Wednesday, December 16, 2015

Being Slow in Math is A Good Thing

"Speed ISN'T important in math. What is important is to deeply understand mathematical ideas and connections. Whether you are fast or slow isn't really relevant." - Laurent Schwartz, mathematician

If you haven't seen the video by Jo Boaler and some of her Stanford students entitled "How to Learn Math: Four Key Messages", you definitely need to. Besides the four powerful messages (which I will list below), it has some great stories and quotes, one of which is the one I have above.  Jo Boaler has done powerful research and written some terrific books on mathematics and learning math (one of my favorites being "What's Math Got to Do with It?" and the video about these four key messages in math is so interesting.

Here are the four key messages about learning math (I highly recommend you watch the video to clarify and define each message a bit more):

  1. Everyone can learn math at high levels
  2. Believe in yourself (your beliefs about your abilities actually changes the way your brain learns)
  3. Struggle and mistakes are really important in learning math
  4. Speed is NOT important
All of these speak directly to the way we currently teach and learn mathematics. One that really struck out for me was #4, speed is not important. I remember my own daughters struggling with the timed math tests - i.e. you have a minute to try and solve 100 times tables, or complete as many addition problems as possible. Very stressful, very ridiculous, and to top it off, they were penalized with poor grades if they couldn't reach the arbitrary goal. It still goes on and students memorize and stress over these timed math drills. Why? It's ridiculous. And, if we continue to do this to students, then they begin to believe they are bad at math (see #2 above), which leads to them thinking they can't learn math (see #1), and therefore giving up when problems get tough (see #3). A self-fulfilling prophecy.

So - I ask those math teachers out there who continue to put pressure on students to perform mathematical skills in a timed matter, where speed is important - stop. Just stop. Focus on what mathematics should be - understanding why those calculations matter, what they are related to, how they help us solve real-world problems. Help students make connections. 

I know I keep coming back to it - but the Common Core Mathematical Practices seem to embody these four key messages. No where in there does it say students have to be able to do ___calculations in _____ minutes. Math is NOT about speed - it's about the struggle, perseverance, conjectures, connections, and applications that help students solve relevant, real-world problems and see the beauty and need for mathematics.

Check out the video here (sorry - it did not allow for embedding):

Friday, December 11, 2015

#HourofCode Sparking the Need for Computer Science Curriculum

It's been great to see all the posts on Twitter this week and the many articles focused on National Computer Science Education Week. The #HourofCode hashtag has been lighting up Twitter this week, which has been really exciting to see (and I have loved reading and sharing all the great links).
Science and teaching students coding. All of course in response to this week being

The idea behind #HourofCode is to expose as many people, especially students, to the basics of coding. The hope is by showing that anyone can learn coding, need for, interest in and involvement with computer science will increase. One of the many articles I read this week was one that suggested Computer Science courses should be considered math credit.  I think this is a great idea - it would provide a valuable math credit option for students, especially those not interested in the traditional Algebra II, pre-calculus, calculus path, which is usually the push for a majority of students. Often a path completely unnecessary for most students, who would benefit more from Probability & Statistics, Computer Science and Financial Literacy courses, which often don't exist as options. In fact, I would go so far as to say, that computer science classes would be far more beneficial for the majority of students as math credit than Algebra II, pre-calculus or calculus. In this technological age, where a majority of jobs require tech skills, coding and/or the ability to understand coding is a necessity.

I look at my recently graduated daughter (from UT Austin), who graduated with an Advertising Degree and is having a difficult time getting a job in large part because the skills many marketing/advertising companies are looking for are coding skills: HTML, Java, Codex - can you design and build a website? An app? None of which was part of her advertising curriculum. And not something that was offered in her high school either. Will she go out and learn it now? Absolutely (hopefully on-the-job training!). But, what this emphasizes for me is computer science is a vital need in our K-12 and higher-ed curriculum and one that is NOT being given enough focus.

Hopefully, #hourofcode is helping to make Computer Science become a required course/subject in schools. But, if it's not in your school, there are so many resources where, no matter what you teach, you could begin utilizing/teaching coding in conjunction with your own topic. Why not start the movement yourself in your own school? The links above provide many resources, but here are some specific resources as well.Obviously there are many more options out there, these are just some of my favorites, with students in mind.

  1. Globaloria - this is an amazing curriculum that can become part of any content area. Students learn coding and content by developing a computer game to teach others. Check it out here:   I've had the privilege of working with these folks, and they do an excellent job connecting Common Core, Next Generation, and ISTE Tech standards to content while students learn to code.
  2. Has some great #hourofcode tutorials kids will love.  Just some listed below:
  3. Tynker has grade-level games and coding and teaching support as well
  4. Khan Academy - coding tutorials among the many free content videos available.

