Category Archives: Common Core

Recap: CA Mathematics Network Forum, 2015

Logo CAMNThe 2015 California Mathematics Network is a community of math education leaders from twelve regions in the State. This Conference focused on the NCTM publication Principles to Actions. The book is an amazing resource that discusses what needs to be done in math classes, and what actions need to be taken by teachers and administrators alike to make that happen. It should be read by anyone who has an investment in math education. A good primer is p 5, 10, & 109-116, or check out the Executive Summary. Following are some terrific ideas from the conference speakers on how to implement these Principles.


The Best of the Common Core Closes the Achievement Gap — Dr. Lee Stiff, former NCTM President

  • Lee StiffThe Achievement Gap can best be narrowed through Effective Teaching of the CCSSM Practices.
  • Where do these effective teachers come from? … “from our good work!” (as instructional leaders)
  • The primary purpose of Principles to Actions is to fill the gap between the adoption of rigorous standards and the enactment of practices, policies, programs, and actions required for successful implementation of those standards.
  • NCTM Guiding Principles
    (from Principles to Action)
    Teaching and Learning
    Access and Equity
    Curriculum
    Tools and Technology
    Assessment
    Professionalism
  • NCTM Teaching Practices
    (from Principles to Action)
    1. Establish mathematics goals to focus learning.
    2. Implement tasks that promote reasoning and problem solving.
    3. Use and connect mathematical representations.
    4. Facilitate meaningful mathematical discourse.
    5. Pose purposeful questions.
    6. Build procedural fluency from conceptual understanding.

    7. Support productive struggle in learning mathematics.
    8. Elicit and use evidence of student thinking.
  • Student placement and support should be based on DATA not DEMOGRAPHICS.
  • We create the gap!!
    Screen Shot 2015-05-21 at 10.50.39 PM

Teaching Practices that Support Student Learning of Mathematics — Peg Smith, University of Pittsburgh

Peg Smith PicDr. Smith had us read through a well-known task, the Hexagon Train, and then analyzed it through the lens of each of the Teaching & Learning Principles in Principles to Actions (Summarized Below):

Hexgon Train

 

 

1. goals
2. tasks
3. representations
4. discourse
5. purposeful questions
6. procedural fluency

7. productive struggle
8. evidence of student thinking

  • It’s all about the task. Choosing the task really matters.
  • “What you put in front of the students frames their opportunity to learn the mathematics.”
  • Have your questions “locked and loaded,” and your responses “in your back pocket.”
  • It’s time to break out of the “postage stamp” lesson plan, (the homework, & examples fit in a little box), and write analytical, anticipatory lesson plans. (This one needs a cute name, too)
  • It’s difficult for teachers to use a high level task. It’s even more difficult for them to use it well.
  • Decrease the complexity of language without decreasing the cognitive demand of the task.
  • “Never Say Anything That a Kid Can Say.” (Article)
  • Writing “SWBT” objectives limit what students learn. Is the goal really to be able to find the length of the hypotenuse or to understand the relationship of the areas of the squares formed by the three sides of a right triangle?
  • Dr. Smith is the co-author of 5 Practices for Orchestrating Productive Discourse in Mathematics Class.
  • Dr. Smith shared this Principles to Action Tool Kit:

Dr. Smith then asked us to restructure a standard series of textbook questions into a more robust task. The conversation at my table was very rich. It was a briefer version of a lesson makeover, and would be an awesome PD activity.


Smarter Balance Update — Mary Tribbey & Jane Liang

This slide makes two BIG statements:

  1. The Red Dot () is along a timeline from the start of the assessment initiative to full implementation. We are still in the early stages of perfecting it.
  2. There do exist Interim Assessments that few schools (including mine) are using to check for student readiness.

Screen Shot 2015-05-19 at 9.52.34 AM

This day was the first I heard of the scaled score for the reporting of the test. It appears that there will now be some reporting on the standards as well as the claims, after all.

Screen Shot 2015-05-19 at 9.53.08 AM

 


Equity-Based Teaching Practices — Karen Mayfield-Ingram, EQUALS Program, UC Berkeley

  1. Mayfield PicGoing Deep with Mathematics
  2. Leveraging Multiple Mathematical Competencies
  3. Affirming Mathematics Learner’s Identity (multiple access points)
  4. Challenging Spaces of Marginality (diminish status within class)
  5. Drawing on Multiple Resources of Knowledge (including culture and experience)

Lesson: “He Was Suspended for Being Mexican” (excerpt from The Impact of Identify in K-8 mathematics Learning and Teaching) This was an anecdote of a teacher who took a students statement, “He was suspended for being Mexican,” and turned into a statistics lesson in which the students had to analyze data to determine if the school policies truly were racist or not. While we can’t tie every topic into a student-oriented context, I think it is a powerful idea that should be done more often.


Technology & Computation — Joe Fielder, Cal State Bakersfield

  • Pic FeidlerAll computation outside the classroom is done by a machine.
  • Machine computation is mostly done with spreadsheets.
  • Hand calculations are only done in math classes. (referenced TED talk by Conrad Wolfram)
  • If we are going to teach students mathematics that is relevant beyond the college entrance exam, we need to give explicit instruction on the tools of computation.
  • TI InspireDr. Fiedler is currently working with the college board to change the SAT to reflect computations done by hand-held graphing calculators.
  • The introduction of the first scientific calculator 1972 was controversial, because teachers were worried that students would no longer be able to use tables.
  • “Students are idle, indifferent, irresponsible in response to absurd work. This is a rational response!”
  • There is no change without a loss. If there is no loss, there is no change. Similarly, literacy diminished the need for memory, but we still teach students to read and write.
  • Yes, part of education’s job is to pass on old knowledge, but it’s not the entire job. It’s time to get with the times.

