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MNPS Numeracy Coaches Ernestine Saville Brock Mathematics Coordinator Teaching Mathematics Through Problem Solving Slide 2 MNPS Vision Metropolitan Nashville Public Schools will provide every student with the foundation of knowledge, skills and character necessary to excel in higher education, work, and life. Slide 3 MNPS Professional Development Vision Metropolitan Nashville Public Schools provides all stakeholders quality professional development for adult learning that results in the growth of the whole child and the improvement of student learning. Slide 4 Norms I commit to beginning and ending on time turning my cell phone to vibrate respecting everyones opinion processing our learning actively participating and having fun learning together I commit to beginning and ending on time turning my cell phone to vibrate respecting everyones opinion processing our learning actively participating and having fun learning together Slide 5 Agenda Problem Solving Activity/Analysis Concerns Related to Teaching Through Problem Solving Problem Solving Activity Focus, Process, Effective Questioning Break Strategies Visualization Exit Ticket Slide 6 Parking Lot Slide 7 Five Easy Steps to a Balanced Math Program for Primary Grades by Larry Ainsworth and Jan Christinson Slide 8 Slide 9 Insert Balanced Math Model Slide Da y 15 minutes40 minutes5 minutes 1Math Review Mental Math Concept Lesson Menu Problem solving, manipulatives, small groups, centers Closure/ math journals 2Math Review Mental Math Concept Lesson Menu Problem solving, manipulatives, small groups, centers Closure 3Math Review Mental Math Concept Lesson Menu Problem solving, small groups, centers Closure 4Math Review Mental Math Concept Lesson Menu Problem solving, manipulatives, small groups, centers Closure 5Math Facts Practice/ Math Review Quiz Problem-based activities, centers, games, small groups 6Math Review Mental Math Concept Lesson Menu Problem solving, manipulatives, small groups, centers Closure 7Math Review Mental Math Concept Lesson Menu Problem solving, manipulatives, small groups, centers Closure 8Math Review Mental Math Concept Lesson Menu Problem solving, manipulatives, small groups, centers Closure 9Math Review Mental Math Concept Lesson Menu Problem solving, manipulatives, small groups, centers Closure 10Assessment/Math Review Quiz Assessment Slide 10 Problem-Solving Strategies Guess and check Act it out Work Backward Draw a picture Make a table, chart, or graph Make a list Write a number sentence Use logical reasoning Find a pattern Slide 11 Slide 12 Riddles with Color Tiles Use the color tiles on your table to solve the following riddle. I have 12 tiles. I used 3 colors. There are no red tiles. There are the same number of green and blue tiles. I have 4 yellow tiles. Slide 13 What did you notice? What standards?What strategies ? Are there different ways to solve the problem? How could students represent the problem? Slide 14 1 st Students will use their prior knowledge to construct concrete representations of math. 2 nd Students must represent their understanding in a reflective &/or symbolic form. 3 rd One or both forms will be a visual reminder for the understanding of the higher-thinking abstract. Concrete Representational Abstract Slide 15 http://www.coach4math.com/wp- content/uploads/2010/03/C-R-A-Model.pdf Slide 16 Slide 17 Slide 18 Slide 19 Slide 20 Slide 21 Slide 22 Slide 23 Slide 24 Two Digit by 1 Digit Multiplication Concrete Representational Abstract Slide 25 Concerns Related to Teaching Through Problem Solving Are young children really able to explore problems on their own and arrive at sensible solutions? Will students sacrifice basic skills if they are taught mathematics through problem solving? How can teachers learn to teach through problem solving? Slide 26 Are Young Children Really Able to Explore Problems on Their Own and Arrive at Sensible Solutions ? Research has indicated that students who used invented strategies before they learned standard algorithms demonstrated better knowledge of base-ten number concepts and were more successful in extending their knowledge to new situations than were students who initially learned standard algorithms. (Carpenter et al, 1998) Student invented strategies may be inefficient and students should be guided to develop more efficient strategies. However, the student invented strategy serves as a basis for the students understanding of the mathematical ideas and procedures. (Cai, Moyer, & Grochowski 1999) Slide 27 Will Students Sacrifice Basic Skills if they are taught mathematics through problem solving? Basic skills and high-order thinking skills in mathematics are important, but having basic math skills does not imply having higher-order thinking skills or vice versa. Research indicates that students learning mathematics through problem solving do at least as well as those students receiving traditional instruction on both basic computation and conceptual understanding. (Carpenter et al, 1998) Slide 28 How Can Teachers Learn to Teach through Problem Solving? Teachers success in teaching through problem solving is related to the encouragement and support they receive from their fellow teachers. (Stigler & Hiebert, 1999) One new role that teachers are asked to play in a classroom based on teaching through problem solving is selecting appropriate tasks. Slide 