Tuesday, October 27, 2015

What People THINK is Common Core ISN'T - It's Misunderstanding, Poor Training, Politics

I swore to myself I wouldn't do my next post on the Common Core, but I just get so irritated by the postings I see out there about it, I can't help myself!

On Facebook, I see silly things like this:

And then the current huge controversy about the math quiz and the teacher grading the problems wrong:

Let's not forget the multitude of articles and storylines on the news talking about parents being angry and states opting out.

Thank goodness there are some people who are trying to bring reason back to this madness about the Common Core. Great response here to the quiz example above by Andy Kiersz. Or this one in response to a parents obnoxious use of what he calls "Common Core Math".  I have already written my own response to my nieces and sisters hatred of what they perceived as Common Core math in a post last year, Common Core: It's Not the Devil.

The problem with all these pictures and stories and examples of problems that are "common core" is that they are taken out of context, or are comparing the wrong things or showing processes wrong, and come from a place of ignorance about the Common Core.  The teacher in the quiz example clearly did not have a good grasp of what the standards were asking or perhaps doesn't know the math well enough herself to help the student make that commutative connection.  As Andy Kiersz points out in his article,
"While this worksheet does present a frustrating situation, it has nothing to do with Common Core. Common Core lays out a set of objectives for what students should be learning in each grade level. It's still up to the states, districts and teachers to come up with the specific curricula and lesson plans to achieve those objectives".
It was bad math on the teachers part, not Common Core. That's lack of training, a lack of mathematical knowledge on the part of the teacher, lack of understanding the building blocks of the common core, and a lack of correct implementation which is what I believe is behind all of this bad-mouthing of the Common Core. People don't really understand it and latch on to ridiculous examples that are not at all Common Core (as in the example on the left above) but are more likely a 'standardized' version or a publishers version or a teachers incorrect perception of Common Core.

I don't know how to stop this seemingly endless and inane bashing of the Common Core. It will only get worse as the Presidential election gets into full swing. And test results coming back showing a drop in scores - forget the fact that that is an expected result anytime something new is implemented.  It's called the implementation dip. Change TAKES TIME and persistence....which if this keeps up, will be a moot point because our culture expects immediate results, expects it to be easy, and doesn't allow for real change or real learning to take place because it's takes too long to see results. It makes me sad, angry, and frustrated since I can't single-handedly knock sense into these people. But - I will keep trying and thankfully there are plenty of other people out there who have educated themselves and are trying to correct the inaccuracies that are out there about the CCSS.

Wednesday, October 21, 2015

Back to the Future Day! Foster Creativity in Students TODAY to Build A Flying Car

Today is October 21, 2015 and Marty McFly is set to arrive this evening, so of course, a post about the technology that actually exists today compared to what the movie predicted is in order! Someone else has done all the work for me and gone through 22 things that the movie got both right and wrong - I will let you check that out on your own.

Here's a movie trailer clip that shows the some of the "things" of the future:

 What I find amazing to consider is that the writers/creators of the movie were making predictions about a future 25 years down the road in a time, 1989, where none of this technology existed.  Heck - the World Wide Web was just being born in 1989. And yet now, 25 years later, some of their predictions are in fact a reality. We have 3D TVs & movies, we have Google Glasses, digital cameras, tablets, talking computers who can do things for us (Siri), and while we don't have flying cars, we do have electric cars. It's like Star Trek technology that was predicted and now exists - i.e. the communicator (cell phone), universal translator (apps that translate phrases into specified languages), the tricorder (Locad and hand-helds that measure microorganisms & blood disorders), and video conferencing (video conferencing!!). The writers and creators of these and other movies and TV shows were thinking outside-of-the-box about technology that did not exist at the time and now, years later, does.  That's creativity.  That's building an impossible solution that became a reality. And THAT'S what we need to foster in students.

Who knows what creative minds and out-of-the-box thinking exists in students today that could change our future? If we stifle student creativity in classrooms, making learning rote and solely focused on passing the standardized tests, then we limit their ability to think, create, expand, and explore. Learning should involve students asking questions, working together, using technology to explore and expand their understandings, building and making things and coming up with impossible solutions or ideas that they can try. I have been in far too many classrooms where learning is drill-and-kill, memorization, and sitting in desks listening. There's no excuse for that. If we want those flying cars, we need to allow students to be creative and deepen their understandings beyond the algorithms and skills to the applications and possibilities.