BREAKOUT: Exploring the Common Core Statistics & Probability Standards — Jim Short, Ventura County Office of Ed

  • Pic Jim Short“Statistics means never having to say your certain.” The irony is that this is what makes math teachers uncomfortable with stats.
  • Teachers are avoiding the teaching of statistics, but the ponderous of the Performance Tasks on State Tests are based on Statistics and Data Analysis.
  • Statistics is more important than Calculus. (referenced TED talk by Benjamin Arnold)
  • From the GAISE Report,
    4 Components of Statistical Problem Solving
    I.   Formulate Questions
    II.  Collect Data
    III. Analyze Data
    IV. Interpret Results
  • You aren’t teaching statistics unless you are teaching modeling.Here are some great tools that we used in the session to generate statistical displays in a spreadsheet:
    g(math) {Google Sheets add-ons}
    Geogebra {box-n-whisker}

    Core Math Tools {NCTM}
    =norminv(rand(), means.d.)” {Excel Macro for generating a set of normalized data}
    Stats vs Prob

BREAKOUT: The Right Answer is Not Enough — Ivan Cheng, Cal State Northridge

  • PIc Ivan ChengWhat the teacher assesses is what the students think that the teacher values.
  • How is “doing math” defined differently under Common Core versus NCLB? How you answer that questions, determines how you teach and assess under the new standards.
  • After a test, if the teacher can’t state what the student misconceptions are, then the teacher needs to do some more digging.
  • Teachers should use assessment questions that intentionally reveal misconceptions.
  • Why “a” student missed a question is as important as which question they missed.
  • Clicking Smarter Balanced ASSESSMENTS (in SBAC navigation bar) will take you to documents that map targets to standards.
  • “Think about getting through to the kids instead of getting through the textbook.”
  • This sample question demonstrated why the students have issues with the new assessments. The students instantly think that the answer is “20,” because x = 20. Since 20 is not a given situation, they often choose “D: Neither.”

Inequality Sample


My Big Take-Aways

  • The achievement gap can be closed by the effective teaching of the Math Practices.
  • It’s all about the task!!
  • Two Big Words kept coming up: Meaningful & Equity. Equity is achieved by giving all students access to meaningful, high-level mathematics.
  • Get with the times, and start using technology in order to move from computation to deeper, higher mathematics.
  • There are some amazing tools available for Statistics tasks. This is a pervasive topic that needs serious attention and support.
  • Our assessments communicate what we value. The assessments are changing, because our goals are changing. Therefore, we teachers must change our values and practices.
  • We should all read Principles to Action.
  • The Region 10 Team is an amazing group of intelligent, passionate people. I look forward to seeing how we will put all these principles into action.

Region 10

Advertisements

Recap: NCTM 2015, Boston

NCTM Boston CropI had the wonderful opportunity of spending a week in Boston for the 2015 NCTM & NCSM conference. I am recapping the NCTM sessions here, and the NCSM sessions in another post.

Since there was so much information, I have summarized each session with some simple (•) bulleted notes and quotes to encapsulate my major take-aways, and occassional a brief italicized commentary.

This was an enormously worthwhile trip. I highly recommend that you get yourself to San Francisco in April 2016, if you can.


NCTM President’s Address: Five Years of Common Core State Mathematics Standards — Diane Briars

  • Diane Briars“College and Career Readiness” in math calls for Statistics, Discrete Math & Modeling.”
  • Standards are not equal to a curriculum.
    We need to pay more attention to the tasks & activities through which the students experience the content, rather than simply focusing on the content itself.
  • 75% of teachers support Common Core, but only 33% of parents support it, and 33% of parents don’t know anything about it.
    So we have to get the word out.

What Decisions — Phil Daro (1 of 3 writers of CCSSM)

  • Phil Daro“Don’t teach to a standard; teach to the mathematics.”
    This was the most challenging statement of the conference for me, mostly because I’m still struggling to wrap my mind around it. I get what he means, but I have been so trained to state an objective on the board and bring closure to that lesson. He shared that Japanese lesson plans are simply descriptions of the math concepts of the unit rather than the typical American model of objective, examples and practice sets.

The Practices in Practice — Bill McCallum (1 of 3 writers of CCSSM)

  • MCCallumStudents understanding what WILL happen without doing the calculations is an example of Using Structure.
    I took this back to my classroom and immediately applied it in the students’ graphing of quadratic functions. One of the more practical things I took from the conference.
  • “A student cannot show perseverance in 20 minutes. It is done day after day.”
  • Noticing & Wondering applies to teachers looking at student work as well.
    Dr. McCallum was referencing an instructional practice made well-known by Annie Fetter of the Math Forum through which students are asked to closely analyze mathematical situations. He was calling on teachers to focus on and analyze student thinking (not simply answers) just as closely.

Five Essential Instructional Shifts — Juli Dixon

  • DixonShift 1: Students provide strategies rather than learning from the teacher.
  • Shift 2: Teacher provides strategies “as if” from student. “When students don’t come up with a strategy, the teacher can “lie” and say “I saw a student do …”
  • Shift 3: Students create the context (Student Generated Word Problems)
  • Shift 4: Students do the sense making. “Start with the book closed.”
  • Shift 5: Students talk to students. “Say Whoohoo when you see a wrong answer, because we have something to talk about.”
    I felt that I do all of these, but that I have been ignoring Sgift #3 this semester. Dr. Dixon compelled me to give this more attention again. 

Getting Students Invested in the Process of Problem-Solving — Annie Fetter & Debbie Wile

  • AnnieTeachers must stop focusing on answer getting before the students will.
  • Honors Students are used to a certain speed and type of outcome, so they need a different type of scaffolding when it comes to problem solving.
  • “If you are focused on the pacing guide rather than the math, you are not going to teach much.”
    This was one of several comments, including Dr. Daro’s, that bagged on the habit of being too married to a pacing guide of standards.

