“About 1 in 5 adults in the U.S. lacks the math competence expected of a middle-schooler, meaning they have trouble with those ordinary tasks [calculating a tip, doing fractions to double a recipe, know how much change to expect from a cashier] and aren’t qualified for many of today’s jobs.” 

I came across this quote in an article called “Early Number Sense Plays a Role in Later Math Skills“. The author attempts to trace this statistic back to a root cause and comes up with a University of Missouri study done with 7th graders who were given a test to assess a variety of math skills needed to function in the world as an adult. They found that the students who were behind on the seventh-grade test were the same students who had the least number sense or fluency in mathematics in the first grade.

The article goes on:

“The gap they started with, they don’t close it,” says Dr. David Geary, a cognitive psychologist who leads the study that is tracking children from kindergarten to high school in the Columbia, Mo., school system. “They’re not catching up” to the kids who started ahead.”

Articles like these are usually written from the perspective of the school children, making the case for why we need to improve the quality of education for our children to prepare them for their future. But they always make me think about our students. How many of our students are those 1 in 5 adults who lack the math competence expected of a middle-schooler? Number sense is something that I have increasingly made more time for in class. I have seen the impact that incorporating it into my instruction has had on my students – both in terms of their sense of themselves as mathematicians and in their abilities. But the quote about the gap that students don’t close really stuck with me. Often we are focused on the very important high school equivalency part of our work, but it is also true that many of us are working with a generation of the “children who were left behind.” We have a real opportunity to help our adult students catch up and give their number sense room to flourish.

How can we improve student number sense?

In About Teaching Mathematics, Marilyn Burns highlights the following six research-based strategies for building number sense¹: (1) model different methods for computing, (2) ask students regularly to calculate mentally, (3) have class discussions about strategies for computing, (4) make estimation an integral part of computing, (5) question students about how they reason numerically and (6) pose numerical problems that have more than one possible answer.

The resource I am reviewing – Estimation 180 – offers opportunities for all of these strategies, but since it’s in the name, let’s talk about estimation.

Isn’t estimation just rounding numbers up or down?

Estimation can be a place where math and wonder meet. Ever been in a crowded subway and wonder just how many people could fit in this train car? Ever sit in a park and try to figure out how tall a tree is? Ever worry if you are putting enough money away into your daughter’s college fund? Ever try to figure out how much you’ll need to retire? Or how long it will take you to get somewhere new?

“Most of the math that we do every day—deciding when to leave for school, how much paint to buy, what type of tip to leave in a restaurant, which line to get in at the grocery store relies not only on mental math but estimations.  However traditional textbook rounding exercises don’t provide the necessary context for students to understand estimating or build number sense.  To do that, estimation must be embedded in problem situations.”¹

Compare that to the “Whole Numbers”  chapter that appears in almost every pre-HSE/HSE math workbook with a page or two on estimation and rounding. Where they describe rounding as the most common estimation strategy and jump right into a disembodied and abstract procedure  – “Underline the digit in the place you are rounding to. If the number to the right of the underlined digit is greater than or equal to five, add 1 to the underlined digit. If the digit to the right of the underlined digit is less than 5, leave the underlined digit as it is. Then change all the digits to the right of the underlined digit to zero.” Then come the practice problems, which ask students to do things like “Round 238,687 to the nearest ten thousand.”

I found some helpful definitions for distinguishing between rounding and estimation. “Rounding means taking a known value and making it less accurate. Estimation means coming up with an inaccurate number you didn’t have in an exact form to start with.”² The first thing that struck me about the definition was they way they used the word “inaccurate.” “Inaccurate” gets a bad rap, especially in math, which is why I really like this definition. One of the things that my students need time to develop is a tolerance and eventually a comfort with the “inaccuracy” of estimation.

So how do we embed estimation in problem situations for our students?

The Simplicity of Estimation 180

Every day of the school year, middle school math teacher Andrew Stadel gives his students an estimation challenge. Estimation180 is the website he set up to share these challenges with the rest of us. The set up is super simple. Students are shown a picture of something and asked to make several estimates about some aspect of the thing – its height, length, weight, etc.

Students are not only asked to make an estimate for what they think the answer is. In addition to coming up with their best estimate, Stadel asks them to come up with a number that is too low and another that is too high (i.e. what they know the answer is not). I love this idea of setting a range because over time it allows students to “triangulate” their guesses. Especially if you encourage students to be bold (I call it “stepping a little further out on the ice”) with their too highs and too lows.