Monday, November 30, 2015

Technology without Training & Sustained Support Will NEVER Succeed

I just read this article by Eric Patnoudes entitled "Beyond the Silver Bullet: Making 1:1 Matter".  As a parent and former teacher, Eric's basically was saying that all the technology in the world and 1:1 initiatives will fail in the classroom if teachers are not provided with the training and support they need to CHANGE their practice. Professional development needs to go beyond how to use the technology and be more about how to teach with the technology in ways that are different and more appropriate for the technology. Using the same old 20th century teaching practices with new technologies is doomed to failure.

Couldn't agree more. My last post speaks to this as well - not only do we need to analyze and plan for WHAT technologies are appropriate, but we need to plan and provide continued training and support to ensure the technology purchased is being used to change teaching and support learning.

In my research regarding implementing technology effectively in the classroom, those teachers who were in fact changing their practice and using technology DIFFERENTLY and APPROPRIATELY had professional development and continued support.  Here are the things that made a difference:

  1. Curriculum & district expectations
    • Technology used actually supports standards and content taught
    • Relevance of professional development content/resources to what teachers actually teach and do in the classroom
    • Providing content-focused, ready-to-use activities/lessons that utilize the technology
    • Clear expectation from administration that using the technology was expected & supported
  2. Teaching practices
    • Professional development emphasized using technology to teach specific content 
    • Professional development provided classroom management and teaching strategies for using the technology 
    • Multiple teaching strategies were modeled in professional development (questioning, collaboration)
    • Teachers had time to collaboratively plan lessons and practice using technology with their content/classroom
  3. Sustained Professional Development 
    • Long-term support was provided
    • Training on technology as well as content-focused implementation of technology
    • Coaching, modeling, active learning all part of sustained professional development
    • Collaboration & time for practice and feedback
  4. Internal & External Factors are accounted for and controlled
    • Access to technology available. Technology integration won't work if students access to the technology is limited.
    • Teachers believe students will benefit from use of technology (so PD emphasizes relevance) and are confident in their ability to use it (so sustained PD is provided and teachers are supported in many ways)
    • Time is provided.  Time for teachers to learn and practice implementation, time for students to learn, and time for changes to take place before judgements/assessments are made
    • Classroom structures support the use of the technology. So - class size, other competing technologies and/or resources are de-emphasized, support for changing classroom teaching strategies, etc. are all considered and addressed prior to and during implementation
As you can see, there is a lot that goes into integrating technology effectively into classrooms if  changes in instructional practices and student achievement are expected. Just providing the technology and a quick how-to professional development training is NEVER enough and doomed to failure. Planning before is vital; time, support, evaluation and feedback are necessary during initial implementation; and continued reassessment and training/support of the implementation are crucial for sustained change and impact on student learning. If you can't provide all of this, then don't expect anything to change.

Thursday, November 19, 2015

Purchasing Digital Resources - Things to Consider

The State Educational Technology Directors Association (SETDA) just released a paper analyzing states' policies in regards to digital materials acquisitions & implementations, along with their recommendations. You can find the report & summary here

In the report, they outline Next Steps: a) Essential Conditions for successful acquisition & implementation; b) suggestions for making the procurement process transparent and easy to navigate; c) the need for strategic short & long-term budgeting; and d) and suggestions for the states to guide schools & districts on best practices for adoption, implementation and vetting of digital resources. You can read the more detailed descriptions of these four "Next Steps" recommended by SETDA here.

While reading the report and the next steps suggestions, it reminded me of my own research about technology acquisition and implementation. I have done several blog posts directly related to technology implementation in the classroom, focused more on a district/school level, and I just wanted to summarize some of my findings. In order for digital resources to be effective in states/districts/schools, there needs to be:

  1. Research about what technologies are currently in use in the schools/classrooms and what the classroom needs are 
    • Examine classroom structures and current resources
    • Examine student data and determine what technology would support standards, classsrooms, content, curriculum goals
    • Research digital resources PRIOR to purchase to determine most appropriate ones that will support determined needs
  2. A clear plan for purchase & implementation
    • What is the budget and will the budget support acquisition 
    • What are the technology requirements - i.e. broadband, hardware, software, and will budget support these requirements
    • Plan for implementation
      • Design appropriate, relevant PD
        • Provide content-related activities
        • provide hands-on learning
        • Provide ready-to-use lessons
        • Provide long-term support & collaboration
      • Plan for continued assessment of implementation and make changes as needed
        • Set clear expectations for use 
        • Have follow-up observations of use and feedback
        • Use data for continuous evaluation
        • Provide continued collaboration & feedback
This is clearly not an exhaustive list but if you look back at my list, it all involves some key components: leadership (administrators); collaboration; planning; funding; evaluation/feedback loop; expectations for use; and TIME.  Effective acquisition and implementation of digital resources can happen and really have an impact on student learning & achievement if done right.

