Using Visualization to Extend Students’ Number Sense and Problem Solving Skills in Grades 4 -6 Mathematics (Part 1) Lou. Ann Lovin, Ph. D. Mathematics Education James Madison University
Number Sense What is number sense? Turn to a neighbor and share your thoughts. Lovin NESA Spring 2012 2
Number Sense “…good intuition about numbers and their relationships. ” It develops gradually as a result of exploring numbers, visualizing them in a variety of contexts, and relating them in ways that are not limited by traditional algorithms” (Howden, 1989). Developing number sense “Two hallmarks of number sense are flexible strategy through use and the ability to look at a computation problem solving and visualization. and play with the numbers to solve with an efficient strategy” (Cameron, Hersch, Fosnot, 2004, p. 5). Flexibility in thinking about numbers and their relationships. Lovin NESA Spring 2012 3
A picture is worth a thousand words…. Lovin NESA Spring 2012 4
Do you see what I see? An old man’s face or two lovers kissing? Cat or mouse? Not everyone sees what you may see. Lovin NESA Spring 2012 A face or an Eskimo? 5
What do you see? Everyone does not necessarily hear/see/interpret experiences the way you do. www. couriermail. com. au/lifestyle/left-brain-v-right-brain-test/story. Lovin NESA Spring 2012 e 6 frer 4 f-1111114604318 6
Manipulatives…Hands-On… Concrete…Visual Lovin NESA Spring 2012 7
T: Is four-eighths greater than or less than four- fourths? J: (thinking to himself) Now that’s a silly question. Four-eighths has to be more because eight is more than four. (He looks at the student, L, next to him who has drawn the following picture. ) Yup. That’s what I was thinking. Ball, D. L. (1992). Magical hopes: Manipulatives and the reform of mathematics education (Adobe PDF). American Educator, 16(2), 14 -18, 46 -47. Lovin NESA Spring 2012 8
But because he knows he was supposed to show his answer in terms of fraction bars, J lines up two fraction bars and is surprised by the result: J: (He wonders) Four fourths is more? T: Four fourths means the whole thing is shaded in. J: (Thinks) This is what I have in front of me. But it doesn’t quite make sense, because the pieces of one bar are much bigger than the pieces of the other one. So, what’s wrong with L’s drawing? Ball, D. L. (1992). Magical hopes: Manipulatives and the reform of mathematics education (Adobe PDF). American Educator, 16(2), 14 -18, 46 -47. Lovin NESA Spring 2012 9
T: Which is more – three thirds or five fifths? J: (Moves two fraction bars in front of him and sees that both have all the pieces shaded. ) J: (Thinks) Five fifths is more, though, because there are more pieces. This student is struggling to figure out what he should pay attention to about the fraction models: is it the number of pieces that are shaded? The size of the pieces that are shaded? How much of the bar is shaded? The length of the bar itself? He’s not “seeing” what the teacher wants him to “see. ” Ball, D. L. (1992). Magical hopes: Manipulatives and the reform of mathematics education (Adobe PDF). American Educator, 16(2), 14 -18, 46 -47. Lovin NESA Spring 2012 10
Base Ten Pieces and Number 4 3 2 1 10 20 30 40 Adult’s perspective: 31 Lovin NESA Spring 2012 11
What quantity does this “show”? ? Is it 4? Could it be 2/3? (set model for fractions) Lovin NESA Spring 2012 12
Manipulatives are Thinker Toys, Communicators n Hands-on AND minds-on The math is not “in” the manipulative. The math is constructed in the learner’s head and imposed on the manipulative/model. What do you see? n What do your students see? n Lovin NESA Spring 2012 13
The Doubting Teacher Do they “see” what I “see”? How do I know? Lovin NESA Spring 2012 14
Visualization strategies to make significant ideas explicit Color Coding ⅓ Visual Cuing Area All Over Perimeter Highlighting (talking about, pointing out) significant ideas in students’ work. 48 + 36 = ? Lovin NESA Spring 2012 48 + 36 70 +14 84 15
Teaching Number Sense through Problem Solving and Visualization Contextual (Word) Problems and Visualization Emphasis on modeling the quantities and their relationships What are the purposes of word problems? Why do we have students work on word problems? (quantitative analysis) Helps students to get past the words by visualizing and illustrating word problems with simple diagrams. Emphasizes that mathematics can make sense Develops students’ reasoning and understanding Great formative assessment tool Lovin NESA Spring 2012 16
