The Leona Group May 23, 2011 Steve Leinwand American Institutes for Research SLeinwand@air

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Translating the Vision of the9-12 Common Core State

Standards into Highly Effective H.S. Math Programs

The Leona GroupMay 23, 2011

Steve LeinwandAmerican Institutes for Research

[email protected]

Agenda8:15 – 9:00 - District Leadership – Perspectives,

leadership opportunities and timelines

9:15 – 11:00 – ALL – Why bother? Glimpses and implications of the CCSSM

11:15 – 12:15 – Coaches and teachers – Doing the math

12:45 – 3:00 – Coaches and teachers – Instruction and professional collaboration

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Opening Gambit

Your knowledge, experience, insights, creativity, energy and

expertise are desperately needed to improve K-12 mathematics education.

(If you don’t feed inadequate…..)

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People won’t do what they can’t envision,

People can’t do what they don’t understand,

People can’t do well what isn’t practiced,

But practice without feedback results in little change, and

Work without collaboration is not sustaining.

Ergo: your job, as leader, at its core, is to help people envision, understand, practice, receive feedback and collaborate.

The key things we know

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And one key perspective:Most teachers practice their craft behind closed doors, minimally aware of what their colleagues are doing, usually unobserved and under supported. Far too often, teachers’ frames of reference are how they were taught, not how their colleagues are teaching. Common problems are too often solved individually rather than by seeking cooperative and collaborative solutions to shared concerns.

(this SOP won’t help us in the coming years)

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Critical next steps• Build familiarity with the CCSSM

• Course by course discussions

• Professional collaboration

• Crosswalks

• The mathematical practices

• Think 15% per year

• Focus on instructional quality/opportunity to learn

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Rollout Timeline, hopefully

• 2010-11: A year of comprehensive planning (clarifying what needs to be done when)

• 2011-12: A year of study (analyzing crosswalks, curricular implications, policy shifts)

• 2012-13: A year of piloting and collaborative discussions

• 2013-14: A year of curriculum and policy implementation and an assessment moratorium

• 2014-15: A year of accountable implementation

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So what can we do?Potential structures for professional sharing:• Structured and focused department meetings• Before school breakfast sessions• Common planning time – by grade and by department• Pizza and beer/wine after school sessions • Released time 1 p.m. to 4 p.m. sessions• Hiring substitutes to release teachers for classroom visits• Coach or principal teaching one or more classes to free up

teacher to visit colleagues• After school sessions with teacher who visited, teacher who

was visited and the principal and/or coach to debrief• Summer workshops• Department seminars

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So what can we do?Potential Strategies for developing professional

learning communities:• Classroom visits – one teacher visits a colleague and the they

debrief

• Demonstration classes by teachers or coaches with follow-up debriefing

• Co-teaching opportunities with one class or by joining two classes for a period

• Common readings and CCSSM sections assigned, with a discussion focus on:

– To what degree are we already addressing the issue or issues raised in this article?

– In what ways are we not addressing all or part of this issue?

– What are the reasons that we are not addressing this issue?

– What steps can we take to make improvements and narrow the gap between what we are currently doing and what we should be doing? 9

So what can we do (cont.)?• Technology demonstrations (graphing calculators, SMART

boards, document readers, etc.)• Collaborative lesson development• Video analysis of lessons• Analysis of student work• Development and review of common finals and unit

assessments• What’s the data tell us sessions based on state and local

assessments• What’s not working sessions• Principal expectations for collaboration are clear and

tangibly supported• Policy analysis discussions, e.g. grading, placement,

requirements, promotion, grouping practices, course options, etc. 10

In other words

• Reduce the professional isolation

• Enhance the transparency of our work

• To raise the quality of what we do

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Discussion Question

Which of these CAN’T you do and why?

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Today’s Little Goals• Provide some perspectives on the array of

problems with the current high school math program.

• Provide more than a glimmer of hope thanks to the new CCSS for Mathematics.

• Provide a range of examples of current and future practice.

• Present some of the implementation issues that will need to be faced.

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Today’s Big GoalTo provoke and inform your thinking about the need and directions for significantly revising the traditional Algebra I, Geometry, and Algebra II courses to ensure relevance, real rigor and fairness that truly meets the needs of all students.

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Economic security and social well-being

Innovation and productivity

Human capital and equity of opportunity

High quality education

(literacy, MATH, science)

Daily classroom math instruction

Where we fit on the food chain

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Opening PremiseMediocre mathematics achievement and unacceptably stark achievement gaps are the symptom – not the problem. If we conceive of it as an “achievement” gap, then it’s THEIR problem or fault.

