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December 1, 2013
Vol. 71
No. 4

Mastery Multiplied

Just getting the right answer in math class isn't enough if students don't know why the answer is the right one.

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Imagine two buzzing classrooms of 2nd graders busily solving mathematics problems that require them to add and subtract numbers under 100, as required in the Common Core State Standards (CCSS.Math.Content.2.NBT.B.5). The teachers from both classrooms have asked students for their solutions to a story problem that could be represented by the equation 100 – 5 = ___. Students in both classes arrived at the correct answer of 95, but how they figured it out was quite different. Here's what Alex did:
I figured it out because I lined up the numbers. You can't do zero minus five and you can't borrow from zero, so I crossed out the one and gave it to the zero to make ten. Then I made that ten a nine so I could give a one to the last zero so it can have ten. I know ten minus five is five and nine minus nothing is nine so the answer is ninety-five.
Bella, in the other class, thought about the problem in this way:
I just know that five less than one hundred is ninety-five. Like if you count backwards by fives. I just took a five out of one hundred.
Now consider two 5th grade classrooms focused on performing operations with decimals to hundredths (CCSS.Math.Content.5.NBT.7). Both classes have solved the equation .19 + .01 to get the solution of .20. Cari explains what she did in this way:
I started on the right and did nine plus one, which makes ten. I wrote the zero and carried the one, so one plus one is two. I wrote two and then put the decimal point in front of the two.
David thought about it differently:
Well, you just add a hundredth to nineteen hundredths and the next number is twenty hundredths. Like on the number line you have nineteen hundredths [.19] and just do a leap of one hundredth [.01] and the next number is twenty hundredths [.20].
The students in all four classes quickly reached their solutions and demonstrated skill at solving the problem. But how did the students' understandings differ?

What Students Understand

Bella and David used their understanding of relationships among numbers to solve the problems. Bella had a "sense" of how far away 95 is from 100, but Alex used the algorithm to figure it out. His algorithm worked and is an excellent and efficient strategy for doing operations with other numbers, but shouldn't he just know that 95 is only five away from 100? Similarly, David understood that if he just added another hundredth to 0.19, the next number is 0.20. Cari performed the algorithm correctly, but did she understand the relationship between 0.19 and 0.20?
Bella and David's strategies demonstrate both mastery and understanding. But Alex and Cari's understanding of number relations isn't clear, despite their mastery of algorithms for solving problems.
Mathematics curriculum reformers have long advocated curriculum, instruction, and assessment centered on meaningful understanding, reasoning, and sense making, with the goal of developing mathematical thinkers like Bella and David. The Common Core standards reinforce these efforts through terms such as understanding and making sense. How does this wording change our perception of mastery? How are teachers shifting their teaching to address new conceptions of mastery? We are two elementary educators who are thinking deeply about these questions. We do not have the answers to all of them, but we do have ideas for how teachers might reconceptualize mastery and teach in ways to meet this goal.

New Goals for Mastery

The Common Core Standards for Mathematical Practice bring attention to how students are understanding mathematics content. Such understanding is difficult to assess through a typical paper-and-pencil assessment or standardized test because students can follow procedures to obtain the correct answer without understanding the number relationships behind the procedure. To assess mastery of the practice standards, we need to explore how students are arriving at their solutions, just as we did with the four students described earlier.
What type of expertise did Bella demonstrate? Bella had shown on other occasions she could perform the algorithm that Alex used. However, she chose to simply use her knowledge of the counting by fives pattern and the relationship between 100 and other numbers. Similarly, David reasoned that 0.20 was only one hundredth away from 0.19, rather than carry out the algorithm, which he too, had used successfully in solving other problems.
Bella and David could be characterized as adaptive experts. They drew on a range of interconnected math knowledge. Hatano and Oura (2003) define adaptive experts as experts who apply their schemas in adaptive and tuned ways. They understand why procedures work or invent new procedures, flexibly apply knowledge, and tend to be creative and innovative within their domain of expertise. This type of expertise goes beyond the routine expertise of speed, accuracy, and automaticity when solving familiar problems.
The concept of adaptive expertise provides an important model of success ful learning (Baroody, Feil, & Johnson, 2007; Verschaffel, Luwel, Torbeyns, & Van Dooren, 2009). But designing and assessing learning experiences that develop adaptive expertise requires a shift in thinking about what mastery means. Is it enough for students to leave our classrooms only knowing the efficient algorithm for solving math problems? According to the Common Core standards, it isn't. We need to address understanding as part of mastery.