Here’s an example for you to try:

How long is “I Got You (I Feel Good)”?

• What’s an estimate you know is too low?
• What’s an estimate you know is too high?
• What is your estimate?
• What is your reasoning? (Do better than “I guessed”. Try “I noticed…”)

Play the answer:

• How did you do? What might account for the difference?
• Did you have a sense of excitement while watching the video – especially during those last few horn riffs – curious about how close you got? (Your students will too!)
• Can you imagine your students working on that for the first few minutes of your class?

6 Things I love about Estimation 180

It doesn’t take much. Only 10-15 minutes a day. Estimation is not a procedure, it is a result of number sense and it needs to be developed in each of us over time. The 200+ estimation challenges here are certainly rich and can go further, but spend 5-15 minutes on them at the beginning of every class and you’ll start to see their impact on your students. They are perfect for the beginning of class when you don’t have enough students to start the lesson of the day but you want to engage/reward the students who are there.

These estimation activities are engaging and accessible for students. The anticipation around the answer, compounded by the collegial competition between classmates, is often palpable and super fun. Also, because of the context, students are more comfortable getting an answer wrong. They are also feel good about being close (which is often not the case). You will also start to notice that you and your students will start to look at things differently afterwards… you start looking for opportunities to estimate size, weight, distance, speed all around you. Students will start to come in with examples of estimations they had to make outside of class.

Students talk about their own thinking and the thinking of their classmates. Students have opportunities to explain their reasoning, construct viable arguments and critique each others’ reasoning.

The power of these activities is not limited to number sense and estimation. They are work as great launches to the math you are teaching. They are accessible to all students and there are so many opportunities to connect to other content. For example, imagine how Day 59’s challenge could introduce a lesson on volume.

Day 59: Order these glasses from least to greatest in capacity.

There are all kinds of interesting connections students will make between the different estimation challenges. For example, here are the three days that come before the James Brown estimation. Can you see what thread Stadel is laying out? Why did he stop each song at the time he did?

The Beatles stops about half way through, the Santana about one-third of the way through, the Queen about one-fifth of the way through and then the James Brown about ¾ of the way through. Students start learning strategies for making the estimations, but they are also using/building content knowledge (fractions and/or proportional reasoning in this case).  These types of related images make for great student conversations about numerical reasoning.

Another kind of connection Stadel makes between the estimation challenges is by showing objects from earlier days next to new objects, building number sense and using units they understand. For example, the first estimation is to calculate Stadel’s height. The next is to calculate his wife’s height, using what you learned about his in the previous class.

Interesting possibilities for data analysis. I’ve only just begun to use the resource in this way, and I would love to hear what other folks do with it.

• Students can use math to determine if their estimations skills are improving (i.e. are their guessing getting closer) – Stadel (with Michael Fenton) created a great Estimation 180 handout to support this. Students can record their answers and find the percent error of their estimates.
• You can also keep track of class high, lows and guesses – find averages and study range.
• Find the mean, median and mode of everyone’s best guess – how does each compare to the actual answer.
• Estimation 180 actually allows you to enter your answers (all three guesses and your reasoning) into the website itself – great to do if you happen to have access to a computer lab with your math students. I usually have students write their answers on paper, but either way you can look at the answers and reasoning of anyone else who has entered their estimates and reasoning on the site, which broadens your pool of data.

Revolutionize the first 10-15 minutes of your math class! (and beyond)]

Please try these daily estimation activities with your students. In the comments below, tell us which ones you try and what your students come up with.

 In addition to the great estimation activities, Stadel has a series of great lessons on the site. They deserve their own review but until then, if you are interested I would recommend HSE teachers check out Stacking Cups (a great way to talk about rate of change and starting amounts). Pre-HSE teachers should take a look at the File Cabinet (looking at surface area). Both of these activities are structured as three-act math tasks.

For a final word on estimation and number sense, here’s a brief Ignite Talk by Andrew Stadel called, “Number Sense: I Don’t Like This Game Anymore” (5:28)

¹ Marilyn Burns on Understanding Number Sense

² Estimation vs. Rounding – Math Forum – Ask Dr. Math

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