Friday, October 16, 2015

Common Core - Final - What Do You Mean Rigorous?

In my final in this Common Core Structure series, I want to just spend a little time discussing the three Key Shifts of the Common Core: Focus, Coherence, and Rigor. The CC are standards - states have always had standards. The difference here is a clearer set of aligned standards, throughout K-12, that ensured the standards built on each other within a grade, between the grades, and provided a cohesive set of understandings, skills and application. Hopefully the previous four posts have given a clearer understanding of how the structure of the CC was designed to support these shifts, so now lets actually look at these three shifts in depth. I am going to use some specific standards to exemplify each shift, as I think it helps make sense of them.

The CC is really focused on students conceptual understanding of mathematics and their ability to apply these understandings to real-world problems. So, within each grade, there are "less" standards,
and more focus to help them build strong foundations before they move on to the next grade level. Each grade goes deeper into fewer concepts, which then, as they move through the grades, they continue to build. So, for example, in grade 3, the focus is on multiplication (because in K-2 they built the foundation of whole numbers and addition & subtraction).

As an example, here is 3.OA.7 standards (Grade 3, Operations & Algebraic Thinking, Standard 7):
Under Cluster Heading of Multiply and Divide within 100: 7. Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g. knowing that 8x5 = 40, one knows 40/5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.
 As you can see, very focused. There is a footnote in Grade 3 that students "need not use formal terms for  the properties of operations", because they are building understanding. Notice, grade 3 is only working with numbers less than 100. And that they are expected to know their times tables between all one-digit numbers.  It's focused, specific, with the idea that if they learn these concepts and skills, and gain true understanding, then they will be able to continue to build on these as they go to the next grade level. Focus is few standards, learned deeply, as a building block for the next level.

If the idea is to deeply understand fewer concepts at each grade and continue to build and add on to
that understanding, then this means between the grade levels there must be connected standards.  This is the coherence. Throughout each grade, the mathematical skills and concepts being focused on are building the foundation for the next grade, where students will continue to develop the understandings and add on to those understandings. What is learned in one grade impacts what comes next and is a result of what came before - coherence. Think of it as a ladder - each grade is a rung, and you must step on that rung (i.e. learn and understand the concepts) in order to get to the next rung, which will eventually get you to the top. Skipping a rung may make you fall because you have missed some important step and aren't secure.

Using the Grade 3 example from above, to demonstrate coherence, lets look at related standards from Grade 2 and Grade 4, so you can see how the standards fit together, or cohere, between grades.

Grade 2
Grade 3
Grade 4
Operations and Algebraic Thinking
Operations and Algebraic Thinking
Operations and Algebraic Thinking
Work with equal groups of objects to gain foundations for multiplication
Multiply and divide within 100
Use the four operations with whole numbers to solve problems
4. Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends
7. Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g. knowing that 8x5=40, one knows 40/5=8) or properties of operations. By the end of grade 3, know from memory all products of two one-digit numbers.
1.Interpret a multiplication
equation as a comparison, e.g., interpret 35=5x7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations

Looking at the coherence - in Grade 2, students are working with rectangular arrays to develop an understanding of multiplication. In Grade 3, they take that further by being able to user properties to see the connection between multiplication and divisions. In Grade 4, they are able to expand on that commutative property and apply it to verbal and written equations. Each grade provides focused understanding that helps them build deeper understanding at the next level - the standards cohere between the grades to reinforce previous understandings and build the foundation for future concepts.

Rigor is probably the most understood of the shifts. Unfortunately, many interpret rigor to mean harder problems or, worse, MORE problems.  Rigor DOES NOT MEAN MORE PROBLEMS OR HARDER PROBLEMS.  I repeat...RIGOR DOES NOT MEAN MORE PROBLEMS OR HARDER PROBLEMS!! It kills me when I go into classrooms, especially those labeled as "gifted" or 'advanced" and see students being assigned 50 homework problems every night because they are "smarter". No, no, no, a thousand times NO! All that does is make students HATE mathematics (I can attest to this with my own daughters, who always had ridiculous amounts of math problems, especially in the higher level classes).  If nothing else, please stop the practice of assigning more homework and thinking this means you are providing rigor.

