Module 5: Problem solving and Metacognition

Worked-out examples are often used to demonstrate the procedural tasks needed to obtain a solution.  If presented effectively, worked examples can also address metacognition skills needed to solve novel problems.

Teaching problem solving is tough.  There are many of lists depicting problem solving processes available for educators to use to teach problem solving skills.  Most of the lists are based on the work of George Polya (1887 – 1885) who believed that the skill of problem solving should be taught.  He identified four principles that form the basis at solving a problem:

1.       Understand the problem

2.      Devise a plan

3.      Carry out the plan

4.      Reflect.

My students often grasp the numerical calculations with ease.  However, their lack of metacognition skills prohibits them from solving similar problems that may include one or two structural changes.

For example, the other day we were covering percent increase and decrease.  Part of the lesson included us working through examples like the following:

a.       What is the percent increase from $345 to $689?

Following five or six problems written as above, the students were asked to flip the worksheet over and complete five problems on their own.  The following represents the type of problems on the back of the worksheet:

a.       Original Amount:  $5000

Final Amount:       $7800

Find the percent increase.

            The students had a very hard time completing the problems on the back of the worksheet on their own.  I found this interesting.  The second type of problem is seemingly more straightforward.  What is it, then, that is keeping the student from being able to solve the problem?

In my opinion the students are lacking mathematical literacy and metacognitive skills.  “What is the problem about?”  "How is this problem similar or different from problems that have already been solved?”  Encouraging students to think this way is a challenge.  Designing instruction to support the development of metacognitive skills in students must include careful planning and monitoring by the teacher. 

Which makes me question why the idea of teaching with minimal guidance became an issue in the first place?  Students struggle with sovling problems on their own. (Why else would we need teachers?)  Kirschner, Sweller, and Clark (2006) found that strong instructional guidance rather than constructivist-based minimal guidance was more effective.  I believe metacognitive skills instruction can be supported by constructivist-based activities and these activities should be monitored closely by the teacher so that students receive ample support for skill development. 

Module 5: Blog Post 1 Metaphor for Learning

“Swimmers, take your mark…” A metaphor for learning.

            The starter calls the field to the blocks.  “Take your mark.”  There is a pause.  I can feel my heart beating in anticipation of the starting device that will be begin the race.  It was March of 1993, I was about to swim my last race of my last meet of my competitive career.  For 18 years, I trained countless hours side by side with my father who was my swim coach.  I dedicated myself mentally, physically, and emotionally for almost two decades to swim this last race.  The starter goes off.  This is it.

Educational metaphors use symbolism to link ideas about teaching and learning to something more familiar.  There exists nothing more familiar to me than the sport of swimming.   The more I contemplated the use of this metaphor, the more similarities between learning and swimming arose.  Next, I will elaborate on one of the similarities that I found. 

Social

Throughout my swimming career, I was surrounded and supported by a team.  We worked, laughed, and cried together every step of the way.  We formed a group of people who shared common goals over an extended period of time.  Our performances were a direct result of premeditated training whether it was for conditioning and strength or drill work for better stroke technique.  However, a fortuitous result occurred in that we learned to function as a team and strive to reach beyond our potential.

Educational theorist, Etienne Wenger defines Communities of Practice (CoP) on his website as “groups of people who share a concern or a passion for something they do and learn how to do it better as they interact regularly.”  Further Wenger points out that his definition of CoP allows for but does not require intentionality.  Learning can be an incidental outcome that accompanies the social interaction within a group that shares common goals.  Wenger (1998) states,  “The point  is not that classroom instruction is to be avoided or that a training function is useless, but that both are supplement, not substitute for, the learning potential inherent in practice” (pg. 250).   Even though, in practice, teachers are required to teach the curriculum, a deliberate set of outcomes, there are times when students create and construct knowledge through engaging in and contributing to the practices of the classroom. 

Wenger’s theory of CoP is rooted firmly in social development theory as discussed by Vygotsky and Piaget.  According to Vygotsky and Piaget, social interaction plays a fundamental role in the process of cognitive development.  Vygotsky felt social learning precedes development.  In his book  Mind and Society: The development of higher mental processes, Vygotsky (1978) notes that a child’s development occurs first on the social level, then on the individual level.  This is in contrast to Piaget’s understanding in that he held that the development of the child comes before learning.  In either case, Vyotzsky and Piaget agree that social interaction plays a key role in learning.  Wenger expands the consensus to include both intentional and non-intentional occurrences of learning. 

