Sunday, October 9, 2016

Why do we estimate?

learning to estimate
With our current unit in math, students are working on addition with regrouping and estimating.  Over the years, I have found that students really see estimating as a task that is a "fill in the blank" activity--not a meaningful math "thinking" experience.

For example, we have pages in our practice book where students are supposed to estimate and then find the exact sum.  Reasonable, right?  After watching a few students working on a page this week asking them to estimate, then add and instead solved the addition problem and then wrote an "estimate" for the exact sum they had found (rather than estimate the two numbers before adding), they shrugged and said that "it was faster".  It was pretty clear that they weren't understanding the purpose of estimating at all...

So--time to back up.

It was time to come back as a whole class and talk about the "why" we estimate--because we DO estimate for different reasons--and I really want my students to NOT think they estimate just because there was an answer blank on a textbook page!

First of all, I asked my students to tell me how many people went to the last Packer game.  They had a few guesses...maybe 70,000.  Maybe 80,000.  I asked them if it was important to know exactly how many, and they thought probably not.  We talked about the food vendors...and the ushers...and the other employees--and that they prepare for a game for "about" how many people will be there..

We then came up with a whole bunch of other times when an "exact" number isn't necessary.  However, this still doesn't really address WHY our textbooks ask students to estimate at seemingly random times.

The simple fact is, one of the BEST uses for estimating involves "checking for reasonableness"--and it's something we should be training our students to do constantly--no matter the content area.

In fact, the Standards for Mathematical Process explicitly state the following as a part of the "Uses Appropriate Tools Strategically" standard.
use appropriate tools
This is a poster from one of my Kid-Friendly Standards for Mathematical Practice posters...available in a number of different designs!  

"They detect possible errors by strategically using estimation and other mathematical knowledge."

This standard is far more than using rulers and's all about making smart mathematical decisions--and estimation is one such way to evaluate those decisions.  Being able to ask the question, "Does this make sense?" is such a critical question we want our students to be able to have in their minds constantly--no matter what unit we are studying!  We can really build upon the Standards for Mathematical Practice by asking students to talk about their estimates and to explain their thinking. This is a huge part of the standard "Construct viable arguments and critique the reasoning of others" as stated below:
constructs reasoning and critiques reasoning

"They justify their conclusions, communicate them to others, and respond to the arguments of others."

That being said, being able to quickly and reasonable estimate with the four operations is critical and provides a great opportunity for students to practice this important skill.  Helping students understand the difference between estimating and rounding (rounding means that you take an exact number and make it "less precise"--by changing to to be the "nearest ten" (or nearest hundred, thousand, etc).  Rounding is a way to estimate--but it isn't the ONLY way to estimate a number!  Rounding can be tricky for students...but it really helps if they understand that it is a way to make a number easier to work with--and that it totally relates to skip counting by a certain place...which "10" is it closest to...or what "100", and so on.  Number lines are the BEST way to show this process, in my opinion!  After teaching it this year, we practiced with these cards in pairs where they had to really work on that math talk where they explained their thinking and had to defend their ideas to their partners.
rounding task cards

So because I want to continue to refine my students' ability to estimate numbers and explain their thinking, I wanted students to have a different type of estimating experience--not based on rounding, but purely based on place value understanding and this sense of "reasonableness".  If you have followed me for any amount of time, you know that concept sorts are a huge part of my math program--for the higher level thinking they require AND for the math talk they generate.  These cards are designed to promote talk--because there are often multiple categories that cards can fit it--and it's up to the students to engage in the discourse that will allow them to make decisions about sorting the cards. I put it together in a resource in case you want to try it with your students...all directions are included.  Just click the image below  if you want to check it out.
estimation lessons

So my challenge to you is this...ask your students some of the following questions:

Why do we estimate?

What is the difference between rounding and esimating?

How do you explain how to estimate?

When do we estimate?

How can estimating help me as a mathematician?

Let's see if we can't get our students really incorporating estimating and checking for reasonableness into their mathematical habits--not just when they are assigned to do it.  I'd love to hear your thoughts!

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