Rigor is understanding mathematical concepts, being able to explain your thinking, being able to apply your understanding to new and different situations. Three things - understand the concept, be able to use the concept to solve problems and explain thinking, and be able to apply the concepts to real-world situations and different situations. So, having a student multiply two digit numbers 50 times for homework is NOT rigorous, because you have left out explaining why and applying to new situations.

How do you create rigor in your math classroom? How do you get away from drill-and-kill and developing fear and hatred of math in your students? My simple suggestion is ask good questions and expect real answers.  Here is where the Mathematical Practices come into play - read those again (or go back to my previous post related to the Practices). Ask questions. Expect students to explain and justify their answers using whatever method works - words, models, similar problems. Give students DIFFERENT types of problems that require them to apply their understanding in new ways. And ask questions!!!  If you don't know what to ask, ask why? Or ask "can you show me what you did"? Or ask "is that true all the time? " To me, rigor means you expect them to explain what they did, why they did it, and can they do it differently or in another way?

Example: Student says 5x8=40, which means that 40/5 = 8.  You ask, why? They start drawing examples of 8 groups of dots five times, showing it equals to forty. They then draw another example of 40 dots, and they circle groups of five dots, and show that their are now 8 groups. You then say, ok...does that work for other problems? (i.e. model it with different numbers) Or, can you give me a real example of this? (i.e. apply what you are learning to a real-world situation).
The CC is not just about standards. Standards are standards. What is to me the most important aspect of the CC is the focus, coherence and rigor that are embedded throughout and that help students build their conceptual understanding, learn appropriate skills at appropriate times, and apply those understanding and skills to new, real-world situations. I know a lot of states are dropping out of CC or changing the name or creating their own standards, much of this due to ridiculous politics and ignorant people not truly understanding the goal and purpose of the CC.  Fine. But - regardless of what state you are in or what mathematical standards you use, if you do nothing else, be rigorous in your instruction. Use the Mathematical Practices and you are helping your students.

Thursday, October 8, 2015

Common Core - #4 Structure of High School Math Content Standards

In my last post, I went in great detail into the structure of the Common Core Math Standards for K-8. 
Long story short, the picture at the right is a visual of the structure showing the funnel effect – where the standard itself is the end product of so much more: Introduction, Domain, Cluster. The gist of the last post was that it is important to look at all the components, not just the specific standards themselves, so that you understand how the standard fits into the learning progression.

When looking at the high school content, the structure of the standards is the same, with an additional component, the conceptual categories.  There are six conceptual categories at the high school level: Number and Quantity, Algebra, Functions, Modeling, Geometry, and Statistics & Probability. Within these
conceptual categories, there are the Introduction, Domains, Clusters and specific standards. 
The idea behind the conceptual categories is that students acquire these understands through repeated exposure across many topics, so Modeling standards are embedded in all high school math courses, not a specific course unto themselves.  Similarly, Functions & Number and Quantity are also standards that will appear in many High School courses, such as Algebra, Geometry and Calculus. The High School standards are not traditional course topics, but rather concepts that students are exposed to throughout their High School mathematics learning.

Here is an example, using the conceptual category of Functions, of the High School Structure. I chose just one standard to highlight again, the structure and overall cohesiveness.

               Conceptual Category: Functions

Relevant Excerpt from the Introduction: Functions presented as expressions can model many important phenomena. Two important families of functions characterized by laws of growth are linear functions, which grow at a constant rate, and exponential functions, which grow at a constant percent rate. Linear functions with a constant term of zero describe proportional relationships.

               Relevant Domain: Linear, Quadratic, and Exponential Models

Relevant Cluster: Construct and compare linear and exponential models and solve problems

Specific Content Standard: CCSS.MATH.CONTENT.HSF.LE.A.2
Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

As you can see from this example, this particular standard could be appropriate in a Pre-Algebra, Algebra, Geometry, or Algebra II course. So the High School standards cross traditional course structures and understanding this and really exploring the High School standards when planning course content and curriculum is an important. These are not isolated standards for a single course, but rather pervasive standards that span several course content areas so that students are continually exposed and using their understanding of mathematics throughout all high school mathematics courses. This helps them see the inter-connectedness of mathematics.

I hope that these last few posts that focused on how the structure of the standards has clarified that just knowing a content standard or knowing the title of a math practice does NOT mean you know enough. It is important to really read through the entire structure of both the practices and the content standards to fully ensure that mathematics instruction is supporting what students need to know and be able to do. It is important to spend time exploring these PRIOR to developing curriculum or lessons, to ensure what is being taught, and HOW it is being taught are truly supporting student conceptual understanding.

