Kimberly Miller, a parent of two elementary students from Centreville, in Queen Anne’s County, Md., writes in a Nov 19 letter to the readers of the Baltimore Sun about the Common Core:
I’ve watched my 3rd grader go from being that child who couldn’t wait for Monday to arrive so that he could get into school to learn to now dreading the start of the school week. … Since (first grade), he has gone through several different vocabulary words for the same math lessons just because he’s being taught to a test. For instance, our math-minded child came home with a 56% on a test because it was focused on “estimation.” When we sat down to discuss what he didn’t understand about the test, he said to us that he didn’t understand estimation. “Mom, I know how to round, but I don’t know what estimation is.” It wasn’t until I explained to him that the only difference was in terminology that he understood…
Third graders doing estimation is something I will address a little later, but first, I wish to compliment Ms Miller on stating the problem so clearly. Perhaps a newspaper isn’t the best place to help one’s children with their homework, but that was her choice, not mine. I am in the position of reading and responding to her letter. My first reaction is that she should speak with her third grader’s teacher about estimation and how it is being taught. The classroom belongs to the teacher, not to me. But, if anyone wants to send me questions about how the Common Core is being taught, I’ll respond within a day or so, either on this blog or by email, depending on the situation. My first response, though, is likely to be something like, “What did the teacher say?” So please, before publishing your child’s school history in a newspaper or online, consult the teacher involved about how the subject is being taught.
That being said and fully disclosed, ahem, let me address estimation in a more general sense, based on Maryland’s adoption of the Common Core for third graders. Looking at the former state learning standards in math, I see objective 6.C.2.a, in third grade, here. There are actually five objectives that address estimation in third grade, but this is the only one that deals purely with estimation. Teachers aren’t using these old standards anymore, but just for a frame of reference to what teachers were doing before this year, let me quote the standard. It simply says students by the end of third grade should be able to “[d]etermine the reasonableness of sums and differences.” The other standards that involve estimation say to “estimate and determine” something, such as temperature on a thermometer to the nearest degree (see 3.A.1.c, for example).
In the Common Core, many of the third-grade standards associated with estimation also involve reading a device of some type, such as a thermometer, a scale, etc., and estimating the temperature, weight, volume, etc., to the nearest unit. For example, standard CCSS.Math.Content.3.MD.A.2 says third graders should be able to “Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l). Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem.” But there’s also this:
CCSS.Math.Content.3.OA.D.8 Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.
There’s that idea of reasonableness coming in again, just as in Maryland’s former state standards in third-grade math. Not much has changed, but in the Common Core, we get some idea of what types of problems third graders will be expected to solve. We also see that estimation strategies include rounding, and there’s a reason for that, considered facilitative by some and evil by others.
From where I sit, which is on the desk at the Maryland State Department of Education that supervises the scoring of the Maryland School Assessment in mathematics—yes, the test used for accountability purposes under the No Child Left Behind federal law—estimation is notoriously difficult, perhaps impossible, to measure on a standardized test. Why? Because kids have plenty of time to work out the arithmetic and pick the answer that’s closest. Consider the following problem:
What is the most reasonable estimate for the sum of 31 + 88?
During the MSA, kids would just do the arithmetic, find the actual sum of 119, round it to 120, and pick (B). That’s not estimation; it’s rounding, which is a perfectly good skill for third graders to have and for the state to test them on among the myriad other skills we test them on, but the question above doesn’t really test students’ knowledge and understanding of estimation, which is what we’re supposed to be testing them on, by federal law. We’re testing them on rounding.
Another factor that makes testing kids on estimation problematic is the idea that several answers are possibly correct. Who am I to say that 119 isn’t a good estimate for the sum? In fact, it’s a very, very good estimate of the sum, but it’s not one of the choices, which might frustrate a kid taking the test, which is a huge problem. Or, when a kid estimates how much liquid is in a beaker when it’s about halfway between 100 and 200, maybe I expect 150 as the answer. But I can’t mark 151, 152, or probably even 160 wrong on a test given to 50,000 third graders in Maryland. These aren’t “bad” estimates. So, on multiple choice, estimation is difficult to test because kids just think it’s about rounding, and on constructed response questions, where kids have to type in the answer, it’s difficult because there are several good answers.
