57. Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.
11The intense study of the last three letters of the alphabet Algebra:The intense study of the last three letters of the alphabet
12Construct viable arguments and critique the reasoning of others Versus:FunctionsModelsStatisticsPersevere solving problemsConstruct viable arguments and critique the reasoning of others
133. Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.a. Make tables of equivalent ratios relating quantities with whole number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.b. Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.d. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.
14Let’s see what you know We drive to Florida at 60 miles per hour. What do you see?- What do you notice when you look:- Left to right between columns?- Top to bottom within columns?- Suppose you went 630 miles?Hours1235____XMiles60540_____
15So what have we gotten? Mountains of math anxiety Tons of mathematical illiteracyMediocre test scoresHS programs that barely work for more than half of the kidsGobs of remediation and interventionA slew of criticismNot a pretty picture!
16If however….. What we’ve always done is no longer acceptable, then… We have no choice but to change some of what we do and some of how we do it.
25The amusement park Ride Time Tickets Roller Coaster 45 4 Ferris Wheel 253Bumper Cars152Rocket Ride40Merry-go-Round10Water Slide30Fun House20
26The Amusement ParkThe 4th and 2nd graders in your school are going on a trip to the Amusement Park. Each 4th grader is going to be a buddy to a 2nd grader. Your buddy for the trip has never been to an amusement park before. Your buddy want to go on as many different rides as possible. However, there may not be enough time to go on every ride and you may not have enough tickets to go on every ride.
27The bus will drop you off at 10:00 a. m. and pick you up at 1:00 p. m The bus will drop you off at 10:00 a.m. and pick you up at 1:00 p.m. Each student will get 20 tickets for rides. Use the information in the chart to write a letter to your buddy and create a plan for a fun day at the amusement park for you and your buddy.
28Why do you think I started with this task? Standards don’t teach, teachers teachIt’s the translation of the words into tasks and instruction and assessments that really matterProcesses are as important as contentWe need to give kids (and ourselves) a reason to careDifficult, unlikely, to do alone!!!
29Today’s GoalTo provoke and inform your thinking about the need to shift our curriculum, instructional practices, professional culture, and mindsets in ways that are aligned with the vision of the new Common Core State Standards and that truly meet the needs of all students.
30(That is, a bombardment to stimulate thinking in 3 parts) Today’s AgendaFour perspectives on our realitySome glimpses of the CCSSMSome fears and challenges(That is, a bombardment to stimulate thinking in 3 parts)
311. What a great time to be convening as teachers of mathematics! Common Core State Standards adopted by 46 statesMore access to material and ideas via the web than everA president who believes in science and dataThe beginning of the end to Algebra II as the killerA long overdue understanding that it’s instruction that really mattersA recognition that the U.S. doesn’t have all the answers
32Impactful Teaching Practices – 4/2014 Establish learning goals. Impactful teaching of mathematics establishes clear goals that indicate what mathematics students are learning and that guide decision-making during a lesson.Implement challenging tasks. Impactful teaching of mathematics engages students in solving cognitively challenging tasks that allow for multiple entry points and varied solution strategies.Connect and use representations. Impactful teaching of mathematics uses connections among multiple representations to deepen understanding and as tools for problem solving..Pose purposeful questions. Impactful teaching of mathematics uses questioning to assess and advance student reasoning and sense making about important mathematical ideas and relationships.
33Facilitate meaningful discourse Facilitate meaningful discourse. Impactful teaching of mathematics facilitates discourse and builds a shared understanding of mathematical ideas by analyzing and comparing student approaches and arguments.Support productive struggle. Impactful teaching of mathematics consistently provides students, individually and collectively, with opportunities to engage in productive struggle and supports them as they grapple with mathematical ideas and relationships.Elicit and use evidence. Impactful teaching of mathematics uses evidence of student thinking to assess the progress students are making toward mathematical understandings and to adjust instruction in ways that support learning.
34Goal:The goal of this slide is to ask students to apply their intuition, make a guess and justify their guess. Because they’ve guessed, students may now be curious to know which of them guessed correctly. and guess an answer. After this slide, students may want to know which of them guessed correctly.Teaching Moves:Ask students to write down guesses. Say, “Would you please write down on your handout which you think has more: the left or the right? Or do they have exactly the same?” [Why?]Ask students to pair and share. Say, “Share your guess with your neighbor. Do you both agree? Do you disagree?”Take a classwide poll of guesses. Ask for a show of hands. “Who thinks the left one has more? The right? The same?” Write down approximate counts of the guesses on the board. [Why?]Interesting Outcomes:The class is divided. The video IS intendED to create controversy. The ideal outcome has the class split into two or three groups of different opinions. At that point, the more you can play up the controversy (“Oooo ... big fight. We can’t all be right, people!”) the more interested students will be in the answer. [Why?]What is your guess? Share your guess with your neighbor and justify your guess.
