Mathematics lesson - Part 2

A sixth-grade teacher asked his students to solve the following question: On one toss of two fair six-sided number cubes, which of the following is the more likely sum of the numbers showing on the cubes: 5, 6, or 7?  The correct answer is 7.  Most of his students gave the correct answer but are uncertain as to why this is the answer.

One major reason for this is that, when it comes to probability problems, students often rely on their intuition.  So when the teacher asks the student why his answer is 7, he could not give out his solution.  Teachers should emphasize that one should beware of solving probability problems with intuition since this more often than not, gives the wrong answer.

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Another reason is that students have a solution but are uncertain if their solution is correct.  To reduce student uncertainties, the teacher should explain the step-by-step procedure on how to solve the problem.  To do this, the teacher may ask his students to follow his step-by-step procedure on how to solve the above problem.

First, ask the students to list the possible outcomes of yielding a sum of, say 5, in tossing two cubes.  For 5, the possible outcomes are 14, 23, 32, and 41, thus giving us 4 possible outcomes for this event.  For 6, the possibilities are 15, 24, 33, 42, and 51, thus we have 5.  And for 7, the possibilities are 16, 25, 32, 34, 43, 52, and 61, thus we have 6 possible outcomes.  Since yielding a sum of 7 has the highest possible outcome.  Then it is more likely that when two fair sided cubes are tossed, the more likely sum out of the three is 7.

Teachers should encourage their students to use the method of science rather than using the method of intuition.  That is, they should always employ a step-by-step procedure to arrive at a solution so that they will always be certain of their answers.  Also, students should not be afraid of coming up with the wrong answer so long as they understand the process involved in finding the solution.  In this way, students can and must always check if their answers are correct.  It is much better than knowing the right answer and not knowing the correct solution.



Achieve Inc - Mathematics lesson introduction? (2002). Foundations for Success: Mathematics Expectations for the Middle Grades.


Ball, D. L., Hill, H. C., & Bass, H. (2005). Knowing Mathematics for Teaching. American Educator.


Conference Board of the Mathematical Sciences. (2001). The Mathematical Education for Teachers. Providence RI and Washington DC: American Mathematical Society and Mathematical Association of America.


Misconceptions in Mathematics: Definite Way to Deal with Probability. (2000).   Retrieved November 1, 2006, from


Patterns of Error. (2002).   Retrieved November 1, 2006, from


Schechter, E. (2006). The Most Common Errors in Undergraduate Mathematics.   Retrieved November 1, 2006, from


Yetkin, E. (2003). Student Difficulties in Learning Elementary Mathematics. ERIC Digest.   Retrieved November 1, from




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