DEEP THOUGHTS Instead of having answers on math tests, can’t we just have opinions, and if my opinion is different than that of everyone else, hey, can’t we all just get along. Jack Handy
THE HORSE PROBLEM A man named Joe buys a horse named Ed for $60 from a woman named Flo. Joe then sells the horse for $70, buys it back again for $80 and sells it again for $90. How much does Joe make or lose in the horse trading business?
POLYA’S Problem-Solving Process: Getting To Know The Problem: This involves making sense of the context of the problem, what information is given (needed and extraneous), and what it is that is being asked? Devising a Plan to Solve the Problem: Students should be encouraged to share possible plans and to discuss alternative approaches. ESTIMATION should be encouraged at this point. Implementing a Solution Plan: (See strategies to follow) Look Back and Beyond: Does the result correlate with the estimation? Were all conditions met and accounted for? Encourage students to extend the problem through “What if…” questions.
Problem: A family has three children, Adam, Beatrice and Caroline. Each child has been given a chore. Adam is to vacuum the carpets in the house every third day. Beatrice must take out the garbage every fourth day and Caroline must mow the lawn every sixth day. In August, on which day(s) will all the children be doing their chores together? Problem: I have an unlimited supply of pennies, nickels and dimes in my pocket. If I take three coins out of my pocket at a time, how much could I have in my hand? Problem: Roll the dice 20 times. Subtract the small number from the large number. If they are the same, count as zero. Plot your results on a line graph.
Problem: You have been given a piece of land for a garden. You must fence this land in totally to keep rabbits out. Assuming you need 24 square metres for your garden, what is the least amount of fence you could use? Problem: You are working in a bicycle repair shop and are in charge of making tricycles and bicycles out of extra parts. You have 18 frames (some for tricycles, some for bicycles) and 46 wheels. Assuming that the wheels on bicycles and tricycles are the same size, how many of each type of cycle can you make (ensure you use all the parts you have). Problem: In the bus loading area there are four school busses lined up, front to back, waiting for students to get on. Each school bus is three metres long and there is a two metre space between each one. How long is the loading area? Problem: A woman appeared on “Who Wants to Be a Millionaire” and won a large sum of money. When she arrived back home she met her three best friends, one at a time. To each friend she met she gave half of the money she had then. After all three meetings she had $8 000.00 remaining. How much did she win on the show?
PROBLEM SOLVING STRATEGIES(Any one or more of these can, and should, be used for a variety of problems) • Dramatize (act it out) or create a model of the situation • Draw a picture • Construct a table or chart • Find a pattern • Solve a simpler problem • Guess and check • Work backwards • Consider all possibilities
LISTEN TO THEM TALK!!!!!!! • LET STUDENTS DEVELOP THEIR OWN STRATEGIES!
CHARACTERISTICS OF A GOOD (RICH) LESSON:(see: Flewelling (2002). Realizing a Vision of Tomorrow’s Classroom, Rich Tasks) • Curriculum relevance • Student relevance • Authentic content and structure • Flexible- for different levels • Problem solving and question posing • Inquiry/exploration/investigation/experimentation • Communication • Reflect on learning • Creative Go to: http://math.unipa.it/~grim/AFlewelling70-72
PROBLEM OF THE DAY/ WEEK/ MONTH/ YEAR? Assuming you have access to pennies, nickels, dimes and quarters, how many combinations of change can you make for a dollar? Come ready to talk about your method of solving this problem and your thinking while you did so!
NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS(NCTM)Ontario Association of Mathematics Educators (OAME) • The primary professional organization for teacher of mathematics of grades K-12 • 1989- released Curriculum and Evaluation Standards • 1991- released Professional Standards for Teaching Mathematics
CURRICULUM STRANDS • Number Sense and Numeration • Geometry and Spatial Sense • Measurement • Patterning • Data Management and Probability
THE ONTARIO CURRICULUM GRADES 1-8 (read p 1-9 of the mathematics curriculum) FIVE STRANDS: • Number Sense and Numeration • Measurement • Geometry and Spatial Sense • Patterning and Algebra • Data Management and Probability
Number Sense and Numeration Counting, numeral representation, more and/or less than, equal to, part-whole relationships, base ten…
Measurement Linear measure, perimeter, area, volume, mass, time, money, comparing sizes of objects, non-standard and standard units of measure…
Geometry and Spatial Sense Simple and complex shapes (two and three dimensional), transformational geometry (flips, slides, turns), attributes of shapes (vertices, sides, faces), graphing coordinates…
Patterning and Algebra Simple repeating patterns, growing patterns, shape designs, sets of numbers, patterns in art, graphs, data collection, equations, relationships, variables…
Data Management and Probability Describing and organizing graphs, statistics, trends, estimations, rations, fractions, collecting, presenting and comparing data…
MATHEMATICS EXPECTATIONS EXPECTATION: What a child should be able to demonstrate ACTIONS:OBJECTS: This is what a child should This is the content that do to demonstrate their learning, will be demonstrated, e.g. e.g. model, compare, build, equivalent fractions, place connect value
FIVE PROCESS STANDARDS 1. Problem Solving 2. Reasoning and Proof (conceptual vs. procedural) 3. Communication (oral, written, drawn, kinesthetic… 4. Connections (within and outside mathematics 5. Representation (symbols, diagrams, graphs, charts, pictures)
SHIFTS IN CLASSROOM ENVIRONMENT • Classrooms as communities and not just collection of individuals • Toward logic and mathematical evidence as verification (away from teacher as authority) • Toward mathematical reasoning (concepts) and away from memorization • Toward conjecturing, inventing, problem solving and creating and away from mechanics of getting the right answer • Toward connecting mathematics to other disciplines