Vygotsky, L.S. (1978). Mind and society: The development of higher mental processes. Cambridge, MA: Harvard University Press.

Wenger, E. (1998). Communities of Practice: Learning, Meaning, and Identity: Cambridge University Press.

Wenger, E. (n.d.).  Communities of Practice.  Retrieved from  http://www.ewenger.com/theory/

Module 4: Worked Examples & Schema Acquisition

A secondary mathematics course, such as algebra, often employs the use of worked examples as an instructional device.  The role of a worked example serves as a key component to understanding the problem type being taught and can be presented to the class in multiple ways.  Under the behaviorist tenet, a teacher presents the worked examples, explains the necessary procedures to obtain the solution, then through guided practice the students utilize the worked examples to work similar problems upon which the students receive feedback regarding their progress.  Using a constructivist approach, the instructor provides students the opportunity to construct mathematical ideas by posing real world problems as examples.  Here the goal would be for the students to realize the procedure to solve the given problem through exploration and discussion as facilitated by the teacher.  In either case, the instructional goal for the lesson would be schema development of the problem type in the student’s long term memory that enables the transfer of information learned to novel problems.

Comprehension of the material is governed by factors such as presentation, relevance, and difficulty level of the content.  However, retention and retrieval are regulated partly by the effectiveness of structure of the lesson design and partly by the individual memory of the student. To design effective worked example presentation it is important to consider the individual memory of the student.  Marsh and Butler (in press) concur in their article stating, “It is key to consider what type of processing different learning strategies encourage.”  The psychological perspective of memory is important to regard when designing instruction using worked examples. 

Atkinson et al. (2000) suggested that highlighting certain subgoals of the problem increase the likelihood that learners will be able to transfer the problem’s structure to novel problems.  Hence implying the problem schema has been formed in the student’s long term memory.    Marsh and Butler (in press) discuss similar strategies in their article.  The human brain has the innate ability to create logical connections between related parts and the bigger picture.  Labeling and visually isolating certain features of the worked examples can direct the learner’s attention to the structural nature of the problem and helped students form connections thus further facilitating schema acquisition.

Regardless of what theoretical approach is taken when designing worked example instruction, the strategies employed should minimize superfluous material and optimize training opportunities to facilitate schema constructions. 

Atkinson, R. K., Derry, S. J., Renkl, A., & Wortham, D. (2000).  Learning from Examples:  Instructional Principles from the Worked Examples Research.  Review of Educational Research, 70(2), 181-214.

Marsh, E. J., Butler, A. C. (in press).  Chapter 29:  Memory in Educational Settings.

Simon, M. A. and Schifter, D. (1991).  Towards a constructivist perspective:  An intervention study of mathematics teacher development.  Educational Studies in Mathematics, 22(4), 309-331.

What should teachers do with learning styles?

What should teachers do with ‘learning styles?’

As a teacher, I have sat through many professional developments designed to teach us how to assess learning styles and how to use them effectively in the classroom.  As a doctoral student, I have learned there are multiple modes under which instruction is more effective.  One of the issues that designers deal with is how to present information in instructional materials in such a way that learning is optimized.  Presenting information in two sensory modalities rather than one leads to a more efficient use of memory resources because the modality-specific subsystems of the working memory are utilized optimally.  Tindall-Ford, Chandler, and Sweller (1997) explored the effect of bimodal instruction in their study that compared student performance using audio text and visual diagrams to students who just used visual-only format.  They concluded that presentations incorporating two sensory modes are more effective than one.  This effect was further explored and named the modality principle by  Mayer (2001). 

Based on my experience, here is what I know to be true:

1.     There are different ways to learn.  The ways are what some deem as ‘learning styles’

2.    There are individual differences in learning

3.    Students learn differently depending on many things.  For example, the topic, the teacher, the type of presentation, the temperature in the room.  (and in high school, the moon phases).

4.    Learning can occur in an infinite number of ways.

In short, I agree with Pashler, McDaniel, Rohrer, and Bjork (2009) and in my opinion, learning ‘styles’ do not exist.  Each student may have a learning preference, per se, but to design instruction based on a single style of learning seems to me to be counterproductive given the many different ways of learning something. 

To answer the introductory question “What should teachers do with learning styles?” I offer two suggestions.  First, teachers need to recognize there are different ways of learning and incorporate a variety of activities in a lesson that support various learning preferences.  Second, while individual learning styles are important, it may be more effective to match instruction to the content that is being taught. 

