Of the 32 students I interviewed, 75% of them gave me numerical responses. Going into this, I predicted it would be closer to 50%. Here are some of my observations from the 32 students:
- 2 students calculated the answer to be 130 (125 + 5)
- 2 students calculated the answer to be 120 (125 – 5)
- 12 students calculated the answer to be 25 (125 ÷ 5)
- 0 students calculated the answer to be 625 (125 x 5)
- 4 students stated that they guessed their answer (90, 5, 42, and 50)
- 4 students tried to divide 125 by 5 but could not correctly implement the procedure
Three particularly interesting students included:
- The student who found the shepherd’s age by using the old “add the sheep and dogs and divide by two” trick and got 65. You didn’t know about that trick?
- The student who got 120, then said that you don’t have enough information to figure out the shepherd’s age, then seemed to feel so uncomfortable with that conclusion that the student decided to guess 90 years old.
- The last student who explained that the reason for dividing was because it didn’t say “sum,” “difference,” or “product” in the problem.
Additionally, the day before I interviewed these students I asked a class of sixth graders the same question and 100% of them gave me numerical answers. That day I also learned that when you turn the microphone on, it is much better at recording audio.
As for Math Practice 1 from the Common Core’s Standards for Mathematical Practice, here are expanded versions of the quotes from the video:
- Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution.
- They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt.
- Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?”
I unfortunately believe that these types of responses are common amongst students. What are your thoughts? What are you doing to help students “make sense of problems” in your classroom? If you try this same question out with your students, please post a comment about how it went along with the students’ grade level.
*The “How Old Is the Shepherd?” question was popularized by an essay written by Professor Katherine K. Merseth in 1993 and was based on research by Professor Kurt Reusser in a paper presented at the 1986 American Educational Research Association annual meeting. It is noteworthy that Professor Merseth wrote that “researchers report that three out of four schoolchildren will produce a numerical answer to this problem.” Twenty years later that statement still held true.