Estimation

UNDERESTIMATING the IMPORTANCE OF ESTIMATION

It probably won't surprise you to learn that in daily life most people typically think in terms of estimations more often than exact values. Despite what our math classes may have led us to believe, exact solutions are not necessary in every mathematical situation. When I go to the grocery store, I know they will expect me to pay the exact amount for my groceries when I get to the register, but do I really need to know the precise cost of each item as I shop to have a sense of whether the $50 in my pocket will be enough? Every day, I drive to and from several schools, but I have no idea of the exact number of miles between the schools, and yet, even without a precise measure of the mileage, I am still able to estimate what time I should leave to arrive at the next school on time (day of the week, construction, weather, and a number of other factors will play into my estimated driving time, anyway!). You get the idea - at times, exact values are necessary, but more often than not, an estimate serves the purpose just fine.  

Let's establish a working definition of estimation:  An estimation is a rough calculation of value based on incomplete or inexact data.  The goal of estimating it to get an answer that is close enough in magnitude to the exact solution to be useful (context is an essential component for defining "close enough"). 

I'm just going to lay it out there: Our students are not good at estimating (and many adults aren't that great at it either!). Because of the way math has been taught for generations, we are much better at calculations than estimations. Despite estimation being a pervasive part of our daily lives, we just haven't given our students enough structured experience with estimation strategies. And now, with a calculation device in the back pocket of nearly every person around, the ability to determine the "reasonableness" of the number on our phone's screen is more important than ever.  Yes, MORE important than ever [That statement may seem backwards at first glance, but pause and take a moment to consider the truth behind it]. Push a button out of order and you may find yourself leaving a $2 tip on a $95 dinner bill and you didn't give it a second thought because the calculator app on your phone showed $1.71 when you typed in .018x95 (see the problem?!) Good for your wallet; bad for the poor server who refilled your water glass all night long.

In the math classroom, the critical step of thinking about an estimated solution prior to each calculation, is often neglected -- and when the skill of estimation is addressed, it is often reduced to an instructional unit that lasts for just a few days and focuses solely on rounding. A few years back, I was proctoring a state test. The students were asked to estimate the most reasonable solution to a multi-digit addition equation. As I watched a student answer the question, I noticed that he solved the problem to get a precise solution and then rounded his precise solution to a less precise solution in order to answer the estimation question. Hmmm??? The student must have been asking himself, "What was the point of that?" As I watched the student, I asked myself, "How can I teach my students a range of estimation strategies and to understand the purpose of each one in a way that is useful, authentic, and purposeful?"   

ESTIMATION IS SO MUCH MORE than simple rounding. And it is definitely not intended to occur after a precise calculation is completed in order to find the correct multiple choice answer on a test. So let's take a closer look at what we can do in our classrooms to develop these essential skills more fully with our students.

There are some very specific estimation strategies that we can teach. You may not know about them because, like me, you may not have been taught them, so let's continue learning together about a few different estimation strategies. Unlike traditional algorithms, estimation strategies go beyond procedural applications and require us to apply mathematical thinking in flexible ways. This type of adaptive problem solving really is the fundamental goal of all mathematics.

Below are FOUR different types of Estimation Strategies. Did we even know estimation had types!? There are probably more than just these four, each with its own nuance, but this list should get us started as we begin to consider how to be more purposeful in our teaching of estimation to students:

~ Putting Estimation Strategies Into Practice in Our Classrooms ~

Remember that question I was asking myself as I watched my students take the state math test?

How can I teach my students estimation skills 

in a way that is useful, authentic, and purposeful?  

Here's the answer I came up with....

1. Make estimation a part of the DAILY routine and not just a unit of instruction that occurs as an isolated event -- consistently ask students to estimate a reasonable answer BEFORE any expectation that they calculate a precise solution.
2. Purposefully (and regularly) teach the various types of estimation strategies and help students to understand when to use each type. 
3. Use estimation activities to build number sense -- Number sense is the foundation of all mathematical thinking. Estimation builds greater number sense and greater number sense leads to better estimates. Check out these sites for some terrific Estimation activities: 
• Steve Wyborney's "Estimation Clipboard" and "Esti-Mysteries" 
• Andrew Stadel's "Estimation 180" 
• Graham Fletcher's "3-Act Task" geared toward elementary 
• Dan Meyer's "3-Act Tasks" geared more to secondary math
4. Accept estimated solutions as the final solution when appropriate -- try to keep in mind that, in the real world outside of the math classroom, an estimated value is often enough to meet the intended needs and an exact solution may be unnecessary - the purpose and context of the situation should be considered.

ONE OF MY FAVORITES: The "Goldilocks" Estimation strategy

Want even more? Try out the Goldilocks estimation strategy that I learned while attending one of Dan Meyer's conference sessions. The strategy encourages mathematical risk-taking and helps students develop their estimation abilities even further. The basics of this routine go something like this:

1. Present an estimation opportunity
2. Ask students, "What estimate is definitely too large? When you look at this jar of pennies, what number do you know is too many to possibly fit in the jar?" --- this gives an entry point for all since 1 million is, in fact, an acceptable "too large" response for the number of pennies in the jar. 
3. Write down all of the "too large" ideas on a list as you encourage students to take bigger risks by estimating numbers that are still too large but perhaps closer to the actual number of pennies in the jar.
4. Through collective agreement, see which "too large" estimate is the LOWEST value that your students can agree is "too large" --- let's say that everyone in room agrees that 35 is too large but several students believe that it could be as low as 30.  Since all of the student agreed on 35 as definitely being too large, you have established a ceiling value of 35.
5. Next ask your students for a value that would definitely be "too low" --- again, anyone can enter the conversation because zero is obviously "too low" and would be an acceptable response.
6. This time, come to a group decision about which number from the list represents the HIGHEST number that everyone can agree is too low --- imagine your students going back and forth between the 14 and the 18 that is listed on the board.  Some think that 18 is too low, but all believe that 14 is too low, so go with the 14 that everyone agrees is too low.  The number 14 is now the group's lowest possible value, the floor value. 
7. We have now reduced an infinite set of numbers to a discreet set of values with everyone in the class agreeing that the "just right" estimate must be somewhere between 14 and 35.
8. At this point, ask students to individually determine their "just right" estimate --- remind them that they have agreed that it is a number between 14 and 35.  If someone does not believe the "just right" amount is within this range, the range needs to be adjusted to fit the number values that EVERYONE can agree upon. 

Think about what just happened in your classroom: 
✅ Students felt safe to enter the conversation because early steps are manageable for all
✅ Students worked together to crowd-source a list of possible values
✅ Students crafted justifications for their ideas and listened to the ideas of others
✅ Students made a final independent estimate based on data (not just a guess!)
✅ Everyone in the room just became a little better at estimating!