A Student’s Guide to Problem Solving Rule 1 If at all possible, avoid reading the problem. Reading the problem only consumes time and causes confusion. Rule 2 Extract the numbers from the problem in the order they appear. Watch for numbers written as words. Rule 3 If there are three or more numbers, add them. Rule 4 If there are only 2 numbers about the same size, subtract them. Rule 5 If there are only two numbers and one is much smaller than the other, divide them if it comes out even -- otherwise multiply. Rule 6 If the problem seems to require a formula, choose one with enough letters to use all the numbers. Rule 7 If rules 1 -6 don't work, make one last desperate attempt. Take the numbers and perform about two pages of random operations. Circle several answers just in case one happens to be right. You might get some partial credit for trying hard. Lovin NESA Spring 2012 17
Solving Word Problems: A Common “Approach” for Learners Randomly combining numbers without trying to make sense of the problem. Lovin NESA Spring 2012 18
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Key Words This strategy is useful as a rough guide but limited because key words don't help students understand the problem situation (i. e. what is happening in the problem). Key words can also be misleading because the same word may mean different things in different situations. Wendy has 3 cards. Her friend gives her 8 more cards. How many cards does Wendy have now? There are 7 boys and 21 girls in a class. How many more girls than boys are there? Lovin NESA Spring 2012 21
Real problems do not have key words! Lovin NESA Spring 2012 22
Teaching Number Sense through Problem Solving and Visualization Contextual (Word) Problems and Visualization Emphasis on modeling the quantities and their relationships (quantitative analysis) Helps students to get past the words by visualizing and illustrating word problems with simple diagrams. Emphasizes that mathematics can make sense Develops students’ reasoning and understanding Great formative assessment tool AVOIDs the sole reliance on key words. Lovin NESA Spring 2012 23
The Dog Problem A big dog weighs five times as much as a little dog. The little dog weighs 2/3 as much as a medium-sized dog. The medium-sized dog weighs 9 pounds more than the little dog. How much does the big dog weigh?
A big dog weighs five times as much as a little dog. The little dog weighs 2/3 as much as a medium-sized dog. The medium-sized dog weighs 9 pounds more than the little dog. How much does the big dog weigh? Let x = weight of medium dog. Then weight of little dog = 2/3 x And weight of big dog = 5(2/3 x) x = 9 + 2/3 x (med = 9 + little) 1/3 x = 9 x = 27 pounds 2/3 x = 18 pounds (little dog) 5(2/3 x) = 5(18) = 90 pounds (big dog)
A big dog weighs five times as much as a little dog. The little dog weighs 2/3 as much as a medium-sized dog. The medium-sized dog weighs 9 pounds more than the little dog. How much does the big dog weigh? 9 9 9 weight of medium dog 9 9 18 weight of little dog 18 18 18 weight of big dog 5 x 18 = 90 pounds 18
A big dog weighs five times as much as a little dog. The little dog weighs 2/3 as much as a medium-sized dog. The medium-sized dog weighs 9 pounds more than the little dog. How much does the big dog weigh? 9 9 x = weight of medium dog 9 x 9 9 2/3 x 18 2/3 x = weight of little dog So…. how do you solve this problem from here? 18 18 5 (2/3 x) 5(2/3 x) = weight of big dog
The Cookie Problem Kevin ate half a bunch of cookies. Sara ate one-third of what was left. Then Natalie ate one-fourth of what was left. Then Katie ate one cookie. Two cookies were left. How many cookies were there to begin with? Lovin NESA Spring 2012 28
Different visual depictions of problem solutions for the Cookie Problem: Sara Sol 1 Kevin Natalie Katie Sol 2 Sol 3 2 Lovin NESA Spring 2012 Katie Natalie Sara Kevin 29
Mapping one visual depiction of solution for the Cookie Problem to algebraic solution: Sara Sol 1 Kevin Natalie ½x ⅓(½x) Katie 1 ¼(⅔(½x)) 2 x Sol 4 ½x + ⅓(½x) + ¼(⅔(½x)) + 1 + 2 = x Lovin NESA Spring 2012 30
