2014 Mathematics Institutes Grade Band K2 1 Making
2014 Mathematics Institutes Grade Band: K-2 1
Making Connections and Using Representations • The purpose of the 2014 Mathematics Institutes is to provide professional development focused on instruction that supports process goals for students in mathematics. • Emphasis will be on fostering students’ ability to make mathematical connections and use effective and appropriate representations in mathematics. 2
Agenda I. III. IV. Defining Representations and Connections Doing the Mathematical Task Looking at Student Work Facilitating the Use of Effective Representations and Connections V. Planning Mathematics Instruction VI. Closing 3
I. Defining Representations Rally Robin 1. 2. 3. Choose a partner Think: When you hear the term representations in mathematics, what does it mean to you? Take turns sharing your thoughts with your partner.
Mathematical Representations Students will represent and describe mathematical ideas, generalizations, and relationships with a variety of methods. Students will understand that representations of mathematical ideas are an essential part of learning, doing, and communicating mathematics. Students should move easily among different representations �graphical, numerical, algebraic, verbal, and physical �and recognize that representation is both a process and a product. Virginia Department of Education. (2009). Introduction Mathematics Standards of Learning for Virginia Public Schools 5
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“Representations are useful in all areas of mathematics because they help us develop, share, and preserve our mathematical thoughts. They help to portray, clarify, or extend a mathematical idea by focusing on its essential features. ” National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics. (p. 206). Reston, VA. 7
Defining Connections Table Discussion Why do we want students to make connections in mathematics? What is an example of a connection we want young students to make?
Mathematical Connections Students will relate concepts and procedures from different topics in mathematics to one another and see mathematics as an integrated field of study. Through the application of content and process skills, students will make connections between different areas of mathematics and between mathematics and other disciplines, especially science. Science and mathematics teachers and curriculum writers are encouraged to develop mathematics and science curricula that reinforce each other. Virginia Department of Education. (2009). Introduction Mathematics Standards of Learning for Virginia Public Schools 9
Planning for the Use of Representations Table Group discussion: What questions should be considered regarding representations and connections when planning for instruction? 10
II. A Mathematical Task Jo made two trays of cookies. One tray had 19 cookies on it. Together, both trays had 43 cookies. How many cookies were on the second tray? Use pictures, numbers and words to show you found the answer. • Work individually. • Solve this task in two different ways. • Be ready to share your ideas.
Mathematical Task • Share your solutions with your table group. – What representations did you use to solve the task and explain it to others? – What connections are evident? • Capture at least 3 different ways that someone in your group solved the task on chart paper.
Mathematical Task Gallery Walk • One person from group stays with group poster to explain why representations were chosen. • Participants ask questions and look for connections between the representations their group chose and the representations used on other charts.
Mathematical Content Large Group Discussion • What is the mathematical content of this task? What big ideas are students working with? • Where do we find this content in the Whole Number/ Number Sense Learning Progression?
Early Number Sense Read pages 107 – 108 (stop at Spatial Patterns) What are the early number relationships that move students from counting to reasoning?
Part-Whole “To conceptualize a number as being made up of two or more parts is the most important relationship that can be developed about numbers. ” Van de Walle, J. A. , Karp, K. S. , Lovin, L. H. & Bay-Williams, J. M. (2014). Teaching Student. Centered Mathematics: Developmentally Appropriate Instruction for Grades K-2 (2 nd ed. ). (Vol. II). (108). Pearson Education Inc.
Part-Whole as a Big Idea Turn and Talk with Shoulder Partner: Thinking about the mathematics that students learn as they go through elementary school and beyond, what connections/applications do you see to this idea of thinking about a number in terms of parts? Jot down a few ideas.
Part-Whole 6
Addition Fact Strategies – Near Doubles 3 + 4 3 + 1
Place Value 2, 134 2 thousands (2000) 1 hundred (100) 3 tens (30) 4 ones (4)
Multi-digit Addition and Subtraction How would you add these numbers mentally? 28 + 26
Multiplication Facts 4 x 7 14 2 x 7
Fractions ½ + ¾ Inside the ¾ is ½ plus an extra ¼
Algebra 2 x + 6 x 2(x + 3) x x x
Part-Whole Decomposing and recomposing numbers is a huge idea and it all starts in kindergarten! What are some additional part-whole connections you and your shoulder partner discussed?