Motivating Our Students with Real-World Problem-Based Lessons — Robert Kaplinksy

  • Kaplinsky CroppedTo students: “I will only give you information that you ask me for.”
  • Chunking tasks (Teacher talks — Students Think/Pair/Share — Repeat) was demonstrated to allow student conjectures, critiquing reasoning and high engagement.
    Robert modeled his “In-n-Out” lesson. I have seen this several times, but I never get tired of it, because it is awesome. Every time I have witnessed this lesson, teachers cheer when the answer to the cost of a 100 x 100 Burger is revealed. I have never heard this from someone looking up an answer in the back of a textbook. Also, Robert expertly demonstrated how a lesson like this should be facilitated in class by chunking and by getting the students to think of what they need to know.

Getting Students to Argue in Class with Number Sense Activities — Andrew Stadel

  • vQWJdnFF“As the teacher, I feel left out if I don’t know what my students are thinking and discussing.”
  • Discussion techniques
    Andrew is known as Mr. Estimation 180.” In this session, he showed how to bring SMP #3 into a number sense activity. The new one that I learned from Andrew here was having students stand up … those that choose A face left, B face right, C face forward. Then find someone near you who agrees and discuss. Find someone who disagrees and discuss. That’s Bomb!
  • Calling for Touch Time with the Tools
    In other words, let’s get the kids measuring with rulers, constructing with compasses, building with blocks, graphing with calculators.
  • Chunking tasks to allow student conjectures, critiquing reasoning and high engagement, as I saw with Kaplinksy.

Using Mathematical Practices to Develop Productive Disposition — Duane Graysay

  • duaneDuane and other educators of Penn State created a 5-week course with the intent of developing a productive disposition in mathematical problem solving.
    There were a lot of data showing the effectiveness of this program which focused on teaching the 8 Math Practices. The most amazing and provocative result was shown by this slide in which student felt that the math was actually harder than they thought before the course, but that they felt more competent.

2015-04-18 08.46.25

(SA = Strong Agree, etc)


Shadow Con — A Teacher Led Mini-Conference

  • Michael pershan-219x181There were six worthy educators from the Math Twitter Blogosphere (#MTBoS) that each offered up a brief 10-minute presentation. The uniquely cool aspect of these talks is the Call to Action at the end of each. In other words, you have to do something with what you learned.
  • Michael Pershan’s talk: Be less vague, and less improvisational with HINTS during a lesson. Instead, plan your hints for the lesson in advance.
    This one resonated most with me, because I once heard that Japanese teachers have a small deck of cards with hints written on them. To draw a hint card, students have to first show effort and progress, then they may draw a hint card. They must use each hint before they may draw another. I accept Michael’s call to action.

 Ignite — Math Forum

  • IgniteThese were a series of 5 minute/20-slides mini-presentations that were more inspirational than informational. Apparently they are part of a larger movement (Ignite Show), but the folks at Math Forum have been organizing these Ignite Math Sessions at large conferences for a few years.
    If want to get fired up about teaching math, these sessions definitely live up to their name. 

Can’t wait for next year!

A Call to Substance, First Interview with Dr. William Schmidt

Schmidt BookcaseIn March of 1998, during the inaugural year of The Math Projects Journal, we had the unique opportunity to publish our interview with Dr. William H. Schmidt, of the University of Michigan. At the time, Dr. Schmidt was the National Research Coordinator and Executive Director of the U.S. National Center which oversaw participation of the United States in TIMSS. The results of the TIMSS report directly led to the developement of the Common Core 20 years later, which is why Dr. Schmidt is nicknamed “the Godfather of the Common Core.” He is also widely published in both journals and books on mathematics education.

We had the opportunity to interview Dr. Schmidt again about the rollout of the new curriculum. Before we post the current interview, we thought it would be valuable to reprint what Dr. Schmidt had to say in the early years of the research. He emphasized focusing instruction on conceptually understanding and higher order thinking skills, rather than on methodology. This is an important message now more than ever with so many untested techniques and ideologies being promoted widely on the internet. This message heavily influenced the trajectory of MPJ‘s lessons and my own classroom teaching. I hope it does the same for yours.
*************

MPJ: Can you give an example of a model lesson from one of the top achieving countries, either Germany or Japan, which are the focus of the videos?

Dr. Schmidt: If you look around the world, there just isn’t a single way to teach that is dominate among the top achieving countries. Some of them are very didactic, lecture-oriented classes. Some of them are like the kind that you see in the Japanese tapes. If teachers know their mathematics well, they can be just as engaging through a lecture format, as they can teaching as the Japanese do. It is very clear to me that there isn’t one way to do this. Instead, the more analysis that I do, the more I believe that there are some principles involved here that just might go across countries.

MPJ: What is the common thread?

Dr. Schmidt: I think the common thread that makes for the top-achieving countries is pure, honest-to-goodness mathematical substance. If the teachers really know and understand the mathematics, then they bring that to the students, through whatever means they know best. Also, a large part of this idea is to develop this stuff conceptually and not just algorithmically. I think many people misunderstand the Japanese videos. It is not so much the methodology, as it is the mathematics. You watch those lessons and the instructor really understands the mathematics, engaging those students in more ways than we do in this country.

MPJ: So, if a teacher were to do a dog-n-pony show lecture with drill-n-kill practice, and do it well, would it work?

Dr. Schmidt: The dog-n-pony show lecture, yes; the drill-n-kill, no. That’s what I said about there being some principles. I think the common element is a clear understanding of the subject matter and then going through it much more conceptually than algorithmically.

MPJ: Can you give us a model to how to teach math conceptually rather than algorithmically?

