Balancing Learning Theories, Instructional Styles and Technology to meet the Demands of

Teaching High School Mathematics in the 21st Century

Angie Kruzich

Kim Hefty Peer Reviewer

EdTech 504

Theoretical Foundations of Education Technology

Dr. K. Diane Hall

Boise State University

April 22, 2013

Abstract

The focus of this paper is to utilize past and present theories of learning and how the

relationship between the theories impact mathematical education in the classroom, specifically

high school. The ideas within this article embrace both the traditional theory of objectivism and

the more modern constructivist learning theory. Included are ideas to incorporate student

centered learning environments, educational technology, and the importance of doing so due to

new teacher evaluation systems. Additionally, you can create a mathematics classroom that

utilizes technology, higher level cognitive student thinking, as well as a student centered learning

environment.

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Introduction

Imagine a math teacher listening to a live symphony concert. The music provides astounding

inspiration to any listener, especially a high school mathematics teacher. The mathematics

combined with creativity needed by Gustav Mahler to write Symphony No. 5 is perplexing.

Orchestra must be a fantastic class to teach. All the students choose to be there as an elective

class. Every student has something in their hands to do at all times and is always participating.

The instructor gives feedback on what students should do and then students have the opportunity

to immediately apply it, continuously participating. Moreover, there is amazing technology

behind such intricate instruments. The thoughts that follow in this article show how any math

teacher can acquire an interactive classroom like orchestra.

Learning Theories of the Past and Present and the Mathematics Classroom

There are many existing learning theories; established theories and emerging theories. The focus

of this paper will pertain to the more traditional objectivism along with the more modern

constructivism. To begin, it is imperative to understand a little behind these two theories.

"Objectivism assumes that learning is the process of mapping...concepts onto

learners. Objectivism...holds that there is an objective reality that we as learners

assimilate. The role of education is to help students learn about the real world.

Students are not encouraged to make their own interpretations of what they

perceive; it is the role of the teacher or the instruction to interpret events for them.

Learners are told about the world and are expected to replicate its content and

structure in their thinking" (Jonassen, 1991).

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Constructivism is a theory that equates learning with creating meaning from experience (Ertmer

1993). It suggests that each listener or reader will potentially use the content and process the

communication in different ways, to construct one's own knowledge. This theory describes

learning as an active process, unique to the individual, which consists of constructing conceptual

relationships and meaning from information and experiences already in the learner's repertoire

(Cooper, 2009). David Jonassen summarizes the ideas within this paper well;

"These two theories are generally described as polar extremes on a continuum

from externally mediated reality (objectivism) to internally mediated reality

(constructivism). Most theorists, however, take positions that fall somewhere in

the middle of the continuum." (Jonassen, 1991).

In one's own life, a person should maintain balance between work and play. Likewise, a teacher

should maintain a balance within the mathematics classroom. A balanced mathematical

classroom occurs when a symbiotic relationship exists utilizing both objectivism and

constructivism learning theories. Typically, objectivism learning is seen when teaching utilizes a

direct instruction approach; whereas constructivism leads to activity based lessons.

Why choose just one theory? The ultimate goal when teaching mathematics should be to balance

learning theories, and therefore balance teaching styles in order to reach the needs of multiple

student learning styles. There are too many students from the past and present that avoid

mathematics because math is taught, all too commonly, using direct instruction. Math should not

be feared by so many people such that it is okay to say "I don't do math." or " I wasn't good at

math." In order to break down these mathematical barriers in the United States for math students,

math teachers must begin to break down their own barriers.

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The New Teacher Evaluation System and the Mathematics Classroom

Throughout the United States, there is a radical change occurring regarding teacher evaluations.

One of the adopted frameworks that will be used by many school districts within Washington

State is called the Danielson Framework. The following is an example of the expectations of all

teachers.

"Virtually all students are intellectually engaged in challenging content through

well-designed learning tasks and suitable scaffolding by the teacher and fully

aligned with the instructional outcomes. In addition, there is evidence of some

student initiation of inquiry and of student contribution to the exploration of

important content. The pacing of the lesson provides students the time needed to

intellectually engage with and reflect upon their learning and to consolidate their

understanding. Students may have some choice in how they complete tasks and

may serve as resources for one another" (Danielson, 2012).

Engaging all students is expected in the Danielson framework. This will affect all teachers, even

math teachers. There are four levels within the framework at which a teacher can be rated;

Distinguished, Proficient, Basic or Unsatisfactory. How will a math teacher attain a

"Distinguished" rating as described above using an objectivism learning theory? It will be crucial

for teachers to begin exchanging many direct instruction lessons for a more constructivism-based

learning style.

Engaging students in learning isn't the only category in which it will be difficult to achieve the

highest rating of "Distinguished". Other categories in which it will be difficult to achieve

satisfactory ratings using pure direct instruction include, communicating with students, using

questioning and discussion techniques, demonstrating flexibility and responsiveness, designing

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coherent instruction, creating an environment of respect and rapport, managing classroom

procedures, managing student behavior, designing student assessments, and showing

professionalism (Danielson, 2012). Although these are the broad titles of the framework, the

detailed expectations described in the Danielson framework document required to be a

distinguished teacher will be essentially impossible to reach when solely using direct instruction.