While researching information for this blog, I came across the following article.  The author, David Glenn, discusses the Pashler article.

http://chronicle.com/article/Matching-Teaching-Style-to/49497/

Mayer, R.E. (2001).  Multimedia learning.  New York:  Cambridge University Press.

Pashler, H., McDaniel, M., Rohrer, D. & Bjork, R.  (2009).  Learning Styles:  Concepts and Evidence.  Psychological Science in the Public Interest.  (9), 105-119.

Tindall-Ford, S., Chandler, P., & Sweller, J. (1997).  When two sensory modes are better than one.  Journal of Experimental Psychology:  Applied, (3), 257-287.

Module 3: Activity 1: Learner Attention

According to information processing theory, humans are processors of information and cognition is a system of brain functions. A student’s contact with the information and knowledge they are expected to learn is through their sense receptors.  The sensory receptors allow the student to make contact with the environment.  While the sensory register has large capacity, the information stored does not last long.  Attention plays a key role in moving information from the sensory registry to working memory.  In working memory, the information is processed.  It is here where students try to make sense of the instructional material and content.  Information in the working memory must be kept active to be retained as activation fades quickly when attention shifts away.  Consequently, maintaining the attention of a student is a key component of effective instructional design.  If nothing is done to gain and keep student attention during the presentation of instructional material, the information to be learned will be lost.  Effective design of instruction plays a key role in gaining and maintaining student attention.

          Robert Gagne, one of the key researchers and contributors to Instructional Systems Design.  His research originated during the behaviorist movement and his focus was on the outcomes, or behaviors, that resulted from training.  However, in his book, The Conditions of Learning (1965), Gagne outlined nine events of instruction which contained the underpinnings of cognitivism and information processing.  His Nine Events of Instruction correlated the conditions that occur during instruction with student outcomes. 

          Gagne noted in order for learning to take place you must first capture the attention of the student as his first of nine steps was “gain attention.”  This is where Gagne believed that instruction should stimulate receptors of students so that they were primed and ready to learn.  Beginning a lesson with an animated multimedia clip that is accompanied by sound is an effective way to stimulate multiple sensory receptors (visual and audio).  Another way to gain student attention is to start a lesson with a thought-provoking question such as one that sparks student curiosity and that motivates the learner.   

          Beyond gaining attention of the learner, it is important to consider strategies to maintain student attention throughout the instruction.  The following is a link to a video in which a college professor discusses strategies he uses to keep students in a large college lecture interested in the material.

Engaging Student Attention Video:  http://www.youtube.com/watch?v=Au_Navo5PCg

          Another way to keep students interested in the material is to use multimedia in presentations.  There has been much research on the effective pedagogy and the use of multimedia presentations during instruction.  Mayer (1997) reviewed eight studies that were conducted to determine whether multimedia instruction was effective.  Overwhelmingly, the evidence showed that using a presentation with verbal and visual multimedia formats was more effective than direct instruction.

          Soon after Microsoft released PowerPoint, it became the preferred method of information delivery in many classrooms.  Educators took advantage of this new media and began using the program to present instructional material.  It rejuvenated what was once considered the ‘boring lecture.’  However, soon thereafter the novelty of the multimedia program wore off and students were once again bound to their seats listening to another ‘boring lecture’ except this time they were looking at a screen instead of just the teacher in the front of the room.  Bartsch and Cobern (2003) investigated whether students liked and learned more from PowerPoint presentations than from overhead transparencies.  They concluded that PowerPoint can be beneficial, but material that is not pertinent to the presentation can be harmful to student learning.

          A couple of years ago, I took a class under the direction of Dr. Gerry Swan where we discussed the downfall of PowerPoint as an instructional tool.  The following two videos put a humorous spin on some of the design flaws when using PowerPoint that can distract or lose the attention of the learner.

 Don McMillan: Life After Death by PowerPoint Video: 

http://www.youtube.com/watch?v=lpvgfmEU2Ck

STOP! You're killing me with PowerPoint (Rap) Video:

http://www.youtube.com/watch?v=_JU48-FVqvQ

          Whether using a Powerpoint or other multimedia tool for presentation, I think a key component of maintaining students attention during instruction is to engage the learner…make them feel a part of the instruction so that they feel ownership.  Students learn by doing and passivity dampens motivation to learn. 

