The Misconceptions and Learning Difficulties on Function of Junior High School Students in Taiwan

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The Misconceptions and Learning Difficulties on
Function ofJunior High School Students in Taiwan

• ByYu-Hsien Chang
• Department of Mathematics
• National Taiwan Normal University

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Abstract

• The main purpose of this study is through the
  view of the ability of the transformation of
  realistic problem in the live to the mathematical
  models and the transformations between the
  different representations of function for the
  junior high school students to study the reasons
  of junior high school students, which composed
  the misconceptions and learning difficulties on
  function.

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LiteraturesLiteratures Review1.doc

• Piaget
• D. Tall
• T. Dreyfus and T. Eisenberg
• B. Schwarz and Dreyfus
• B. Schwarz and R. Hershkowitz
• K. Hart
• C. L. Yan
• Anna Sfard Sfard ???1.doc, E. Dubinsky

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Procedure of this study

• Define the scope of the target conceptions in
  terms of propositional statements and concept
  maps representing the knowledge required to
  understand the function concepts Function concept
  map.doc
• Develop the open-end items writing questionnaires
  and test 6 classes students

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• (3) Analyze the students responses to find out
  the possible commonly held misconceptions and
  difficulties of learning function and
  interviewing some representative students to
  obtain further explanation and clarification
  Abbreviate record of interview.doc
• (4) Develop three multiple-choice test
  instruments (one for grade 7, one for grade 8 and
  one for grade 9 the adjacent grades have half
  questions are the same) and examining the
  students random sampled from the North area of
  Taiwan to identify the possible commonly held
  misconceptions and difficulties of learning
  function of the junior high school students
  (around 300 students)

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• (5) Analyze the students responses again and
  modifying the multiple-choice test instruments
  into two questionnaires (one for grade 7, 8 and
  one for grade 9 two questionnaires have half
  questions are the same)
• (6) Test the junior high school students random
  sampled from the all area of Taiwan (around 3000
  students)??????(??).doc ??????(??).doc
  ??????(??).doc
• (7) Analyze the students responses to identify
  the commonly held misconceptions and difficulties
  of learning function of the junior high school
  students

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Statistical Analysis

• Reliability The values of coefficient alpha (a)
  for each questionnaire were 7th grade a 0.5796,
  8th grade a 0.7063 and 9th grade a 0.7806
  Reliability R1.doc Reliability R2.doc Reliability
  R3.doc
• Validity
• (1) Build up two-way specification tables
• (2) Factor analysis 7th grade factor analysis
  F1.doc 8th grade factor analysis F2.doc 9th grade
  factor analysis F3.doc

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Mistake types of difficulties for learning
function concept

• Completely no idea
• Confuse with the independent variables and
  dependent variables
• Ratio is not concern with the function
• Any equation is a function
• Function must contain variables x and y

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• 6. y constant is not a function
• 7. Confused the graph of function with the number
  line
• 8. Have difficulty to connect the real live
  problem with the function
• 9. Function has to be linear function
• 10. It is a function only if there has some
  special corresponding relations between numbers,
  but it is a function as long as there has a table
  for corresponding objectives

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• 11. Any figure is the graph of certain function
• 12. Some of students couldn'tt make the judgment
  from the given figure is a graph of some function
  or not, they still depends on the algebras
  calculation
• 13. Function must have some corresponding formula
  
• 14. Couldn't apply the vertical line test
• 15. Couldn't handle unfamiliar problem

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• 16. Do not have shifting concept (composition of
  functions)
• 17. The bar chart has to be linear
• 18. Not clearly understand the discrete types
  function
• 19. Ignore the domain and range during the stages
  of solving problems
• 20. Transferring the table to formula is much
  easier than transferring the graph to formula

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Examples

• (a) Some students lack the ability of connecting
  the described information with given graph of a
  same function.

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• (b) Some students ignore the domain and range
  during the stages of solving problems.
• The following example shows that some of the
  students careless of the domain and range while
  they are solving problems.

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• (c) Some students lack the ability of deduction
  (or analogy)
• The results of following two questions showed the
  ability of deduction (or analogy, or caution) to
  most of junior high school students is weak. The
  result of first question showed that they are
  easily neglects some elements in an unspecified
  domain while they need to do some deductions.
  For the second question, one may easily solve the
  problem reversal. But most of students were going
  to set up a very complex equation, and trying to
  solve their complex equation by a sequence of
  multitudinous calculations.

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• (d) Some students could not distinguish the
  differences of geometry figure and the graph of
  function

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• (e) Composition and decomposition of function is
  hard for junior high school students
• Some of the junior high school students do not
  have the concept of composition of functions.
  They also do not understand the meaning of
  shifting a graph. This caused some of students
  although remember the shifting graph formula, but
  they still cannot apply it. The following
  results of questions showed that some of students
  could not see the transformation of a straight
  line to another straight line was a function.

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• ( f ) Culture influence the function concept
  learning
• The result of following question showed the
  culture influences. The phenomenal of nature
  world described in Chinese sometimes is very
  difficult to understand and formulize. If some
  students were weak in the syntax (of Chinese),
  then they were short of the ability of
  transferring a realistic problem in the live to
  the mathematical formula and they could not
  clearly understand (or memorize) the definition
  of function (which causes them ambiguous with the
  independent variable and dependent variable).

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• (g) Some of the students have difficulty to
  connect the real live problem with the function.
• The result of following question showed some of
  the students have difficulty to connect the real
  live problem with the function. The phenomenal
  of nature world described in Chinese sometimes is
  very difficult to formulize.

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These mistakes might cause by the following
reasons

• 1. Not clearly understand the definition of
  function.
• 2. Not globally understand the representations of
  function.
• 3. Do not have original view, so cannot solve
  unfamiliar problem .
• 4. Couldn't apply the techniques, which learned
  before for the problem solving.

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Possible reasons of learning difficulties

• Weak ability of language
• Weak sense of connection
• Influenced by the old concept image
• Weak ability to reverse a process as a new
  process

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Comparison

• Ambiguous of the hierarchy for the development of
  function concept Ambiguous of the hierarchy for
  the development of function concept.doc Ambiguous
  hierarchies of function concept.doc
• Old Concept image influenced new learning Old
  Concept image influenced new learning.doc
• Individual achievements Individual
  achievements.doc

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Suggestions

• Teachers may consider making the connection of
  sketching a graph of a given function and
  determining (or analyze, or explain) a given
  figure is a graph of function or not by certain
  examples as long as the students learned both
  skills.
• Teachers may consider summarizing all
  representatives for function as long as the
  students learn all representatives of function.
• Teachers may lead students consider the problems
  reversal sometimes.

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• 4. Teachers should not only teach the
  mathematical manipulation skills (for the concept
  of function) but also should cultivate the
  students ability of the transformation of
  realistic problem in the live to the mathematical
  models.
• 5. Teachers should recall the materials which the
  students leaned before teach anything new.