Dr. Schmidt: A U.S. lesson typically starts out with the algorithm. For instance, there is the example in the videos of a guy teaching geometry. He says to the kids, “Here are two supplementary angles, one is thirty, how much is the other?” A student says, “a hundred and fifty.” And the teacher says “Good, now why is it that?” And her response is, “Because they are supplementary.”

Instead, conceptually, you could show them that if they measure a straight line, it’s always one hundred and eighty degrees. Then they realize that if they put a line anywhere its going to cut it into two parts. That’s conceptual; you start with understanding why, so if you forget the stupid name, supplementary, and you see a line with an angle you’ll know what the other one is. That’s the difference.

MPJ: How is a strong conceptual understanding of the mathematics developed among teachers?

Dr. Schmidt: It comes from two sources. In some countries, they must major in these fields. The other thing we don’t think about is that they are products of their own systems. For instance, Japanese teachers don’t necessarily take more mathematics at the university level than we do. But look at what they already know before going to the university. They are already ahead.

MPJ: In regard to the things that our readership is looking at, active learning, projects, manipulatives, do you have any models from these other countries, or that you think could be done here?

Dr. Schmidt: You don’t find very much of that anywhere else. They seem to be uniquely American inventions, especially the cooperative learning. We asked teachers how much they use groups, and it’s pretty much nonexistent. We are too much into the methodology in this country, and we miss the substance. We start talking about small groups and manipulatives and it just becomes process. Therefore, the substance behind it gets lost in the shuffle. And for a lot of these ill-prepared teachers, that’s what they grab onto because that’s what they understand.

MPJ: We hear that the US teachers assign more homework and spend more class time dealing with homework than the top achieving countries.

Dr. Schmidt: The dominate activities in the U.S. lessons were reviewing homework and doing seat work. One thing that was startling is that the typical American lesson had only 10 minutes or less of instruction.

MPJ: What role does homework play in other countries?

Dr. Schmidt: It varies a lot. Japan doesn’t give a lot of homework, but the kids study for the next lesson. There’s a difference, of course. Studying is what you do at the university, and homework is what you do in grade school. But Japan is unique. Worldwide, homework and seat work are still the dominate activities. I think if you do that and you do it well, and develop the topics conceptually, it can work.

MPJ: Is this a curriculum issue instead?

Dr. Schmidt: It is the core issue, but just putting that in place by itself wouldn’t work. You have to help teachers teach in ways that engage kids.

MPJ: So, that is something that teachers could start doing today. We could focus on engaging students and developing topics conceptually?

Dr. Schmidt: That is my point. We must start paying much more attention to the subject matter and teach it more conceptually and less algorithmically. And that is why we are in a catch-22. The Japanese teachers grew up in their system seeing math developed conceptually, no matter what they learned at the university level. For our teachers it is a lot more difficult; they have to break out of a mold that they’ve been put into. But I think that is something that teachers can do — Get off the algorithmic side. Don’t just give an equation and when a kid asks why say, “Because that’s the equation.” Try to get them to understand what lies underneath some of this stuff.

MPJ: It seems that, chronologically, you are suggesting a lesson should move from concept to algorithm to application.

Dr. Schmidt: A lot of the lessons that we’ve seen, like in France and such, start out with an application as a motivator. An example is a science one about transformers. They started out by looking at a map of the city and looking how electricity would flow. This got them hooked on the issue, then they hit them with some good hard science about the transformer. That’s very often how it happens: hook them with some kind of application, then take them into it conceptually, let them flounder — that’s where I think what the Japanese do is a good idea — let them talk about some of their ideas, then give them an algorithm, a formula and a few examples. Whereas we typically start with the formula with a few sentences about it, and then have them do worksheets.

MPJ: The report states that American textbooks cover too many topics, yet they typically have only fifteen chapters.

Dr. Schmidt: That is mistaking the notion of what a topic is. The definition of topic has to do with the substance of the mathematics, and when we defined it that way, the measurement across all these topics is not how many chapters are in each book.

MPJ: Can you give us an example of four or five topics?

Dr. Schmidt: Congruence and similarity, three-dimensional geometry, linear equations, and fractions. We actually tested 44 topics and determined how many of these topics were in any given textbook. Our 700 page books address about 35 topics. The Japanese, on the other hand, spend half of the eighth grade year on congruence and similarity alone, and their gain in that year is higher than in any other country. The dilemma I have in telling you what to do is that the teacher shouldn’t decide which five to ten topics should be studied in a year. It only works if somebody coherently lays this thing out as to what needs to be done.

MPJ: Do you have any last things to add?

Dr. Schmidt: People still think that there are general things a teacher should do, like cooperative learning. That’s what people push. We push all the things that have nothing to do with subject matter. I’d like to challenge the notion that there is a single way to do things. If you listen to the ideological left, they say that there is only one way to teach. And the data just do not support that. Among the top achieving countries you cannot find one dominate way of teaching. On the other hand, the ideological right are calling for “the basics.” Yet, the latest analysis shows that the United States, through 8th grade, does average or above average in all the standard arithmetic skills. This is not the place were we are hurting the most. That is all we teach. That is what’s wrong, we never go beyond the basics.

If I wanted to become rich and be an advisor to schools to jack their scores up, I know how to do it. We have certain areas of math that we have the international comparisons on. I can tell you the seven items that we are the weakest on, and if schools just did something in those areas, we’d go up in the international rankings. None of those areas is anything that we would consider the basics.

MPJ: What are those area of weakness?

Dr. Schmidt: Measurement, error analysis, geometric shapes, perimeter, area and volume, congruence, similarity, vectors, geometric transformations, and three-dimensional geometry. These are not the basics.

MPJ: Tomorrow, our readers will not be able to change the textbooks or create national standards. What can a teacher do in the classroom today that will model the type of change that you and the TIMSS report call for?

