learning through problem solving – studying mathematics instruction in japan intermediate / senior...
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Learning through Problem Solving – Studying Mathematics
Instruction in Japan Intermediate / Senior Mathematics Winter 2011SESSION 10 – Jan 31, 2011
Upcoming DatesActions• Monday Jan 31 – Assessment and Evaluation
− Bring Growing Success electronic or paper version• Wednesday Feb 2 – Assessment and EvaluationDue Dates• Monday Jan 31 – Micro Teaching due and Lesson
Plan due
Let’s Do Math as a Teacher!
• Pick a number.
• Double it.
• Add 6.
• Double again.
• Subtract 4.
• Divide by 4.
• Subtract 2.
• What’s your final number?Analyze and Explain
Why did you end up with that number?
How does it work?
32
26
13
?
What Does It Look Like Algebraically?
X2 + 62 ( )- 4_____________4
( ) - 2
Pick a number, double it, add 6, double again, subtract 4, divide by 4, and subtract 2.
= X
Understanding the Problem … The Teaching Gap
“As important as it is to know how well students are learning, examinations of achievement scores alone can never reveal how the scores might be improved.
We also need information on the classroom processes – on teaching – that are contributing to the scores.”
“Teaching is a cultural activity. We learn how to teach directly, through years of participation in classroom life, and we are largely unaware of some of the most widespread attributes of teaching in our own culture.”
“If we wish to make wise decisions, we need to know what is going on in they typical classrooms.”
“Video information can shake up the way we think and let us take a fresh look at classrooms.”
The Teaching Gap - TIMSS Video Study
(Stigler & Hiebert, 1999)
Background – the TIMSS StudyWhy U.S., Japan, Germany?
• Goal – Through video study, describe and compare 8th grade mathematics instruction in United States, Germany, and Japan
• TIMSS study largely funded by US gov’t• Japan – scored near the top in all
international comparisons of mathematics achievement for decades
• Germany – an important comparison country, because like Japan, it is a major economic competitor of United States
(Stigler & Hiebert, 1999)
Background – the TIMSS StudyWhy Videotaping … How?
• problem definition – − “If we wish to make wise decisions, we
need to know what is going on in typical classrooms.”
• different teachers use the same words to mean different things
• data collected – random sample of 238 grade 8 mathematics classrooms− videotape of a grade 8 lesson− teacher questionnaire about their day
before and day after teaching plans− collection of teaching materials used
(textbook pages, worksheets)(Stigler & Hiebert, 1999)
Background – the TIMSS StudyObservation Codes and Analysis
• taped a typical lesson scheduled for that teaching day
• to avoid bias – they developed a set of standard observation codes to identify and quantify the frequency of specific teaching events
• a team of 6 coders independently watched the 238 videos and identified images of teaching and assigned observation codes and recorded frequency
(Stigler & Hiebert, 1999)
A mathematician’s perspective• “In Japan, there is the mathematics on one hand and the
students on the other. The students engage in the mathematics and the teacher mediates the relationship between the two.” – structured problem solving
• “In Germany, there is the mathematics as well, but the teacher owns the mathematics and parcels it out to students as she sees fit, giving facts and explanations at just the right time.” – developing advanced procedures
• “In the U.S. lessons, there are the students and there is the teacher. I have trouble finding the mathematics. I just see interactions between the teacher and the students.” – learning terms and practising procedures
(Stigler & Hiebert, 1999)
What is the Teaching Gap?Placemat Activity
Individual ideas
Individual ideas
Individual ideas Individual ideasCommon thoughts
What is the Teaching Gap?Describe and compare teaching and learning mathematics in Japanese, German, and US schools
• How is content−developed?−elaborated?−coherence
maintained?
• What is the nature of −the work done in the
classroom • by the students?• by the teacher?
−the mathematics?−the learning?
What is teaching and learning through Problem Solving?
Criteria for a Problem Solving Lesson
Content is • challenging• rationale and reasoning used to derive understanding
through teacher and student discussion • developed as the teacher weaves together ideas and
activities making certain the relationships between the learning goal and the lesson task are explicit−92% of Japanese teachers −76% of Germany teachers −45% of U.S. teachers
Criteria for a Problem Solving LessonNature of mathematical work by students • equal time practising procedures and inventing new
methods • seeing new relationships between mathematical ideas• learning by first struggling to solve math problems
then participating and discussing how to solve the problem – listening to ideas of others pros and cons, adapting thinking, constructing connections between methods and problems
• they use their time to explore, invent, make mistakes, reflect, and receive needed information just in time
What Can We Learn From TIMSS?Problem-Solving Lesson Design
BEFORE • Activating prior knowledge; discussing previous days’
methods to solve a current day problemDURING • Presenting and understanding the lesson problem • Students working individually or in groups to solve a
problem• Students discussing solution methodsAFTER• Teacher coordinating discussion of the methods (accuracy,
efficiency, generalizability)• teacher highlighting and summarizing key points• Individual student practice (Stigler & Hiebert, 1999)
BEFORE – Angle Measures Problem
1. Show these angles in any way that you know: 0o, 90o, 180o , 270o, 360o
2a. Fold paper along 180o in halves, to create different angles.
2b. Label the measure of each angle you make.
2c. What’s the relationship between the angles you made and 180o.
DURING - Angle Relationship ProblemWhat are 2 possible solutions for x?
AFTER – Angle Relationship Problem
What should be the next problem for students to practise?• Different angle measures• Real context/situation
What Can We Learn From TIMSS?Problem-Solving Lesson Design
BEFORE (ACTIVATING PROBLEM 10 min) • Activating prior knowledge; discussing previous days’
methods to solve a current day problemDURING (LESSON PROBLEM 20 min) • Presenting and understanding the lesson problem • Students working individually or in groups to solve a
problem• Students discussing solution methodsAFTER (CONSOLIDATION the REAL teaching 30 min) • Teacher coordinating discussion of the methods (accuracy,
efficiency, generalizability)• Teacher highlighting and summarizing key points• Individual student practise
(Stigler & Hiebert, 1999)
Compare your solutions. How are they similar? How are they different?
There are 36 children on school bus.There are 8 more boys than girls.How many boys? How many girls?
a) Solve this problem in 2 different ways. b) Show your work. Use a number line, square grid, picture,
graphic representation, table of values, algebraic expression
c) Explain your solutions. 1st numeric; 2nd algebraic
Bus Problem
There are 36 children on school bus.There are 8 more boys than girls.How many boys? How many girls?
a) Solve this problem in 2 different ways. b) Show your work.c) Explain your solutions using one or more operations.
What’s the Mathematical Relationship to the Previous Problem? …. Knowing MfT
Did you use this mathematical approach?
(Takahashi, 2003)
(Takahashi, 2003)
Did you use this mathematical approach?
Did you use this mathematical approach?
(Takahashi, 2003)
(Takahashi, 2003)
Did you use this mathematical approach?
(Takahashi, 2003)
Did you use this mathematical approach?
How are these algebraic solutions related to the other solutions? … Knowing Math on the Horizon
(Takahashi, 2003)
Which order would these solutions be shared for learning? Why? ... Knowing MfT
(Takahashi, 2003)
Growing Success: Assessment, Evaluation, and Reporting in Ontario Schools. 2010
Seven Fundamental Principles – What do they mean for Mathematics Education?
Work with a partner and complete this form. Be specific about mathematics tools and strategies.
Assessment and Evaluation
Success Criteria based on Expectations
Practicum
Practicum
Practicum
Practicum
See WIKI
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