Teaching Mathematics Through Problem Solving

Teaching Mathematics Through Problem Solving MNPS Numeracy Coaches Ernestine Saville Brock Mathematics Coordinator

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.

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.

Norms I commit to… • beginning and ending on time • turning my cell phone to vibrate • respecting everyone’s opinion • processing our learning • actively participating and having fun learning together

Agenda • Problem Solving Activity/Analysis • Concerns Related to Teaching Through Problem Solving • Problem Solving Activity – Focus, Process, Effective Questioning • Break • Strategies • Visualization • Exit Ticket

Parking Lot

Five Easy Steps to a Balanced Math Program for Primary Grades by Larry Ainsworth and Jan Christinson

Elements of Balanced Math

Insert Balanced Math Model Slide

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

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.

What did you notice?

C R A Model 1st Students will use their prior knowledge to construct concrete representations of math. 2nd Students must represent their understanding in a reflective &/or symbolic form. 3rd One or both forms will be a visual reminder for the understanding of the higher-thinking abstract. Representational Abstract Concrete

http://www.coach4math.com/wp-content/uploads/2010/03/C-R-A-Model.pdfhttp://www.coach4math.com/wp-content/uploads/2010/03/C-R-A-Model.pdf

C R A Model Activity Two Digit by 1 Digit Multiplication Representational Abstract Concrete

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?

Are Young Children Really Able to Explore Problems on Their Own and Arrive at Sensible Solutions? • Research has indicated that students who used inventedstrategies 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)

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)

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.

Break

Selecting appropriate tasks 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

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?

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.

Gradual Release of Responsibility Explicitly Teaching Problem Solving Whole Class  Cooperative Groups/Teams  Partner  Independent

Gradual Release of Responsibility Gradual Release of Responsibility https://mnpstube.mnps.org/watch_video.php?v=06ad360fe025e63

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.

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.

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

Student Problem Solving Rubric

Magic Squares Problem Solving Activity

Visualization is the key It is more important than learning key words. Many assessment items contain “misleading key words” or there are “no words associated with the operations.

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

Model Drawing Example • http://thesingaporemaths.com/P4math2f.swf

Teaching Mathematics Through Problem Solving