporating inquiry- and problem-based learning into my classroom. It just means

so much more to students than a traditional lecture experience (which was how I started teaching

six years ago). I participated in a week-long camp for exceptional students last January. The

coordinator would not let the three lecturers allow us to just talk in front of the class the whole

week. We had to come up with projects for the students the whole time. I gave a PowerPoint

presentation on teamwork, and then had them work on a team-building exercise (see Appendix -

Figure 1). My collaborators worked on a water bottle bungee exercise to illustrate linear

equations (see Appendix – Figure 2). By the end of the week, the students really were getting

into the whole problem-based learning experience and I began to really see the value in this type

of teaching and learning. I could see that these exercises really went a long way in teaching

them about teamwork and linear relationships; far more than us just standing up in front of the

class talking leadership and plotting points on a plane.

In addition to these teaching strategies, I learned about metacognition and its importance

for students’ learning. Good learners combine cognitive strategies and metacognitive awareness.


Livingston (1997) asserts, “those with greater metacognitive abilities tend to be more successful

in their cognitive endeavors”. Cognitive strategies include taking notes and asking questions.

Metacognitive awareness is the “learner’s awareness of the learning process and what it takes to

achieve good results in a specific learning task” (Luke, 2006, p. 2). These include self-

evaluation, setting goals for learning, using self-instruction, self-questioning, monitoring

comprehension and progress, and self-rewarding for success. As instructors, I think it is easy to

forget these since we get so bogged down in focusing on the content material, but ideally, these

metacognitive abilities should be touched upon throughout the course. Learners should be given

time to discuss and model strategies which they can then take and use on their own.

I learned the value of emotions and the role they play in cognition. Emotions are

“specific and intense psychological and physical reactions to a particular event” (Advameg,

2014). According to Dr. Luiz Pessoa on Scholarpedia.org, it has been shown that humans

remember better “emotionally arousing information” and that “…emotion and cognition

conjointly and equally contribute to the control of thought and behavior” (2011). This has

always been a significant thing for me in my classroom. If you make the classroom an inviting

and fun place to be, students will want to show up and participate in class. I just did not realize

that emotions played that huge a part in thinking and learning. Ahmad and Rana (2012) assert,

“Cognition and emotions interact and influence human behavior…A person who is in [a] good

mood is generally productive and vice versa. Anxiety as an emotion bears on educational

performance in affecting students’ attention and memory processes hampering the cognitive

functioning and consequently academic output. This suggests that understanding and regulating

emotions can help in promoting efficient intellectual functioning” (p. 109).

Along the lines of emotions and mood, I also learned that our brains can build new


neurons and connections and that “new neurons are highly correlated with memory, mood, and

learning” (Jensen, 2008, p. 6). The process of building new neurons can be enhanced by

exercise, good nutrition, and lower levels and stress. This can increase students’ mood and help

with memory. Jensen (2008) also writes that the brain can “rewire and remap itself by means of

neuroplasticity” (p. 7). A study done by scientists at Brown University in 2000 looked at

changes in the brain during learning processes. The researchers did studies on rats and found

“learning engages a brain process called long-term potentiation (LTP), which in turn strengthens

synapses in the cerebral cortex” (Turner, 2000). Learning, therefore, produces actual physical

changes in the synapses between neurons. Students (once they find that learning style that best

suits them) can perfect that learning style, and just by repeatedly using their sensory system, can

increase the sensitivity of their neural networks in the brain. They can begin to process data more

efficiently and make more skilled responses to questions asked of them. In other words,

neuronal connections change and grow in that part of the brain that you use the most.

Fascinating! This speaks to what I wrote before, in that once students find that learning style that

works for them, to really stick with it and sharpen it, since that will only help them become a

stronger learner. That being said, however, students must also realize that other skills are

important, especially when they move out into the workforce. If they are verbal learners, they

should try to work on their visual skills. If they are reflective learners, they should try to work

on becoming a little bit more active. If you are a sequential thinker, try to think globally once in

a while. While strengthening your personal learning style is important, it also helps to Play Out

of Town and think “outside the box”, since different situations will arise which will necessitate

the use of other learning styles. The more prepared students are for different situations, the

better off they will be. Giving them experiences Out of Town will help students stretch their


minds to allow this to happen.