Bartsch, R. A. & Cobern, K. M. (2003).  Effectiveness of PowerPoint presentations in lectures.  Computers & Education, 41(1), 77 – 86.

Gagne, R.M. (1985).  The conditions of learning and theory of instructions (4^th ed.).  New York:  Holt, Rienhart, and Winston. 

Mayer, R.E. (1997).  Multimedia learning:  Are we asking the right questions? Educational Psychologist. 32(1), 1-19.

Activity 5: Old Fashioned Play Builds Skills

I am going to have to take a break from my attempt at academic writing as I was moved personally by this segment.  My daughters, Isabell (11 years old) and Riette (10 years old) are always outside playing when the weather allows.  We live on a farm and there is plenty of room to play :).  Their imaginations take over. I sit on the side of the porch (so they cannot see me) and watch them play "teacher", "singer-star", "Harry Potter", and the like.  I cannot express in words how much joy it brings. Just one of those things.   They play for hours and complain when I call them inside to complete various chores or eat. 

Then, I read the excerpt from NPR.  "According to Berk, one reason make-believe is such a powerful tool for building self-discipline is because during make-believe, children engage in what's called private speech: They talk to themselves about what they are going to do and how they are going to do it."  That is so powerful. And i promise to let them play for just a little bit longer next time. 

Riette & Isabell are happy that Dr. Usher
assigned the excerpt from NPR.

Self-regulation is an important notion for Vygotsky.  He believed that self-regulation gave children opportunites to make choices and decisions to make rules within their zone of proximal development. As a mother, I am enjoying this 'zone' immensely, and often ponder how long will it last?

 

Group Q6: A Constructivsm Video

How to Teach Area and Perimeter using Google Earth

 VIDEO LINK  http://www.youtube.com/watch?v=PHwrehm6HO8

This instructional video demonstrates how to use Google Earth to teach area and perimeter.  The video serves as a tutorial for teachers, and even though it is not a "classroom" example, it is an excellent application under the constructivist tenet.  Students can use the program to construct viable knowledge about area and perimeter.

     A lesson plan such as this supports the constructivism tenet in that learning is an active process of constructing meaning.  Further, the lesson is engaging and provides a worldly experience under which a student can learn about area and perimeter. 

                                                                                                ~Ellen Bloomfield

This video is a tutorial for teachers on teaching area and perimeter using Google Earth.  The narrator points out that you’ve “gotta make learning fun,” and having students measure familiar landmarks can be a way to do that.  In the video, he shows two examples: one of measuring the Pentagon and another of measuring the students’ own school.  These learning activities are good examples of active learning exercises which could be useful for a teacher interested in incorporating more constructivist activities into the math classroom

                                                                                                ~Stacey Greenwell

The following video provides a visual of how a constructivist elementary classroom is guided by the teacher with questioning and exploring and discovering of the objects with peers. There are a total of three videos that accompany the example above.  This video illustrates the concept of constructivism because it shows the teacher encouraging the students to be engaged as they construct meaning with the different shapes.

                                                                                    ~Renee Kaufmann

Activity 4: Vygotsky's contributions

Vygotzky’s theory is one of the foundations of constructivism.  He asserted that social interaction plays a fundamental role in the process of cognitive development.  In contrast to Piaget, Vygotsky felt social learning precedes development.  Vygotsky’s theory emphasized community, and the important role it played in the process of meaning making and knowledge construction. 

Vygotsky defined the “Zone of Proximal Development” (ZPD) as the distance between a student’s ability to perform a task under supervision or with collaboration and the student’s ability to solve the same problem independently.  The ZPD is on a continuum between what is known and unknown.  This is unlike Piaget’s developmental description where it is strictly based on chronological age.

Vygotsky recognized dialogue as a key influence in developing our understanding of the social foundations of learning and thinking.  Specifically, he believed the deployment of socially transmitted language contributes to student understanding of the concepts presented (Renshaw, 2004). I am in the beginning stages of planning for my dissertation and my tentative proposal of study is influenced by Vygotsky’s theory.  To summarize, I want to examine the association between the features of the SMARTboard™ interactive whiteboard an display of worked examples.  I also want to explore the dialogic interaction in an IWB classroom that occurs during the presentation of worked examples.  Specifically, I want to find out how the SMARTboard™is used to direct learner attention to the problem state and solution procedures.  In addition,  I want to know how it is used to support dialogic interactivity between the teacher and student during the presentation of worked examples.

