1. THE REFERENT UNIT:A DIFFERENT WAY OFLOOKING AT MATH(K – 6)Tim Whiteford PhDSt. Michael’s CollegeColchester, Vermonttwhiteford@smcvt.eduhttp://academics.smcvt.edu/twhiteford/Math/Math.htm
2. What Are Referents?• They are things numbers refer to.• How large is Ted Turner’s ranch in “Rhode Islands” ?• “That’s very nearly an armful” said a British comedian while giving blood.• They’ll need the whole 9 yards – how much is that?• It’ll take about a fortnight!• How many holes does it take to fill the Albert Hall?
3. Uses of Number• Number can be used in the cardinal, counting sense • “I have one daughter”.• Number can be used in the nominal, naming sense • “I have the ONE “• Number can be used in the ordinal, sequential sense • “We are number one”
4. Referents give Numerals Magnitude• Numerals are graphemes until they have a referent. 5• The referent affects the value of the numeral.• Numbers can change their referents (and magnitude) depending on the context or reference point. We each can have many referents.• “Naked numbers” should be avoided in the elementary school.• “One World” is the biggest number she could think of.
5. Bushels, pecks, pottles, firkins, tierces Bytes, megabytes, pixels, ram There’s Dictionary of Referent Units at: http://www.unc.edu/~rowlett/units/
6. Referents can be implicit or explicit•Implicit; • It cost 4.99 • It’s about 65 today • She must have been doing 110! • He did a 1080 • I feel like I’m 32 again • It cost three and six •Explicit • He’s 5 feet 9 inches tall. • They cost 75 cents each.
7. Continuous and Discrete Referents• Continuous - time, length, money, weight, • Much – more or less• Discrete – minutes, yards, dollars, grams • Many – more or fewer • “How much M&Ms” or “How many M&Ms” or are both correct?
8. Place Value Referents• Base Ten is groups of groups of tens• 0 – 9 is repeated in each place• 10 ones make (or are regrouped) as one ten etc• The 100, 10, 1 pattern is repeated in ones, thousands, millions and so on. Millions Thousands Ones 100 10 1 100 10 1 100 10 1
9. Addition and Subtraction Problems• We can join, separate and compare groups of things that have different referents. • If there are 3 girls and 4 boys at the playground how many children are there altogether? • I have a coin jar full of 110 dimes and quarter. If I gave you all the quarters and kept the 50 dimes, how many quarters did I give you? • My son has 125 Hotwheels and 30 Matchbox cars. How many more Hotwheels does he have than Matchbox cars? •
10. Multiplication and Division• Repeated addition • How many roses do I have if I have four bunches and there are a dozen roses in each bunch? 4 x 12 = 48• Area Concept • What is the area of a carpet that is 8 feet long and 4 feet wide? 4 x 8 = 32• Cartesian Product • How many options do you have if there are 4 Ben and Jerry’s flavors and 3 containers, a waffle, a cone or a cup? 4 x 3 = 12
11. Remainders in Division• The nature of the referent determines what happens to the remainder. • Whole number • How many candies will each person get if 25 candies are shared between 4 people? • Fractional part • If you can drive 400 miles a day, how long will a 1000 mile journey take? • Forced • How many 60-seater school buses will you need to take 200 students on a field trip? • Discarded • How many 6-foot skipping ropes can you make from a 50 foot length of rope?
12. Referents and Fractions• Would you rather have half the money I’m holding in my right hand or a quarter of the money I have in my left hand?• Addition and subtraction of fractions. The 1 must remain the same. • 1/5 + 3/5 = 4/5• Multiplication and division with fractions. The 1 changes. • What is the referent for each fraction in: • ½ x ½ = ¼ and ½ ¼=2
13. Measurement Referents• All measurement is the repetition of the referent unit. • E.g. in measuring distance all the referents have to be in a line with no gaps or overlaps.• Converting between one system and another involves changing the size of the referent unit. • E.g from Metric to Imperial measures.• Sometimes selection of the correct referent is critical to the development of understanding. • E.g. Degrees are measures of rotation and not distance.
14. Culturally Defined Referents• The Metric System• Different Monetary Systems• Cultural Math Games• Implicit referents in sub-cultures • She rode a 750 • He made a 360• Durham reached 124 for seven off 34 overs compared to Worcesters 128 for six, but the tail subsided... David Byas took his seasons tally to 702, passing John Hampshires 684 set in 1976, by hitting 54 as Yorkshire posted 214 for six. Then Darrin Gough took a competition-best five for 13 on his 24th birthday as Sussex were dismissed for 177. Captain Alan Wells top-scored with a battling 64 including five fours and a six off 70 balls. Mike Watkinson, with four for 32, led Lancashire to a 47 run win over Leicestershire despite a broken thumb".
15. How far is it to Boston? How heavy is this box? How many marbles in the jar? How many leaves on a tree? We needs to know the size and nature of the referent as well as a strategy for estimation. K – 6 Math Education resource website; http://academics.smcvt.edu/twhiteford/Math/M ath.htm