Wednesday, November 11, 2015

#Edchat Discussion - Politics, Religions & Education

I participated in an interesting #edchat this past Tuesday, as I try to do every Tuesday at noon (Eastern time) if it fits into my schedule.  The topic for this hour long chat was:
Education should reflect culture of the country, but do politics and religion have too much influence in American education?
My immediate reaction and response was yes, religion and politics have way too much influence in  American education. Two prime examples are the current hot-button issues of The Common Core Standards and whether the word "God" should be included in the Pledge of Allegiance. Here are my personal opinions on both:

  1. "God" in schools/Pledge of Allegiance - 
    • First of all, God was not included in the original pledge, written by Francis Bellamy in 1892. "under God" was added in 1954 in response to the Communist threat of the times
    • But, regardless of when "under God" was added, this is just a political & religious ploy to get everyone up in arms over nothing. In all honesty, the pledge, with "under God" included is said in most schools to this day - I travel all over the country and have yet to be in a school where this is taken out. I am sure there are probably some, but I have yet to be in one. 
    • Whether the words are there are not, most schools (public) give the option for students to say the words. In all the schools I taught in, it was respectful to stand, but you didn't have to say the words.  Same with those moments of silence - which students could do anything they want - pray, think about the upcoming math test, worry about the football game, etc. Let's move on and get to learning!
    • Public schools should be about educating all students, regardless of their religion - its America after all, and we have a mix of everything. This continued push, mostly from the religious right/Christians, is an example of too much religion influencing public, state-run education system, and there should be a separation of church and state. 
  2. The Common Core Standards
    • If you haven't read my previous postings regarding the Common Core Standards, suffice it to say I am a big fan. The fact that there is now a huge push from many politicians, particularly those running for President, to do away with the Common Core is a clear example of too much politics influencing education
    • First - The Common Core State Standards were NOT an Obama or Federal initiative, and have nothing to do with No Child Left Behind, despite so many politicians on the right blaming the Obama and Federal Government/administration. It was a state-led initiative, with teachers and education experts and state leaders developing standards, based on research.
    • Second - The Common Core is NOT a curriculum, so it does NOT define what states, schools and teachers teach. Rather it is a set of goals demonstrate the skills and understandings students need to be successful. How these goals are taught and reached are determined by the individual school districts within each state. 
    • What I see are politicians and media stories that are full of inaccuracies about The Common Core, that are then used to fuel the fear of parents about what their students are learning, fuel the fear of teachers worried about standardized test scores and their jobs, and used to basically halt any change in the education system and bring us back to where we started - nowhere. Yes - definitely too much politics in education!
So - religion and politics are too much of an influence in our education system, in my opinion. Unfortunately - I have no answers on how to make this stop. As my examples show, politics and religion regularly influence American education and unfortunately, are part of the reason there is so little change in the educational system. We are stagnating and our students are suffering and it needs to change. I wish I knew how because the small pockets of change and innovation I do see do not seem to be enough to become systemic. If I knew what it took, I'd be a rich woman and schools would be teaching and students would be learning. In the #edchat discussion, the idea of revolution came up - Dr. Mark Weston sent me a link to his thoughts on that - great read. Like Mark, I want to revamp the whole system. Guess I will keep looking for ways to do so...if you have ideas, let me know. I've thought about becoming Secretary of Education, but then I think politics would make it impossible to really do anything, right?!! It's a catch 22.....

Wednesday, November 4, 2015

What if? A simple question to engage students.

I am often asked by teachers how do I engage my students? What questions can I ask? Questioning is a skill that many teachers struggle with, as we are often prone to ask simple yes/no questions or one-word response questions (i.e. what's the answer?).  It takes effort and practice to ask questions that help students think, analyze their thoughts, make conjectures, etc. But, as I often tell teachers, if there a few questions that you can ask that force students away from the simple responses or the yes/no answers. Questions such as Why? of Is that always true? or, my favorite's What if? or What do you wonder?

Annie Fetter, of Math Forum fame, did an incredible Ignite Talk at NCTM one year entitled "What Do You Notice or Wonder" and it really opened my eyes to the power of observation & wondering. I share it here as a here as inspiration to help you ask the questions that will invoke wonder, inquiry and thinking in students.

Additionally, Randall Munroe did a great TedTalk on the question "What If..." that explores a wonderful math/physics problem.  Again, to demonstrate the power of a simple question to help engage students in thinking.

These are both examples of how a simple question can open up the math ideas that students have and encourages them to explore to find the answers. Ask a simple question and open up the door!