29 Break Slide 30 When selecting a particular problem for students to solve, we look for one that will allow them to demonstrate their ability to apply the math they are learning to a real-world problem or situation. ~ Five Easy Steps to a Balanced Math Program pg. 33 Selecting appropriate tasks Slide 31 Guiding Questions Does this problem promote application of the mathematical ideas currently being studied? Does this problem match the students current instructional level? Is this problem relevant, engaging, and accessible to all students? Does this problem require students to stretch their thinking? Does this problem involve more than one strand or standard of mathematics? Is there more than one way to solve the problem? Can the problem be extended or enriched? Does the teacher fully understand the mathematics of this problem? Slide 32 Teaching Students to Solve the Problem-Solving Task Once we select the problem that will become the Problem- Solving Task, we follow a specific instructional sequence to teach students how to mathematically solve an application problem and communicate orally and in writing the process they used. Our ultimate goal for our students is for them to be able to solve independently a two-step or multistep problem and to communicate verbally (kindergarten) and in writing (first through fourth) the process they used. Slide 33 Gradual Release of Responsibility Explicitly Teaching Problem Solving Whole Class Cooperative Groups/Teams Partner Independent Slide 34 Gradual Release of Responsibility https://mnpstube.mnps.org/watch_video.php?v=06ad360fe025e63 Slide 35 Slide 36 Planning and Planting Your Garden Suppose you were going to plant a rectangular garden that covered 24 square feet. The side lengths are whole numbers. What might it look like? You can use manipulatives to model your thinking. Draw all possible outcomes of your garden. Write and explain why you think you found them all. Slide 37 Getting to the Math Ask questions that focus on the process. Focus on what student did right and their ability to explain their thinking. Honor more than one way to solve the problem. What other questions could you ask? Allow students to evaluate their own work. Slide 38 Problem-Solving Task Write-Up Guide How do you want your students to show you what they know? R.A.P Restate, Answer, Prove Problem-Solving Recording Sheet Teacher-created Recording Sheets Slide 39 Slide 40 Slide 41 Slide 42 Slide 43 Slide 44 Slide 45 Slide 46 I get aIf I 0--- do nothing -or- just copy the prompt. 1--- write a number sentence that doesnt go along with the prompt. do not write an explanation. 2--- use numbers from the prompt with an operation included. write an explanation telling what I did. 3--- use numbers from the prompt. write number sentences that make sense, even if they are not the right answer. write a simple paragraph explaining what I did. 4--- correctly answer most of computation that I write. write a paragraph explaining what I did. It can have minor mistakes, but it still makes sense. Anyone should be able to understand how I solved the prompt. 5--- get the correct answer. make tables, diagrams, charts, etc. write a detailed paragraph telling step-by-step what I did to solve the prompt. My paragraph tells all my steps! 6--- get the correct answer. make more than one table, diagram, chart, etc. write a detailed paragraph telling step-by-step what I did to solve the problemand include what I was thinking as I solved it! (I did this because) tell what I did so well that I dont think my teacher could explain it better! Student Problem Solving Rubric Slide 47 Magic Squares Problem Solving Activity Slide 48 It is more important than learning key words. Many assessment items contain misleading key words or there are no words associated with the operations. Visualization is the key Slide 49 Bar Diagrams A new approach to solving word problems is to use bar diagrams as visual representations that show how quantities in a word problem are related. Seeing those relationships and connecting those to operation meanings enables one to select an appropriate operation for solving the problem. A diagram can serve to unpack the structure of a problem and lay the foundation for its solution. Randall Charles Slide 50 Model Drawing Example http://thesingaporemaths.com/P4math2f.swf Slide 51 Bar Diagram example Grace has 113 ducks. Susan has 45 more ducks than Grace. How many do they have in all? Carrie has 125 U.S. stamps. She has 3 times as many foreign stamps as U.S. stamps. How many stamps does she have altogether? Slide 52 Resources Step by Step Model Drawing by Char Forsten 8 Step Model Drawing Singapores Best Problem Solving Math Strategies by Bob Hogan and Char Forsten Writing Strategies for Mathematics by Tricia Brummer and Sarah Kartchner Clark. Marilyn Burns http://illuminations.nctm.org National Library of Virtual Manipulatives www.nlvm.usu.edu www.nlvm.usu.edu Figure This! Math Challenges for Families www.figurethis.org www.figurethis.org Slide 53 1. What is the most significant thing you learned today? 2. What component will you implement first and with what math standard? 3. What support do you need next as you implement Balanced Math? 1. What is the most significant thing you learned today? 2. What component will you implement first and with what math standard? 3. What support do you need next as you implement Balanced Math?