35What information is important here? How would you get it? Goal:The goal of this slide is for students to see that diameter and height are the essential variables for finding volume and also the easiest to measure with a ruler.Teaching Moves:Ask students what information is important. Ask, “What information would be useful to know here and how would you get it? Would you write down some ideas for yourself on your handout?” [Why?]Ask students to pair and share. Say, “Share your ideas with your neighbor. Do you both agree? Do you disagree about what’s necessary?”Make a list of information requested. Ask students to share the information they want. Write suggestions on the board next to the owner’s name or type them in a document, depending on which is faster for you. But record everything. The goal here isn’t to pass judgment on good and bad suggestions. It’s to collect student ideas. There’s no need to collect every student’s idea, though. Once you’ve recorded a broad cross-section of the class’ ideas, cut the process short. [Why?]Push students for precision. See the interesting outcomes below and how you might respond. As you pull and tug on certain students’ responses, put a star or some marking next to the suggestion to illustrate that you’ve addressed it. It isn’t necessary to address all of them. This is a chance to redirect students to more precise mathematical vocabulary. For example, some teachers would start a lesson by introducing the vocabulary. However we didn't, so when students say, "We need to know how tall and wide the glasses are" teachers can say something, "When we are talking about cylinders, we call how tall they are their height and how wide they are their diameterInteresting Outcomes:Students ask for a measuring cup so they can find volume. Take this one first. Say something like, “Great idea. That’d be so easy and we could all get out of here for break. But unfortunately I don’t have a measuring cup and all I’m working with at the moment is a ruler.”Students ask for “height.” This isn’t very precise so we’ll push on it. Say, “Okay, when you ask for height? You’re looking for this height here, right?” Then put your fingers from the very top of the glass to the very bottom of the glass. They need to tell you, “No, we want the height of the soda.”Students ask for “radius” or “diameter” or “circumference.” Talk about how we know we can turn any one of those facts into any of the others. But which of those three is easiest to get and which is hardest? Have them think about the question for a minute and write down a response, then share it with a neighbor.Radius. Radius is difficult because it relies on finding the center of the glass, which is invisible.Circumference. Circumference is difficult with a ruler, of course. But even if you allow a piece of string (or a seamstress’ tape) it doesn’t help to pull the string taut around the outside of the glass because that will include the thickness of the glass in with the soda. It’s difficult to pull the string taut around the inside of the glass, which is what’s required.Diameter. Diameter may seem easiest but it too relies on finding the center of the circle. We know a curious fact about circles that may be very helpful, though: The diameter is the longest chord in a circle. So fix one end of the ruler to a point on the circumference of the glass and then rotate the ruler around the rest of the glass. The longest measurement has to be the diameter.Students ask for some extremely precise information. For example, “Is it really a cylinder?” or “What about the thickness of the glass?” or “Is the bottom of the glass really flat?” It may seem initially that these students are being disruptive or disagreeable but every other student ought to emulate this attention to precision. Answer these questions directly and with sincerity, “It’s really a cylinder.” and “We’ll get measurements that take the thickness of the glass into consideration.” and “The bottom of the glass is really flat.” Then throw a lot of praise on them.’’What information is important here?How would you get it?
365.5 cm7 cm10 cmGoal:The goal of this slide is to give students the information you’ve settled on as most precise and easiest to measure with the tools on hand, all without implying a specific solution strategy yet. The goal of this slide is that students struggle productively, and come to so that they crave more information so they can know what to do with this information and worry abouat what to do with this information, all before you teach them the formula for volume.Teaching Moves:Give them some time to struggle. Say, “Okay, you guys asked for this information because you had some rough, fuzzy plan for it. I want you to take a minute to work on your own and put that plan into action.” [Why?]Ask them to share ideas with their neighbors. Don’t expect your students to just “discover” the formula for cylinder volume. Don’t expect all this understanding to arise without some intervention on your part. But if students invest time and struggle in the question, they’ll want the explanation even more and be in a better place to appreciate it.Interesting Outcomes:They don’t solve the problem. In this case we’ll move to the lecture in the next slide.Some students solve the problem. This is always a possibility. If a student solves the task quickly, pose one of the extension questions we’ve included in later slides.3 cm
37DOESN’T THE ACT 3 VIDEO RESOLVING THE ISSUE GO HERE??? Goal:The goal of this slide is to verify that math is a functional model for describing the world. [Why?]Teaching Moves:Discuss math as a model. Say, “Now the math shows that the left glass has more. But turning the world into mathematics isn’t always a super precise process. It can result in some error. Let’s find out how close we were.”Give status to students who guessed correctly. Ask the students who originally guessed “Left.” Have the rest of the class give them exactly one clap.