Instruction for Understanding

The changes in classroom practice that meet our new mastery goals can overwhelm any teacher. One manageable way to begin changing classroom practice is by using daily number-sense experiences to develop students' understanding. These could be quick 5-minute, 10-minute, or 15-minute experiences at the beginning of a math lesson (Shumway, 2011). For example, Count Around the Circle, one of many daily warm-ups to any math lesson, involves using a counting sequence, such as counting around by tens starting at 54. One student begins by saying "fifty-four," and then, going around the circle, each student says the next number in the sequence (64, 74, 84, and so on).
Let's peek into Joan's classroom to get a sense of how we used this instructional practice to lead to mastery with understanding. At the time of this lesson, we had been consulting together to guide Joan's class toward the 5th grade Common Core goal of using the relation ship between decimals and whole numbers "to understand and explain why the procedures for multiplying and dividing finite decimals make sense" (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010, p. 33). Mastery of this standard involves not just performing operations on decimals to hundredths, but also understanding the place-value system, using patterns in the number system, understanding and explaining relationships among numbers, and justifying why the solution method works.
With this conception of mastery with understanding, we planned to use Count Around the Circle daily for several weeks as the warm-up to Joan's mathematics lessons to develop students' adaptive expertise through experiences with number magnitude, estimation, and mental math.
Our first Count Around the Circle warm-up was to count by tenths starting at zero. The first student said "one tenth," the next student said "two tenths," and so on around the circle while Jessica wrote the sequence on the whiteboard. After counting around the circle by tenths, the 5th graders discussed the leaps from nine tenths (0.9) to one whole (1.0) and from one and nine tenths (1.9) to two wholes (2.0).
Jessica wondered whether students' knowledge of these leaps would transfer to their understandings about hundredths. So she asked the students, "What if we count by hundredths? What will Alex [the 11th person in the circle] say if we start at zero and count by hundredths?" A student estimated "one and one hundredth" (1.01), and many students agreed. Students defended this estimate by stating that they will have to leap to one whole again, just like they did when they went from nine tenths to one whole when counting by tenths. But one student said, "I think we will land on eleven hundredths" (0.11). Many students were not convinced that this estimate would pan out, so we counted to find out.
Students counted "nine hundredths" (0.09) to "ten hundredths" (0.10), and the next person said "one and one tenth" (1.1). Jessica stopped the counting there and said, "This is where we had some disagreement about what number we'd land on. Do you agree or disagree that the next number after ten hundredths [0.10] is one and one tenth [1.1] if we count by hundredths?"
Shari said, "I'm not so sure because if you look at a hundreds flat from the base 10 blocks, and one cube is one hundredth [0.01], it takes a hundred of those to make one whole." Shari was getting at the important idea that .01 × 100 = 1.
Jordan repeated Shari's thinking in his own words: "I agree the number might be eleven hundredths [0.11] instead of one and one tenth [1.1]—it can't be one whole yet because you need one hundred hundredths to make a whole … just one more hundredth is eleven hundredths (0.11). We are making little jumps." The students were critically thinking about place value and applying their understandings to this discussion.
We counted around the circle several more times by hundredths. Students noticed that they could use regrouping to figure out the next number and described changes in place-value patterns. Several students stated that 20 hundredths is the same as two tenths (.20 = .2) and 30 hundredths is the same as three tenths (.30 = .3). The students were developing a stronger understanding of the relationships among tenths, hundredths, and whole numbers.

A Flexible Understanding

In reflecting on several consecutive days of Count Around the Circle, we pinpointed several big ideas in students' discussions. Specifically, students reasoned about "the whole" and the difference in how long it takes to get to the whole when counting by tenths versus counting by hundredths. Also, students' discussion highlighted their developing understanding of place value as it relates to decimal numbers and their relationships.
Such understanding is achieved through multiple experiences in a mathematics classroom that emphasizes discussion, reasoning, and justification. This is why we focus on developing students' number sense every day. Sequences on subsequent days involved having students count backward by a variety of different numbers or having them count forward by the same number multiple times, starting on different numbers each time. With each sequence, students built expertise that they can apply to future problems. Sometimes they might use the standard algorithm. Sometimes they will use compensation, changing the numbers to make the problem easier to solve mentally. Sometimes they will reason about the relationships between numbers to determine the solution.
The point is that they have a toolbox of strategies and can choose the strategy that suits the type of problem, the context, and the numbers. We view these students as adaptive experts who demonstrate mastery with understanding. Their work is an example of the outcomes we seek as we teach mathematics.

Baroody, A. J., Feil, Y., & Johnson, A. R. (2007). An alternative reconceptualization of procedural and conceptual knowledge. Journal for Research in Mathematics Education, 38(2), 115–131.

Hatano, G. & Oura, Y. (2003). Reconceptualizing school learning using insight from expertise research. Educational Researcher, 32(8), 26–29.

National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington, DC: Authors.

Shumway, J. (2011). Number sense routines: Building numerical literacy every day in grades K–3. Portland, ME: Stenhouse.

Verschaffel, L., Luwel, K., Torbeyns, J., & Van Dooren, W. (2009). Conceptualizing, investigating, and enhancing adaptive expertise in elementary mathematics education. European Journal of Psychology of Education, 24(3), 335–359.

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