Find the next term in the sequence: 9, 4, 1, 8, 27,

When I’ve proposed this problem before, the most common response I get is “64” (do you see how someone might make that guess?), but it’s wrong.  Would you have guessed that the next two numbers after the blank are 6 and 14?

The number that correctly fills in the blank is provided (and explained) in the following paragraph.  Trust me, you’re probably not going to guess the answer – nobody ever has (who hasn’t known – or figured out – the “trick”).  But if you insist on thinking about it, don’t go to the next paragraph until you give up.

The following excerpt from a team chronology that used to be available on the official Boston Red Sox website gives away the answer:

1931: The Red Sox player first wear numbers on their uniforms (in 1931). Since then, the Red Sox have retired five uniform numbers: Ted Williams’ No. 9 and Joe Cronin’s No. 4 officially retired May 29, 1984; Bobby Doerr’s No. 1 retired May 21, 1988; Carl Yastrzemski’s No. 8 retired August 6, 1989; and Carlton Fisk’s No. 27 retired September 4, 2000. Major League Baseball retired the No. 42 of Jackie Robinson for all teams.

Subsequently, the numbers of Johnny Pesky (6), Jim Rice (14), Pedro Martinez (45),Wade Boggs (26), and David Ortiz (34) have also been retired.

“But wait a second”, you say, “this problem isn’t fair. I have to know how you picked the numbers in order to correctly guess the next term in the sequence.”

Exactly!

In fact, if we’re honest, every problem of the type, “What’s the next term in the sequence?” is really a mind reading problem!.

I could go even further and point out that, on an all-or-nothing high-stakes test, all “what’s the next term” problems are inherently unfair, since they require the student to read the mind of the test writer.

Let me be clear.  By no means do I think “guess the next term” problems are bad classroom problems, especially if students are asked to explain the pattern(s) they observe and how their pattern gives them their “answer”.  These types of problems provide excellent opportunities for students to flex their problem solving muscles as well as giving them a chance to explain their solution method. The unfairness arises when we insist that everyone notice the same pattern.

A “real-life example”, shown below, provides a cautionary tale from the 2002 grade 10 Massachusetts Comprehensive Assessment System (MCAS – the test that must be passed before a student is allowed to graduate from a public high school in Massachusetts). While the story had a happy ending, think about what would have happened had the young woman not questioned AUTHORITY and spoken out about what she thought was a correct solution?

From the Massachusetts Department of Education press release (December 4, 2002)

 Points Awarded To Students Who Selected Alternate Answer On Grade 10 MCAS Exam MALDEN – An additional 449 students in the classes of 2003 and 2004 have earned a competency determination, thanks to the ingenuity of a Whitman-Hanson Regional High School student who found a unique method of answering a math question on the 10th grade exam, education officials announced on Wednesday. Because of this finding, a second answer on the question is now counted as correct. As a result, an additional 136 students in the class of 2003 and 421 students in the class of 2004 have now passed the math exam, and will not have to take the MCAS math retest being given next week. “Although the answer we had marked as the right one is correct, it’s clear now that what this student found is also right, and I think that’s terrific,” said Education Commissioner David P. Driscoll. “This girl was able to take a typical math question and come up with a completely unique method of solving it that even our math experts, teachers in the field and our test reviewers never considered. This is a great example of just how creative our students can be, and I applaud her efforts.” The question presented a real-world application of the binary, or base two, number system, such as the one used by computers. The numbers zero through 10 were shown as a sequence of on and off switches in four-switch panels. Students were asked to identify the panel that would represent number 11, the next in the sequence. The correct answer could be found either by knowing the binary system, or by recognizing changes in place value throughout the series. However, when the student looked at the panels, she saw something different: a spatial sequence in the on and off switches, rather than the numeric pattern envisioned by the developers and reviewers of the item. After conferring with mathematicians, education officials this week determined that the student’s answer is also a viable second solution to the question.

See below for the problem in question