**Steve Leinwand thoughts on estimation**

I am surprised and disappointed by how little estimation I see in the mathematics classes I observe. We seem to be so compulsive about exact right answers that we forget the commonality and importance of ballpark answers, reasonable estimates and rough approximations. A key requisite for the effective use of calculators is having a sense of a reasonable answer to ensure that a button was not inadvertently pressed. A sense of what is “close” is essential when reviewing a set of data or checking that the spreadsheet formulas seem right. And every time we employ the power of mathematics to create possible models, it is the ability to estimate that gives us our first sense that our model seems viable.

But I rarely see estimation of calculations, quantities or solutions on our tests. I rarely hear or see the question “about how much is…” or the request for a “ballpark solution” before a word problem or application is attempted.

This is disappointing, first of all, because number sense – a comfort with numbers that includes estimation, mental math, numerical equivalents, a use of referents like ½ and 50%, a sense of order and magnitude, and a well-developed understanding of place value—is one of the overarching goals of mathematics learning. Instruction that fosters the development of number sense should be an ongoing feature of all instruction. I believe that we should be asking and hearing such questions, along with their justified answers, as:

- Which is most or greatest? How do you know?
- Which is least or smallest? How do you know?
- What else can you tell me about those numbers?
- How else can we express that number? Is there still another way?
- About how much would that be? How did you get that?
- What’s a ballpark solution? How did you get that?

For example, as I’ve suggested in

*Accessible Math*:

- Given a problem with 1534, 4933, and 588, you can ask: Which is the greatest and which is least? About how much is the sum? How did you get that? Which number is different from the other two? In what ways?
- Given a table with 2 ½, 5, and 11/2, you can ask the same questions and then ask: What’s the sum? What’s the difference between the greatest and least? How else can you express 11/2 and how do you know and what do you call those numbers (5 ½, 5.5, 22/4, 5.50, etc.)
- Given the number 0.2, you can ask “how else can you express that? (2/10, 1/5, 20%, .20) and show me where it goes on a number line.
- Given a page of subtraction exercises like 1539 – 612 or percent exercises like 15% of $26.75, ask students for a reasonable estimate and a justification and encourage multiple reasonable estimates and alternative approaches.

Beyond these strategies, estimation is also how many students get to outsmart the multiple-choice tests they face. Any time, from a high-stakes grade 3 assessment to the SAT or ACT, that a student can use estimation to reject two of four possible answers, the benefits of a guess become decidedly worth considering.

Try it. Take a page of 10 word problems related to whatever topic you are teaching and ask student to set their calculators and scratch paper aside, read each problem and propose and justify a reasonable estimate. Then have students compare and critique their estimates and the diversity of approaches they use to arrive at their estimates. With little effort your class opens doors to SMP#3: constructing viable arguments and critiquing the reasoning of others.

Submitted: April 21, 2013

## Thanks Steve for sharing.

You continue to inspire! I love the challenge you present us with at the end about proposing and justifying estimates for 10 word problems! Thanks again for sharing with us. It means a lot!

~ Andrew

~ Andrew