In my next post, I am going to spend a little time discussing the key shifts that are at the heart of the Common Core Standards. I think there is a lot of misconceptions about coherence, focus and rigor, which leads to much of the confusion that teachers, parents, the media and the politicians have about the standards.

Friday, October 2, 2015

Common Core - #3 Structure of K-8 Content Standards: Footnotes Matter!

My last two posts focused on the structure of the Common Core Standards of Mathematical Practice. The big idea of those posts was the title of the practice is NOT enough - you need to read the narrative to get to what students should be doing and saying.

The theme of this post is much the same - the Common Core Math Content standard alone is NOT enough to truly
understand what it is students should know and be able to do if they have mastered the content. The content standard, which is often what is posted in textbooks or put on the board, is in fact, a small part of the big picture and without seeing the big picture, we end up teaching isolated skills and facts. Understanding the structure of the content standards provides a big picture at each grade level (focusing on K-8 right now) of where students are going, and how the standards, as a whole, are continuing to develop and expand mathematical content knowledge. It is a "learning progression".I am going to focus on K-8 structure now, as it is the same for all these grades. My next post will focus on the High School Content Standards, which have an additional structure.

For all grades K-8, there is a grade introduction, which gives the BIG IDEA of the whole grade and what, by the end of that grade, students should know and understand about mathematics. I am pulling the introduction from Grade 3 as an example:

In Grade 3, instructional time should focus on four critical areas:
  1. developing understanding of multiplication and division and strategies for multiplication and division within 100;
  2. developing understanding of fractions, especially unit fractions (fractions with numerator 1;
  3. developing understanding of the structure of rectangular arrays and of area;
  4. and describing and analyzing two-dimensional shapes.
  1. 1) Students develop an understanding of the meanings of multiplication and division of whole numbers through activities and problems involving equal-sized groups, arrays, and area models; multiplication is finding an unknown product, and division is finding an unknown factor in these situations. For equal-sized group situations, division can require finding the unknown number of groups or the unknown group size. Students use properties of operations to calculate products of whole numbers, using increasingly sophisticated strategies based on these properties to solve multiplication and division problems involving single-digit factors. By comparing a variety of solution strategies, students learn the relationship between multiplication and division.

  2. 2) Students develop an understanding of fractions, beginning with unit fractions. Students view fractions in general as being built out of unit fractions, and they use fractions along with visual fraction models to represent parts of a whole. Students understand that the size of a fractional part is relative to the size of the whole. For example, 1/2 of the paint in a small bucket could be less paint than 1/3 of the paint in a larger bucket, but 1/3 of a ribbon is longer than 1/5 of the same ribbon because when the ribbon is divided into 3 equal parts, the parts are longer than when the ribbon is divided into 5 equal parts. Students are able to use fractions to represent numbers equal to, less than, and greater than one. They solve problems that involve comparing fractions by using visual fraction models and strategies based on noticing equal numerators or denominators.

  3. 3) Students recognize area as an attribute of two-dimensional regions. They measure the area of a shape by finding the total number of same-size units of area required to cover the shape without gaps or overlaps, a square with sides of unit length being the standard unit for measuring area. Students understand that rectangular arrays can be decomposed into identical rows or into identical columns. By decomposing rectangles into rectangular arrays of squares, students connect area to multiplication, and justify using multiplication to determine the area of a rectangle.

  4. 4) Students describe, analyze, and compare properties of two-dimensional shapes. They compare and classify shapes by their sides and angles, and connect these with definitions of shapes. Students also relate their fraction work to geometry by expressing the area of part of a shape as a unit fraction of the whole.
As you can see, the introduction is important because it gives the critical areas (i.e. BIG IDEAS) that are the focus for the entire grade and all the standards in that grade. The critical areas are highlighted at the beginning and then there is a more detailed description of each critical area. The introduction provides the end result, what students in that grade should be able to say, do, know and understand if all the standards for that grade have been effectively taught and learned. How many people read the introductions? If you just look at the standards alone for a specific grade, you miss many key components - such as, in this example, the idea that you should be developing the idea of multiplication and division using equal size groupings in various arrangements? Or that you begin fraction understanding with unit fractions?