For example, a perfectly reasonable way to estimate this answer, depending on how close you had to come, is to say 88 is a little less than 100 and 31 is a little more than 30. Add 100 + 30, which can be done without the help of scratch paper, to get 130, and either say that’s the estimate, which would have to be counted as correct, or say it’s probably a little less than that, like 120 or so, because 100 was a very high way to round 88. But on tests, most kids just do the addition first and then round, which is, from a teacher’s perspective, different from rounding the numbers to make the problem easier and then adding. On a test, there’s just no way to tell the difference between the multitude of appropriate strategies.
Notice how, in the Common Core, the words “including rounding” have been added to the estimation standard in third grade. This makes it easier to test the standard—or at least to claim we’re testing it. Students can now use rounding, which they would use anyway, to “estimate” the answer to a sum or difference, and we can still believe statisticians when they say they’re testing students on their estimation skills. In the Common Core, we have added two-step word problems, which means word problems involving two arithmetic operations, and the list of operations has been expanded from sums and differences under Maryland’s old third-grade standard to include products and quotients. Students, probably those a little higher than third grade, might now see a test question similar to the following:
Sapna has 3 friends and Jingo has 8 friends. Each of their friends decorated 81 t-shirts for a 5K run to cure cancer. About how many t-shirts did their friends decorate?
This allows us to combine a few operations like addition and multiplication, but on a test, kids have plenty of time to simply multiply 81 by 8, multiply 81 by 3, add 648 and 243, come up with 891, round to 890, and correctly pick (D). Again, we haven’t really tested the estimation part of the standard, but standardized tests are extremely limited in terms of what standards can actually be assessed. For this reason and a few others, I have called for a period of testing for the standards in the Common Core, some sort of a “grace period” as teachers find out which ones work and which ones don’t. Then, we can have a discussion about whether we want to revise the standards that don’t work, eliminate them, or simply mark them in such a way that they won’t appear on a standardized test.
But Ms Miller, the standards aren’t likely to be revised in any meaningful way, and that is one of the biggest complaints among educators about the standards. From your kids’ point of view, it means, as you have found, he’s going to waste a lot of time revisiting math concepts over the years, as we educators get our act together. It might not be too bad for your son, since kids tend to be pretty resilient, but we still need to address this issue for teachers.
Anyway, back to estimation. The above question would technically “align” to the Common Core standard cited (3.OA.D.8), plus or minus a few references to cancer, which might offend some students, and the fact that one of the operations may be multiplying two-digit numbers. I just made up the question to illustrate the difficulties of testing estimation across an entire state. The question may be illegal for other reasons.
However, we definitely haven’t measured the student’s estimation skills or his ability to assess the reasonableness of an answer. Even Maryland’s old standards required us to test that, and we have done an abysmal job in third grade. Let’s hope future test designers do it better or just come to the realization that 3.OA.D.8 can’t be tested. This would avoid kids’ feelings of frustration in a high-pressure test on a problem they should never have been asked, but it would leave it open for teachers to explore the concept of estimation in the comfort of their own classrooms. A teacher, with the freedom to teach this standard, might explore the following scenario to illustrate estimation in action:
Rebecca walks into a Subway restaurant and orders a 12-inch turkey sandwich, along with the Fresh Value meal. The sandwich costs $6.25, and the Fresh Value meal add-on of chips and a soft drink costs $2.30. The Subway in this scenario charges a 6% sales tax, because it’s in Maryland, where Rebecca lives.
When she gets to the register, there are five people waiting impatiently behind her. The cashier tells Rebecca she owes $9.70. Should she complain and make the people wait while the cashier calls the manager over to investigate? She thought the total price shouldn’t have been much more than $9.
This is a real-world scenario where estimation skills are needed. Maybe she can do the addition in her head: 6.25 + 2.30 = 8.55. But she doesn’t really have time to take out some scratch paper and multiply 1.06 by $8.55. Rebecca wouldn’t want to annoy everyone behind her in line if she was wrong, yet she doesn’t want to pay more than she owes for her sandwich, chips, and beverage. But what’s the tax on $8.55?