Dr. Schmidt: That’s a tough question, because most of what I have argued is, based on the data, these really are systemic issues. However, the data also shows that how we teach is as important as what we teach. Teachers should challenge students with more mathematical substance and develop the ideas more conceptually rather than algorithmically.

Get to the Core of The Core

apple coreThe Common Core curriculum can basically be summed up in the following sentence:

Teach your students to THINK and COMMUNICATE their thinking.

Thinking and communicating are the 21st Century skills. Many people believe that the skills of the future involve the competent use of technology. While it is true that using digital tools in school and the work place is the new reality, it is actually the proliferation of technology that makes thinking and communicating imperative in the information age. When all the knowledge of humankind is available at anyone’s fingertips, memorizing information becomes far less important than being able to construct, evaluate and apply it. You can Google information; you cannot Google thinking.

So the core of the Core truly is Thinking & Communicating.

To make my case for this, I would like to pose that the following equation

6 + 4 + 4 + 8 = 22

be adjusted to

6 + 4 + 4 + 8 = 21

Before you start shouting that everything you have read on Facebook about the Common Core is true, let me declare that I am using this equation simply as a teaching device, not a true mathematical statement. You will understand what I mean after I present my evidence.

6 Shifts

Let me start my case that the core of the Core is Thinking & Communicating with the 6 Shifts, which are best represented by the following document found at Engage NY.

6 Shifts

In essence, these shifts are redefining rigor. Old school rigor was defined as sitting quietly taking notes, and completing long homework assignments in isolation. The new school definition of rigor envelops the last 4 shifts on the list: Fluency, Deep Understanding, Applications, and Dual Intensity. The rigor is now placed on the students’ minds instead of on their behinds.

The shifts are also calling for balance. Dual Intensity insists on both procedural fluency AND critical thinking by the students at a high level. It is not about dual mediocrity or about throwing the old out for the new, but a rich coupling of both mechanics and problem solving.

Therefore, I make the case that:
                     6 Shifts = 21st Century Skills,
which are to
                     Think & Communicate.

4 C’s

Another list that is framing much of the Common Core dialoge is the 4 C’s. Resources for this list can be found at Partnership for 21st Century Learning (p21.org).

4 C'sThese C’s redefine school…

The old school definition: A place where young people go to watch old people work.

The new school definition: A place where old people go to teach young people to think.

… and they redefine learning.

The difference of old school vs new school learning can best be contrasted by the following images of the brain.

Brain Chillin   Brain Build

The image on the left shows a passive brain that just hangs out as we stuff it with esoteric trivia. The image on the right shows a brain being built, symbolizing its plasticity. We now know that when the brain learns, its neurons make new connection with each other. In other words, learning literally builds the brain. The 4 C’s  claim that this building involves the capacity of the students’ brains to Critically Think, Communicate, Create and Collaborate.

Therefore, I make the case that:
                     4 C’s = 21st Century Skills
which are to
                     Think & Communicate.

4 Claims

Smarter Balance creates it’s assessments based on 4 Claims. (I teach in California. PARCC has 4 Claims that closely align to those of SBAC.)

SBAC

4 Claims

Notice that Claims #2 & 3 are explicitly stated as Thinking & Communicating, which also overlaps with two of the 4 C’s. Mathematical modeling is #4, which will be discussed later. I want to point out here that Claim #1 reinforces our idea of Dual Intensity from the 6 shifts.

There are two important notes for teachers about this first claim. 1) It says Concepts and Procedures, not just procedures. The students need to know the why not just the how. 2) The Procedures alone account for about 30% of the new state tests, so if we continue to teach as has been traditionally done in America, we will fail to prepare our students for the other 70% of the exam which will assess their conceptual understanding as well as their abilities in problem solving, communicating and modeling.

Therefore, I make the case that:
                     4 Claims = 21st Century Skills
which are to
                     Think & Communicate.

8 Practices

If you open the Common Core Standards for Mathematics, the first two pages of the beastly document contain a detailed description of the Standards of Mathematical Practice. Then at the beginning of each of the grade level sections for the Standards of Content you will find 8 Practices summarized in the grey box shown below.
8 practicesWhat do you notice about the list? Indeed, these habits of mind all involve Thinking & Communicating. While the content standards change with each new grade level, the practice standards do not. With each year of school the students are expected to get better at these 8 Practices. Notice that the first half of the list has already been included in the ones discussed previously: Problem Solving, Communicating Reasoning, Constructing Viable Arguments and Modeling. A case is often made that the other four are embedded in these first four. However one might interpret the list, “Memorize and Regurgitate” is not on there.

Therefore, I make the case that:
                     8 Practices = 21st Century Skills
which are to
                     Think & Communicate.

The Sum of the Numbers

So, as you can now see, the 6 Shifts, the 4 C’s, the 4 Claims and the 8 Practices are all focused on the 21st Century Skills of Thinking & Communicating. Therefore, I can finally, explain my new equation …

Since,

    6 Shifts
    4 C’s
    4 Claims
+  8 Practices
= 21st Century Skills

then 6 + 4 + 4 + 8 = 21!

None of these numbers represents a list of content, because the content changes brought on by the Common Core, while significant, are actually no big deal in the long run. A few years from now we won’t remember all the fuss regarding Statistics and Transformations, but we will all spend the rest of our careers learning how to teach kids to Think & Communicate.

I rest my case.

Common Core and The Land of Oz

Oz FourThe Common Core is a noble cause. Who would argue that teaching kids to think and communicate their thinking is anything but a virtuous goal? It’s like the Emerald City in the Land of Oz, and standing between us and that bright shining city is a Wicked Witch and a bunch of Flying Monkeys. We know how the movie ends, though; we will melt that witch and make it down the Yellow Brick Road.

