Ed Tech 504 Peer Review by Kim Hefty

8/9/2019 Ed Tech 504 Peer Review by Kim Hefty

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

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

centered learning environments, educational technology, and the higher demands of Bloom's

Taxonomy in the mathematics classroom and the importance of doing so due to new teacher

evaluation systems. And by doing so, you can create a mathematics classroom that utilizes

technology, higher level cognitive student evaluation and synthesis, as well as a student centered

learning environment.

Comment [H1]: Starting sentence with word

And may not be most effective. Consider using

‘Additionally, by doing so, …’


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Introduction

As I am sitting here listening to a symphony concert, I am astounded at the inspiration the music

provides me to begin writing down ideas for this paper. The mathematics combined with

creativity needed by Gustav Mahler to write Symphony No. 5 is perplexing to a simple

mathematics high school teacher. If only we could get young minded teenagers to understand the

mathematics involved in writing such a masterpiece. Or can we?

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.

There are times when the instructor gives feedback on what students should do and then students

have the opportunity to immediately apply it. And there is amazing technology behind such

intricate instruments. I wonder how we can get more math teachers to apply such ideas in a high

school mathematics classroom?

Learning Theories of the Past and Present and the Mathematics Classroom

There are many existing learning theories including established theories and emerging theories.

The focus of this paper will pertain to the more traditional objectivism and constructivism. So

first it would be 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).

Comment [H2]: Consider changing out of first

person since this is an professional paper. This is

great hook but …Be sure you really want first pers

for an academic paper.

Comment [H3]: Eliminate ‘I wonder’ start with

How can we …

Comment [H4]: Eliminating ‘I wonder’ will ma

this a question.

Comment [H5]: Add comma

Comment [H6]: Consider, “To begin, it is

imperative …

Comment [H7]: Watch your verb tenses

Comment [H8]: Should you use a colon?

Comment [H9]: Comma between name and

year? Double check all your citations.


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States where students do not follow a career path that requires a mathematical background, then

math teachers must begin to break down their own barriers.

The New Teacher Evaluation System and the Mathematics Classroom

"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).

This is just one sub category, engaging students in learning, in the Danielson framework that will

be used to evaluate all teachers in Washington State's new teacher evaluation system. This will

affect all teachers and especially traditional math teachers when the system goes into place. In

the near future, evaluations will involve much more specific targets that was not part of the

previous evaluation system. To reach a distinguished category rating, it will require a teacher to

do what is described in the above quote. How will a teacher accomplish all this using objectivism

learning theory and direct instruction? It will be crucial for teachers to begin exchanging many

direct instruction lessons for a more constructivism based learning style.

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

the highest rating of distinguished. Other categories that will be more difficult to accomplish a

distinguished rating using purely direct instruction in a math classroom include communicating

with students, using questioning and discussion techniques, demonstrating flexibility and

Comment [H22]: In order to break down the

barriers in the United States for students who do

not follow a career path that requires a

mathematical background, math teachers must

begin to break down their own barriers.

Comment [H23]: Need a topic sentence to beg

this paragraph. And a lead in to the quote.

Comment [H24]: Make this the start of the

sentence.

Comment [H25]: …teachers, especially math

teachers, …

Comment [H26]: …were not previously includ

… try changing the wording around

Comment [H27]: …requires …

Comment [H28]: Hyphenate

Comment [H29]: Remove

Comment [H30]: … is not …

Comment [H31]: If it is a rating category, add

and capitalize.

Comment [H32]: Add semicolon after

distinguished then …other difficult categories

include …


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responsiveness, designing 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,

to be a distinguished teacher within these areas will be essentially impossible when solely using

direct instruction.

The Danielson framework clearly calls for some constructivism learning style 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). Without a doubt these expectations identify more than just an

objectivism learning environment. The constructivism learning theory would be more supportive

by implementing both student centered activities and educational technology. Designing

coherent instruction directly refers to applying technology in the classroom as well.

Complex Instruction and the Mathematics Classroom

"Student centered learning environments (SCLEs) provide interactive

complimentary activities that enable individuals to address unique learning

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

understanding" (Land 2012).

Complex instruction is a fantastic way to implement student centered activities and involve more

of the constructivist learning theory into your classroom. Unfortunately, when student centered

activity based instruction first came out, many math teachers failed to make it successful within

their own classroom. What was still missing was how to implement group work in an effective

manner; the organization to make activities work was missing. The teacher has a role to make it

Comment [H33]: I would remove without a

doubt and instead say, ‘these expectations clearly

Comment [H34]: Of what?

