NCDPI Curriculum and Instruction Mathematics “Teaching for Understanding” Posted on January 28, 2013 by Bill McCallum Once every few months or so I receive a message about the following standard: 6.G.2. Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V=lwh and V=bh to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems. Guess what people think the problem is before reading on? http://commoncoretools.me/2013/01/28/to-b-or-not-to-b/ Norms • Listen as an Ally • Value Differences http://thebenevolentcouchpotato.wordpress.com/201 1/11/30/norm-peterson-bought-the-house-next-door/ • Maintain Professionalism • Participate Actively Who’s in the Room? “Teaching for Understanding” Phil Daro Math SCASS February 12, 2013 Dr. Phil Daro “In Person” (Almost) Problem: Cause: Cure: Mile wide –inch deep curriculum Too little time per concept More time per topic “LESS TOPICS” Why do students have to do math problems? a. To get answers because Homeland Security needs them, pronto. b. I had to, why shouldn’t they? a. So they will listen in class. d. To learn mathematics. To Learn Mathematics • Answers are part of the process, they are not the product. • The product is the student’s mathematical knowledge and know-how. • The ‘correctness’ of answers is also part of the process. Yes, an important part. What is learning? • Integrating new knowledge with prior knowledge; explicit work with prior knowledge • Prior knowledge varies across students in a class (like fingerprints); this variety is key to the solution, it is not the problem. • Thinking in a way you haven’t thought before and understanding what and how others are thinking. “Answer Getting vs. Learning Mathematics” United States: • “How can I teach my kids to get the answer to this problem?” Japan: • “How can I use this problem to teach the mathematics of this unit?” “The Butterfly Method” Discussion • How might these ideas challenge teachers in your district or school? • How can we move from “answer getting” to “learning mathematics”? • What evidence do you have that teachers might not know the difference? “The Butterfly Method” Blogstop.com “Faster Isn’t Smarter” by Cathy Seeley “Hard Arithmetic is not Deep Mathematics” p. 83 “Hard Arithmetic is not Deep Mathematics” • What issues or challenges does this message raise for you? • In what ways do you agree or disagree? • What barriers might keep students from reaching these standards, and how can you tackle these barriers? Let’s Do Some Math! Area and Perimeter • What rectangles can be made with a perimeter of 30 units? Which rectangle gives you the greatest area? How do you know? • What do you notice about the relationship between area and length of the sides? Instructions • Discuss the following at your table – What thinking and learning occurred as you completed the task? – Would this task be considered “Deep Mathematics”? Why or why not? Compared to…. 5 10 What is the area of this rectangle? What is the perimeter of this rectangle? Traveling With Graphs Bing.com Carl’s bike trip takes 3.5 hours. During each hour of the trip, Carl notes the speed at which he is traveling. The trip is over level ground, and for most of the trip Carl maintains a constant speed, since he is a good rider. Based on the data Carl collected, what is the total distance he covered during his trip? Hours Speed (mph) 0 0 1 8 2 8 3 8 3.5 0 For these data, he plots a speed/time graph. Compared to…. 1. What is the area of a rectangle with a length of 2in. and a height of 8in.? 2. What is the area of a triangle whose base is 3 units and it’s height is 8 units? 3. If Carl rode a bicycle for 3 hours and traveled 40 miles, what was his average speed? Traveling With Graphs • What concepts are addressed in this situation? • What strategies could be used to develop conceptual understanding? • At what level could this task be used as a lesson task? How is this task foundational for future concepts? “Who’s doing the talking, and who’s doing the math?” Cathy Seeley, former president, NCTM How do we move from a culture of “answer getting” to one of “learning mathematics”? “Modeling in Mathematics” by CCSSO and Math SCASS (Council of Chief State School Officers) (The State Collaborative of Assessment and Student Standards) What is modeling? A word with different meanings 1. “Modeling a Task” - An instructional strategy where the teacher shows step by step actions of how to set up and solve the task Mathematical Task: 2 + ___ = 8 Use step by step actions to “model” how to solve this task What is modeling? A word with different meanings 2. “Model with Manipulatives” - Start with the math then use manipulatives to demonstrate and understand how to solve the problem. math Toothpicks as a model What is modeling? A word with different meanings 3. “Model with Mathematics” - Start with the task and choose an appropriate mathematical model to solve the task Four birds sat on a wire, 2 flew away. How many birds remain on the wire? Choose a grade appropriate mathematical model to solve the task: e.g. writing the number sentence 4 – 2 = 2 What is modeling? A word with different meanings 4. “A Model with Mathematics” What is modeling? A word with different meanings 1. 