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Number and Number Sense

Mark Roddy, PhD Seattle University. Number and Number Sense. A lecture for Primary Mathematics and Numeracy 2 (101584) 11 September 2014 University of Western Sydney. Session Objectives . The Plan. Students will develop their grasp of:

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Number and Number Sense

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  1. Mark Roddy, PhD Seattle University Number and Number Sense A lecture for Primary Mathematics and Numeracy 2 (101584) 11 September 2014 University of Western Sydney

  2. Session Objectives The Plan Students will develop their grasp of: • the concepts of number and number sense as these terms are used in mathematics education; • the benefits of number sense; • Effective pedagogy; • Relevant resources. • Introduction • Pig • What is number and what is number sense? • How do I teach it? • Where are the resources? • What are the benefits of number sense?

  3. PIG and Number Sense …a nice opportunity for students to generate data through their own activity and to do meaningful, self-directed maths with these data – to make sense of their experience and to understand it more fully through mathematics.

  4. ¿Number? ¿Number Sense? OED - number: “an arithmetical value, expressed by a word, symbol or figure, representing a particular quantity and used in counting and making calculations.” 77 out of 217 pages devoted to “Number and Algebra”

  5. Number:the kitchen sink of mathematics Statistics Probability ELSE! EVERYTHING Algebra N umber Measurement Geometry

  6. Number Sense “…Number sense is also about knowing that 6 + 7 and 7 + 6 both produce a sum of 13; that 25 × 7 is less than 200; that the quotient for 1/2 ÷ 1/4 is larger than 1/2; that the stadium could not seat 400,000 people; that tripling the square footage wouldn’t make for the most economical home to heat; etc.” As students estimate, talk about numbers, compute, use mental math, and judge the reasonableness of their results, they become more flexible in working with numbers. A sense of number emerges that is built on [solid] foundations, which yields responses such as, ‘I knew 3/4 was more than 3/5 because the pieces were bigger in fourths.’ This is what all math teachers want.” Number Sense—Right Now! Francis (Skip) Fennell (2008). When you have number sense, mathematics becomes another way of making sense of the world.

  7. Number sense begins at home…. “How old are you?” “We’re halfway there!” “The dishwasher is only half full.” “Only 5 more days.” “You can invite three friends.” “How much does it cost?” “Which is more?” “Go two steps forward.” “You can have three more.” “Get five spoons, please.” “How tall are you?” “How much more is there?” “You can play for 15 more minutes …” “How long will it take us to get there?” “There’s a half moon out tonight.”

  8. Number sense is developed at school. “Students develop efficient strategies for numerical calculation, recognize patterns, describe relationships, …” • Whole Numbers; • Addition and Subtraction; • Multiplication and Division • Fractions and Decimals; • Patterns (and Algebra). patterns and patterning place value What do teachers do in order to enable students to develop number sense? composing and decomposing numbers commutative, associative, distributive laws facts and fluency NSW Syllabus (2012)

  9. MA2-6NA uses mental and informal written strategies for multiplication and division “If I know … then I know …” • e.g. you draw 6 and 4 => 6x4=24 so, …. • 4x6=24 and, • 4+4+4+4+4+4=24 and, • 6+6+6+6=24 and, • (4+4+4)+(4+4+4)=(12)+(12)=24 and, • (6+6)+(6+6)=(12)+(12)=24 and (6+6)=(4+4+4)! and, • (2x6)+(2x6)=24 and, • 2x2x6=24 and ….! Teachers orchestrate experiences that allow students to construct understanding.