Comment [H35]: How? Why? elaborate

Comment [H36]: Create an intro … see

comments previously

Comment [H37]: Make a better word choice

Comment [H38]: Comma after centered


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work, but what is it? Before complex instruction, student centered learning environments didn't

work for me either. With every new curriculum came my hope that that text would provide

activities that would engage all students. However, the typical complaints by teachers was that

one or two students in the group do all the work held true in my classroom as well. I had

basically given up. Then I had the opportunity to attend a complex instruction training in the

summer of 2011. It not only changed the learning environment in my classroom, it also created

something I did not expect; all kids were participating.

"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).

My brief summary of complex instruction doesn't do justice to the process. 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.

Removing the stigma set forth by students that math is boring is important. And to do so math

teachers must begin to break down their own barriers that is preventing them from using SCLEs.

Educational Technology and the Mathematics Classroom

Comment [H39]: Personal pronouns are bette

for reflections

Comment [H40]: See above

Comment [H41]: I would change the voice and

make it more observational

Comment [H42]: Quote to indicate that this is

how they feel

Comment [H43]: …that are… maybe add a

comma


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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 student's 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, as a teacher you will have a much higher probability of engaging

every student in your classroom. When each student has 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. When teachers have to compete for student attention

against video games, you need every tool you can gather.

"Government documents America 2000 and Goals 2000...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 is introduction of the use of a

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

mathematics and students understanding why we do this mathematics. The graphing calculator

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. It

Comment [H44]: Add semicolon

Comment [H45]: Either ‘a student’ ‘ or

‘students’ ‘ plural

Comment [H46]: … take out as and you … it w

clarify

Comment [H47]: change each student to

students, all the rest of words in sentence are plu

Comment [H48]: In order to compete with

video games, teachers need every tool they can ge

Comment [H49]: Awkward … reword

Comment [H50]: Consider linking all the

technology paragraphs … each paragraph needs atopic sentence and a concluding sentence


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

basic information of geometry and into a deeper understanding of geometry.

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.

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 text books (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 many different subjects and allow for internet research.

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

run it could save school districts a lot of money. Districts would not be purchasing individual

textbooks or spending money on computer labs and the maintenance of such labs. Schools would

save space by not needing multiple labs within a building.

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 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

Comment [H51]: Add a comma

Comment [H52]: Make this a complex senten

Comment [H53]: At this time …


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great step towards progress. But, until there are more applicable student centered activities, many

higher level mathematics classrooms will remain direct instruction.

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

telling teachers to make use of technology isn't enough, teachers need appropriate training on

how to effectively utilize technology. And math teachers must begin to break down their own

barriers preventing them from moving forward with technology.

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 your students and training of yourself. This

applies to classroom management as well as integrating a successful SCLE. Just like when my

family took our dog to obedience school, the training was more about training the humans than

Comment [H54]: However

Comment [H55]: …to be as authentic …

Comment [H56]: Remove, redundant

Comment [H57]: Is not

Comment [H58]: Add semicolon after

Comment [H59]: remove

Comment [H60]: Both students and teachers

Comment [H61]: Don’t use personal pronoun

make more generic … just like when training a

family dog etc


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the dog. Likewise, changing your teaching technique from direct instruction to a balance

between direct instruction and activities, takes training, teacher commitment and faith in the

process. School districts must begin to spend money on more authentic 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?

So when does 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 is too

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

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

is not everyone in the group is building the same initial math concept, but then all four group

member results help to build a pattern followed by a group conjecture.

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

too simple and can be too easily completed, then it isn't a group worthy concept. This could also

apply to review concepts and so students already know the outcome. Or the opposite of this is if

a mathematical concept was so complex it would take days to establish an outcome by a group.

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 2-4 activities, but

then students are given the next GSP activity in as CI. 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

Comment [H62]: …a teacher changing …

Comment [H63]: …commit… to spending mon

…

Comment [H64]: What purpose do these

questions serve in your paragraph. Make them

statements.

Comment [H65]: comma

Comment [H66]: must be

Comment [H67]: reword, it’s not clear

Comment [H68]: is not

Comment [H69]: clarify

Comment [H70]: include in previous paragrap


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Conclusion

How do you go about doing all this? First and foremost, there needs to be a shift in how a district

spends money and trains teachers, especially math teachers. And math teachers need to believe

that there is a better way. Without teacher buy-in, he/she will not change. Excellent training and

immediate positive results can help to adjust a teacher's outlook towards student centered

learning. With organization, excellent training opportunities, and appropriate technology, then all

students learning will follow.

