· Number Magnitude
· Mental Computation
· Mathematical Properties
· Effects of Operations
As crucial as estimation is to our daily lives, it bears little resemblance to the way it is presented in mathematics classrooms. Textbooks give passing attention to it. Some may ask students to estimate the answer to a multiplication problem by rounding to significant digits, but these are isolated lessons. Douglas Owens stated, “For students to become highly skilled at estimation, it had to be incorporated into their regular instruction over several years.”[i] Typically number sense is embedded within a lesson, but it can stay in bed if we don’t awaken it.
Frequently I ask my students to estimate problems before allowing them to pick up a pencil, paper, or calculator. For example, I recently used the following question as a warm-up.
Which is greater: 86 x 38, 88 x 36, or are they equal?
Then I asked my students to simply think about the problem without writing anything or discussing it. This allowed them to approach the problem in their own way. After about a minute, I asked them to stand on one side of the room if they believed that 86x38 was greater and the opposite side if they thought 88x36 was greater. If they believed the two products were equal, they stood in the center of the class.
Many students said they changed their mind during the thinking time; their initial thought was superseded by more advanced thinking. Some students actually solved the problem mentally. Though many thought the products were equal, some noticed that the first problem has cross products that are greater than the second one. That is, during the multiplication process when you have to multiply the diagonals, the first problem yields six groups of thirty and eight groups of eighty. The second problem however gives eight groups of thirty and only six groups of eighty.
Other problems we have used as warm-ups to develop number sense include these:
Estimate 0.52 x 789
Estimate 40 x 26.7
Estimate 4953 ÷ 68
Estimate 715/1866 as a percent
As my students gained more experience with estimation, we noticed an increase in their accuracy. For example, if a student estimated the last problem as 35% we compared it to the actual answer of approximately 38%. Dividing their answer by the actual answer shows they achieved 92% accuracy.
I also gave my algebra students some uniform motion word problems. These are the typical “a train leaves Baltimore” problems that give students nightmares. Before allowing them to get their calculators, I asked them to read each problem and write their best estimate next to it. After solving them, many students had come very close, and four had “estimated” every problem exactly correct.
In order for students to estimate, they need two precursor skills: the ability to approximate and the ability to compute mentally.[ii] This means that we must give them opportunities to practice these skills concurrently. It also means that the “guess how many jellybeans are in the jar” problem is not estimation but “guesstimation”. Unless the student is mentally approximating the volume of the jellybean compared to the jar, he is simply making a guess based on number magnitude. The winner is likely more lucky than mathematically proficient.
In the past two years as I’ve placed an increased emphasis on number sense, I’ve seen an accompanying proficiency as students tackle tough problems. After they completed the high school placement exam this year, I asked them if they encountered any uniform motion problems. “Yeah,” there was one, a student responded flatly, “but you could just see the answer was two hours.”
Brad Fulton is a veteran math teacher and nationally recognized educational consultant and keynote speaker. In 2005, he was selected as the C.L.M.S. middle school educator of the year.
[i] Owens, Douglas, Research Ideas for the Classroom: Middle Grades Mathematics, 1993
[ii] Case, R., Intellectual Development, 1985