Part–Whole: Stages in Thinking • Recognizing number as a quantity • Seeing and describing smaller parts inside a number, but not remembering them • Realizing and remembering all the different parts that make up a number • Decomposing numbers in flexible ways to facilitate computation Adapted from: Richardson, K. (2012). How Children Learn Number Concepts: - A Guide to the Critical Learning Phases. Bellingham, WA: Math Perspectives Teacher Development Center, .
III. Looking at Student Work With a partner(s), choose a set of student work K, 1, or 2. Be sure someone at each table looks at each collection. – What representations are evident in the student work? Can you find similarities and differences? – What does the student work tell us about their understanding? How do the representations give us clues to understanding? Note that K and 1 use the same problem, but with different numbers.
Looking at Student Work Do we see evidence of the different types of representation in student work?
Looking at Student Work • Do you see evidence that students are making connections within their own work? • How could you use the student work to help all students make connections? – Among representations – Among strategies – Among mathematical ideas
IV. Representations for Part-Whole Guiding Questions • What representations can be used to help students develop strong part-whole ideas? • What are the benefits and limitations of these representations?
Dot Representations • • Dot cards Dot plates Dominoes Dice
Subitizing • Perceptual subitizing – recognizing small quantities without counting • Conceptual subitizing – recognizing patterns and groups to help determine a quantity
Subitizing “Children use counting and patterning abilities to develop conceptual subitizing. This more advanced ability to group and quantify sets quickly in turn supports their development of number sense and arithmetic abilities. ” Clements, Douglas H. 1999. “Subitizing: What Is It? Why Teach It? ” Teaching Children Mathematics 5 (7); 401
Subitizing Questions to ask children when using Dot Representations 1. How many dots did you see? 2. How did you see it? 3. What did the pattern look like? 4. Did you see any parts that you know?
Subitizing - Let’s Try Some Examples
Activities for Using Dot Representations • Activity 8. 8 Find the Same Amount • Activity 8. 9 Learning Patterns • Activity 8. 10 Dot Plate Flash What are the benefits and limitations of using dot representations?
Five-Frames and Ten-Frames How do these representations help anchor numbers to 5 and 10?
Ten Frames – Number Talk VIDEO Kindergarten Class How is the ten-frame helping students? What questions is the teacher asking to help students focus on parts within a quantity?
Activities for Using Five- and Ten-Frames • Activity 8. 13 Five-Frame Tell-About • Activity 8. 14 Number Medley • Activity 8. 15 Ten-Frame Flash Cards What are the benefits and limitations of five- and ten -frames?
Rekenreks How is this representation similar to the other representations explored? How is it different?
Rekenreks Let’s make a rekenrek! You need: • A piece of cardboard • Two pipe cleaners • 10 red beads • 10 black beads
Rekenrek VIDEO Click Here How is this number talk different from the previous number talk? How does the teacher honor and value various strategies while encouraging part-whole thinking? What are the benefits and limitations of rekenreks?
Rekenrek Your turn!!!! 1. Choose one person to be the teacher. Everyone else turns back to screen. 2. Teacher uses rekenrek to show the combinations below. After each combination, ask… How many beads did you see? How did you know there were ___ beads? 6 on the top, 6 on the bottom 6 on the top, 7 on the bottom 6 on the top, 5 on the bottom
Additional Physical & Visual Representations Activity 8. 16 Activity 8. 17 Activity 8. 18 Activity 8. 19 Activity 8. 20 Build It in Parts Covered Parts Missing-Part Cards I Wish I Had Number Sandwiches
Part-Whole Mat Use the mat and some counters to model the problems. 1. Sam had 6 red balls and 5 blue balls. How many balls did Sam have? 2. Sam had 12 stickers. Four of his stickers were torn. How many were not torn? 3. Sam had 14 goldfish. He gave some to Mary. Now he has 7. How many goldfish did he give to Mary?