I made this comparison for a news reporter after my keynote address at the Idaho State Math Conference last fall. My analogy made NPG News at the same time that my math coaching colleagues and I back at Temecula Valley Unified were developing a four-year plan for professional development and student support in our district. So we wove the Wizard of Oz theme into our plan.

It turned out to be more than a catchy metaphor. The theme is actually quite symbolic to the trials and potentials of rolling out the common core.

4 Year PlanLet’s begin with the Emerald City. The Common Core claims to teach students 21st Century skills. In our district, we have summed up those skills as the ability to “Think and Communicate.” This, then, is our noble cause, our shining city.

Along the Yellow Brick Road is the infamous Wicked Witch and her Flying Monkeys. Our number one issue for teachers in Year 1 of the roll out was the lack of resources, and therefore, the demand upon them to find and create their own curricula. We did not anticipate this phenomenon, but it quickly consumed our role as math coaches. Our first year will end (hopefully), with Units, Pacing Guides & Model Lessons in place, and with an infrastructure to share them among the 130 secondary teachers in our district. Since this is by far the biggest obstacle facing us, and the ugliest work to overcome, establishing the content, scope and sequence gets the tag as the Wicked Witch. In Year 2 (the first of the Flying Monkeys) our primary purpose is to change our method of first instruction. The Common Core is calling for radical shifts in how we teach as well as what we teach, so that will be the focus of Year 2. Year 3 then focuses on what to do for those students who don’t get it (Tier 2 intervention). Finally, while we continue with the work that we laid out in the first three years, Year 4 will emphasize enrichment for students who easily learn the material and on implementing student use of technology.

Reflection FrameWhile many of the obstacles listed above deal with the work of us math coaches, the work of the teachers is personified by the four main characters of Oz: Dorothy, Tin Man, Cowardly Lion and Scarecrow. Their training is structured around the four Essential Questions of a PLC (Professional Learning Community). Dorothy must go first, because she was all about direction (“There’s no place like home.”)  So she asks the question, “What do we want the students to know and be able to do?” The Common Core has defined this question very clearly for us, particularly when it comes to the Mathematical Practices. We summed up these practices on a Reflection Fame that we use to debrief with teachers after our elbow coaching sessions. Year 2 calls upon the Tin Man, because it takes a heart to care for those students who don’t get it, especially in secondary schools. We are now commissioned to deliver a “guaranteed and viable curriculum to ALL students.” Year 2 will focus then on Tier 1 interventions … reaching and teaching ‘those kids’ … within the classroom. In order to do this we must have formative assessment and data collection protocols in place to be able answer the question “How do we know if they know it?” The Lion personifies Year 3, because it will take Courage to deliver Tier 2 intervention in response to “What do we do when they don’t know it?” Then, to answer the question “What do we do when they do know it?,” the Scarecrow and his brain will be employed in Year 4, when all the mighty work of the first three years is in place, and we can focus on the needs of the advanced students and on teaching all students to Think with and Communicate through technology.

Finally, and most importantly, we turn our attention to the students results. These are personified by who else, but the Munchkins. We plan to establish Student Mile Markers. These will be Performance Task benchmarks that will be given each year with the Final Exams (but not necessarily counted in a grade) to be used as a gauge to our collective progress (that of students, teachers, coaches and administrators) down the Yellow Brick Road.

The Wizard of Oz gives us a nice frame to dialogue within, but it also offers an important lesson for all teachers. The Wizard gave Dorothy and her friends absolutely nothing, other than the realization that they already had inside each of them that which they had been seeking all along. As do we. Brains, Courage, a Heart, and a Direction Home.

Common Core Pathways: Redefining Algebra

PathwaysI have fielded a great many questions lately regarding the creation of Common Core Pathways (course sequences), especially in regards to the big question: to accelerate or not to accelerate. I appreciate the curiosity, because in this last year I did a great deal of investigating in order to help my school district develop our own pathways. I recently had a request to share our pathways “with commentary.” This makes sense, since there are many misconceptions of the Common Core out there that I had to sort through, and the rationale for these pathways will help others decide if these will work for their system. So I share four things:

1) A primer for the Common Core Pathways, particularly in terms of Algebra content.
2) The needs of my district that led to the development of three pathways.
3) The actual Pathways that my district decided upon, with links to resources that helped us get there.
4) Student placement.

I hope this helps.

A Common Core Pathways Primer

The Common Core spells out clearly what students are expected to know at each grade level K-8. Then for high school it lumps the standards together in High School Domains (Number & Quantity, Algebra, Geometry, Functions, Statistics & Probability and Modeling). This is done in order to allow high schools to structure courses in a Traditional Model (Algebra 1, Geometry, Algebra 2) or an Integrated Model (Math 1, 2, 3). At first glance it looks like CCSS is now delaying Algebra until 9th grade, after years of states pushing it in the 8th grade. This is because CCSS does not define Algebra as a course, but rather a domain across grade levels. Understanding this is key to creating accelerated pathways.

Traditionally, an Algebra course is seen as starting with the arithmetic (integers & fractions) and the simplifying of expressions (which many consider to be Pre-Algebra), followed by solving of equations, then moving onto linear equations and systems by the end of first semester, with polynomials, quadratics and rational expressions rounding out second semester. In other words, we go from balancing a check book to racing cars to launching rockets in a single year. However, the Core spreads these concepts out over several years. Arithmetic, simplifying and basic solving is mastered in 6th grade. Solving multi-step equations and deeply understanding rates and ratios is the focus of 7th grade. The 8th grade standards focus on linear equations and systems. While Geometry topics like surface area, volume and transformations are spread throughout the middle school grade levels, along with probability & statistics, the key here is to see that the entire first semester of a traditional algebra course is covered by the end of 8th grade. This way, the students can be handed an exponential function when they walk in the door on the first day of their freshman Algebra class. So don’t get it wrong; students under the common core are still learning Algebra in middle school; they are just not finishing it. The Common Core does not delay the Algebra course for students; it simply redefines Algebra.