2. 3. 4. “Modeling a Task” “Modeling with Manipulatives” “Model with Mathematics” “A Model with Mathematics” What is modeling? A word with different meanings 1. 2. 3. 4. “Modeling a Task” “Modeling with Manipulatives” “Model with Mathematics” “A Model with Mathematics” What makes something a modeling task? • Are there criteria for “modeling tasks”? • What are the skills involved? Let’s Do Some Math! The Ride; Task 932: (unpublished) Alysha really wants to ride her favorite ride at eh amusement park one more time before meeting her parents for lunch at 12:20 pm. There is a pretty long line at his ride, which Alysha joins at 11:50 am (point A in the diagram below). Alysha is nervously checking the time as she is moving forward in the line. By 12:03 she has made it to point B in line. What is your best estimate for how long it will take Alysha to reach the front of the line? Can she ride one more time before she is supposed to meet her parents? How well posed is well enough? • Should a student still have questions after they read the task? • Should students have to find their own information outside of what is given in the problem? • Should assumptions be stated, or reasoned differently by each individual student? Problems to Ponder • Painting A Barn Think about…… • The Ice Cream Van How each problem is posed. • Birthday Cakes How much information is provided and when it’s provided? • Graduation • Sugary Soft Drinks How much information is needed and how will they find it? Painting A Barn Alexis needs to paint the four exterior walls of a large rectangular barn. The length of the barn is 80 feet, the width is 50 feet, and the height is 30 feet. The paint cost $28 per gallon, and each gallon covers 420 square feet. How much will it cost Alexis to paint the barn? Explain your work. Ice Cream Van You are considering dividing ice cream van during the summer vacation. Your friend who “knows everything” tells you that “its easy money.” You make a few inquiries and find that the van costs $600 per week to rent. Each ice cream cone costs 50 cent to make and sell for $1.50. Birthday Cakes Would all the birthday cakes eaten by all the people in Arizona in one year fit inside the University of Phoenix football stadium? Cody Patterson Original Graduation The SLV High School graduation started at 1:00 pm. After some speeches, the principal started reading off the names of the students, alphabetically by last name. When he finishes, the graduation will end. Sugary Soft Drinks How many packets of sugar are in a 20 ounce bottle of soda? http://threeacts.mrmeyer.com/sugarpackets/ Lunch Collecting and Selecting Information “Modeling Information Descriptors” All and only relevant information is given Brainstorm what you need and then are given it Told what you need, you go and find it Given information, but you decide what is useful Determine what information is needed and find the information yourself Collecting and Selecting Information “Modeling Information Descriptors” Use the contents of the envelope on your table to: • Match each task with it’s aligned “Collecting and Selecting Information” description. Collecting and Selecting Information “Modeling Information Descriptors” 1. All and only relevant information is given 2. Brainstorm what you need and then are given it 3. Told what you need, you go and find it 4. Given information, but you decide what is useful 5. Determine what information is needed and find the information yourself Collecting and Selecting Information “Modeling Information Descriptors” 1. All and only relevant information is given 2. Brainstorm 3. Told what you what you need need, you go and then are and find it given it 4. Given information, but you decide what is useful 5. Determine what information is needed and find the information yourself Matching Activity Use the contents of the envelope on your table to: • Match each task with it’s aligned “Collecting and Selecting Information” description. • Order the tasks on a continuum based on the amount of information needed? Collecting and Selecting Information How much information is needed? 1 3 4 2 5 Collecting and Selecting Information How much information is needed? 1 3 4 2 5 How much information needs to be found? 