392. Where we live on the food chain Economic security and social well-being Innovation and productivityHuman capital and equity of opportunityHigh quality education(literacy, MATH, science)Daily classroom math instruction
403. 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.
41(That’s the hope of the CCSSM for Math) 4. 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 CCSSM for Math)
42Non-negotiable take-away: Make no mistake, for K-12 math in the U.S., this (the advent of Common Standards) IS brave new world.
43Full disclosureFor better or worse, I’ve been drinking the CCSSM Kool-aid. Leinwand on the CCSSM in the 2011 Heinemann catalog.
44A Long Overdue Shifting of the Foundation For 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 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
45Kool-aid (for you youngsters) The flavored crystals you mix with water and put in the large pitcher because it’s cheaper than soda.As in Tom Wolfe’s “The Kool-aid Acid Test”, the liquid used to mix the LSD among Ken Kesey and the Merry Pranksters.The liquid Jim Jones and his cult members used to ingest the mass suicidal poison.In all three cases, nothing to be proud of.
46So finally, let’s take a look at the game changer: The Common Core State Standards for Mathematics
47PromisesThese 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. — CCSSM (2010, p.5)
48Some design elements of the CCSS Fewer, clearer, higherFairer – rational grade placement ofproceduresNCTM processes transformed intomathematical practicesLearning trajectories or progressionsSpirals of expanding radius – lessrepetitiveness and redundancyA sequence of content that results in all studentsreaching reasonable algebra in 8th gradeBalance of skills and concepts – what to know andwhat to understand
49CCSSM 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.
508 CCSSM Mathematical Practices Make sense of problems and persevere in solving them.Reason abstractly and quantitatively.Construct viable arguments and critique the reasoning of others.Model with mathematics.
518 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.
52Some glimpses of the future (the real hope and implications)
53Dream with me Our reality in 18 months! Assessment as a process for gathering evidence to support claims about studentsComputer adaptive assessments3 week testing windowsComputer-scored constructed response itemsItem banks for formative, benchmark AND summative assessmentsStudent, class, school, district, state and nation results just days after the window closesOur reality in 18 months!
56In a sale, all prices are reduced by 25%. PARCC/SBACIn a sale, all prices are reduced by 25%. 1. (Claim 1) Julie sees a jacket that cost $32 before the sale. How much does it cost in the sale? Show your calculations.This item with multiple independent parts provides evidence for claims 1, 2, and 3. Question 1 contributes to Claim 1, Question 2 contributes to Claim 3 and Questions 3 and 4 contribute to Claim 2. [Simple procedural applications, even those embedded in context like #1 above, are considered Claim 1 items, NOT Claim 2 problem solving.]Smarter Balanced Content Specifications
57In the second week of the sale, the prices are reduced by 25% of the previous week’s price. In the third week of the sale, the prices are again reduced by 25% of the previous week’s price. In the fourth week of the sale, the prices are again reduced by 25% of the previous week’s price.2. (Claim 3) Julie thinks this will mean that the prices will be reduced to $0 after the four reductions because 4 x 25% = 100%. Explain why Julie is wrong.3. (Claim 2) If Julie is able to buy her jacket after the four reductions, how much will she have to pay? Show your calculations.4. (Claim 2) Julie buys her jacket after the four reductions.What percentage of the original price does she save? Show your calculations.
59What’s to be afraid of? Ok – we’ve got the standards, go do ‘em Another fad with political intrusionDoomed by the same lack of capacity that got us into this mess in the first placeAssessment compromisesNot enough time gelWe forget that it’s instruction stupidWe ignore the essential roles of school and department culture
60But what’s to be so hopeful about? Systemic alignment of standards, materials, assessments and professional developmentLess attention to WHAT math and more attention to HOW to best teach itGreater collaboration around clearer and common goals and better dataMarket incentives for technology and videoA chance to finally focus primarily in instruction and learning
61It really is a brave new world https://labs.desmos.com/pennies
62So….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 mathematics program to better serve our students, our society and our future.