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CCSS mathematics A System of Courses Phil Daro

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Phase-in We are making long overdue improvements in mathematics instruction AND Shifting to a new technology AND New tests Times like these need leaders who …..

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Evidence, not Politics High performing countries like Japan Research Lessons learned

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2011 © New Leaders | 4 Mile wide –inch deep causes cures

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2011 © New Leaders | 5 Mile wide –inch deep cause: too little time per concept cure: more time per topic = less topics

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Two ways to get less topics 1.Delete topics 2.Coherence:A little deeper, mathematics is a lot more coherent a)Coherence across concepts b)Coherence in the progression across grades

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Silence speaks no explicit requirement in the Standards about simplifying fractions or putting fractions into lowest terms. instead a progression of concepts and skills building to fraction equivalence. putting a fraction into lowest terms is a special case of generating equivalent fractions.

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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? c)so they will listen in class d)to learn mathematics

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Why give students problems to solve? 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.

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Three Responses to a Math Problem 1.Answer getting 2.Making sense of the problem situation 3.Making sense of the mathematics you can learn from working on the problem

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Answers are a black hole: hard to escape the pull Answer getting short circuits mathematics, making mathematical sense Very habituated in US teachers versus Japanese teachers Devised methods for slowing down, postponing answer getting

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Answer getting vs. learning mathematics USA: How can I teach my kids to get the answer to this problem? Use mathematics they already know. Easy, reliable, works with bottom half, good for classroom management. Japanese: How can I use this problem to teach the mathematics of this unit?

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Butterfly method

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Two major design principles, based on evidence: –Focus –Coherence

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The Importance of Focus TIMSS and other international comparisons suggest that the U.S. curriculum is ‘a mile wide and an inch deep.’ “On average, the U.S. curriculum omits only 17 percent of the TIMSS grade 4 topics compared with an average omission rate of 40 percent for the 11 comparison countries. The United States covers all but 2 percent of the TIMSS topics through grade 8 compared with a 25 percent non coverage rate in the other countries. High-scoring Hong Kong’s curriculum omits 48 percent of the TIMSS items through grade 4, and 18 percent through grade 8. Less topic coverage can be associated with higher scores on those topics covered because students have more time to master the content that is taught.” Ginsburg et al., 2005

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Grain size is a major issue Mathematics is simplest at the right grain size. “Strands” are too big, vague e.g. “number” Lessons are too small: too many small pieces scattered over the floor, what if some are missing or broken? Units or chapters are about the right size (8-12 per year) Districts: –STOP managing lessons, –START managing units

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What mathematics do we want students to walk away with from this chapter? Content Focus of professional learning communities should be at the chapter level When working with standards, focus on clusters. Standards are ingredients of clusters. Coherence exists at the cluster level across grades Each lesson within a chapter or unit has the same objectives….the chapter objectives

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What does good instruction look like? The 8 standards for Mathematical Practice describe student practices. Good instruction bears fruit in what you see students doing. Teachers have different ways of making this happen.

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Mathematical Practices Standards 1.Make sense of complex problems and persevere in solving them. 2.Reason abstractly and quantitatively 3.Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6.Attend to precision 7.Look for and make use of structure 8. Look for and express regularity in repeated reasoning. College and Career Readiness Standards for Mathematics

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Expertise and Character Development of expertise from novice to apprentice to expert –Schoolwide enterprise: school leadership –Department wide enterprise: department taking responsibility The Content of their mathematical Character –Develop character

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What does good instruction look like? Students explaining so others can understand Students listening to each other, working to understand the thinking of others Teachers listening, working to understand thinking of students Teachers and students quoting and citing each other

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motivation 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.

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Students Job: Explain your thinking Why (and how) it makes sense to you – (MP 1,2,4,8) What confuses you – (MP 1,2,3,4,5,6,7,8) Why you think it is true – ( MP 3, 6, 7) How it relates to the thinking of others – (MP 1,2,3,6,8)

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What questions do you ask When you really want to understand someone else’s way of thinking? Those are the questions that will work. The secret is to really want to understand their way of thinking. Model this interest in other’s thinking for students Being listened to is critical for learning

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Explain the mathematics when students are ready At the end of the lesson Prepare the 3-5 minute summary in advance, spend the period getting the students ready, get students talking about each other’s thinking, quote student work during summary at lesson’s end

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Students Explaining their reasoning develops academic language and their reasoning skills Need to pull opinions and intuitions into the open: make reasoning explicit Make reasoning public Core task: prepare explanations the other students can understand The more sophisticated your thinking, the more challenging it is to explain so others understand

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Language, Mathematics and Prior Knowledge

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Develop language, don’t work around language Look for second sentences from students, especially EL and reluctant speakers Students Explaining their reasoning develops academic language and their reasoning power Making language more precise is a social process, do it through discussion Listening stimulates thinking and talking Not listening stimulates daydreaming

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Prior knowledge There are no empty shelves in the brain waiting for new knowledge. Learning something new ALWAYS involves changing something old. You must change prior knowledge to learn new knowledge.