At this point in the game of my doctoral pursuit, my research will be based on activity theory.  Activity theory presumes that learning and behavior can be controlled by mediating artifacts.  This theory also has its roots in Vygotsky and his studies of cultural-historical psychology.  Activity theory relates the incident of learning to a facilitating tool.  The tool of my study will be the interactive whiteboard.  The goal of my study will be too see how effective this tool is during a series of instructional events. 

Van der Linden, J. & Renshaw, P. D. (2004).  Dialogic Learning: Shifting Perspectives to Learning Instruction, and Teaching.  Dialogic Learning Teaching and Instructions:  Theoretical Roots and Analytical Frameworks.  Kluwer Academic Publishers:  Dordrecht, Boston, London. 

Nardi, B. A. (1997).  Context and Consciousness.  Massachusetts Institute of Technology

Activity 3: Bart the Genius

Airasian & Walsh (1997) state that constructivism is an epistemology, a philosophical explanation about the nature of knowledge.” They emphasize that constructivism is not an instructional method but rather it “describes how one attains, develops, and uses cognitive processes (p. 445).  The authors, Airasian & Walsh, would say that Bart was transferred from a behavioral environment to one in which students were able to construct their own knowledge.  They would also note that the teacher’s role was one of a facilitator which guided students learning experience.  Each student was able to explore and create at their own leisure.

This is very much an example of constructivism.  Further it shows the difference between education based on behaviorism and constructivism.  (Even the first principal's name was Skinner!).  Students were able to construct their own knowledge through the series of activities provided by the teacher.

I am a HUGE fan of constructivism.  I love to watch students create meaning out of an experience.  That is the kind of knowledge that sticks!  I believe constructing one’s own knowledge empowers an individual.  Allowing students to be in charge of their own learning can result in self-affirmation and encourage students to continue to learn. 

Activity 2: Piaget and James

Q2:  Piaget vs. James

The video displayed students completing various exercises and demonstrated the developmental levels of the children to the viewer.  In one exercise, the same amount of juice of is placed in glasses of the same size and then again in glasses that vary in size -- One glass is skinny, the other is wide.  The younger of the children believes that when the size of the glass is changed, the amount of liquid inside the glass is changed also.  This is unlike the older child who recognizes that the amount of liquid does not change even though the glass size does. 

I think a Piagetian approach to developing the cognitive level in students for this particular problem would have the students continue to explore with the glasses until it is recognized that the liquid amount did not change.  Contrastingly, James would have the teacher show the student the difference and then proceed to go on to the next lesson. 

Module 2-Question 1: Piaget

Q1:  Piaget’s contributions to cognitive constructivist theory. 

Jean Piaget’s research in the fields of psychology, sociology, and education has greatly influenced the field of education.  After years of research in developmental psychology and genetic epistemology, Piaget came to believe that intelligence is a form of adaptations.  He claimed that the child adapts to his physical and social environment thought the processes of assimilation and adaptation (Wadsworth, p.13-14).  It is through the processes of assimilation and adaptation that a child constructs his own intellectual world.  He believed that a child’s development occurred in four distinct stages:  Sensory Motor (Birth to approximately 2 years), Preoperational (until ages 6 or 7), Concrete Operations (10-12 years) and then finally the Formal Operations stage.

When designing instruction it is important to consider the sequence in which the learning events occur.  Of course, the sequence is dependent upon the desired objectives; however, the goal of designing instruction is to select the most appropriate sequence for the given content.  In some cases, the sequence is determined by the objectives.  This is especially true when the objective is performance based.   Other topics, however, have a less obvious sequence.  In this case, sequencing instruction based on developmental stages of the learner, such as Piaget proposed, would be beneficial.  Further, by utilizing Piaget’s processes of adaptation, instruction can address learning events that contain experiential and real world content.  Instructional design of world related applications is also parallel to Piaget’s thoughts on schema.  Suppose you are designing instruction for a lower level Algebra 1 class on two variable linear equations.  There are several different ways to teach graphs of lines and their equations, some of which laborious and full of procedures.  However, the relationship between two variables can be explained easily by using real life examples.  By means of the real world application, a student can develop a schema under which instruction is developed.   

Group Q7: Theory to Practice

Our group watched http://timssvideo.com/45 on polygons.  The following are our responses to question 7. 

