Place value is one of the key concepts in mathematics curriculum and though it is only explicitly in the standards in the lower grades, understanding place value (or not understanding it) will follow students through their mathematics journey. It is essential that students understand the meaning of a number. For example, in the number 635, the 6 represents 600. Without this understanding, students often struggle with when to regroup ones and tens or “borrow,” and algorithms for adding and subtracting multi-digit numbers make little sense. However, being able to say that 635 is 600 + 30 + 5 does not, in and of itself, mean that a student understands place value. Place value encompasses not only position and value of digits but also decomposition of numbers and a number’s relationship to others in the number system.
For students who struggle with understanding place value, zeros are especially difficult to comprehend. Take for instance the number 406. Many students read this as “four, oh, six”. While this does not necessarily signify a lack of understanding place value, it does make it more difficult to understand what those three numbers mean when grouped in that order. Suppose the problem is 406 – 32; for a student that sees the middle digit just as zero, it becomes a puzzle as to how you could “take 3 away”. On the other hand, a student who recognizes that 6 hundreds is the same as 60 tens, is able to make sense of taking three of those tens away.
The struggle with place value (and zero specifically) carries into multiplication and division. Many students try to use the standard algorithm for multiplication after being asked what 30 times 10 is. This is a good example of how strongly fluency and efficiency are related to understanding. If a student understands place value and base ten, they can quickly recognize that 10 groups of 30 is equivalent to 1 group of 300. Alternatively, students may know the “shortcut” of “adding a zero” without having an understanding of the underlying mathematics.
While these “shortcuts” may be helpful to students, they often break down when students are faced with more difficult problems, or numbers involving decimals. Decimals rely heavily on a solid understanding of base ten. Even a problem that seems to involve only whole numbers like 378 ÷ 30, requires students to understand where to record numerals in the quotient and how to record a decimal. In many cases, students will get 126 as their quotient (instead of 12.6) and not recognize that it is not a reasonable answer.
In order to help students develop an authentic understanding of place value, it is crucial that they construct meaning for themselves through applicable, hands-on activities. This understanding needs to begin with repeated and prolonged exposure to the base ten system. Using ten-frames and Rekenreks help students build their comfort with tens.
The use of manipulatives is extremely important in moving past working with just tens. Place value blocks that help student see magnitude of numbers help them visualize the differences among place values. When students are asked to regroup, it is important that the manipulatives and a context support this. For instance, The T-Shirt Factory problem from Contexts for Learning, focuses on regrouping, equivalence, and place value using a t-shirt factory inventory as the premise. For an outline of this problem and more information, see http://www.contextsforlearning.com/gradesK_3/default.asp
Helping students build an authentic understanding of place value through base-ten work and interesting problems, will help them be more successful as mathematicians. Recognizing that numbers can be broken apart, rearranged, and re-formed, gives students a better understanding of how addition, subtraction, multiplication, and division work. This is especially true when students have a sound understanding of what each part of a whole number represents.
To see Matific’s list of place value activities, click here.
Elizabeth Masalsky is a 6th, 7th, and 8th grade mathematics teacher at Battery Park City School in Manhattan. She has a post-baccalaureate in mathematics from Brandeis University and her master’s in secondary math education from Bard College. Elizabeth is a Math for America Master Teacher and continues her professional development through workshops with Math for America, Metamorphosis and Math in the City.