Alternatively, it is a system failure, the heart of which is modal instruction that fails to provide adequate opportunity to learn, that is the problem. If we conceive of it as an “instruction” gap, then it’s OUR problem or fault.

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The System ProblemA depressingly comprehensive, yet honest, appraisal must conclude that our typical math curriculum is generally incoherent, skill-oriented, and accurately characterized as “a mile wide and an inch deep.” It is dispensed via ruthless tracking practices and focused mainly on the “one right way to get the one right answer” approach to solving problems that few normal human beings have any real need to consider. Moreover, it is assessed by 51 high-stakes tests of marginal quality, and overwhelmingly implemented by under-supported and professionally isolated teachers who too often rely on “show-tell-practice” modes of instruction that ignore powerful research findings about better ways to convey mathematical knowledge.

Quite a mouthful and not a pretty picture.

But when little of what we do works for more than 30% of our students;

And when most of us really aren’t comfortable putting our own kids in many more than 30% of available classrooms;

We’ve got work to do!

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So let’s take a look at some context-setting sound bites or perspectives:

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1) Let’s be clear:We’re being asked to do what has never been done before:

Make math work for nearly ALL kids and get nearly ALL kids ready for college.There is no existence proof, no road map, and it’s not widely believed to be possible.

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2) Let’s be even clearer:Ergo, because there is no other way to serve a much broader proportion of students:

We’re therefore being asked to teach in distinctly different ways.Again, there is no existence proof, we don’t agree on what “different” mean, nor how we bring it to scale.

(That’s the hope of the CCSS for Math)

3) The pipeline perspective:

1985: 3,800,000 Kindergarten students

1998: 2,810,000 High school graduates

1998: 1,843,000 College freshman

2002: 1,292,000 College graduates

2002: 150,000 STEM majors

2006: 1,200 PhD’s in mathematics

(the best case I know for 113 9-12 math for normal standards vs. the 43+ math for nerds)

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4) A critical perspective

As mathematics colonizes diverse fields, it develops dialects that diverge from the “King’s English” of functions, equations, definitions and theorems. These newly important dialects employ the language of search strategies, data structures, confidence intervals and decision trees.

- Steen

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5) Another critical perspective

Evidence from a half-century of reform efforts shows that the mainstream tradition of focusing school mathematics on preparation for a calculus-based post-secondary curriculum is not capable of achieving urgent national goals and that no amount of tinkering in likely to change that in any substantial degree.

- Steen

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Non-negotiable take-away

Make no mistake,

for K-12 math in the U.S., this IS brave new world.

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Full disclosure

For better or worse, I’ve been drinking the CCSSM Kool-aid.

Leinwand on the CCSSM in the 2011 Heinemann catalog.

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A Long Overdue Shifting of the FoundationFor as long as most of us can remember, the K-12 mathematics program in the U.S. has been aptly characterized in many rather uncomplimentary ways: underperforming, incoherent, fragmented, poorly aligned, unteachable, unfair, narrow in focus, skill-based, and, of course, “a mile wide and an inch deep.” Most teachers are well aware that there have been far too many objectives for each grade or course, few of them rigorous or conceptually oriented, and too many of them misplaced as we ram far too much computation down too many throats with far too little success. It’s not a pretty picture and helps to explain why so many teachers and students have been set up to fail and why we’ve created the need for so much of the intervention that test results seem to require.

But hope and change have arrived! Like the long awaited cavalry, the new Common Core State Standards (CCSS) for Mathematics presents us – at least those of us in the 44 states that have now adopted them (representing over 80% of the nation’s students) – a once in a lifetime opportunity to rescue ourselves and our students from the myriad curriculum problems we’ve faced for years.

COHERENT FAIR TEACHABLE 28

So to high school and the task at hand:

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My First Premise

As currently implemented, high school Algebra 1, Geometry, and Algebra 2 are not really working and meeting neither societal or student needs.

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A brief justification for my premise:

• Increasingly obsolete and useless symbol manipulation focus at the expense of functions, models, applications, big ideas and statistics

• An impossible to “cover” scope and sequence• 1200 page tomes• 1st half of Algebra 2 = 2 to 1 dilation of Algebra 1• Extraordinarily high failure rates• Turns millions off to mathematics• Designed for a very narrow slice of the cohort• Neither relevant, rigorous, nor fair!

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In blunter terms:

No wonder HS math is often referred to as “a rotten

geometry sandwich stuffed between two stale slices of

algebra”

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My Second Premise

The 9-12 Common Core State Standards have the potential to resolve many of these problems.

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Some perspectives to support my first

premise

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Just for fun…

Simplify:

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√2 + √7

- 9 (√2 - √7)

Actual retail value: 11.08

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Versus substance too rarely addressed

If 0 < x < 1, which of the following is greatest?

a) 1/xb) √xc) xd) x2

- What if x > 1?- What if x < -1?