According to some researchers and scientists, cognition is not a separate, localized entity

within our skulls or brains, but it is integrated with our bodies (specifically our sensory systems -

vision, sound, smell, etc.) and integrated with how it interacts with the environment (sensing and

reacting to it). This constitutes the embodiment view of cognition. Since the environment is

composed of many different types of variables and is always changing, they call it a “dynamic”

system. This concept is still a little strange to me, and I would really need to study this more in

detail before I felt comfortable discussing any concepts related to this topic in my classroom.

Finally, the Mickey McManus video, Innovate: Education really struck a chord with me,

since I am not only interested in mathematics, but filmmaking as well (it was my minor as an

undergraduate student). I truly believe that human beings cannot live without something that

fills their spirit and soul. We have the core subjects, of course, and I am so lucky to love algebra.

But, I also really love film, because it allows people to be creative and highlights people’s talents

in such a way that sitting in a classroom just is not the same. The moving image is amazing.

The filmmaking process is so much fun, and collaborating with a team to produce and make a

film was one of the most challenging, yet rewarding experiences as an undergraduate student.

Whenever I got stressed taking my science classes, my film classes were there to relieve me.

They helped stir my soul, and I was able to be creative and take ideas from my head and translate

them into something visible that really held meaning for me. We all played a part to make this

movie, but we all contributed our own vision and creativity to it. Students truly learn by doing

and we all took away something from that experience that we will treasure for the rest of our

lives. As instructors, you can inspire students and reach inside them in such a way that is deeper

than just giving them a lecture. What do they want to do with their lives? How can we help


them get there? How can we compose a lesson plan such that it incorporates creative design in

its layout? How can we help make students ready for the 21st century? This course truly has

inspired me to find out.


APPENDIX


BROKEN SQUARES EXERCISE (TEAM BUILDING ACTIVITY) Cheryl Fox

Objectives (NOT to be shared until the exercise has been completed): Students will be able to

1. Analyze certain aspects of cooperation in solving a group problem.2. Become sensitive to some of their own behavior which may contribute toward or obstruct the solving of a

group problem.3. Identify the role of trust building in cooperation situations.

Group size: any number of groups, 5 people per group.

Time required: 20 minutes for the activity, 20-25 minutes for discussion.

Materials needed: one set of broken squares for every five members of a group.

Instructions:

1. Divide participants into groups of five.2. Each group should form a small circle.3. Each group will introduce all members of the group to each other.4. One person in each group will be designated the “observer” of the group.5. Give each group an envelope containing one set of squares. Each set is broken down into five sets of pieces.

Do not let the group open the envelope until the instructions are read.6. Read aloud:

“The game you are about to play is a learning experience that will be discussed later. In each

envelope there are five sets of pieces of paper for forming squares. When I give the signal to begin, the task

of your group is to form five squares of equal size. Therefore, by the end of the exercise each individual in

you group will have a separate square in front of them. You will be given 20 minutes to complete this task.

Specific limitations are imposed upon your group during this exercise. They are:

No member may speak during the entire exercise. No member may ask another member for a piece or in any way signal (i.e. point, nod head) that an-

other person is to give him / her a piece. No member may take a piece from another member. Members may, however, give their pieces directly to other members – not put them in the middle of

the group. It is permissible for a member to give away all the pieces to his/her square, even if he has already

formed a square.7. The observer in the group will watch to ensure that all members follow the rules mentioned above.8. ARE THESE INSTRUCTIONS CLEAR?” (Questions are answered at this time). 9. Call an end to the game after 20 minutes.10. Show the players who were unable to complete the squares how to do so.

Discussion:

1. How did you feel during this exercise? How many of you were frustrated? Why?

*The usual answer to this is, “I couldn’t communicate.” Suggest this isn’t true, but rather


normal patterns of communication were disrupted. They could communicate by giving away

appropriate pieces.

2. Ask the observers if anyone mentally/physically dropped out when they had completed their square? Why? *For westerners with an individualistic orientation, we hear the instructions as

individuals. Not everyone hears them this way.