4 1 2 3 5 “All Around the School” A class was studying metric and customary measurement, comparing quantities of one unit of measure to quantities in the other. (2003) Question: If all the students in the school hold hands, will they create a chain long enough to circle the school? Compared To…… Our school is 485 meters around. There are 535 students in the school, and the average arm span of a child is 2 meters. Can we circle the school if we hold hands and make a human chain? Blogstop.com “Faster Isn’t Smarter” by Cathy Seeley “Constructive Struggling” p. 88 Let’s Do Some Math! Show 15  3 =   1. As a multiplication problem 2. Equal groups of things 3. An array (rows and columns of dots) 4. Area model 5. In the multiplication table 6. Make up a word problem Show 15  3 =   1. As a multiplication problem (3 x [ ] = 15 ) 2. Equal groups of things: 3 groups of how many make 15? 3. An array (3 rows, ? columns of 3 make 15?) 4. Area model: a rectangle has one side = 3 and an area of 15, what is the length of the other side? 5. In the multiplication table: find 15 in the 3 row 6. Make up a word problem Show 16  3 =   1. As a multiplication problem 2. Equal groups of things 3. An array (rows and columns of dots) 4. Area model 5. In the multiplication table 6. Make up a word problem 16 3 =   • What concepts are addressed in this situation? • What strategies could be used to develop conceptual understanding? “Who’s doing the talking, and who’s doing the math?” Cathy Seeley, former president, NCTM Blogstop.com “Faster Isn’t Smarter” by Cathy Seeley “Faster Isn’t Smarter” p. 93 Personalization The Tension: personal (unique) vs. standard (same) Why Standards? • Social Justice • Good curriculum for all students • Start with the variety of thinking and knowledge students bring • On-grade learning in the cluster of standards Standards are a Peculiar Genre 1. We write as though students have learned approximately 100% of what is in preceding standards. This is never even approximately true anywhere in the world. 2. Variety among students in what they bring to each day’s lesson is the condition of teaching, not a breakdown in the system. We need to teach accordingly. 3. Tools for teachers…instructional and assessment…should help them manage the variety. What is learning? • Integrating new knowledge with prior knowledge; explicit work with prior knowledge • Prior knowledge varies across students in a class (like fingerprints); this variety is key to the solution, it is not the problem. • Thinking in a way you haven’t thought before and understanding what and how others are thinking. Minimum degree of varying prior knowledge in the average classroom Student A Student B Student C Student D Student E Lesson START Level Degree of prior knowledge in the average classroom Planned time Student A Student B Student C Student D Student E Needed time Lesson START Level CCSS Target Level I - WE - YOU CCSS Target Student A Student B Student C Student D Student E Lesson START Level I - WE - YOU CCSS Target Student A Student B Student C Student D Student E Answer-Getting Lesson START Level You - We – I Instruction based on prior knowledge Student A Student B Student C Student D Student E Lesson START Formative Assessment Level Four Levels of Learning I. Highest Standard: Understand well enough to explain to others II. Good enough Standard: Understand enough to learn the next related concepts III. Low Standard: Can get the answers IV. No Standard: Noise Four levels of learning The truth is triage, but all can prosper I. Understand well enough to explain to others As many as possible, at least 1/3 II. Understanding enough to learn the next related concepts Most of the rest III. Can get the answers without understanding Sometimes we have to settle for low, but don’t aim low IV. Noise Aimless Blogstop.com “Faster Isn’t Smarter” by Cathy Seeley “Crystal’s Calculator” p. 159 Illustrative Mathematics Example Problems illustrativemathematics.org Teach at the speed of learning • Not faster • More time per concept • More time per problem • More time per student talking • Fewer problems per lesson The Mathematical Practices develop character: the pluck and persistence needed to learn difficult content. We need a classroom culture that focuses on learning…a try, try again culture. We need a culture of patience while the children learn, not impatience for the right answer. Patience, not haste and hurry, is the character of mathematics and of learning. www.ncdpi.wikispaces.net Nominate An Outstanding Teacher Website: www.paemst.org [email protected] What questions do you have? DPI Mathematics Section Kitty Rutherford Elementary Mathematics Consultant 919-807-3841 [email protected] Johannah Maynor Secondary Mathematics Consultant 919-807-3842 [email protected] Ashton Megson Secondary Mathematics Consultant 919-807-3934 [email protected] Barbara Bissell K – 12 Mathematics Section Chief 919-807-3838 [email protected] Susan Hart Mathematics Program Assistant 919-807-3846 [email protected]