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## Teaching for Understanding: What Will It Take?

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**Teaching for Understanding:What Will It Take?**Jim Stigler UCLA May 29, 2015**What’s at stake**• Why it’s hard • What it will take • Understanding statistics Plan for Today**What’s at stake**Community college developmental mathematics students**View math as rules/procedures to be memorized;can’t**“figure it out” • When students were asked, “What does it mean to be good at math?”77% gave answers like these: • “Mathisjustallthesesteps.” • “Inmath, sometimesyouhavetojustacceptthatthat’sthewayitisandthere’snoreasonbehindit.” • “Idon'tthink [beinggoodatmath] hasanythingtodowithreasoning. It'sallmemorization.” Developmental Math Students: Views of Math**Which is greater: or ?**• a • 5 • a • 8 • 53% correct, but very few could explain why • Compulsion to calculate… 5a = a8 5 = 8 ✔ Developmental Math Students**462**+ 253 715 Interviewer: “How could you check that your answer is correct?” 715 – 253 = 462 Interviewer: “Why did you subtract 253? Could you have subtracted 462 to check the answer?” Developmental Math Students**Flexible vs. routine expertise (Hatano & Inagaki)**• What is flexible expertise? • Procedural fluency • Conceptual understanding • Disposition to think/make sense of mathematics • Ability to nimbly bring knowledge to bear across a wide array of new situations What Do We Want Our Students to Learn?**Why it’s hard**Teaching is a cultural activity**Teaching is a cultural activity; varies more across than**within cultures (TIMSS 1995 video study; goals/scripts) TIMSS Video: Teaching is Cultural**US: “Quick and snappy”**Japan: “Slow and sticky” SOURCE: Hess & Azuma (1991)**Cultural activities are:**• Learned implicitly (ritual learning) • Multiply determined • Hard to see (e.g., confusion) • Hard to change: culture wins TIMSS Video: Teaching is a Cultural Activity**What it will take**A focus on learning opportunities**What do high-achieving countries have in common?****Do they all teach like Japan?**Teaching looks different in every country – no one way to**teach.**But: despite using different strategies…**…high-achieving countries create similar learning opportunities for students.**For deep learning, with understanding, students need**recurring and sustained opportunities for: • Productive struggle – with important mathematics • Explicit connections – among concepts, procedures, problems, situations • Deliberate practice – increasing variation and complexity over time Research Indicates Three Types of Learning Opportunities Required for Deep Understanding**Students randomly assigned to two conditions to study**science text: • Four study periods • Read passage 14.2 times • Predicted good recall • One study / 3 recall periods • Read passage 3.4 times • Predicted poor recall Productive Struggleand Desirable Difficulties Source: Roediger & Karpicke (2006)**Students randomly assigned to two conditions to study**science text: • Four study periods • Read passage 14.2 times • Predicted good recall • One study / 3 recall periods • Read passage 3.4 times • Predicted poor recall Productive StruggleandDesirable Difficulties Source: Roediger & Karpicke (2006)**“I never enlighten anyone who has not been driven to**distraction by trying to understand a difficulty or who has not got into a frenzy trying to put his ideas into words. When I have pointed out one corner of a square to anyone and he does not come back with the other three, I will not point it out to him a second time” (不愤不启, 不悱不发. 举一隅不以三隅反, 则不复也.) (Lau, 7:8) Confucius**Few Connections**Many Connections B B C C A A D D F F E E G G H H The Power of Connections**What fraction of the rectangle is shaded brown?**(a) (b) ⅓ ¼ ⅓ ½ Example: Struggle to Make Connections SOURCE: Deborah Ball (2013)**Different from repetitive practice**• Constantly increasing variation, complexity, challenge • With feedback • Staving off premature automaticity • Understanding takes practice, too! Deliberate Practice**Explicit Connections**− + − + Productive Struggle Learning Opportunities**Deliberate Practice**✪ ✪ Explicit Connections − + ✪ ✪ − + ✪ Maintaining struggle and connections through time Productive Struggle Learning Opportunities**Understanding statistics**Making connections to core concepts**Related Concept**Core Concept 1 Core Concept 2 Core Concept 3 L3A L3B Example: Connect concept of mean to that of best fitting regression line. L3C Connections Map**Statistics is the study of distributions.**• The “five chords”: the Distribution Triad. • Hypothesis: practicing these connections over time will result in more coherent and flexible understanding. Working Hypothesis**The whole is more than the sum of the parts**• Describing/comparing distributions Developmental Progression**Working from sample to population, and vice versa**• Random sampling, and bias • Problem of inference; informal approaches • Sampling variation Developmental Progression**A very abstract idea, hard to understand with or without**simulation • Confidence intervals, hypothesis tests • Wait: I thought there were only three distributions? Developmental Progression**Final Question:**How do you make this happen?**“What was the main thing you wanted students to learn from**today’s lesson?” Teacher’s Goal for the Lesson Back**“Every system is perfectly designed to achieve the results**it gets.” -Paul Bataldan