P1:  Two elements of instruction that relate to the theory of behaviorism

1) Positive Reinforcement --> Operant Conditioning

2) Classical Conditioning

Reflection:

1) According to Skinner (1953), the only defining characteristic of a reinforcing stimulus is that it reinforces" (p.72).  Positive reinforcement is seen in this classroom.  For example, the instructor poses  a question to the students, a student comes to the board, responds to the question by drawing on the board and the instructor responds with, "Okay, very good" after a student successfully completed the task of drawing a polygon.  The positive reinforcement establishes that the student's actions were done correctly by praising the student, which exemplifies operant conditioning.

2) Classical conditioning is seen a couple of times within this video.  One example is when the instructor enters the classroom. The students all stand and respond to the students' greeting in unison. The students then sit down and the class begins.  The second example of classical conditioning is when the teacher says, "Class dismissed." The students response is to stand-up and say goodbye, sit down and collect their belongings to leave.

                                                                                                ~Renee Kaufmann

P2:  Two elements of instruction that relate to the theory of cognitive constructivism.

According to cognitive constructivism, knowledge is built through experience from which schemas are formed.  Piaget believed that schemas could be changed through assimilation and accommodation.  Under this theory, the role of the teacher is to provide exploration activities.  In this video. after a teacher led discussion of the definition of polygons, students were asked to complete a series of drawings to determine the similarities and differences between different types of polygons.  By using this instructional activity, the teacher provides a directed way for students to explore the characteristics of polygons.  As noted by Piaget, the students exploring and categorizing the shapes by characteristic enables schema to be constructed.

Reflection:  Due to the nature of the content, cognitive constructivism is a viable theorectical support for geometric instruction.  By exploring the theorems and relationships in geometry, students are able to construct their own knowledge.
                                                                                                ~Ellen Bloomfield

P3:  Two elements of instruction that relate to the theory of social constructivism. 

In the polygons lesson, the best example of social constructivism in practice are the two one-minute discussion with a neighboring student. The first one considered the difference between two kinds of polygons, and the second asked students to discuss how to find a pentagon. Each student discussed the concept with a partner. After the discussion, some students were asked to come to the board to demonstrate their findings through drawing the appropriate type of polygon. The important part is the social aspect--the students work in groups to learn.

                                                                                                ~Stacey Greenwell

 


Swimmer Lainie showing
Piaget &Vygotsky around their
first swim meet. 





Vygotsky & Piaget experiencing
the starting blocks



Vygotsky & Piaget discussing
their observations of the
young swimmers.







Module 1: IAT

I took the Male/Female  Career/Family IAT test.  I found taking the test troublesome, as it did not allow me to answer the questions as I wanted.   Both of my parents were teachers and my dad was also a swim coach.  Growing up, my mother did a majority of the rearing as my dad was busy with the swim team.  However,  the swim team was a part of my family.  We celebrated holidays as a family, we mourned losses as a family.  What I experienced was a large diverse group of people that helped raise me and influence who I am today.  Further, for most of my life at home, my family was the only Jewish family in the entire county.  When we first moved there, we were asked to leave by people who thought that Jews should not be a part of the community.   Dad ignored the requests (and in some cases, threats) and to this day my mom is an integral part of my hometown.  This, along with my swim team family, has helped define the lenses in which I look through, and it is one of extreme tolerance for individuals , their individual beliefs, and the roles they chose.    I did not understand why the IAT was so “black-white”  You were not allowed to choose which category you associated the word given.  I do not necessary associate female with family and  male with career.  As a matter of fact, my husband is (and has always been) a stay-at-home dad.  While he does have a job coaching the swim team, we as a family unit view the swim team as a family endeavor.   Needless to say, my results reflected my thoughts as I had a moderate association to female with career and male with family.

The process of learning is a culmination of associations.  We associate items to remember them.  I often have my students use acronyms to help them remember how to complete a certain mathematical procedure.  For example, GLAM is the acronym we use for solving multi-step equations.  G – grouping symbols.  L – like terms, A – undo addition, M – undo multiplication.  It works well for the students and I believe it is due to the fact they do not have to spend as much cognitive energy memorizing the acronym as they due the lengthy steps needed to solve a multi-step equation.   

James states that the teacher’s job is to build up useful associations.  “The more copious the associative systems, the completer the individual’s adaptations to the world “ (p. 42).  I believe the background in which I was raised enables me to adapt to various ‘bumps in the road’ most of the time, fairly easily.  Further,  I try to provide as many associative systems for my students when designing instruction in order for them to be able to apply what was learned to novel problems.