Visual and Graphic Depictions of Problems Research suggests…. . It is not whether teachers use visual/graphic depictions, it is how they are using them that makes a difference in students’ understanding. Students using their own graphic depictions and receiving feedback/guidance from the teacher (during class and on mathematical write ups) Graphic depictions of multiple problems and multiple solutions. Discussions about why particular representations might be more beneficial to help think through a given problem or communicate ideas. (Gersten & Clarke, NCTM Research Brief) Lovin NESA Spring 2012 31
Supporting Students Discuss the differences between pictures and diagrams. Ask students to Explain how the diagram represents various components little medium big of the problem. Emphasize the importance of precision in the diagram (labeling, proportionality) Discuss their diagrams with one another to highlight the similarities and differences in various diagrams that may represent the same problem. Discuss which diagrams are most appropriate for particular kinds of problems. Lovin NESA Spring 2012 32
Visual and Graphic Depictions of Problems Singapore Math Meilin saved $184. She saved $63 more than Betty. How much did Betty save? $184 Meilin Betty ? $63 184 – 63 = ? Singapore Math, Primary Mathematics 5 A Lovin NESA Spring 2012 33
Visual and Graphic Depictions of Problems There are 3 times as many boys as girls on the bus. If there are 24 more boys than girls, how many children are there altogether? girls 12 24 boys 12 12 12 4 x 12 = 48 children Singapore Math, Primary Mathematics 5 A Lovin NESA Spring 2012 x = # of girls 3 x = x + 24 2 x = 24 x = 12 34
Contextual (Word) Problems Use to introduce procedures and concepts (e. g. , multiplication, division). Makes learning more concrete by presenting abstract ideas in a familiar context. Emphasizes that mathematics can make sense. Great formative assessment tool. Lovin NESA Spring 2012 35
Multiplication A typical approach is to use arrays or the area model to represent multiplication. 4 3× 4=12 Why? 3 Lovin NESA Spring 2012 36
Use Real Contexts – Grocery Store (Multiplication) Lovin NESA Spring 2012 37
Multiplication Context – Grocery Store How many plums does the grocer have on display? plums Lovin NESA Spring 2012 38
Multiplication - Context – Grocery Store apples lemons Groups of 5 or less subtly suggest skip counting (subitizing). Lovin NESA Spring 2012 tomatoes 39
How many muffins does the baker have? Lovin NESA Spring 2012 40
Other questions How many muffins did the baker have when all the trays were filled? How many muffins has the baker sold? What relationships can you see between the different trays? Lovin NESA Spring 2012 41
Video: Students Using Baker’s Tray (4: 30) What are the strategies and big ideas they are using and/or developing How does the context and visual support the students’ mathematical work? How does the teacher highlight students’ significant ideas? Video 1. 1. 3 from Landscape of Learning Multiplication mini-lessons (grades 3 -5) Lovin NESA Spring 2012 42
Students’ Work Jackie Edward Counted by ones Lovin NESA Spring 2012 Skip counted by twos 43
Wendy Students’ Work Sam Skip counted by 4. Used relationships between the trays. Saw the middle and last tray were the same as the first. Amanda Decomposed larger amounts and doubled: 8 + 8 = 16; 16 + 4 = 36 Lovin NESA Spring 2012 Used relationships between the trays. Saw the right hand tray has 20, so the middle tray has 4 less or 16. 44
Area/Array Model Progression Ar y n pe O rra a ea m od el u sin g g r id pa p er Context (muffin tray, sheet of stamps, fruit tray) Lovin NESA Spring 2012 45
4 x 39 How could you solve this? (Can you find a couple of ways? ) Video (5: 02) (1. 1. 2) Multiplication mini-lessons Lovin NESA Spring 2012 46
Number Sense “…good intuition about numbers and their relationships. ” It develops gradually as a result of exploring numbers, visualizing them in a variety of contexts, and relating them in ways that are not limited by traditional algorithms” (Howden, 1989). “Two hallmarks of number sense are flexible strategy use and the ability to look at a computation problem and play with the numbers to solve with an efficient strategy” (Cameron, Hersch, Fosnot, 2004, p. 5). Flexibility in thinking about numbers and their relationships. Lovin NESA Spring 2012 47
Take a minute and write down two things you are thinking about from this morning’s session. Share with a neighbor. Lovin NESA Spring 2012 48
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