There are 11 domains in the K-8 content standards, some of which appear only in certain grades (i.e. Counting & Cardinality is only in Kindergarten). Domains are the groups of related standards, which help give focus. The domains in K-8 are as follows (with the grades they appear in listed):
  • Counting & Cardinality (K)
  • Operations & Algebraic Thinking (K-5)
  • Number & Operations in Base Ten (K-5)
  • Number & Operations-Fractions (3-5)
  • Measurement & Data (K-5)
  • Geometry (K-8)
  • Ratios & Proportional Relationships (6-7)
  • The Number System (6-8)
  • Expressions & Equations (6-8)
  • Functions (8)
  • Statistics & Probability (6-8)
It makes sense that kindergarten is the only place we see counting & cardinality - remember the more time in kindergarten should be spent on number than anything else. Which means when students leave kindergarten they should have a clear understanding of counting and cardinality that will help them as they move forward into the other grades. Notice that fractions don't come along until third grade - because K-2 you should be building the foundation of whole numbers, operations, algebraic thinking first.  The domains give you a broad sense of each grade levels focus and how the progression of standards is occurring as we move vertically in the grades. Students are building their knowledge as they go through the grades, and even though a domain may not appear in a grade, it doesn't mean that students are no longer using those skills or content - they are applying what they learned in new and different ways.

Clusters group related standards within a domain and are a summary of those standards that explains how they relate. Let's look at the clusters within the Grade 3 domain "Operations & Algebraic Thinking":
Domain: Operations & Algebraic Thinking (Grade 3)
  • Represent and solve problems involving multiplication and division (cluster)
  • Understand properties of multiplication and the relationship between multiplication and division (cluster)
  • Multiply and divide within 100 (cluster)
  • Solve problems involving the four operations and identify and explain patterns in arithmetic. 
So, if we look at the first cluster, taken into account with the domain it falls under, all the content standards within this cluster relate to representing and solving problems that have to do with multiplication and division, which help students understand algebraic thinking (domain) and the operations themselves. 

Content Standards 
Have you noticed all the information that has occurred before we even reach the specific content standards? The content standards are the SPECIFIC skills students should understand and be able to do within the broader context of the critical areas, domain and cluster. Let's look at the specific content standards for 3rd grain under the following Domain & Cluster:
Domain: Operations & Algebraic Thinking (Grade 3)
  1. Cluster: Solve problems involving the four operations and identify and explain patterns in arithmetic
  •  3.OA.D.8  Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.3
  •  3.OA.D.9  Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends.
As you can see, these specific content standards are not just isolated skills but are supporting the cluster, where students are able to solve problems using the four operations and explain patters, and are also part of students overall operations and algebraic thinking. All of these components in the structure of the standards work together. The standards are specifics that are just a piece of the big idea for the whole grade level.

Footnotes and Examples
Additionally, there are footnotes and examples embedded throughout the content standards, appearing in all parts (Introduction, Domain, Cluster & Standards). These are often overlooked but are key to really helping focus the learning at each grade level.  The examples provide insight into what the standard might look like at that grade. The footnotes have several purposes - referencing the Glossary for terms, or further explanations, or...most importantly, provide additional parameters.  In the example above, you will note that standard 3.OA.D.8 (3 is grade level, OA is domain, D is cluster, and 8 is standard) has footnote 3, which states:
This standard is limited to problems posed with whole numbers and having whole-number answers. Students should know how to perform operations in conventional order when there are no parentheses to specify a particular order.

Um...WOW!!  So, students should be solving two-step word problems with the four operations, use equations, etc. but with WHOLE NUMBERS and with WHOLE NUMBER answers. So, even though 3rd grade is introducing fractional numbers, this particular Grade 3 standard says students should work with whole numbers, as they are just developing fractional understanding. Footnotes are important!  How many teachers are reading them?

I am not sure I need to reiterate, but I will - the content standard is NOT ENOUGH!!  The individual content standards themselves are just a small, specific skills that are part of much larger overarching content. The standards all relate back to the critical ideas for each grade level . It is important to familiarize yourself with your grades' overall big idea, which includes not just the standards, but where they fit into the domain and critical areas (big ideas) for your grade. And don't forget your footnotes!!!

(In my work with The Charles A. Dana Center at UT, Austin, they have this nice visual to the right of the K-8 content standards structure).