T-algebra

1
T-algebra Intelligent Environment for
Expression Manipulation Exercises

• Rein Prank, Marina Lepp, Dmitri Lepp, Vahur
  Vaiksaar, Eno Tõnisson
• University of Tartu, Estonia
• rein.prank_at_ut.ee
• http//math.ut.ee/T-algebra/
• http//math.ut.ee/prank/publikats.htm

2
Abstract

• T-algebra is an interactive learning environment
  for exercises in four areas of school
  mathematics - calculation of the values of
  numerical expressions- operations with
  fractions - solving of linear equations,
  inequalities and linear equation systems -
  operations with monomials and polynomials
• For grades 5-8
• T-algebra implements 61 task types

3
Starting point

• For learning of technical side of expression
  manipulation
• For assessment and diagnosis of knowledge
  gapswe need an environment where
• all the decisions and calculations would be made
  by the student,
• the environment would be able to understand the
  mistakes and give feedback.

4
Three-stage model

• When the students solve an expression
  manipulation task, they should at each solution
  step
• choose a certain operation in the algorithm (or
  some simplification or calculation rule known
  earlier),
• select the operands (certain parts of expressions
  or equations) for this operation,
• replace them with the result of the operation.
• Some task types (such as factorisation) are
  solved without having a textbook algorithm
• but the solutions steps require from the student
  to make the same three decisions

5
Available environments

• Rule-based environments (MathXpert, EGPY
  (Stanford))
• Student selects a rule and subexpression,
  computer applies the rule.
• Good for complex nonalgorithmic tasks
  (factorisation, solving complex equations,
  integration)
• - Not appropriate for initial learning of new
  operations and algorithms - learning of details
  is passive
• Input-based environments (Aplusix).
• Student enters the result of solution step,
  computer checks the input.
• It is very hard to diagnose the errors more
  precisely than not equivalent
• T-algebra is an attempt to integrate the two
  approaches

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Solution Step in T-algebra

• Each step in T-algebra consists of three
  actions1) selection of the operation from the
  menu,2) marking the operand(s) in expression,3)
  entering the result of the operation
• First two operations are mixed
• T-algebra checks the input at two points- after
  first two operations - after input of result

7
Input modes

• T-algebra has three input modes for entering the
  results of operations - free input (one box),
  - structured input (structure of boxes),-
  partial input (some boxes are filled by program)

8
Task types

• T-algebra implements 61 types of expression
  manipulation tasks. Some of them are used to
  practice a specific skill (Combine like terms,
  Open parentheses), some integrate the content of
  a whole chapter (Solve linear equation).
• The task type determines - allowed form(s) of
  initial expression, - set of rules available for
  solution steps,- required final form of
  expression,- possible additional parameters
  (Check the solution)
• For each task type, T-algebra has a built-in
  solution algorithm

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Error diagnostics (operation)

• The menu for each problem type contains
  operations from the textbook algorithm,necessary
  calculation and simplification rules
• It is quite easy to diagnose that application of
  selected operation
• a) is impossible
• b) does not correspond to the algorithm
• T-algebra does not give error messages about
  selected rule

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Error diagnostics (marking of operands, 1)

• T-algebra diagnoses the following errors in the
  selection of operands
• Marked part is not a syntactically correct
  expression,
• Marked expression is not a proper subexpression,
• Number of marked operands does not correspond to
  the selected operation,
• Operand does not have the form required for the
  selected rule,
• Operands do not satisfy the compatibility
  requirements (are not like terms, etc),
• Operands do not satisfy the location requirements
  (do not belong to the same sum, etc)
• The actual error messages depend on the selected
  operation

11
Error diagnostics (marking of operands 2)

• T-algebra requires precise marking of operands
• Correspondingly, Editor allows to mark more than
  one separate piece
• For diagnostic purposes, T-algebra does not allow
  parallel application of selected operation to
  two or more groups of operands
• T-algebra counts with the sign before marked
  subexpression as it were marked

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Error diagnostics (result of operation)

• The following errors are diagnosed
• Entered term is not a syntactically correct
  expression
• Entered subexpression should be preceded by a
  sign or should be put in parentheses
• Entered subexpression does not have the structure
  required for the current operation and operands
  (is not a proper fraction, is not a monomial,
  wrong number of members, etc)
• Specific parts of the input (sign, coefficient,
  variables, exponents, denominator, etc) do not
  have the correct value
• Entered expression is not equivalent with the
  marked part (if T-algebra does not find more
  precise reason)
• Selected operation with marked terms is not
  performed (nothing reduced, terms are not moved
  to other side, etc)

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Error diagnostics (general)

• Using three-stage dialogue we can diagnose
  (without guessing) errors made in first two
  stages of step
• Verification before input of result prevents
  needless work by the students for calculation and
  entering of meaningless expressions

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Recording the Solutions and Error/Hint
Statistics

• The solution file contains
• the final solution path (after correction of
  mistakes and/or possible undoing of steps)
• records of error situations - previous
  expression together with marking, - selected
  operation, - entered expression - error message
• table of errors by categories
• records of hint use (in a similar manner)

15
Some conclusions

• Three-stage dialogue for expression manipulation
• Is intuitively understandable for the students
• Requires the same amount of keyboard/mouse work
  as pure input
• Provides the program with information about
  intentions of the student
• Allows to point in error messages to real places
  of mistakes
• Requires
• Standardization of understanding the rules

16
Experimental result

• Linear equations, spring 2007
• Number of solved tasks was equal to
  paper-and-pencil work
• Already two hours of systematic feedback lead to
  a clear decrease in some technical mistakes -
  application of minus before the brackets or
  before the fraction only to the first member -
  forgetting to multiply the member that is not a
  fraction when multiplying an equation by common
  denominator

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Is our three-stage scheme very expensive?

• In comparison with pure-input interface
• Selection of operation - is only one selection
  in menu
• Input of changed part - is included in
  pure-input dialogue too
• Marking of operands - is necessary by pure-input
  too (if we want the unchanged parts to be copied
  automatically)
• In general our scheme requires practically the
  same amount of input as pure-input scheme