Technology Enhanced Mathematics Instruction: An Action on Objects Approach

1
Technology Enhanced Mathematics
InstructionAn Action on Objects Approach

• Michael L. Connell, Ph.D.
• MKahnl_at_aol.com
• 281-218-9700

2
Overview

• Introduction
• A lesson from history
• Information Societies
• Technology Replacement Models
• Technology Enhancement Models
• An Action on Object Model
• Object and Action Classes and Pedagogy
• Conclusions

3
Historical Thoughts On Educational Technology and
Change

• Students today can't prepare bark to calculate
  their problems. They depend upon their slates
  which are more expensive. What will they do when
  the slate is dropped and it breaks? They will be
  unable to write.
• ...Teachers Conference, 1703

4

• Students today can't depend upon store bought
  ink. They don't know how to make their own.
  When they run out of ink, they will be unable to
  write words or cipher until their next trip to
  the settlement. This is a sad commentary on
  modern times.
• ...Rural American Teacher 1829

5

• Ball point pens will be the ruin of education in
  our country. Students use these devices and then
  throw them away! The American virtues of thrift
  and frugality are being discarded. Business and
  banks will never allow such expensive luxuries.
• ...Federal Teacher, 1959

6

• Calculators and computers will be the ruin of
  Math education. Students will never learn math
  concepts. How will they calculate when they
  don't have their calculator with them.
• ...Many Teachers Today

7
Changes in Society

• Society is rapidly changing from an industrial
  focus emphasizing manufacturing to a highly
  information-centered orientation. This change in
  focus has created dramatic changes in the nature
  of life and environment in the society we live
  in.

8
Information Overload

• As information management tools become more
  common in the work place, in school and at home,
  the time that is spent in dealing with
  information will become greater.
• Furthermore, we are often at a loss to evaluate
  the reliability and accuracy of the information
  we are buried under.

9
Who will teach in an Information Society?

• Some people still imagine a teacher-replacement
  scenario when thinking of technology in the
  classroom. In this view discussion concentrates
  on replacing teachers with technology, and the
  perceived gains in efficiency of teaching this
  provides

10
The Human Touch

• Such an approach comes from a basic
  misunderstanding of the foundational role of
  teacher. In this simplified perspective a
  teacher is viewed as a mere provider of stimulus,
  evaluator of results, and guide to next stimulus
  in other words a human Skinner box.

11
Does this look familiar?
12
Teaching Thinking

• However, education involves human growth. Thus
  the mere development of new information sources
  and presentation schemes does not immediately
  lead to improvements in wisdom -- or pedagogical
  methods.
• We cannot limit ourselves to simplified models of
  teaching which reproduce a Skinner box!

13
Expert Teachers in Technology Enhanced Classrooms

• The role of the teacher, far from being replaced
  by a simple instructional delivery system, is
  actually more critical in a technologically
  enhanced classroom.
• The tremendous flow of information possible in a
  technology enhanced classroom, coupled with the
  immense modeling and tools which technology
  enables, makes the role of teacher of critical
  importance.

14
Premises of Technology Enhanced Mathematics
Instruction

• Mathematics education has seen a marked
  transformation in its relation to, and use of,
  educational technology.
• Todays technologies support markedly different
  mathematical objects of thought.
• Using these objects, questions can be asked and
  explored that are foundationally different -not
  just psychologically, but also mathematically -
  than technology enabled 20 short years ago

15
An Action on Objects Model
Connell, M. L (2001). Actions upon objects A
metaphor for technology enhanced mathematics
instruction. In D. Tooke N. Henderson (Eds).
Using information technology in mathematics (pp.
143-171). Binghamton, NYHawarth Press.
16
Class One Interactions

• Using this Action on Objects model an Object (of
  any abstraction class) might be experienced by an
  Actor (in this case a student) via their senses.
  As results of this experience an Emergent (in
  this example an awareness of the information
  presented) is generated.
• http//www.math2.org/math/general/multiplytable.ht
  m

17
Class Two Interactions

• In a Class Two Interaction the Actor (student)
  now acts directly upon the object. In this case
  the Emergent which results was created by the
  Actor using actions of their own choice upon an
  Object whose properties were both perceived and
  presented in a form allowing for manipulation by
  the Actor.
• http//naturalmath.com/mult/mult2.html

18
Class Three Interactions

• In a Class Three Interaction the direction of the
  interaction still originates with an Action being
  performed by an Actor (student), generally in
  response to a problem situation or problem
  solving goal. As was the case for a Class Two
  Interaction the Actor (student) performs Actions
  of their choice directly upon the Object. The
  Object may, depending upon the supporting
  programming or context, spawn further Objects for
  additional processing or representational
  purposes. In this case the Emergent which
  results is created by both the Object and the
  Actor using Actions originating from the Actor
  and mediated by the Object based tools.
• http//matti.usu.edu/nlvm/nav/frames_asid_192_g_1_
  t_1.html

19
Object Classes and Pedagogy Two Examples

• As these descriptions illustrate, current
  teaching with technology must support dual
  reification of both procedural and
  representational elements.
• This ability requires a shift in pedagogy to a
  level not seen in previous technology
  implementations.

20
The Calculator Case

• Calculator usage in the classroom did not
  constitute a revolution of instruction requiring
  a new pedagogy.
• Rather, it supported faster ways of doing the
  same old tasks.
• This can be understood by considering two
  factors
• the information-processing based pedagogy of the
  time
• and the procedural nature of the tasks the
  calculator is best suited for.