The final benefit to mention when applying both learning theories in your classroom is how your

classroom will be more appealing. By alternating between activities and direct instruction, it

keeps it more interesting for students. The instructional technique delivery system will depend

upon the math teacher and the topic. Which will work best for today's concept, objectivism or

constructivism? As John Dewey said,

"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 such that you can create a more balanced high school math classroom that utilizes

different learning styles and technology. If you are still teaching solely using a direct instruction

technique, then take a look around your classroom to truly analyze your results. When direct

Comment [H71]: create a statement and

combine with next sentence


Comment [H73]: clarify more directly


Comment [H75]: …to make a classroom…


Comment [H77]: will keep it …

Comment [H78]: …in order to





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instructing, are all of your students paying attention? Are all students participating? Are all

students learning well?

References

Angeli, C. (2009). Epistemological and methodological issues for the conceptualization,

development, and assessment of ICT TPCK: Advances in technological pedagogical contentknowledge (TPCK). Computers and Education (0360-1315), 52 (1), 154-168.

doi:10.1016/j.compedu.2008.07.006

Cohen, E. G., Lotan, R. A., Scarloss, B. R., & Arellano, A. R. (1999). Complex instruction:

Equity in cooperative learning classrooms. Theory into Practice, 38(2), 80-86. Retrieved from

http://search.proquest.com/docview/909852737?accountid=9649

Cooper, S. (2009). Constructivism: Constructivist learning theory. Retrieved February 16, 2013

from http://www.lifecircles-inc.com/Learningtheories/constructivism/constructivism.html

Danielson, C. (2012). Danielson framework for teaching rubrics by Washington state criteria(Version 1.1) Retrieved from Office of the Superintendent of Public Instruction Washington

State http://tpep-wa.org/wp-content/uploads/Danielson-Rubrics-by-criteria.pdf

Dewey, J. (1938). Experience and education. (p. 17) New York, NY: Macmillan.

Ertmer, Peggy and Newby, Timothy J. (1993). Behaviorism, Cognitivism, Constructivism:

Comparing critical features from an instruction design perspective. Performance ImprovementQuarterly 6(4), 50-71. Retrieved February 16, 2013 from

http://ocw.metu.edu.tr/file.php/118/Week_6/Ertmer-Newby-beh-cog-const.pdf

Hannan, J. (2012). Top 12 math iPad apps for students and teachers [Web log post]. Retrieved

from http://www.teachhub.com/top-12-math-ipad-apps-students-and-teachers

Haselton, T. (2013). Apple:More than 8 milion iPads sold to education institutions [Web log

post]. Retrieved from http://www.technobuffalo.com/2013/03/03/apple-8-million-ipad-school/

Heick, T. (2012). 12 of the best math iPad apps of 2012 [Web log post]. Retrieved from

http://www.teachthought.com/apps-2/12-of-the-best-math-ipad-apps-of-2012/

Johnston, C. (2013). At long last, TI releases graphing calculator for the iPad. Retrieved fromhttp://arstechnica.com/apple/2013/02/at-long-last-ti-releases-graphing-calculator-for-the-ipad/

Jonassen, D. H. (1991). Objectivism versus constructivism: Do we need a new philosophical paradigm? Educational Technology Research and Development, 39(3), 9-10. doi:

10.1007/BF02296434.

Land, S., Hannafin, M., & Oliver, K. (2012). Student centered learning environments. In D.Jonassen & S. Land (Eds.), Theoretical foundations of learning environments (pp. 3-25). NewYork, NY: Routledge.

Comment [H82]: make these statements

http://dx.doi.org.libproxy.boisestate.edu/10.1016/j.compedu.2008.07.006http://dx.doi.org.libproxy.boisestate.edu/10.1016/j.compedu.2008.07.006http://dx.doi.org.libproxy.boisestate.edu/10.1016/j.compedu.2008.07.006


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Woodward, J. (2004). Mathematics education in the united states: Past to present. Journal of Learning Disabilities, 37 (1), 16-31. Retrieved from

http://search.proquest.com/docview/194228235?accountid=9649

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Ed Tech 504 Peer Review by Kim Hefty