No More Than 3, Sometimes 4

Tim Kanold once shared with me the pathways created at Stevenson HS in Illinois. He claimed that they had two pathways… one pathway led to Calculus, another to Pre-Calculus. It was actually one pathway: Algebra 1, Geometry, Algebra 2, Pre-Calculus, Calculus. What made this sequence look like two pathways was the course that students enrolled in as freshmen (Algebra 1 vs Geometry). Stevenson HS offered a ride on a single train; the only variation was which boxcar a student boarded when arriving at high school. I ask Tim if every student graduated with a minimum of Pre-Calculus. He said that while 58% of the seniors graduated with Calculus, some only took three years of math. When I pressed for the pathway offered for special education students and the like, he conceded that those rare few were allowed to deviate from the given path. He stated, “Create only 1 path, no more than 2, and sometimes 3.”

My district embraced this idea, but we have one more level of need. My high school has an International Baccalaureate (IB) Program. In order for students to be able to reach its “Higher Level,” we need some students to come into high school taking Algebra 2 as freshmen. Furthermore, while California only requires two years of math, my district requires three, and the state still only requires Algebra 1 to graduate, not Algebra 2. Therefore, students on an IEP may take Middle School math classes through our Special Education Department, and anyone passing Algebra 1 may take Accounting to complete the third year.

With all that, my district adopted a “No more than 3, sometimes 4,” policy. These  3+ pathways are shown below.

The Pathways for Temecula Valley Unified

Pathways Math

Our district decided to stick with our traditional model. The scope and sequence of our “Common Core Pathway” is very similar to what the Dana Center of Texas produced. We also took some inspiration from Montgomery Schools in Maryland (scroll to the bottom of their page, and you will see a graphic very similar to ours) and Tulare County in California which beautifully laid out the scope and sequence for both the traditional and integrated models.

The Traditional Pathway allows students to reach Pre-Calculus or other similar 4-year college options. There are two keys to notice here. One, there is no remedial track. All mainstreamed students will be taking Algebra and Geometry. This is freaking out teachers who are anticipating having a significant number of “those kids” in college prep classes. They have told me that the kids won’t be properly prepared. I pushed back claiming that the kids will be ready, but I am not sure we teachers will be ready. (side note: Professional development training is imperative to make this work.). The second key to notice is that there are two types of Algebra 2 courses. Our Pre AP course was designed with the Common Core plus standards (+) included, for those students who intend to go beyond Calculus AB (Calculus BC or IB). For details on other courses shown in the diagram visit the Math Department at Great Oak High School.

The Accelerated Pathway was an easy adjustment. If we note the definition of Algebra explained above, then in 8th grade we teach a traditional Algebra course, substituting the Geometry and Stats topics for the Pre-Algebra topics. 6th and 7th grade remain untouched. Two years of math is condensed into one.

The Compacted Pathway was a bit trickier to create. In the past, students who wanted to take Geometry as 8th graders, simply skipped 6th grade math and got to Algebra 1 by 7th grade. That’s no so easily done now under the Common Core. So we have to compress 4 years of courses (6th, 7th, Algebra & Geometry) into 3 years as shown.

NOTE: Now that we have implemented these three pathways, I would only recommend the first two. Unfortunately, the Compacted Pathway is too much for both students and teachers. Since our high schools still need a means for students to reach Calculus B/C and beyond, it appears best to have that relatively small and uniquely talented population to accelerate in high school, through summer school, online options or Junior College courses.

Choosing a Pathway

The big question that follows after creating these pathways is “Which students are assigned which pathway?”  Or more to the point “Who gets to Accelerate?” We actually would like to see the majority of students follow the Traditional Pathway. For our upper level high school math programs to thrive, we need at least 20% of the middle school students on the Accelerated Pathway, and a little under 10% for the Compacted. Of course, we shouldn’t fit students to the needs of the school. The students are to be recommended by ability based on assessments and teacher recommendations. Our schools need to be watchful, though, because our community has parents who feel their child won’t be able to compete for a top college if they are in the bottom track. While some vigilance will be necessary, we also have an open access policy… students/parents may take any courses they wish. I am curious how these pathways portion out.

Furthermore on placement, another of Stevenson High’s policies that my district is adopting next year is the practice of moving students onto the next course … even if they flunk. I have also heard this same pitch from Bill Lombard. So, if a student flunks Algebra, the student enrolls in Geometry the following year, and makes up the class in summer school, online remediation or concurrently. Same thing is true when going from Geometry to Algebra 2. However, if a student fails Algebra 2, they may repeat, since these students have multiple options at this level (Trig, Stats, PreCalc, etc.). Needless to say, our teachers have a great deal to get done in terms of Intervention and Standards Based Grading to make this work.

I hope this helps those of you that are planning ahead. My district and its teachers still have a great deal of work ahead of us, so please share here what you learn in the construction of your own pathways.

Rich and Robust

Coffee beansI recently had the pleasure of learning from Tim Kanold of Stevenson High School fame. I heard him speak on several occasions last fall, and he kept saying that we need to involve students in “rich and robust tasks.” He was addressing the Common Core‘s call to the Standards of Practice. These practices can be summarized by saying that the Common Core is demanding students to think and to communicate their thinking. This can’t get done by taking notes from an overhead and doing the odd problems in the textbook. It gets done by purposefully deciding that students are going to solve rich and robust problems rather than simply watch their teacher complete examples of algorithms.