  10. MAe-6NA – groups, shares and counts collections of objects … x + 4 The Join Machine (for addition) The 4x Join Machine (for multiplication) Concrete -> Symbolic -> Abstract

  11. MA1-6NA - uses a range of mental strategies & concrete materials for mult. and division. 3 x 4 = ? Concrete -> Symbolic -> Abstract

  12. MA2-6NA - uses mental and informal written strategies for multiplication and division … 12 x 13 = ? Concrete -> Symbolic -> Abstract

  13. MA3-6NA - selects and applies appropriate strategies for multiplication and division … 26 x 48 = ? partial products algorithm standard algorithm Concrete -> Symbolic -> Abstract http://thinkmath.edc.org/resource/multiplication-and-division

  14. MA2-7NA - represents, models and compares commonly used fractions and decimals Etcetera … Volunteers? Gilbert, A. (2002). Teaching the Three R's: Through Movement Experiences

  15. 1 Locate the running time on a DVD. Convert this into hours and minutes. c) You arrive at your friend’s house on Saturday, 2PM, to watch the movie. What time should your mum pick you up? d) Pick some of your favourite DVDs. How many could you watch in 12 hours? Explain your thinking. e) Pick 6 different DVDs and create a timeline based on the year each one was made. 5 cm per Second Princess Bride The Castle Off the Map 1985 1990 1995 2000 2010 2015

  16. 2 Sit down and think about this question: What is your absolute favorite activity to play at the park? Circle one of the activities:1) Slides 2) Swings 3) Monkey bars 4) Climbing nets 5) Tunnels IIII II III I Now survey the students in your maths trail group about which of the five above activities is their favorite and record it using a tally. Now, in the space below, graph your findings. (Be sure to label your graph.) • Which is the most popular? • Which is the least popular? • How many people did you survey? • How many people liked the same activity as you?

  17. 2 What fraction of the people liked the slides best? Write down and draw as many ways as you can think of, to represent the amount of people who like the slide over the total amount of people you surveyed.

  18. 3 1) Estimate the length of the Flying Fox, then use the trundle wheel to check your guess. 2) Working in pairs, one person will run and another will keep time, race to the end of the Flying Fox. Document your times. 3) Now, working in pairs again, race on the Flying Fox. Compare the times. 4) Was there much difference from running to using the Flying Fox? Why would one be faster than the other? 5) Can two people on the Flying Fox go faster than one? Why or why not?

  19. 4 If the T stand (for the swings) is rolled 20 times, what position will it be in? T T T T T T T T T T

  20. MA2-9MG Measures, records, compares and estimates lengths, distances and perimeters in metres,cmsand mms, …. Grow Beasts! MA2-18SP Selects appropriate methods to collect data, & constructs, compares, interprets and evaluates data displays, including tables, picture graphs and column graphs. MA2-2WM (Problem Solving) Selects and uses appropriate mental or written strategies, or technology, to solve problems.

  21. How does a Grow Beast grow? Let’s take a look, shall we? http://growbeast.wikispaces.com Concrete -> Tabular -> Graphical -> Abstract (After AIMS “Model of Learning” http://www.aimsedu.org/resources/modelL.html)

  22. Self-contained SPED classroom grades 1-3 Ms. True set the stage for meaningful engagement => durable learning!

  23. http://roddyatuws.wikispaces.com Students are … • Making choices; • Making predictions; • Measuring; • Recording data; • Constructing charts; • Drawing conclusions; • And enjoying the process! In short, they are using maths in making sense of their experiences with a fun toy.

  24. Sample Resources • NLVM (NationalLibrary of VirtualManipulatives) • NCTM > Illuminations > Interactives • Arcademics • 101Questions http://RoddyAtUWS.wikispaces.com/

  25. 10 cm in 10 minutes => 1 cm/min Wow! … really?? = ~1.4 cm/minute! 2 highs and 2 lows/day => 6 hours between Each high and low …. 15 ft high and -1 ft low => 16 ft change 16 ft is ~5m = 500 cm and 6 hours is 6 x 60 = 360 minutes So 500 cm/360 min … and ….

  26. So …. For too many of our students, particularly those in the upper grades, maths is just a subject to be endured, or worse, avoided. With number sense, mathematics can be a stimulus to pay attention to the world all around us. It can enable us to pose questions and to look for answers that make sense. If we can help our students see mathematics as a way of making sense of the world, we have done them a service that can last a lifetime.

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