Not convinced?

Feast your eyes on the Algebra 2 Final Exam

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A little synthetic division perhaps?

Or perhaps you would prefer ignoring all of the technological advances of the

past 25 years and just do some factoring for fun?

Talk to engineers about what math they use and how they do it

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To summarize our expectations….

Simplify Solve Factor Graph

vs.Simplify Solve

Factor Graph Find Express

Display ModelRepresent Solve

Predict Demonstrate

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And what are the outcomes?Achieve ADP Algebra I 2009 Exam

33,446 students (KY, OH, RI and NJ)

Level Scale Score % of Students

Advanced 850-575 1.6%

Proficient 574-450 16.4%

Basic 449-387 26.2%

Below Basic 386-300 55.8%

Ave. Scale Score: 384 (850-300)43

And what are the outcomes?Achieve ADP Algebra II 2009 Exam

102,396 students (13 states, 60% AZ & IN)

Level Scale Score % of Students

Well –prepared 1650-1275 3.5%

Prepared 1274-1150 11.1%

Needs preparation 1149-900 85.4%

Ave. Scale Score: 1032 (900 - 1650)

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Additional ADP Findings

• On both the Algebra I and Algebra II exams, students earned, on average, only 11% and 14%, respectively, of the possible points available on constructed-response items.

• On the Algebra II exam in both 2008 and 2009, nearly one-third of the students earned no points on the 2-point or 4-point constructed response items.

• And we claim they’re ready for the 21st century???

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So Why Bother? Look around. Our critics are not all wrong.• Mountains of math anxiety• Tons of mathematical illiteracy• Mediocre test scores• HS programs that barely work for half the

kids• Gobs of remediation• A slew of criticism

Not a pretty picture and hard to dismiss

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Yes, Virginia or Houston or whoever,

WE HAVE A PROBLEM!!!

(and no amount of tinkering around the edges is going to fix it)

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Enter the new Common Core State Standards for

Mathematics(www.corestandards.org)

Not perfect, but clearer, fairer, and more coherent

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The Math Field of Activity

The heart of ensuring instructional quality and producing high levels of student achievement includes four key elements:

• A coherent and aligned curriculum that includes a set of grade level content expectations, appropriate print and electronic instructional materials, with a pacing guide that links the content standards, the materials and the calendar;

• High levels of instructional effectiveness, guided by a common vision of effective teaching of mathematics and supported by deliberate planning, reflection and attention to the details of effective practice;

• A set of aligned benchmark and summative assessments that allow for monitoring of student, teacher and school accomplishment at the unit/chapter and grade/course levels; and

• Professional growth within a professional culture of dignity, transparency, collaboration and support.

(What, how, how well and with what support to do it better)49

But….as we need to acknowledge

• Our curriculum is stale,

• Our instruction is underperforming,

• Our assessments are mediocre, and

• Our professional development is essentially useless!

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BUT…Great News• Our curriculum is stale – enter CCSSM• Our instruction is underperforming,• Our assessments are mediocre – enter

SBAC/PARCC• Our professional development is

essentially useless!

Welcome to a far more simplified world.

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The Reduced Field of Activity(in the CCSS Era)

The heart of ensuring instructional quality and producing high levels of student achievement includes four key elements:

• A coherent and aligned curriculum that includes a set of grade level content expectations, appropriate print and electronic instructional materials, with a pacing guide that links the content standards, the materials and the calendar;

• High levels of instructional effectiveness, guided by a common vision of effective teaching of mathematics and supported by deliberate planning, reflection and attention to the details of effective practice;

• A set of aligned benchmark and summative assessments that allow for monitoring of student, teacher and school accomplishment at the unit/chapter and grade/course levels; and

• Professional growth within a professional culture of dignity, transparency, collaboration and support.

(What, how, how well and with what support to do it better)52

So finally, let’s take a look at the game changer:

The Common Core State Standards for Mathematics

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Promises

These Standards are not intended to be new

names for old ways of doing business. They are

a call to take the next step. It is time for states

to work together to build on lessons learned

from two decades of standards based reforms.

It is time to recognize that standards are not

just promises to our children, but promises we

intend to keep.

— CCSS (2010, p.5)54

Characteristics of the CCSS• Fewer, therefore teachable

• Fair, therefore expectable

• Aligned with college and career expectations

• Internationally benchmarked

• Rigorous content and application of higher-order skills – rigor as depth, not complexity

• Learning progressions that build coherence and mastery expectations

• Research-based

• Common – finally!

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CCSSM Mathematical Practices

The Common Core proposes a set of Mathematical Practices that all teachers should develop in their students. These practices are similar to NCTM’s Mathematical Processes from the Principles and Standards for School Mathematics.