There is still a time and place for direct instruction and guided practice; but that should not be the complete experience for students, which is what we unfortunately find in the vast majority of American classroom instruction. For quite sometime, MPJ has been producing what we hope to be rich and robust tasks. Due to the growth of the internet, the availability of such rich and robust tasks has expanded tremendously. There are many exotic islands of innovation among the seas of tradition, but the blogosphere has made these islands less remote. Below I have listed a few, alongside my paraphrasing of the some of the Common Core Practices. This is not a comprehensive list by any means, however, I encourage you to take a few minutes and peruse these lessons in order to get a quick taste of what I think Kanold means by Rich and Robust.

Listed here are some additional sites that offer rich and robust tasks. {Note: I will be happy to update this list with any reader-submitted links, subject to review.}

The activities listed above obviously are not your typical math lessons. For good or for bad, the mathematical frontier created by the experiences highlighted here would make for a far different academic education than the gauntlet of lectures that most of us remember from school.

Now, I am going to assume that while the thought of introducing these large-scale examples into one’s repertoire is exhilarating for many, it may be terrifying for some. Let me ease those hearts by saying that rich and robust can be done on a much smaller scale. For example, we could simply ask the students: “Is x times x equal to two x or x squared. In other words, which of the following statements do you think is always true, if either: x·x = 2x  or x·x = x2?”

The CC Practices call for students to construct viable arguments and critique the reasoning of others. If your students stare back at you in silence with this question, then you will know why the Common Core Practices are so needed. If you answer the question for them, then they will watch you participate in a rich and robust activity, while they again participate in mundane note taking. For those that believe that this prompt is too elementary for any course above Algebra, let me assure you that it is not. I posed this very question to my International Baccalaureate students. A handful chose incorrectly, while several “could not remember.” When I asked the rest of the class, which was comprised of some of the brightest seniors on campus, no one could justify their correct answer. The best I got was that they “remember someone teaching us that once.” A simple question turned out to be far more rich and robust than it should have been, but it was a worthwhile day. {Try this one and get back to me.}

I must say here that I am grateful for my math education; it was far better than not having one at all. However, admittedly it was not rich and robust. The question is: Will we make it so for our students? It will take a conscious decision on our part to give our students a different educational experience than most of us had. So ask yourselves: When was the last time that you immersed your students in a rich and robust task? When is the next one planned? Has the time between those two dates been far too long? Are we up to the rich and robust task of offering rich and robust tasks?

7 Lessons Learned about the Common Core

Here is my list of lessons learned at The California Math Council’s South Conference, in Palm Springs. The overall theme of the conference was implementing the Common Core State Standards.

1) The new standards are truly world-class. I wish my own children went through school in the common core era. The expectations of the next generation of students are far higher than anything the state expected of my son and daughter. If our nation can rise to this new bar, we will finally be on par with the top performing countries around the world.
Message for teachers: If we only meet a small fraction of these expectations, the students of the future will still receive a better education than those sitting in our classrooms now.

2) Leaders are optimistic about the train wreck that is about to happen. There is no way that teachers will get students ready in time for the first wave of common core assessments. The change is too great, too quickly. The initial results promise to be abysmal. The question is from that point whether we will rise from the wreckage and move forward, or back pedal towards the old ways. Dr. Bill Schmidt and Tim Kanold are both optimistic that states, schools and teachers will continue to train, learn and eventually meet expectations.
Message for teachers: Brace for impact, then take advantage of the special opportunity that is being presented to us.

3) “We have been doing the common core long before there was a common core.” I heard this phrase from Brad Fulton and several of the reform leaders and innovators at the conference. The Math Projects Journal could make the same claim. For the 14 years of our publication, we have pushed for limited topics, conceptual understanding, higher order thinking and holding students accountable to all the above (4.5 Principles). This is not because anyone of us had unique ideas on education. We were just following the international research studies. I have heard that education is always 20 years behind the research. Any coincidence that California with conduct its first Common Core assessment 18 years after the publication of the TIMSS report?
Message for teachers: Educate yourself; what seems new really is old school.

4) The answers are out there. There was plenty of information on instruction, assessment and professional development. There are no secrets on how to implement the new standards. The only real question is how quickly and pervasively will that information find its way into classrooms. Tim Kanold claims, with evidence, that the best vehicle to deliver the information and practices is teacher collaboration.
Message for teachers: Have your school adopt a PLC model.

5) The Common Core emphasizes teaching practices as much as content. I was aware of the practices listed in the Common Core documentation, and though I regularly implement most (but admittedly not all), they always seemed more like guidelines than rules. (Problem-Solving, Reasoning, Modeling, Arguing, Tools, Precision, Structure, Patterning) In all the featured presentations, there was the consistent message that students learn as much from how we teach as from what we teach. Dr. William Schmidt of the University of Michigan claims that our current practices lack the logic and structure that is inherent in our subject matter.
Message to teachers: You will not only change what you teach, you will change how you teach.

6) One-third of the content of most textbooks can be thrown out. Dr. Schmidt led a study in which teachers corresponded their lessons and accompanying textbook pages to the Common Core standards. The study discovered that on average, one-third of the textbook content was avoided. There really will be time to slow down and to teach problem-solving.
Question for teachers: How well are you going to use the extra three months?

7) “Technology needs to mean more than paper on an iPad.” Dan Meyer gave a compelling presentation on the use of technology to push students to higher levels of thinking. He said that currently many teachers and companies are simply moving textbooks and lesson plans over to electronic devices as scanned material. Much of Mr. Meyer’s presentation was on the unique ways in which videos and photos can be used to perplex students.
Message for teachers: Times are a changin’. It’s time to catch up.

For the alarming number of those who had not heard much of the Common Core, it was a terrifying weekend. For those of us who have embraced its values over the last two decades, it was an exhilarating, yet still disconcerting, conference. Big kudos to the committee for putting on such a hugely successful event. Thank you for helping point us all in the right direction.

P.S. Catch ’em 2012 from Chris Shore’s presentation at CMC.