Computational Fluency Connections to MPS Math Programs & Foundational Experiences for Grouping and Ungrouping Math Alliance April 13, 2010 Beth Schefelker and DeAnn Huinker.
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Computational Fluency Connections to MPS Math Programs & Foundational Experiences for Grouping and Ungrouping Math Alliance April 13, 2010 Beth Schefelker and DeAnn Huinker Computation Strategies from Students Addition Strategies • • • • Subtraction Strategies • Subtract each place • Add on the other number in parts Subtract the number in parts • Use a nice number and compensate Add up from the subtracted number • Change to an easier equivalent problem Use a nice number then compensate • Change to an easier equivalent problem Add each place from left to right Homework: Subtraction Strategies 2–3 Minute Table Discussion Reflect on how the information in the article helped you think further about fluency with subtraction. Reflect on your own growing fluency with addition and subtraction strategies. How do MPS Math Programs promote computational fluency? WALT We Are Learning To… Connect computational fluency strategies for addition and subtraction to current MPS textbook programs. We will know we are successful when… We are able to apply and explain the different strategies found in the programs, and discuss their value for student learning. Read, Try, Compare, Discuss Split your table into two groups. One group will read about algorithms in Everyday Math and the other will read HM Math Expressions. As you read, make note of these ideas: Algorithms discussed Representations suggested Vocabulary used to develop the ideas Textbook Programs: Algorithms, Representations, & Vocabulary 1. Discuss these features with your small “math program” group. Use sticky notes to write the important examples in each area. 2. Discuss as a table group, with each math program group presenting to each other. Use the sticky notes to help explain the nuances of each program. Add examples of the algorithms to help develop understanding. Reflecting on your learning… How are the programs similar? different? In what ways would these strategies support struggling learners? How do they support our district learning targets and descriptors? Mathematically Desirable and Accessible Methods for Computation These more accessible methods fit students’ thinking better, so they are easier for students to understand and to explain. Most are easier to do procedurally and are less prone to errors than are the Current Common methods. --Fuson (2007, p. 77) What might be some advantages of New Groups Below method? 189 + 157 1 1 346 New group is near the digit you record in the answer space, so it is easier to see the partial sum more easily. You add the numbers you see in the problem and have less to hold “in your mind.” Students say the digit above changes the problem; again must hold more in one’s mind. Students confuse the digits above for addition with the crossing out of top numbers for subtraction, and later for multiplication. --Fuson (2007, p. 81, 82) Try It: 576 + 238= ? Use the “New Groups Below” method to solve the problem. Consider how to explain this approach to students with language that reflects the quantities and not merely the digits. A volunteer in your group talks through the process using “quantity” language. Learning Computation Strategies Concrete & Slow Informal Methods ? Jump to the Standard Algorithm Students do not jump from “Concrete and Slow Informal” methods to rote formal Current Common methods as in traditional teaching but to methods they can relate to visual supports and come to explain as well as carry out. (Fuson, 2007, p. 73) Fluency with Computation Strategies Fluency includes being able to explain and relate computational methods to a visual or situational support. --Fuson (2007, p. 77) Visual and Verbal Supports: Importance of Math Drawings & Explanations It is important to link math drawings or other visual support to the numeric method for each step. It is such tight linking that enables the meanings for the visual or contextual supports to become attached to the notations and thus to take on those meanings. We found that moving rapidly to having students make math drawings with their solution methods was extremely powerful in supporting everyone. --Fuson (2007, p. 79, 83) Visual and Verbal Supports Math Drawings Explain with “quantity” language Students (and teachers) must use quantity language (hundreds, tens, ones) when adding tens or hundreds…. This helps the numerical methods to take on these quantity meanings, which will remain when students no longer need to make the drawings. --Fuson (2007, p. 83) Show All Totals or Partial Sums: 189 + 157 = ? Try It: 247+ 155= ? Make a step-by-step proof drawing using the “Show All Totals” or “Partial-Sums” method. Include both a math drawing and the numeric notation. A new volunteer in your group talks through the process using “quantity” language. Ungroup First or Trade-First Subtraction What are the key ideas in using this method? What might be some advantages of using this method with students? Try It: 432 – 276 = ? Make a step-by-step proof drawing using the “Ungroup First” or “Trade-First” method. Include both a math drawing and the numeric notation. A new volunteer in your group talks through the process using “quantity” language. Student 4: “Dariah’s Method” What is this student thinking? Try It: 64 – 37 = ? Solve this problem using the same reasoning as in “Dariah’s Method.” Include both a math drawing and the numeric notation. A new volunteer talks through the reasoning using quantity language. Base-ten Number System: Place Value Learning about whole number computation must be closely linked to learning about the base-ten number system The heart of this work is relating the written numeral to the quantity and to how that quantity is composed and can be decomposed. Teacher Note, Computational Fluency and Place Value, Investigations Grade K-5. TERC, 2007 Base Ten Blocks Hundred (Flat) Ten (Stick) One (Unit) The Place Value Mat Hundreds Tens Ones Race to 100 Two teams of two players each. Goal: Get enough Tens and Ones to trade for a Flat worth 100. One team rolls two number cubes, finds the sum of the numbers, represent it with Ones, then puts it on their place value mat. When a team gets 10 Ones or more, trade 10 Ones for 1Ten. Teams take turns rolling, finding the sum, putting Ones on their mats, and trading Ones for Tens. As soon as a team gets blocks worth 100 or more, they make a trade for 1 Hundred (Flat). The first team to do this wins. Be ready to talk about what you did to get a Hundred (Flat). Teacher’s Role: Ask Key Questions What number is on your mat? How far are you from 30? 50? (multiple of 10) How far are you from 100? What are you hoping to roll next and why? What math are you learning from this? Go for Broke Two teams of two players each. Start with 1 hundred (flat), 2 tens (sticks/rods), 5 ones (units) on your mat. One team rolls two number cubes, finds the sum, and removes that amount from their mat, “ungrouping” as needed. The first team to “be broke” (has no more blocks on their place value mat) wins. Teacher’s Role: Ask Key Questions What number is on your mat? How much would you have on your mat if you rolled 10? Rolled 5? How far are you from 80? 50? (multiple of 10) What are you hoping to roll next and why? What math are you learning from this? Computational Fluency Flexibility Comfortable with more than one approach. Chooses strategy appropriate for the numbers. Efficiency Easily carries out the strategy, uses intermediate results. Doesn’t get bogged down in too many steps or lose track of the logic of the strategy. Accuracy Can judge the reasonableness of results. Has a clear way to record and keep track. Concerned about double-checking results. Source: Russell, S.J. (2000). Developing computational fluency with whole numbers. Teaching Children Mathematics, 7, 154 - 158. Thinking Back on Your Learning How has your definition of computational fluency changed from when we first conversation of class? Thinking About Your Work… What is one goal you have for yourself or a student related to improving computational fluency? Closing Thought . . . . Computation should “make sense.” Students should flexibly choose computation methods appropriate to the numbers and context. Traditional algorithms are “digit-oriented” and “rigid” and rely on memorizing rules without reasons, and can lead to common errors. Alternative strategies are “number-oriented” and “flexible” and rely on making sense of working with numbers, and build confidence in students. Homework (a) Read “Math for Teachers” Textbook, section 4.2 (p. 139-146). (b) Problems for Section 4.2 (p. 148-149): Complete Problems 1, 2, 3, & 12 (c) Problems for Section 4.2 (p. 149): Problem #14. Complete parts (a) and (b). This is the Equal Additions method for subtraction commonly used in many countries in Europe and Latin America, as well as Australia.