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8 CCSSM Mathematical Practices

1. Make sense of 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.

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8 CCSSM Mathematical Practices

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.

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HS: Six conceptual categories

• Number and Quantity

• Algebra

• Functions

• Modeling

• Geometry

• Statistics and Probability

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Common Core Format

High School

Conceptual Category

Domain

Cluster

Standards

K-8

Grade

Domain

Cluster

Standards

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Common Core Format

Domains are large groups of related standards. Standards from different domains may sometimes be closely related. Look for the name with the code number on it for a Domain.

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Common Core Format

Clusters are groups of related standards. Standards from different clusters may sometimes be closely related, because mathematics is a connected subject.

Clusters appear inside domains.

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Common Core Format

Standards define what students should be able to understand and be able to do – part of a cluster.

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Format of High School Standards

StandardStandard

ClusterCluster

DomainDomain

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High School Pathways

• Pathway A: Consists of two algebra courses and a geometry course, with some data, probability and statistics infused throughout each (traditional)

• Pathway B: Typically seen internationally that consists of a sequence of 3 courses each of which treats aspects of algebra, geometry and data, probability, and statistics.

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CCSS 9-12 AlgebraSeeing Structure in Expressions

• Interpret the structure of expressions

• Write expressions in equivalent forms to solve problems

Arithmetic with Polynomials and Rational Expressions

• Perform arithmetic operations on polynomials

• Understand the relationship between zeros and factors of polynomials

• Use polynomial identities to solve problems

• Rewrite rational expressions

Creating Equations

• Create equations that describe numbers or relationships

Reasoning with Equations and Inequalities

• Understand solving equations as a process of reasoning and explain the reasoning

• Solve equations and inequalities in one variable

• Solve systems of equations

• Represent and solve equations and inequalities graphically

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A closer glimpse at algebra

Create equations that describe numbers or relationships

1. Create equations and inequalities in one variable and use them to

solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on

combinations of different foods.

4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.

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CCSS 9-12 FunctionsInterpreting Functions

• Understand the concept of a function and use function notation

• Interpret functions that arise in applications in terms of the context

• Analyze functions using different

representations

Building Functions

• Build a function that models a relationship

between two quantities

• Build new functions from existing functions

Linear, Quadratic, and Exponential Models

• Construct and compare linear, quadratic, and exponential models and solve problems

• Interpret expressions for functions in terms of the situation they model

Trigonometric Functions

• Extend the domain of trigonometric functions using the unit circle

• Model periodic phenomena with trigonometric functions

• Prove and apply trigonometric identities

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A closer glimpse at functions

Understand the concept of a function and use function notation

1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

3. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.

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CCSS 9-12 Geometry

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A closer glimpse at geometry 1

Similarity, Right Triangles, and Trigonometry Understand similarity in terms of similarity transformations1. Verify experimentally the properties of dilations given by a center and

a scale factor:a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

2. Given two figures, use the definition of similarity in terms of similaritytransformations to decide if they are similar; explain using similaritytransformations the meaning of similarity for triangles as the equalityof all corresponding pairs of angles and the proportionality of allcorresponding pairs of sides.

3. Use the properties of similarity transformations to establish the AAcriterion for two triangles to be similar.

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A glimpse at geometry 2Prove theorems involving similarity4. Prove theorems about triangles. Theorems include: a line parallel to one

side of a triangle divides the other two proportionally, and conversely; thePythagorean Theorem proved using triangle similarity.

5. Use congruence and similarity criteria for triangles to solve problemsand to prove relationships in geometric figures.

Define trigonometric ratios and solve problems involving right triangles6. Understand that by similarity, side ratios in right triangles are

properties of the angles in the triangle, leading to definitions oftrigonometric ratios for acute angles.

7. Explain and use the relationship between the sine and cosine ofcomplementary angles.

8. Use trigonometric ratios and the Pythagorean Theorem to solve righttriangles in applied problems.

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CCSS 9-12 Statistics and Prob

Interpreting Categorical and Quantitative Data

• Summarize, represent, and interpret data on a single count or measurement variable

• Summarize, represent, and interpret data on two categorical and quantitative variables

• Interpret linear models

Making Inferences and Justifying Conclusions

• Understand and evaluate random processes underlying statistical experiments

• Make inferences and justify conclusions from sample surveys, experiments and observational studies

Conditional Probability and the Rules of Probability

• Understand independence and conditional probability and use them to interpret data

• Use the rules of probability to compute

probabilities of compound events in a uniform probability model

Using Probability to Make Decisions

• Calculate expected values and use them to solve problems

• Use probability to evaluate outcomes of

decisions

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A closer glimpse at StatisticsUnderstand and evaluate random processes underlying statistical

experiments

1. Understand statistics as a process for making inferences about population parameters based on a random sample from that population.

2. Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model?

Make inferences and justify conclusions from sample surveys, experiments, and observational studies

3. Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

4. Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.

5. Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.

6. Evaluate reports based on data.74

But the challenge is how we translate this vision

and these words into daily classroom reality

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Problem 1

Tell me five things you see.

f(x) = x2 + 3x - 5

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Problem 2Siti packs her clothes into a suitcase and

it weighs 29 kg.Rahim packs his clothes into an identical

suitcase and it weighs 11 kg.Siti’s clothes are three times as heavy as

Rahims.What is the mass of Rahim’s clothes?What is the mass of the suitcase?

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The old (only) way:

Let S = the weight of Siti’s clothes

Let R = the weight of Rahim’s clothes

Let X = the weight of the suitcase

S = 3R S + X = 29 R + X = 11

so by substitution: 3R + X = 29

and by subtraction: 2R = 18

so R = 9 and X = 2

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Or using a model:

11 kg

Rahim

Siti

29 kg

Problem 3F = 4 (S – 65) + 10

Find F when S = 81

Vs.

First I saw the blinking lights… then the officer informed me that:

The speeding fine here in Vermont is $4 for every mile per hour over the 65 mph limit plus a $10 handling fee.

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Connecticut: F = 10 ( S – 55) + 40

Maximum speeding fine: $350

• Describe the fine in words

• At what speed does it no longer matter?

• At 80 mph how much better off would you be in VT than in CT?

• Use a graph to show this difference81

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Problem 4You may or may not remember that the formula

for the volume of a sphere is 4/3πr3 and that the volume of a cone is 1/3 πr2h.

Consider the Ben and Jerry’s ice cream sugar cone, 8 cm in diameter and 12 cm high, capped with an 8 cm in diameter sphere of deep, luscious, decadent, rich triple chocolate ice cream.

If the ice cream melts completely, will the cone overflow or not? How do you know?

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Problem 5Solve for x: 16 x .75x < 1

Vs. You ingest 16 mg of a controlled

substance at 8 a.m. Your body metabolizes 25% of the substance every hour. Will you pass a 4 p.m.

drug test that requires a level of less than 1 mg? At what time could you

first pass the test?

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Big Ideas 1– Functions are the mathematical rules for taking input

(independent variables) and producing output (dependent variables).

– Algebraic properties allow the generation of equivalent forms of most expressions and equations.

– Linear functions are additive (the dependent variable increases additively); exponential functions are multiplicative (the dependent variable increases multiplicatively).

– There is a direct relationship, for any function, among a point on a graph of the function, an ordered pair in a table of the function, and a solution to the symbolic form of the function.

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Big Ideas 2– Solving equations is an undoing process of

equivalent equations based on inverse to isolate the variable, where addition undoes subtraction and vice verse, multiplication undoes division, square rooting undoes squaring.

– The graphs of functions can be visualized and predicted based on transformations of a parent function (all linear functions are transformations of the parent function or line y = x; all quadratic functions are transformations of the parent function y = x2).

– The graph of the inverse of a function is its reflection across the line y = x.

– Just as subtraction undoes, or is the inverse of, addition, the square root is the inverse of squaring and finding a log is the inverse of raising to a power.

Big Ideas 3

• Similarity

• Proof

• Dimension

• Model

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From Traditional Algebra 1Unit 1: Expressions and Equations

Chapter 1 The Language of Algebra

Chapter 2 Real Numbers

Chapter 3 Solving Linear Equations

Unit 2: Linear Functions

Chapter 4 Graphing Relations and Functions

Chapter 5 Analyzing Linear Equations

Chapter 6 Solving Linear Inequalities

Chapter 7 Solving Systems of Linear Equation and Inequalities

Unit 3: Polynomials and Nonlinear Functions

Chapter 8 Polynomials

Chapter 9 Factoring

Chapter 10 Quadratic and Exponential Functions

Unit 4: Radical and Rational Functions

Chapter 11 Radical Expressions and Triangles

Chapter 12 Rational Expressions and Equations

Unit 5: Data Analysis

Chapter 13 Statistics

Chapter 14 Probability91

To sensible Algebra 1

Unit 1: Patterns

Unit 2: Equations

Unit 3: Linear functional situations

Unit 4: Representing functional situations

Unit 5: Direct and indirect variation

Unit 6: Data

Unit 7: Systems of equations

Unit 8: Exponential functions

Unit 9: Linear programming

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From Traditional(and Unteachable) Algebra 2

