Reconceptualizing Mathematics Part 1: Reasoning About Numbers and Quantities Judith Sowder, Larry Sowder, Susan Nickerson CHAPTER 2 – NUMERATION SYSTEMS 1 © 2010 by W. H. Freeman and Company. All rights reserved. The need to quantify and express values of quantities have led to the invention of numeration systems. A variety of words and symbols, called numerals, have been used to communicate number ideas. Our Hindu-Arabic numeration system uses ten digits, 0–9. 2-2 ACTIVITY Below are different ways of representing twelve. Can you deduce what each individual mark represents? How would the number ten have been written in each case? 2-3 Some ancient cultures did not need many number words. For example, in a recently discovered culture in Papau New Guinea, the same word—“doro”—was used for 2, 3, 4, 19, 20, and 21. By pointing also to different parts of the body and saying “doro,” these people could tell which number was intended. 2-4 DISCUSSION 2-5 2.1 2-6 Example: 2-7 In our base ten system, the whole-number place values result from groups of ten—ten ones, ten tens, ten hundreds, and so on. This is because our system works fine until we have ten of something because there is no single digit meaning ten. So, for example, when we reach ten ones, that’s when we name one ten, with zero ones left over. 2-8 EXAMPLE 2-9 DISCUSSION 2-10 The decimal point indicates that we are breaking up the unit one into tenths, hundredths, thousandths, and so on. The number one is the focal point of the system. For instance, .642 is 642 thousandths of one. Example: .6 is six-tenths of one 6 is six-tenths of ten 70 is seven-tenths of one hundred .007 is seven-tenths of .07 .08 is eighty thousandths 2-11 2.2 2-12 2-13 Suppose we live in “cartoon land,” and instead of having ten digits, we only have eight (0, 1, 2, 3, 4, 5, 6, 7). Then, counting would look like: 0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, 20, 21, 22, 23, 24, 25, 26, 27, 30, 31… To let everyone know what base one is working with, we would write our numbers in this system with a subscript of “eight.” For example, 10eight (we would say “one-zero”) 2-14 ACTIVITY 2-15 EXAMPLE 2-16 DISCUSSION How could we find a way to write the number 6072 in base eight? 2-17 ACTIVITY What are the first 20 numbers in base three? What is 1000three in base ten? Write 547 in base five. What are the place values in a base “b” system? What digits would be needed? What is 18 in base two? 2-18 BASE BLOCKS It is helpful to think of other bases using base blocks. We represent them as follows: These are labeled as “small cube,” “long,” “flat,” and “large cube,” respectively. Base four, for example, would have zero, one, two or three of any of the given types of blocks. 2-19 DISCUSSION 2-20 2-21 2.3 2-22 2-23 EXAMPLE Consider adding in base eight: Make sure you understand the methods used on both sides here. 2-24 EXAMPLE 2-25 2-26 2-27 ACTIVITY 2-28 EXAMPLE 2-29 EXAMPLE 2-30 ACTIVITY Subtract 231four from 311four in base four using only drawings. 2-31 2.4 2-32 Understanding place value is absolutely foundational for elementary age children. Even so, oftentimes children lack true understanding of what we even mean by “base ten”. By working with other bases, hopefully you’ve come to a deeper understanding of place value. One activity centered program asks children to begin grouping by twos, threes, and so on even before extensive work in the traditional base ten. 2-33 The manner in which we vocalize numbers is also important. Some young U.S. children will write 81 for eighteen. Place value instruction in schools has become a matter of memorizing. For instance, the 7 in 7200 is known to be the “thousands place,” but children are not able to answer the question of how many hundred dollar bills are in $7200. Using base blocks has been a successful way for children to get a better feel for place value. 2-34 Also, children often don’t understand the true meaning of decimal places. They don’t understand that, for instance, .642 is 642 thousandths of one. Further, that we use tens and tenths, hundreds and hundredths, causes children to lose sense-making when it comes to decimals. When teachers say, instead, “two point one five” for 2.15, it removes any sense for the number itself. Plan to give additional emphasis when you use the “th” sound with children. 2-35 DISCUSSION When comparing .4 and .40 one student stated that it was necessary to, “...add a zero to the end of .4 so that the numbers are the same size.” What was the misconception behind the student’s thinking? Which do you think the student thought was the larger of the two numbers? How would you correct this error when working with such a student? 2-36 If teachers postpone work with operations on decimals until students conceptually understand these numbers, students will be much more successful in the long run. Research has shown that once students have learned only rote rules for calculating decimals, that it is extremely difficult for them to re-learn how to calculate them meaningfully later on. 2-37 continued…. 2-38 2-39