How does this affect the team?3. Did dominant individuals emerge, or did everyone seem to participate equally?4. How willing were people to give away pieces of their puzzle? Were participants more interested in getting

than in giving?5. Did anyone violate the rules by talking or pointing as a means of helping fellow members solve their puz-

zle?6. Was there any critical point at which the group started to cooperate? What was the cause?7. What are some principles for successful group cooperation?

Each individual must understand the total problem Each individual should understand how he/she can contribute (sharing what they know) toward

solving the problem Each individual should be aware of the potential contributions of other people There is a need to recognize the problems of other individuals, in order to aid them in making their

maximum contribution8. Questions:

What happens if you ignore another person’s task? What lessons did you learn about organization? What lessons did you learn about being a more effective team member? How was trust developed or broken down within the whole group? What was necessary to build trust within the group?

Here are the original squares, which are cut along the lines and split up into five different envelopes:

(Picture from Broken Squares, 2012)

Figure 1


Bungee Jumping – Water Bottle (Colm Duffin & Anna Sicko - used with permission)

READER “The legend says that in the village Bunlap a man called Tamalie had a quarrel with his wife. She

ran away and climbed a Banyan tree where she wrapped her ankles with liana vines. When Tamalie came up to her,

the woman jumped from the tree and so did her husband not knowing what his wife had done. So he died but the

woman survived. The men of Bunlap were very impressed by this performance and they began to practice such

jumps in case they got in similar situations. This practice transformed into a ritual for rich yam harvests and also for

proving manhood.”

READER: “To honor this ritual, we will have a competition in the classroom. The rules are as follows:

a. Select a bottle and make a bungee cord by connecting rubber bands to the bottle.b. Drop the bungee bottle from the top of the stairs… 594 in.c. The winning group will come as close to the ground as possible without hitting it.”

MAKE AN ESTIMATION (before we start the experiment)

1) Estimate the number of rubber bands you think it will take to get as close to the ground as possible from a height of 594 in:

I think we will use ______ rubber bands

READER: “You will collect data in your classroom first.

Hold the end of the 5th rubber band at the jump line with one hand, and drop the water bottle from the line with the other hand.

Mark to the lowest point where that water bottle reaches on this jump. Measure the distance in inches. Record the value in the data table below. The bungee jumper’s life could depend on your accuracy!

The teacher will do a demonstration and then we will now break into groups to complete the experiment.”

2) Complete the table below 3) Create a scatter plot of the data on the grid below.


NUMBER OF

RUBBER BANDS (X)

JUMP

DISTANCE IN INCHES (Y)

5

8

10

12

15

4) Come up with an equation for your line of best

fit. Remember it is possible that all members

in your group have different equations

My Equation: ______________________________

5) Before we make a prediction for the stairwell, predict the height the water bottle should bungee jump if 20 rubber bands are used. (Show your work). Does this answer seem reasonable?

6) Work as a group to predict the number of rubber bands it will take to have the best bungee jump from 594 in. off the ground. (It’s possible that all members in your group will have different equations, and therefore have slightly different answers. Remember to help each other!) Show your work and put your answer in a sentence!

7) What is the equation the graphing calculator gives for your line of best fit? (Enter the rubber band data in L1, and enter the jump distance data for L2)

Calculator equation: _____________________________

8) How does the rate (in the calculator equation) relate to the situation?

________________________________________________________________________

________________________________________________________________________

9) Using the equation from the graphing calculator, predict the maximum number of rubber bands so that the water bottle could safely jump from the top of the stairs, which is 594 in. Show your work and put your an-swer in a sentence.

10) Which answer do you think will be more accurate, the number of rubber bands you calculated in number 6 or number 9? Explain why.


________________________________________________________________________

________________________________________________________________________

11) What are some of the reasons your prediction may not end in a perfect jump?

________________________________________________________________________

Now that we finished the math, let’s drop our bottles from the stairwell and test your prediction!

12) Was your group accurate? Describe what happened when you dropped your water bottle:

________________________________________________________________________

13) Is there a correlation or causation between the number of rubber bands & the jump? Explain.

Figure 2

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