Chapter 1: Analyzing Equations and Inequalities

Chapter 2: Graphing Linear Relations and Functions

Chapter 3: Solving Systems of Linear Equations and Inequalities

Chapter 4: Using Matrices

Chapter 5: Exploring Polynomials and Radical Expression

Chapter 6: Exploring Quadratic Functions and Inequalities

Chapter 7: Analyzing Conic Sections

Chapter 8: Exploring Polynomial Functions

Chapter 9: Exploring Rational Expressions

Chapter 10:Exploring Exponential and Logarithmic Functions

Chapter 11: Investigating Sequences and Series

Chapter 12: Investigating Discrete Mathematics and Probability

Chapter 13: Exploring Trigonometric Functions

Chapter 14: Using Trigonometric Graphs and Identities

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To sensible Algebra 2

Unit 1: Review and reinforce big ideas and key skills of Algebra 1

Unit 2: Quadratic functions

Unit 3: Polynomials and polynomial functions

Unit 4: Patterns, series and recursion

Unit 5: Exponential and logarithmic functions

Unit 6: Rational and radical functions

Unit 7: Probability and statistics

Unit 8: Optimization, graph theory and topics in discrete mathematics

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Dream with me• Computer adaptive assessments with routers

and appropriate items for stage 2• 4 week testing windows• Computer-scoreable constructed response

items• Item banks for formative, benchmark AND

summative assessments• Student, class, school, district, state and

nation results three days after the window closes

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Outstanding Issues• Some vs. All

– 113 for all + 43 (*) for STEM nerds

• Traditional vs. Integrated– “Pathways”

• Grade 8 Algebra– When can we skip? Or should we double

up?

• Catching up– Boost-up

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Math for Normals and Nerds

Grade Traditional Normal

Integrated Normal

AcceleratedNerd

Double-up Nerd

6 6 6 6/7 6

7 7 7 7/8 7

8 8 8 Alg 1/Math 9 8

9 Alg 1 Math 9 Geom/Math 10

Alg 1/Math9

10 Geom Math 10 Alg 2/Math 11 Geom/Alg 2 or

Math 10/11

11 Alg 2 Math 11 Pre-calc Pre-calc

12 (Pre-college) (Math 12) Calc/Stat Calc/Stat

85% → 10% 5%

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Long Reach HSHoward County (MD) recognized that there were a significant number of 9th graders who were not being successful in Algebra 1. To address this problem, the county designed Algebra Seminar for approximately 20% of the 9th grade class in each high school. These are students who are deemed unlikely to be able to pass the state test if they are enrolled in a typical one-period Algebra I class. Algebra Seminar classes are:

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• Team-taught with a math and a special education teacher;

• Systematically planned as a back-to-back double period;

• Capped at 18 students;

• Supported with a common planning period made possible by Algebra Seminar teachers limited to four teaching periods;

• Supported with focused professional development;

• Using Holt Algebra I, Carnegie Algebra Tutor, and a broad array of other print and non-print resources;

• Notable for the variety of materials and resources used (including Smart Board, graphing calculators, laptop computers, response clickers, Versatiles, etc.);

• Enriched by a wide variety of highly effectively instructional practices (including effective questioning, asking for explanations, focusing of different representations and multiple approaches); and

• Supported by county-wide on-line lesson plans that teachers use to initiate their planning.

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Jo Boaler’s WorkMultidimensional classes

“In many classrooms there is one practice that is valued above all others – that of executing procedures (correctly and quickly). The narrowness by which success is judged means that some students rise to the top of classes, gaining good grades and teacher praise, while other sink to the bottom with most students knowing where they are in the hierarchy created. Such classrooms are unidimensional.”

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Jo Boaler’s WorkMultidimensional classes

“At Railside the teachers created multidimensional classes by valuing many dimensions of mathematical work. This was achieved, in part, by having more open problems that students could solve in different ways. The teachers valued different methods and solution paths and this enabled more students to contribute ideas and feel valued.”

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When there are many ways to be successful, many more students are successful.

“When we interviewed the students and asked them “what does it take to be successful in mathematics class?” they offered many different practices such as: asking good questions, rephrasing problems, explaining well, being logical, justifying work, considering answers…

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“When we asked students in “traditional” classes what they needed to do in order to be successful they talked in much more narrow ways, usually saying that they needed to concentrate, and pay careful attention.”






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Let’s do some math!

• Skin

• Pancakes

• Peas

• Why Roads are Crowded

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TIMSS Video Study 1

• Teacher instructs students in a concept or skill.

• Teacher solves example problems with class.

• Students practice on their own while the teacher assists.

• In other words……

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Putting it all together one way

Good morning class.

Today’s objective: Find the surface area of right circular cylinders.

