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Using Visualization to Develop Children's Number Sense and Problem Solving Skills in Grades K-3 Mathematics (Part 2) Lou. Ann Lovin, Ph. D. Mathematics Education James Madison University

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 2

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 3

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) Discussions about why particular representations might be more beneficial to help think through a given problem or communicate ideas. Graphic depictions of multiple problems and multiple solutions. (Gersten & Clarke, NCTM Research Brief) Lovin NESA Spring 2012 4

Supporting Students Discuss the differences between pictures and diagrams. Ask students to Explain how the diagram represents various components Katie Natalie Sara Kevin of the problem. Emphasize the importance of precision in the diagram. 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 5

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 6

Summary of A Common “Approach” for Learners to Solve Word Problems Randomly combine numbers without trying to make sense of the problem. Lovin NESA Spring 2012 7

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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. There are 7 boys and 21 girls in a class. How many more girls than boys are there? Wendy has 3 cards. Her friend gives her 8 more cards. How many cards does Wendy have now? Lovin NESA Spring 2012 10

Real problems do not have key words! Lovin NESA Spring 2012 11

Teaching Mathematical Concepts and Skills through Word Problems Contextual (Word) Problems Introduce procedures and concepts using contextual problems (e. g. , subtraction; multiplication). Makes learning more concrete by presenting abstract ideas in a familiar context. AVOIDs the sole reliance on key words. Lovin NESA Spring 2012 12

Visual and Graphic Depictions of Word Problems Quantitative Analysis Visual models (like Singapore Math, Vande. Walle) Helps children to get past the words by visualizing and illustrating word problems with simple diagrams. Emphasis is on modeling the quantities and their relationships. Difference between pictures and diagrams. Lovin NESA Spring 2012 13

Visual and Graphic Depictions of Problems Ben has 5 cats and his cousin, Jerry, has 3 cats. How many cats do they have together? 5 Ben Jerry How would you write this computation as an equation? 3 Jerry has 3 cats. Ben has 5 more cats than his cousin Jerry. How many cats does Ben have? Lovin NESA Spring 2012 14

Visual and Graphic Depictions of Problems Jerry has 3 cats. Ben has 5 more cats than his cousin Jerry. How many cats does Ben have? Ben 8 5 Jerry 3 How would you write this computation as an equation? Lovin NESA Spring 2012 15

Visual and Graphic Depictions of Problems Meilin saved \$184. She saved \$63 more than Betty. How much did Betty save? \$184 Meilin Betty ? \$63 How would you write this computation? (Primary Mathematics volume 3 A, page 21, problem 7. ) Lovin NESA Spring 2012 16

Solve these problems: Jacob had 8 cookies. He ate 3 of them. How many cookies does he have now? Jacob has 3 dollars to buy cookies. How many more dollars does he need to earn to have 8 dollars? Nathan has 3 dollars. Jacob has 8 dollars. How many more dollars does Jacob have than Nathan? How did you find your answer?

Perspectives Most adults think 8 – 3 = 5, because it’s the most efficient way to solve these tasks. Young children see these as 3 different problems and use the action or situation in the problem to solve it – so they solve each of these using different strategies. (Unfortunately, too often children are told to subtract – because that’s how we interpret the problem. )

A first grader… Jacob has 8 cookies. He ate 3 of them. How many cookies does he have now? X X X 1 2 3 4 5 Jacob has 3 dollars. How many more dollars does he need to earn to have 8 dollars? 1 2 3 1 4 2 5 3 6 4 7 5 8 Nathan has 3 dollars. Jacob has 8 dollars. How many more dollars does Jacob have than Nathan? 1 2 3 4 5

Rekenrek While students can use the rekrenrek to generate different strategies for solving basic facts, they can also use it to solve story problems such as the ones below. Visualization is key to helping find a solution. Together, Claudia and Robert have 7 apples. Claudia has one more apple than Robert. How many apples do Claudia and Robert have? Claudia had 4 apples. Robert gave her some more. Now she has 7 apples. How many did Robert give her? Lovin NESA Spring 2012 20

Town Sports ordered 99 scooters. They have received 45 scooters. How many scooters is Town Sports waiting on? 45 ? 99 99 – 45 = ______ Lovin NESA Spring 2012 OR 45 + ____ = 99 21