The Danielson framework clearly calls for constructivism when it states "Students contribute to

extending the content and help explain concepts to their classmates." or "Teacher persists in

seeking effective approaches for students who need help, using an extensive repertoire of

instructional strategies and soliciting additional resources from the school or community."

(Danielson, 2012). Furthermore, it also directly refers to the use of technology in the classroom

by stating, "Plans represent the coordination of in-depth content knowledge, understanding of

different students’ needs, and available resources (including technology),..." (Danielson, 2012).

All of these expectations clearly identify going beyond objectivism. The constructivism learning

theory will be more supportive of meeting the Danielson framework expectations by

implementing both student centered activities and educational technology.

Complex Instruction and the Mathematics Classroom

Complex instruction is an organized way to successfully implement the constructivism learning

theory in a mathematics classroom. Complex instruction is one way to implement student

centered learning environments and embrace more of the constructivist learning theory in a

classroom.

"Student centered learning environments (SCLEs) provide interactive

complimentary activities that enable individuals to address unique learning

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interests and needs, study multiple levels of complexity, and deepen

understanding" (Land, 2012).

Unfortunately, when SCLE's first came out, many math teachers failed to make it successful

within their own classroom. What was missing was how to implement group work effectively;

the organization to make group activities work was missing. The typical complaints by teachers

was that one or two students in the group do all the work. Lack of experience, that the teacher

has a special role to make it work, was missing.

"Complex Instruction (CI) is an instructional approach that allows educators to

address these questions successfully. In CI, teachers use cooperative group work

to teach at a high academic level in diverse classrooms. They assign open-ended,

interdependent group tasks and organize the classroom to maximize student

interaction. In their small groups, students serve as academic and linguistic

resources for one another. When implementing CI, teachers pay particular

attention to unequal participation of students and employ strategies to address

such status problems" (Cohen, 1999).

This brief summary of complex instruction doesn't do justice to the process. CI not only changes

the learning environment in a classroom from objectivism-based to constructivism-based, it also

created something unexpected; all kids were engaged in the activity. This is a difficult

requirement to meet in the Danielson Framework. Teachers that want to make CI group work

and constructivism victorious in their own classroom really need to attend training and observe

other teachers using the process. It is the employment of the strategies from CI training that

make SCLEs and constructivism flourish. Most importantly, CI can also remove the stigma set

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forth by students that "math is boring" or "I can't do math." by dealing with preconceived student

status issues. To initiate CI, math teachers must break down their own barriers that are

preventing them from using SCLEs.

Educational Technology and the Mathematics Classroom

Under the structure of CI, a math teacher can find many ways to alter a direct instruction lesson

into a more constructivist activity via technology. It is a very natural transition; placing

technology into students' hands immediately engages students. Today's students do not know the

world without technology and by giving them technology to work with in the form of computers,

graphing calculators, or iPads, a teacher will have a much higher probability of engaging every

student in the classroom. When students have a piece of technology in their hands, they will

naturally start pushing buttons and discovering how the technology works. Already,

constructivist learning is taking place.

Technology is necessary in today's classroom as stated by both the Bush and Clinton

administrations in their documents titled America 2000 and Goals 2000.

"These documents focused on the need for education to produce knowledge

workers who were proficient in the uses of technology and communication skills

and who possessed high levels of mathematical literacy. It was evident that

computer technology was reshaping the mathematics that students needed to

know now and in the future" (Woodward, 2004).

A perfect technology example in the high school math classroom was introduction of the

graphing calculator. This helped many math teachers bridge the gap between students doing

mathematics and students understanding why we the mathematics. The graphing calculator

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technology helped to reframe how mathematics was taught but still remains in a mostly direct

instruction venue.

Geometer's Sketchpad is another fantastic piece of technology that can be used in the

mathematics classroom at many different levels, from elementary math through calculus. This

program allows shapes to be constructed, measured and analyzed such that students can move

beyond the basic information of geometry and into a deeper understanding behind the

mathematics.

Recently, graphing calculators made another technological leap by developing wireless

capabilities in the TI-Nspire. This allows math teachers to be more interactive with students.

Teachers can immediately send data back and forth between student and teacher and check for

student understanding.

Utilizing iPads in the high school math classroom is also occurring. Some school districts are

now issuing an iPad to every student instead of checking out textbooks (Haselton, 2013). There

is a natural integrated use of an iPad in a school as it can work as a replacement for textbooks,

download many different apps for a variety of subjects and allow for internet research. Imagine

the joy by students, parents and teachers of a de-cluttered student backpack. In the long run,

implementing iPads could save school districts a lot of money. Districts would not be purchasing

individual textbooks, spending money on computer labs, and maintaining these labs. Schools

would also save space by not needing classrooms for labs in each building.