Open to page 384-5.3

Example 1: S.A.= 2πrh + 2 πr2

4

Find the surface area. Page 385 1-19 odd

TIMSS Video Study 2

• Teacher presents complex, thought-provoking problem

• Students struggle with the problem individually and in groups

• Student present their work

• Teacher summarizes solutions and extracts important understandings

• Students work on a similar problem108

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Putting it all together another way

Overheard in the ER as the sirens blare:

“Oh my, look at this next one. He’s completely burned from head to toe.”

“Not a problem, just order up 1000 square inches of skin from the graft bank.”

You have two possible responses:- Oh good – that will be enough.

OR- Oh god – we’re in trouble.

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• Which response, “oh good” or “oh god” is more appropriate?

• Explain your thinking.• Assuming you are the patient, how

much skin would you hope they ordered up?

• Show how you arrived at your answer and be prepared to defend it to the class.

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Peter Dowdeswell of London, England holds the world record

for pancake consumption!

62 6” in diameter,

3/8” thick pancakes, with butter and syrup

in 6 minutes 58.5 seconds!SO?

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So?

• About how high a stack? Show and explain• Exactly how high?• How fast?• How much?• Could it be, considering the size of the

stomach?• What’s radius of single 3/8” thick pancake of

same volume?• Draw a graph of Peter’s progress.

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Valid or Invalid?Convince us.

• Grapple

• Formulate

• Givens and Goals

• Estimate

• Measure

• Reason

• Justify

• Solve114

Why ARE roads crowded?

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Powerful Teaching

• Provides students with better access to the mathematics:– Context

– Technology

– Materials

– Collaboration

• Enhances understanding of the mathematics:– Alternative approaches

– Multiple representations

– Effective questioning

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So it’s instruction, silly!Research, classroom observations and common sense provide a great deal of guidance about instructional practices that make significant differences in student achievement. These practices can be found in high-performing classrooms and schools at all levels and all across the country. Effective teachers make the question “Why?” a classroom mantra to support a culture of reasoning and justification. Teachers incorporate daily, cumulative review of skills and concepts into instruction. Lessons are deliberately planned and skillfully employ alternative approaches and multiple representations—including pictures and concrete materials—as part of explanations and answers. Teachers rely on relevant contexts to engage their students’ interest and use questions to stimulate thinking and to create language-rich mathematics classrooms. 118

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Breakfast or dessert?

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NCTM Standards

Process Standards

Content Standards

• Problem Solving• Reasoning and Proof• Communication• Connections• Representations

• Number• Measurement• Geometry• Algebra• Data

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All the standards rolled up into one:

• Problem Solving: What is this? What’s that white thing?

• Communication: Tell the person sitting next to you.

• Reasoning: How do you know?

• Connections: A real rip-off ad.

• Representations: A picture

But look at what else this example shows us:

Consider how we teach reading: JANE WENT TO THE STORE.

• Who went to the store?

• Where did Jane go?

• Why do you think Jane went to the store?

• Do you think it made sense

for Jane to go to the store? 123

Now consider mathematics:TAKE OUT YOUR HOMEWORK.

- #1 19

- #2 37.5

- #3 185

(No why? No how do you know? No who has a different answer?)

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Strategy #1

Adapt from what we know about reading

(incorporate literal, inferential, and evaluative

comprehension to develop stronger neural connections)

The bottom line

Good math teaching BEGINS with an answer

(often a wrong answer)

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Number from 1 to 6

1. What is 6 x 7?

2. What number is 1000 less than 18,294?

3. About how much is 32¢ and 29¢?

4. What is 1/10 of 450?

5. Draw a picture of 1 2/3

6. About how much do I weight in kg?

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Good morning Boys and GirlsNumber from 1 to 5

1. What is the value of 3P – 5R when P=4 and R=-6?

2. What are the slope and y-intercept of the line represented by y = -5x + 10?

3. If 140 pounds of rice cost $240, how much rice can you buy for $360?

4. Solve for M: 17M – 12 = 4M + 40

5. About how much do I weight in kg?

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Strategy #2

Incorporate on-going cumulative review into instruction every day.

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Implementing Strategy #2

Almost no one masters something new after one or two lessons and one or two homework assignments. That is why one of the most effective strategies for fostering mastery and retention of critical skills is daily, cumulative review at the beginning of every lesson.

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On the way to school:

• A term of the day

• A picture of the day

• An estimate of the day

• A skill of the day

• A graph of the day

• A word problem of the day

What about homework?

• Typical practice: 356 349 341

1-19 2-24 31, 34, 37

odd even

• Alternative practices:– Max of 10– Answers on board/answer sheets and self-check– Check your partner– Which 2 need more attention and why– Document camera to save time

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Great

Take a deep breath!