Joining Physical Action Taiwan’s Cookie Problem Separate Part-part Whole No Physical Action Lovin NESA Spring 2012 Comparing 22

Start Unknowns Bear Dog had some cookies. Taiwan gave him 8 more cookies. Then he had 13 cookies. How many cookies did Bear Dog have before Taiwan gave him any? ? 8 13 Lovin NESA Spring 2012 23

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 24

Use Real Contexts – Grocery Store (Multiplication) Lovin NESA Spring 2012 25

Multiplication Context – Grocery Store How many plums does the grocer have on display? plums Lovin NESA Spring 2012 26

Multiplication - Context – Grocery Store apples lemons Groups of 5 or less subtly suggest skip counting (subitizing). Lovin NESA Spring 2012 tomatoes 27

How many muffins does the baker have? Lovin NESA Spring 2012 28

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 29

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 30

Students’ Work Jackie Lovin NESA Spring 2012 Edward 31

Students’ Work Sam Wendy Amanda Lovin NESA Spring 2012 32

Area Model Grid Paper Show a 2 x 3 rectangle Show a 4 x 5 rectangle Lovin NESA Spring 2012 33

Open Array 12 5 Lovin NESA Spring 2012 34

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 35

2 x 30 How do you think about determining what 2 x 30 is? What do we mean by “adding a zero”? Video 1 (: 19) (1. 1. 5) and Video 2 (3: 59) (1. 1. 6) Multiplication mini-lessons Lovin NESA Spring 2012 36

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 37

Take Aways Help children create diagrams to represent the quantities and their relationships in problems. Children can solve the same problem using different operations. Take advantage of children’s tendencies to subitize (rekenreks and arrays) Use real world contexts to introduce arrays (multiplication) Lovin NESA Spring 2012 38

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 39

References Carpenter, Fennema, Franke, Levi, Empson. (1999). Children’s Mathematics: Cognitively Guided Instruction. Heinemann: Portsmouth, NH. Diesmann, C. , & English, L. (2001). Promoting the use of diagrams as tools for thinking. In A. Cuoco & F. Curcio (Eds. ), The Roles of Representation in School Mathematics, pp. 77 -89. Reston, VA: NCTM. Dolk, M. , Liu, N. , & Fosnot, C. (2008). The Double-Decker Bus: Early Addition and Subtraction. Portsmouth, NH: Heinemann. Fosnot, C. & Dolk, M. (2001). Young Mathematicians at Work: Constructing Number Sense, Addition, and Subtraction. Portsmouth, NH: Heinneman. Fosnot, C. (2008). Bunk Beds and Apple Boxes: Early Number Sense. Portsmouth, NH: Heinemann. Fostnot, C. & Cameron, A. (2007). Games for Early Number Sense. Portsmouth, NH: Heinneman. Gersten, R. & Clarke, B. (2007). Research Brief: Effective Strategies for Teaching Students with Difficulties in Mathematics. NCTM: Reston, VA. Ministry of Education Singapore. (2009). The Singapore Model Method. Panpac Education: Singapore. NCTM (2000). Principles and Standards of School Mathematics. NCTM: Reston, VA. Parrish, S. (2010). Number Talks: Helping Children Build Mental Math and Computation Strategies. Math Solutions: Sausalito, CA. Storeygard, J. (2009). My Kids Can: Making Math Accessible to All Learners. Heinemann: Portsmouth, NH. Wright, R. , Martland. J, Stafford, A. , & Stanger, G. (2006). Teaching Number: Advancing Children’s Skills and Strategies. London: Sage. Using the Rekenrek as a Visual Model for Strategic Reasoning in Mathematics by Barbara Blanke (www. mathlearningcenter. org/media/Rekenrek_0308. pdf) Vande. Walle, J. & Lovin, L. (2005). Teaching Student-Centered Mathematics: Grades K-3. Boston: Pearson. Lovin NESA Spring 2012 40

Cognitively Guided Instruction Strategies Direct Modeling Strategies Counting Strategies Derived Number Facts Known Number Facts (as in recall) return Lovin NESA Spring 2012 41