21
Calculators and Pedagogy

• Instruction during the period of initial
  calculator implementation was heavily influenced
  by a brand of information-processing coming from
  instruction design and computer science.
• The emphasis in this information-processing
  instructional model was upon efficient and
  accurate processing of optimized algorithms
  leading to a single correct answer.

22
An Illustrative Example
23
Student Actions Within this Framework

• Student actions consisted of entering information
  into working memory, performing simple operations
  without regard to underlying context or meanings,
  storing information and temporary results as
  necessary, and outputting correct answers.
• Each of these operations were of a class and type
  ideally suited for performance by the electronic
  calculator.

24
Calculators and Pedagogy Conclusion

• The existing information-processing pedagogy was
  perfectly adequate for use in applying this new
  tool.
• I am not suggesting that the calculator has not
  had a deep and significant impact upon
  mathematics education. I am suggesting that the
  calculator, in of itself, did not require a
  change or shift in existing pedagogy.

25
Technology Learning Objects The impact of dual
reification

• We are now able to design and implement
  intelligent objects with more number-sense than
  the beginning students who will be utilizing
  them.
• The creation of new objects of thought or tools
  to think with can become very powerful
  pedagogically, assuming we understand the
  concepts underlying them.

26
Learning Objects and Pedagogy

• It is not possible to effectively implement
  modern object oriented technologies while
  simultaneously keeping a traditional pedagogy.
• The teacher must recognize that mathematics
  contains both representational and procedural
  elements.
• Furthermore, from a psychological perspective the
  teacher must guides the student to act upon these
  mathematical models and representations to build
  personally relevent meanings.

27
Some Illustrative ExamplesWeb-Based
Mathematical Objects

• www.matti.usu.edu

28
A Calculator/Computer ExampleTI-InterActive

• TI-InterActive! has many capabilities, but one of
  its greatest strengths rest in the ability to
  pose problems with multiple representations of
  mathematical ideas that otherwise would be
  abstract and to then enable the students to
  easily perform repetitive and exact action on
  these representational objects.

29
A Factoring Example

• An activity written by Bos (2005) on factoring
  using TI-Interactive begins with the familiar
  representation of two binomials being multiplied.

30
Box Form

• When a new value replaces the existing value the
  representations of different ways to factor are
  shown demonstrating how the values affect the
  factoring process. The processes are represented
  in box form, graph, table, and symbolically
  through algebraic properties shown in steps.

31
Graph and Table Form
32
Symbolic With Steps
33
Actions on Active Objects
34
Student Actions Within this Framework

• To be effective in this technology-enhanced
  environment students must understand the concepts
  upon which the mathematical objects we are to act
  upon are based.
• If this is not done all of our lovely correct
  answers are meaningless.
• Unlike a traditional calculator computer enabled
  learning objects in many cases make suggestions
  to the student at a level much higher than they
  are able to function.

35
Technology and Learning Objects

• One of the first shifts which must take place
  lies in a reconceptualization of what mathematics
  is.
• Bob Davis once told me, "mathematics often acts
  like a verb - something that you do. However,
  mathematics is also a thing - in and of itself.
• Modern object oriented programs enable both of
  these divergent features of mathematics to emerge
  naturally and powerfully.

36
Conclusions

• Current technology enhanced objects support both
  procedural and representational components
• this enables dual reification of mathematical
  object to occur to an unprecedented level (Sfard
  and Thompson)
• and enables students to think about and with
  mathematical objects in new ways
• www.matti.usu.edu
• These new ways of expression allow for strong
  mathematical understandings to be developed -
  but require a significantly different pedagogy
  and teacher preparation.
• These understandings lead to a more flexible
  student understanding of content.

37
References

• Bos, B. Connell, M. L. (In Press).
  TI-InterActive! An Action on Object Approach to
  Learning. Technology and teacher education
  yearbook 2005. Charlottesville, VAAssociation
  for the Advancement of Computing in Education.
• Connell, M. L. (2003). Preparing teachers for
  object-oriented and technology-enhanced
  classrooms. In Crawford, C., Davis, N., Price,
  H., Weber, R., and Willis, D. Information
  Technology and Teacher Education Annual 2003.
  (pp. 2877-2891). Norfolk,VAAssociation for the
  Advancement of Computing in Education.
• Connell, M. L (2001). Actions upon objects A
  metaphor for technology enhanced mathematics
  instruction. In D. Tooke N. Henderson (Eds).
  Using information technology in mathematics (pp.
  143-171). Binghamton, NYHawarth Press.
• Connell, M. L. (1998). Technology in
  constructivist mathematics classrooms. Journal
  of Computers in Mathematics and Science Teaching.
  17(4), 311-338.
• Connell, M. L. (1995). Technology and the
  elementary mathematics methods course An effort
  to build a technology enhanced mathematical
  community. Journal of Technology and Teacher
  Education. 3(2/3), 251-266.
• Harnisch, D. L. Connell, M. L. (1991). An
  introduction to educational information
  technology. 3rd Edition. NEC Technical
  CollegeKawasaki, Japan.
• Connell, M. L. (1988). Using microcomputers in
  providing referents for elementary mathematics.
  In M. Miller-Gerson (Ed.), The emerging frontier
  Interactive video, artificial intelligence and
  classroom technology (pp. 55-60). Phoenix, Az
  Arizona State University.