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Tell me what you see.

2 1/4

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•

135

Tell the person sitting next to you five things you see.

136

137


1

)2

1(

n

n

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Strategy #3

Create a language rich classroom.

(Vocabulary, terms, answers, explanations)

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Implementing Strategy #3Like all languages, mathematics must

be encountered orally and in writing. Like all vocabulary, mathematical terms must be used again and again in context and linked to more familiar words until they become internalized.

Sum = both Difference – bigger thanArea = covering Quotient = sharingPerimeter = border Mg = grain of sandCos = bucket Cubic = SCircumference = a belt Surface area = skinTan = sin/cos = y/x for all points on the unit circle

Ready, set, picture…..

“three quarters”

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The radius and the tangent at point S on a circle with

center at point O

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y = sin xy = 2 sin xy = sin (2x)

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Strategy #4

Draw pictures/Create mental images/

Foster visualization

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Thinking vs. Math

24.The number of boys attending Fairfield High School is twice the number of girls.

If 1/6 of the boys and 1/4 of the girls are in the school band, what fraction of the student are in the school band?

5/36 7/36 2/9 7/24 5/12

The “right” way:Let B = the number of boys

Let G = the number of girls

So: 2G = B

Find: 1/6 B + 1/4 G = 1/6 (2G) + 1/4 G)

B + G 2G + G

= (2/6 G + 1/4 G)/3G

= 7/12 G / 3G = 7/36

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The Stanley Kaplan Approach

Boys Girls 2 : 1

1/6 1/4

Try: 100 students – 2 to 1 – no

Try: 90 students – 60 and 30, 6 and oops

Try: 6ths & 4ths – 24 and 48 – 8 + 6 out of 72

14/72 = 7/36

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Or even better for some:• K kids, ergo boys = 2/3K and girls =

1/3K

Band = 1/6 (2/3K) + 1/4 (1/3K)

= (1/9 + 1/12) K or 7/36 K

• OR 2/3 1/3

Boys Girls 147

1/6 of 2/3 1/4 of 1/3

So how do we get there?

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Critical next steps• Build familiarity with the CCSSM

• Course by course discussions

• Professional collaboration

• Crosswalks

• The mathematical practices

• Think 15% per year

• Focus on instructional quality/opportunity to learn

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Rollout Timeline, hopefully

• 2010-11: A year of comprehensive planning (clarifying what needs to be done when)

• 2011-12: A year of study (analyzing crosswalks, curricular implications, policy shifts)

• 2012-13: A year of piloting and collaborative discussions

• 2013-14: A year of curriculum and policy implementation and an assessment moratorium

• 2014-15: A year of accountable implementation

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So what can we do?Potential structures for professional sharing:• Structured and focused department meetings• Before school breakfast sessions• Common planning time – by grade and by department• Pizza and beer/wine after school sessions • Released time 1 p.m. to 4 p.m. sessions• Hiring substitutes to release teachers for classroom visits• Coach or principal teaching one or more classes to free up

teacher to visit colleagues• After school sessions with teacher who visited, teacher who

was visited and the principal and/or coach to debrief• Summer workshops• Department seminars

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So what can we do?Potential Strategies for developing professional

learning communities:• Classroom visits – one teacher visits a colleague and the they

debrief

• Demonstration classes by teachers or coaches with follow-up debriefing

• Co-teaching opportunities with one class or by joining two classes for a period

• Common readings and CCSSM sections assigned, with a discussion focus on:

– To what degree are we already addressing the issue or issues raised in this article?

– In what ways are we not addressing all or part of this issue?

– What are the reasons that we are not addressing this issue?

– What steps can we take to make improvements and narrow the gap between what we are currently doing and what we should be doing? 152

So what can we do (cont.)?• Technology demonstrations (graphing calculators, SMART

boards, document readers, etc.)• Collaborative lesson development• Video analysis of lessons• Analysis of student work• Development and review of common finals and unit

assessments• What’s the data tell us sessions based on state and local

assessments• What’s not working sessions• Principal expectations for collaboration are clear and

tangibly supported• Policy analysis discussions, e.g. grading, placement,

requirements, promotion, grouping practices, course options, etc. 153

In other words

• Reduce the professional isolation

• Enhance the transparency of our work

• To raise the quality of what we do

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Discussion Question

Which of these CAN’T you do and why?

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So….While we acknowledge the range and

depth of the problems we face,

It should be comforting to see the availability and potential of solutions to these problems….

Now go forth and start shifting YOUR school’s high school mathematics program to better serve our students, our society and our future.

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Thank you!

[email protected]

Documents

The Leona Group May 23, 2011 Steve Leinwand American Institutes for Research SLeinwand@air