Assessing Place value Understanding

Assessing Place value Understanding It is an enormous leap from operating with units of one to multi-digit computational procedures that use units of...
Author: Lenard Lang
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Assessing Place value Understanding

It is an enormous leap from operating with units of one to multi-digit computational procedures that use units of tens, hundreds, thousands and so forth, as well as units of one. To work with units of different values it is necessary to first sort out the complicated ways that each is related to the other. The ten for one trade structure of our number system is quite complex. Being able to label the tens place and the ones place, or even being able to count by tens, does not, necessarily signal an understanding that 1 ten is simultaneously 10 ones. Becoming mindful of this relationship between tens and ones, or staying mindful of it, is neither simple nor trivial. Asking students to represent numbers with concrete objects or pictures and carefully examining their use gives an insight into their conceptions. Task: Use Dienes blocks to represent the number 426

Mark First Year

Task: Use Dienes blocks to represent the number 107 in as many ways as you can Susan First Year

Task: Rename or represent 476 in as many ways as you can Sarah Louise 1st year

Task: Represent 99 with Dienes blocks William First Year

Examine the student work    

What does each student’s work tell you about their understanding of Place Value? What questions would you like to ask each student to find out more about their understanding of Place Value? How would you use Diene’s blocks in your class to help your students develop their understanding of Place Value? What questions would you ask to ‘dig deeper’ into student thinking about place value?

A teacher’s reflection …I was very impressed with Sarah Louise’s work.I asked her .How would you represent the 42600 hundreths and the 213 twos with Dienes blocks. She very quickly held up two little unit blocks and said well this is a ‘2’ so I woud get 213 of them. She had to think a little longer about the 42600 hundreths and said ….Well I would call this a one [holding up the 100 square] this a tenth [holding up the 10 stick]and this a hundredth [holding up a unit cube] and then I would need 42600 of them but I don’t think we’ll have enough.I thought Sarah Louise has a well- developed concept of place value, she is able to look at numbers as separate ‘units’ and is able to confidently rename; this understanding will be great when we move on to operate with rational numbers..

As students become adept at breaking apart and recombining numbers, they often invent multi-digit addition and subtraction procedures. These can be the starting places for deeper understanding of the tens structure itself and how it behaves in computation. Consider the first piece of work below, the two students drew out a pile of 38 cubes counting in ones each time. Then they drew another pile of 25 cubes starting again at one. Next they counted both sets together, starting at one until all the cubes were counted. This was in contrast to other students in the class who worked more abstractly with number and made use of groups of ten, generating solutions such as the one shown. Like many students they chose to work with the larger numbers first;in this case tens.

What students do with the objects they use for modelling mathematics situations reflects their understanding of the structure of the situation. In the word problems here, for example, the structure of addition is understood from physically joining quantities.

While the second style solution certainly demonstrates some ability to decompose and recombine numbers using groups of tens and ones, we can’t really tell from this example whether any of these students understand 30 +20 =50 to be equivalent to 3 tens + 2 tens = 5 tens, and therefore to be both similar to, and different from, 3 ones + 2 ones = 5 ones. Certainly the students solving this problem by drawing out cubes and counting them one by one are not looking at numbers in this way. As they grow beyond the need to represent all of the amounts and actions in problems, they no longer rely entirely upon counting to determine the results of joining or separating sets, beginning instead to reason numerically about the quantities involved.

After students have modelled many situations in which they represent all the amounts in the addition and subtraction problem with concrete objects, they develop a more abstract concept of number and begin to use counting up and counting back strategies. Fuson (1992) Carpenter et al (1996)

When students are able to pay attention to how all the amounts in a problem are related to one another, they can combine and separate them more flexibly. They often use strategies based on facts they already know, when they get to this stage they take apart numbers and recombine them to form new quantities that they find easy to work with (Fusion, 1992).

When presented with the problem 39+18 Sarah changed it to involve numbers she found easier to work with.

Problem solving reminder: If you are going to use these tasks remember, answers are important but what is more important is the mathematics students can learn from engaging in the tasks.