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You must change a brain full of answers To a brain with questions. Change prior answers into new questions. The new knowledge answers these questions. Teaching begins by turning students’ prior knowledge into questions and then managing the productive struggle to find the answers Direct instruction comes after this struggle to clarify and refine the new knowledge.

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Variety across students of prior knowledge is key to the solution, it is not the problem

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15 ÷ 3 = ☐

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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

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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 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

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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

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Teach at the speed of learning Not faster More time per concept More time per problem More time per student talking = less problems per lesson

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Personalization The tension: personal (unique) vs. standard (same)

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Why Standards? Social Justice Main motive for standards Get good curriculum to all students Start each unit with the variety of thinking and knowledge students bring to it Close each unit with on-grade learning in the cluster of standards Some students will need extra time and attention beyond classtime

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2011 © New Leaders | 40 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

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Grain size is a major issue Mathematics is simplest at the right grain size. “Strands” are too big, vague e.g. “number” Lessons are too small: too many small pieces scattered over the floor, what if some are missing or broken? Units or chapters are about the right size (8-12 per year) Districts: –STOP managing lessons, –START managing units

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Unit architecture

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Four levels of learning I.Understand well enough to explain to others II.Good enough to learn the next related concepts III.Can get the answers IV.Noise

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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.Good enough to learn the next related concepts Most of the rest III.Can get the answers At least this much IV.Noise Aim for zero

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Efficiency of embedded peer tutoring is necessary Four levels of learning different students learn at levels within same topic I.Understand well enough to explain to others An asset to the others, learn deeply by explaining II.Good enough to learn the next related concepts Ready to keep the momentum moving forward, a help to others and helped by others III.Can get the answers Profit from tutoring IV.Noise Tutoring can minimize

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When the content of the lesson is dependent on prior mathematics knowledge “I do – We do– You do” design breaks down for many students Because it ignores prior knowledge I – we – you designs are well suited for content that does not depend much on prior knowledge… You do- we do- I do- you do

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Classroom culture: ….explain well enough so others can understand NOT answer so the teacher thinks you know Listening to other students and explaining to other students

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Questions that prompt explanations Most good discussion questions are applications of 3 basic math questions: 1.How does that make sense to you? 2.Why do you think that is true 3.How did you do it?

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…so others can understand Prepare an explanation that others will understand Understand others’ ways of thinking

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Minimum Variety of prior knowledge in every classroom; I - WE - YOU Student A Student B Student C Student D Student E Lesson START Level CCSS Target Level

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Variety of prior knowledge in every classroom; I - WE - YOU Student A Student B Student C Student D Student E Planned time Needed time Lesson START Level CCSS Target Level

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Student A Student B Student C Student D Student E Variety of prior knowledge in every classroom; I - WE - YOU Lesson START Level CCSS Target Level

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Student A Student B Student C Student D Student E Variety of prior knowledge in every classroom; I - WE - YOU Lesson START Level CCSS Target Answer-Getting

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You - we – I designs better for content that depends on prior knowledge Student A Student B Student C Student D Student E Lesson START Level Day 1 Attainment Day 2 Target

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Differences among students The first response, in the classroom: make different ways of thinking students’ bring to the lesson visible to all Use 3 or 4 different ways of thinking that students bring as starting points for paths to grade level mathematics target All students travel all paths: robust, clarifying

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The arc of the lesson Diagnostic: make differences visible; what are the differences in mathematics that different students bring to the problem All understand the thinking of each: from least to most mathematically mature Converge on grade -level mathematics: pull students together through the differences in their thinking

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Next lesson Start all over again Each day brings its differences, they never go away

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School Leaders and CCSS Develop the Mathematics Department as an organizational unit that takes responsibility for solving problems and learning more mathematics Peer + observation of instruction Collaboration centered on student work Summarize the mathematics at the end of the lesson

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What to look for Students are talking about each other’s thinking Students say second sentences Audience for student explanations: the other students. Cold calls, not hands, so all prepare to explain their thinking Student writing reflects student talk

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Look for: Who participates EL students say second sentences African American males are encouraged to argue Girls are encouraged to engage in productive struggle Students listen to each other Cold calls, not hands, so no one shies away from mathematics

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Shift from explaining to the teacher to convince her you are paying attention To explaining so the others understand From Just answer getting To the mathematics behind and beyond the answers

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Stay calm, teach on This whole thing is a shift from a peculiar world that failed large numbers of students. We got used to something peculiar. To a world that is more normal, more like life outside the mathematics classroom, more like good teaching in other subjects.

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NGSS and Common Core State Standards. Mathematical Practices Make sense of problems and persevere in solving them Reason abstractly and quantitatively.

NGSS and Common Core State Standards. Mathematical Practices Make sense of problems and persevere in solving them Reason abstractly and quantitatively.

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