Creativity Module 15:
Creative Place Value Pedagogy
Background Information:
It is no wonder that the traditional approach to teaching place value causes confusion to the learner because it is wrong. Place value is a pervasive difficulty in learning arithmetic, which if not corrected early, affects understanding computation well into adulthood—and is especially apparent in those students with math anxiety and dyscalculia. A creative new approach to learning place value uses Place Value/Counting Boards, which represent a physical embodiment of the rationale for moving from the units (ones) place to the tens place to the hundreds place, etc. In the beginning stages of learning, children do not think, “I have ten units; that’s one group of ten and no units.” Rather, initial learning takes the form of a count-on-by-one model and incorporates a horizontal move to the next position to the left after the count of nine, not ten. The Place Value/Counting Boards are a truer physical embodiment of our numeration system.
The Place Value/Counting Boards were created by Reisman (Reisman & Severino, 2020) after many years of diagnosing why students had so much difficulty learning place value. Place Value Charts, Dienes Blocks, Cuisenaire Rods, the abacus, and bead charts have been used for years. All these methods rely on the “exchange model” where multiples of ten are exchanged for the next higher place value. On the contrary, each board has nine spaces with increasing values from 1 through 9 (see Figures 9.2 through 9.7). An empty board has a value of zero. The counting boards represent the finite set of digits in our Hindu/Arabic numeration system (0,1,2,3,4,5,6,7,8,9). The following activities address correctly teaching place value (Torrance & Reisman, 2000).
Activities
In using the boards, a value of one is obtained by jumping a chip onto the board to land on space one. It is important to represent the board’s initial value as zero by starting with an empty board.
By the time students are in eighth through twelfth grade, a different physical representation is needed if they still do not understand place value, which underlies the algorithms for operations on whole numbers, as well as computation with decimals. An analysis of place value reveals that two very different relationships are involved in understanding place value—a “count-on-by-one” model and an “exchange” model (bundling-of-tens).
The count-on-by-one model emphasizes the fact that when recording the counting sequence, a change in thinking from units to tens occurs after the count of 9. On the other hand, the bundling-of-tens model involves a shift in thinking after the count of 10. We count ten objects and then bundle them to represent 1 ten. In fact, we count to 9 and then make our move to the tens place; we change place values after the count of 9, not 10. Therefore, the bundling-of-tens model does not accurately represent our notational system. However, we will see later in this chapter that the exchange/bundling model is appropriate for teaching addition and subtraction of whole numbers with renaming (carrying and borrowing).
Have students practice counting board values from zero to nine by jumping a chip onto the Board to count “one,” moving the chip up 1 space to show “2,” and so on up to “9”. Continue counting one more beyond 9 and call out “ten” as the chip is moved up and off the top of the counting board. Although there is now no chip on the Board, set a rule that once the count starts, the counting board continues to have value, especially when the count is greater than nine and the Units Board appears empty. Otherwise, students might focus on the visual emptiness of the Board and forget that they had been counting upward beyond nine to ten, with the chip now sitting in the space just above the Board
After the student has had practice using one board, provide a second counting board. Position the second board (“Tens Board”) to the upper left of the “Units Board” to continue the upward counting sequence beyond 9.
The board to the upper left represents the “tens” place value. Each space on the “tens” board is worth 10, just as one space on the “hundreds board” is worth 100, etc. Remind students that as the chip is moved up a space, the board value increases by its value. Thus, moving up a space on the Tens Board is 20, 30, 40, etc.
Present the problem-solving situation of what will the count be if the chip is moved up one move beyond 9; it is now just above the Units Board. Move the chip horizontally to the left onto the bottom space of the board to the upper left. Since only a horizontal move and no upper move occurred, there was no increase in value. Consequently, the value of the bottom space on the board in the “tens” place is 10; the boards show 1 “ten” and 0 “units” or 10.
Repeat the process as students count to 19. At the count of 20, the counting boards show a chip in the ones space on the “Tens” board with a second chip sitting atop the “Units” board. Make sure that students understand that just as two digits cannot be written in one space, two chips (representing digits) cannot be in one space.
The chip with a count of 10, sitting above the Units Board needs to move horizontally to the Tens Board, maintaining the value of 20, which results in two chips in the bottom Tens Board space. This cannot be, just as we do not write two digits, one over the other, in the same number position. So how to solve the dilemma?
There now is the opportunity to allow a 2-for-1 exchange, by exchanging the two chips worth 10 each for one chip placed on the second space whose value is 20.
Assessment
Creative Place Value Pedagogy Assessments
Place Value items with answer in bold.
1. Circle the place value that tells the value of 3 in the number 3,520.
ones
tens
hundreds
thousands
2. Round 193 to the nearest hundred. Circle your answer.
200
190
210
100
3. Circle the number 3,243 rounded to the nearest hundred.
3,000
3,200
3,240
3,300
4. Circle the order from least to greatest.
7980, 7890, 7750
5780, 5879, 5889
6980, 6978, 6877
5780, 5680, 5670
5. Circle the number with the GREATEST value that can be made with the digits 4, 8, and 5 using all the digits only once.
854
584
548
845
Other Place Value Questions using the Computational Counting Boards
6. Teacher asks student to count from 1 to 9 on the units counting board
7. Teacher asks student to count from 1 to 10 on the units and tens counting boards
8. Teacher asks student to count from 1 to 13 on the units and counting boards and write the digit 1 under the Tens Board and the digit 3 under the ones board to show 13
9. Teacher says numbers from 1 to 99 for students to show on the units and tens counting boards
10. Ask students to show 111 on the counting boards. Allow creative problem solving discussion until someone says they need a third board to show hundreds. Teacher asks students to explain why they need the third counting board.
11. Teacher asks students to show numbers 1 to 999 on the counting boards.
12. For 5th or 6th graders, extend the counting boards to the right to show decimal place values. Place the board worth tenths to the lower right of the Units Board and continue with hundredths, thousandths, etc. Finally, have all boards moved to the same level so that digits written below their respective board will appear as a normal number, e,g,,13.1