At this time however, there is simply a lack of high school math apps available. Most

mathematical apps are oriented towards elementary and junior high math (Heick, 2012). What

about high school? Without these resources, it explains why so many mathematics classrooms

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are still operating using a direct instruction technique and not integrating more technology. There

is a serious lack of technology applications above the geometry level (Hannan, 2012). When

some well-written apps are developed for the high school level including calculus, then more

teachers will be apt to utilize technology in the classroom. Finally, just two months prior to this

paper, Texas Instruments released a TI-Nspire graphing calculator app for the iPad (Johnston,

2013). This is a great step towards progress. However, until there are more applicable student

centered activities, many higher level mathematics classrooms will remain direct instruction with

limited technology.

Mathematics teachers need the help from the private sector to develop iPad applications but also

need school districts to support them with the training it will take to successfully implement

technological activities.

"...teacher educators need to explicitly teach how the unique features of

affordances of a tool can be used to transform a specific content domain for

specific learners, and that teachers need to be explicitly taught about the

interactions among technology, content pedagogy, and learners." (Angeli, 2009)

According to Angeli's research, new and experienced teachers that had been trained to properly

incorporate technology into their specific content areas had students outperform students whose

teachers were without training (Angeli, 2009). The training days provided by districts also need

to be as authentic for teachers as student centered activities need to be for students. Just telling

teachers to make use of technology is not enough; teachers need appropriate training on how to

effectively utilize technology. Again, math teachers must begin to break down their own barriers

preventing them from moving forward with technology.

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Applications in a Mathematics Classroom

When applying the use of technology in a mathematics classroom, it seems like a perfect time to

remove the direct instruction reins and let students begin to construct their own knowledge. The

first three weeks are critical training times for both students and teachers. This applies to

classroom management as well as integrating a successful SCLE. For example, when training a

family dog in obedience school, the training is more about training the humans than the dog.

Likewise, a teacher changing from direct instruction to a balance between direct instruction and

SCLE's, takes teacher training, teacher commitment and faith in the process. School districts

must commit to spend money on more genuine teacher trainings, rather than spending money on

another ineffective training day. Look around the room on these days. Is every teacher paying

attention? Are all teachers participating? Are all teachers learning well? The Danielson

Framework should apply to teacher learning environments as well.

So when should a math teacher use an objectivist or constructivist approach in their classroom?

First of all, a complex instruction type SCLE is not always appropriate. In order for group work

to be successful, the activity needs to be interdependent. In other words, the activity must be too

complicated for one or even two group members to complete by themselves. This helps to draw

all group members into the process. A common technique that helps to draw in all group

members is to give all members a different problem to complete. From this, a pattern can be

found using at least three members' results. Then the group can build a conjecture that results

from the pattern.

It is still acceptable to use direct instruction within a high school math classroom. If a concept is

too simple and can be too easily completed, then it is not group worthy. Likewise, review

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concepts are not a good choice for CI because students already know the outcome. The opposite

is true for a mathematical concept that is too complex and takes days to establish an outcome.

Technology would be another example of applying both direct instruction and SCLEs. Perhaps

teachers give students the skills they need by guiding students through GSP for two to four

activities, but then students are given the next GSP activity in a CI format. Now students are

following the instructions on their own to develop something as complex as the proof for the

Pythagorean Theorem. This could also work using the TI-Nspires. Math teachers must begin to

break down their own barriers to allow for such student growth in a math classroom.

Conclusion

The biggest barrier that math teachers must overcome, is that a teacher teaches math how they

were taught math, using direct instruction. However, to achieve a well-balanced math classroom

utilizing objectivism, constructivism, SCLE's and technology, first and foremost, there needs to

be a shift in how a district spends money on training teachers, especially math teachers. Math

teachers need to believe that there is a better way. Without buy-in, teachers will not change.

Excellent training and immediate positive results can help to adjust a teacher's outlook towards

SCLE's. With organization, first-rate training opportunities, and appropriate technology, a

teacher can successfully engage all students when learning math and help students learn it well.

The final benefit when applying both learning theories in a classroom is how the classroom will

be more appealing. When alternating between activities and direct instruction, the day-to-day

variety will keep the classroom more interesting for students. From day-to-day, the instructional

technique delivery system will depend upon the math teacher and the topic. The instructor must

decide which will work best for today's concept, objectivism or constructivism? As John Dewey

said,

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"Mankind likes to think in terms of extreme opposites. It is given to formulating

its beliefs in terms of Either-Ors, between which it recognizes no intermediate

possibilities. When forced to recognize that the extremes cannot be acted upon, it

is still inclined to hold that they are all right in theory but that when it comes to

practical matters circumstances compel us to compromise. Educational

philosophy is no exception" (Dewey, 1938)

Math teachers must begin to break down their own barriers by devoting the time to be properly

trained, to create a more balanced high school math classroom that utilizes different learning

styles and technology. If an instructor is still teaching solely using a direct instruction technique,

then take a look around the classroom to truly analyze the results. When direct instructing, are all

of the students paying attention? Are all students participating? Are all students learning well?

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