Part- Whole Mat How does modeling problems using a Part-Whole mat help students? What are the benefits and limitations of part-partwhole mats?
Number Lines Number Path – discrete counting model Number Line – length or distance model
Challenges or Limitations with Number Lines Young children try to use a number line as a counting model – – They count the numbers or tic marks, not the segments – They don’t start from 0, because they typically begin counting from one
Can you identify the misconception?
Number Lines • Number Lines are not recommended as a representation at K and 1 because of the conceptual difficulties they present • Introduce at grade 2 with an emphasis on ‘hops’ or lengths, but be cognizant of the difficulties young children have with number lines. Fuson, K. , Clements, D. , & Beckman, S. (2010). Focus in Kindergarten: Teaching with Curriculum Focal Points. Reston, VA: National Council of Teachers of Mathematics.
Open Number Lines Read page 135. What is an open number line? How do you introduce open number lines? Going back to our student work, which students used open number lines while solving the problem? What are the benefits and limitations of number lines?
Representations “Representations do not “show” the mathematics to the students. Rather the students need to work with representations extensively in many contexts as well as move between representations in order to understand how they can use a representation to model mathematical ideas and relationships. ” National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics. (p. 208). Reston, VA.
Role of the Teacher • Create a learning environment that encourages and supports the use of multiple representations • Model the use of a variety of representations • Orchestrate discussions where students share their representations and thinking • Support students in making connections among multiple representations, to other math content and to real world contexts Van de Walle, J. A. , Karp, K. S. , Lovin, L. H. & Bay-Williams, J. M. (2014). Teaching Student-Centered Mathematics: Developmentally Appropriate Instruction for Grades 3 -5 (2 nd ed. ). (Vol. II). Pearson. 53
Role of the Student • Create and use representations to organize, record, and communicate mathematical ideas • Select, apply, and translate among mathematical representations to solve problems • Use representations to model and interpret physical, social, and mathematical phenomena Van de Walle, J. A. , Karp, K. S. , Lovin, L. H. & Bay-Williams, J. M. (2013). Teaching Student-Centered Mathematics: Developmentally Appropriate Instruction for Grades 3 -5 (2 nd ed. ). (Vol. II). Pearson. 54
Students must be actively engaged in developing, interpreting, and critiquing a variety of representations. This type of work will lead to better understanding and effective, appropriate use of representation as a mathematical tool. National Council of Teachers of Mathematics. (2000) Principles and Standards for School Mathematics. (p. 206). Reston, VA. 55
"Students representational competence can be developed through instruction. Marshall, Superfine, and Canty (2010, p. 40) suggest three specific strategies: 1. Encourage purposeful selection of representations. 2. Engage in dialogue about explicit connections among representations. 3. Alternate the direction of the connections made among representations. " National Council for Teachers of Mathematics. (2014). Principles to Actions. (p. 26). Reston, VA 56
V. Planning for Instruction Let’s revisit our list of questions regarding representations and connections when planning. Do you have any additions or revisions?
Planning Mathematics Instruction: Essential Questions • Work with a partner. • Highlight questions you already think about when planning. • Which questions are new for you to think about? • Are the questions on our list reflected in this document?
Planning a Lesson • Revisit the student work for student 1 - D. • Assuming that student 1 -D’s work is representative of your class, plan a short lesson that will continue to move the students forward in their thinking. Consider the representations that you will use and the connections you want to have students make.
Representation should be an important element of lesson planning. Teachers must ask themselves, “What models or materials (representations) will help convey the mathematical focus of today’s lesson? ” - Skip Fennell, F (Skip). (2006). Representation—Show Me the Math! NCTM News Bulletin. September. Reston, VA: NCTM 60
VI. Closing 3 Name three representations you will use to develop part-whole thinking. 2 Describe two ways that you will help your students make connections between representations. 1 Identify one key question that you will incorporate into your planning process to focus on representations and connections.
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