Here’s where estimation comes in. Is the total price of $9.70 “reasonable”? She might think of it like this: “I can see that $9.70 means I’m being charged more than a dollar in tax, and a whole 10% of $10 is a dollar in tax, and my subtotal was less than $10 and the tax in Maryland is less than 10%. Therefore, I know the total is wrong because I’m being asked to pay more than a dollar in tax.”
Sure enough, the cashier added a cookie by mistake bringing it to $9.70, as shown on the receipt at right, even though Rebecca didn’t order a cookie. She was therefore justified in her questioning of the cashier and didn’t waste people’s time on her error.
Of course, we can’t put this type of task on a test that will not allow kids to perform the arithmetic first, but in a classroom, teachers can certainly play the role of the Subway cashier. Furthermore, under the Common Core, students would more likely be called upon to explain their reasoning in determining that the $9.70 was not a reasonable total. I’m sure kids will make up as many strategies for Rebecca as there are kids in any given classroom, and that’s what gets a little exciting about the Common Core.
Now, multiplying decimals isn’t something we teach third graders, even under the Common Core, so this scenario wouldn’t work in a third-grade classroom. But use your imagination. I’m sure you can come up with real-world settings, such as buying things at a dollar store or pouring two measuring cups of liquid into a third when baking a cake. You could even eat the cake after you played out the estimation scenario.
Or, an example might be: “This beaker has between 700 and 800 ml of water, this one has a little less than 200 ml of water, and this empty one holds 1000 ml of water. Can I pour the water from the first two into the third, or will it overflow?” In other words, without worrying about the exact volumes, is it reasonable to pour the water from the first two beakers into the third? These scenarios work on a standardized test, but they don’t really test how skilled kids are at estimating the reasonableness of an answer.
I have reprinted the corrected receipt below. Feel free to use it in your lesson plans if you want to play an estimation game at a higher grade level, or make up your own. The point is, “estimation” has to do with reasonableness and only includes rounding as one possible strategy, which was added to the Common Core ostensibly to make it easier to write standardized test items. Teachers, parents, students: Do not concern yourself with anything in the Common Core strictly to impose limits on test question writers. In fact, don’t trust anything in the Common Core or that anyone tells you about it that would seem to restrict or limit your exploration and discovery. Such phrases aren’t there for you, if they’re there at all.
Now, many have argued that if a standard can’t be tested, why is it even in the Common Core? Aren’t we supposed to test all the standards in the Common Core? Wasn’t that the point?
Let me put it this way: I hope that wasn’t the point. And even if it was the point, it’s not the reality of the Common Core. Many of the standards in the Common Core cannot be tested, except in real-world scenarios. On this standard, all the interaction and all the computer graphics in the world won’t negate the fact that kids solve these types of problems on tests by doing the arithmetic and rounding, which is not the same thing as estimation. We can’t claim to test estimation if kids are doing rounding.
But I actually thought the point was to improve classroom instruction by encouraging students to think a little deeper about problems they might encounter in the real world, not to perform better on a test. That’s what I hope this diversion into the detail beneath one of the third-grade standards has shown.
I have said from the beginning, the devil’s in the details. As we implement the Common Core—a process that should take a few years but has, in reality under federal law, been given much less time because of standardized tests, teacher evaluations, student data collection, and narrow but imposed curriculums—we will discover nooks and crannies that open doors in the classrooms and shut the door on standardized tests. I hope we discover the fallacy of standardized testing and get on with the business of school.
Ms Miller, thank you for writing a very interesting letter. As an afterthought, I find it troubling that a teacher put a “56%” on a third grader’s test. Please reassure your 8-year-old that nobody really knows how this will work yet, and that mark of “56%” was simply a guess based on a set of standards that we haven’t tested enough to have even the slightest confidence that it’s the correct score at all. It’s a guess, an estimate, if you will, and we have no measuring cup yet for estimating the amount of knowledge in kids’ brains on the Common Core. I would therefore call it a “wild guess.”