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Factoring

Evolution of Prototypes

 

Initial Designs

Design 1

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Provide branching lines and binomial form structure. When the user types in one factor into a binomial, the calculator automatically completes the corresponding factor (in the other binomial) for the user.

Branching lines intended to support understanding of which numbers should be factors of a particular coefficient.

Both designs have an answer-checking animation.

Design 2

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Design 2 automatically calculates factors of the first and last term in the trinomial. The user selects a pair of factors to use in the binomial using the up/down and left/right arrows.

Branching lines intended to support understanding of which numbers should be factors of a particular coefficient.

After the user chooses a pair of factors and presses Enter, a check animation would be shown

(Note: this is not functional in the prototype)

 

Iteration 1

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Design change: Drop Design 2

  • Keep the binomial structure.
  • Drop design 2 (calculating all factors automatically and showing them in a list)
   

Rationale

3/4 students

Most students mentioned that the lines and the binomial form were helpful.

1/4 users

One user indicated that some students may assume the initial selection is the correct answer of the problem. It is also unclear which box is in focus.

Design Change: Provide less help

  • Provide less help for the user by not completing the second factor
  • Hide the factors by default. Add an option to "Show Factors" in the "Options" menu
  • Keep the binomial structure.
   

Rationale

1 teacher

For design 1, calculating the corresponding factor might be too much help.

2/4 students

In design 1, when the user enters one factor, the corresponding factor is filled in for the other binomial. Some students appeared to be using this as part of a guess and check strategy (instead of strategically thinking of numbers and entering them). They ultimately gave up when they reached trinomials with first term greater than 1.

 

Iteration 2

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Design change: Immediate feedback

  • Instead of having a checking animation after the student has entered in factors, always show the checking step
  • Give instant feedback about whether the answer is correct/incorrect by showing which parts end up being correct or incorrect
   

Rationale

5/7 students

Most students could not factor trinomials with the first term >1. We took this as a sign that our designs were not as effective as they should be. One user said, “with this one, if you don’t know what you are doing it won’t help because it just gives you the factors", indicating that this design did not assist in the understanding of the relationship between coefficients and factors.


3/3 users

All users indicated that showing the factors was not helpful in solving factoring problems

3/6 experts

Half of our experts suggested providing more feedback

  

Design change: Don't automate factors

  • Don't automatically calculate factors, and remove the option to show factors for the first and last terms of the trinomial
   

Rationale

4/6 experts

Most experts are concerned about giving too much help for students. There is a trade-off between performance and learning, while performance is helping the students now, learning helps them in the long run.

 

2 experts

Two experts mentioned the importance of automating only the steps of factoring that students can do easily

  

 

Iteration 3

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Design change: Simple/Advanced Modes

  • If the user enters a trinomial with the first term equal to 1, then a simpler checking interface will be provided.
  • If a>1, the checking interface with lines for the first and third coefficients of the input trinomial will be provided.
  • Regardless of the mode, the checkmarks and X's are still provided.
   

Rationale

1/2 students

Of 2 students, 1 found the lines confusing and the other understood the meaning of the lines clearly. 1 of 2 students were not sure what to do with the first coefficient of the trinomial.

4 experts

4 experts suggested to include modes with varying levels of help.

 

2/2 students

2 of 2 students mentioned that the checkmarks are helpful, so the checkmarks will be kept in this interface.