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).

Friday, September 25, 2015

Common Core - #2 of Studying Structure - More on the Practices

In the first post of my Common Core series, I discussed the importance of really looking at the Standards of Mathematical Practices (SMP), because the title alone is not enough. You cannot assume that each title of the 8 SMPs gives enough information to truly know what students should be saying and doing to enhance their mathematical understanding. The descriptive narratives for each practice are necessary reading to clearly make good instructional decisions that will truly support student understanding of mathematics

My last post focused on SMP #4, Model with Mathematics. In this post, I'd like to choose another of the SMPs, and be a little more explicit in breaking down the structure to show how both the title and the narrative help inform instruction. Then, hopefully, you can do your own investigation of the remaining SMPs.  This is a great exercise to do with collaborative groups of teachers - maybe your next team/faculty meeting?! All of this is from work I have done being part of the International Fellows working with The Charles A. Dana Center, UT Austin.

I am choosing SMP #1, Make Sense of Problems and Persevere In Solving Them. Every teachers dream that students approach problems and understand them and not give up, right?!  But what does that look like in class? How do teachers support students so that they reach this point of being able to take a problem, make sense of it, and continue to try to solve it even when the going gets tough?  Here's where the narrative of the SMP comes into play - title is not enough, remember?!!

Narrative of SMP #1Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.
That's a lot of stuff going on there!  I have underlined some key components related to STUDENT actions.  Notice the narrative is about the students - what they should be doing, saying, considering. Taking this to the next step, reflect on students actions and write down (T-chart works) what students might be saying and doing if they are demonstrating this practice. Create a list, like the one below (this is NOT an exhaustive list, just some examples.  See posted pictures for more suggestions).

Students should saying/doing
Teacher should be saying/doing
-asking questions to their peers

-highlight key words or numbers to understand the problem

-making decisions about a plan to solve

-trying out solutions and CHANGING their plan if it is not working

-continually ask themselves “does this make sense”

-drawing pictures, getting tools (i.e. rulers, calculators, manipulatives) to help make sense and plan a solution

-checking their solution using a different method

Once you've generated several things students are saying and doing, THEN look at teachers actions. What you want students to do informs the actions of the teacher, so this step is done after getting a clearer understanding of the student actions.  Again, here's a list of teachers actions/behaviors, but it is NOT an exhaustive list.

Students should saying/doing
Teacher should be saying/doing
-asking questions to their peers
-provide a classroom environment that encourages conversation and student dialogue
-highlight key words or numbers to understand the problem
 -teach strategies to help students identify key words, numbers and questions in problems
-making decisions about a plan to solve
 -model and demonstrate various problem solving strategies
-trying out solutions and CHANGING their plan if it is not working
 -Allow for productive struggle and act as a facilitator
-continually ask themselves “does this make sense”
 -Asking students questions like "is there another way?" "Can you explain how you did this?"
-drawing pictures, getting tools (i.e. rulers, calculators, manipulatives) to help make sense and plan a solution
 -Provide multiple resources 
-checking their solution using a different method
 -Encourage collaboration and sharing of solutions; require students to explain their thinking

This process, of really studying the structure of the SMPs, the title AND the narrative, and then focusing on student actions first allows for a clearer path of what teacher actions are necessary to help students achieve the SMP skills.

I encourage you to do this yourself, but better yet, do some collaborative learning with other teachers around the structure of the SMPs and you will get a better understanding of what students need in order to become the mathematicians and problem solvers that the Common Core Standards strive to achieve.

Next in my continuing series will be looking at the structure of the content standards - a much more complex structure than the SMPs.

Monday, September 21, 2015

Common Core - Studying Structure & A Personal Fight Against Politicking

I am tired of the #CommonCore bashing, so I am just going to start writing some posts about things I have learned that have helped me realize how powerful and SUPPORTIVE of student learning these standards and practices can be.

Don't get me wrong - I never needed to be convinced that the Common Core State Standards for Math (CCSSM) were a good thing for math education.  I cheered their adoption because to me, these standards supported what I had been practicing and preaching, both as a teacher and a professional development provider. But - even though I have spent the last several years studying and coaching others in implementing the Common Core in a variety of ways, I just recently learned more to solidify, for me, their power. It was sort of an ah-ha moment: if I, who consider myself very familiar with the CCSSM, just learned something new, imagine how those who have spent little, if any, time with these standards (i.e. parents, news reporters, politicians) could benefit from looking a little more deliberately at the standards instead of relying on the rhetoric or a cursory glance to make their decisions?!

I've been working with The Charles A. Dana Center at UT Austin supporting math teacher training. As part of this endeavor, I participated and delivered some training that focused on looking at and understanding the structure of the CCSSM and alignment of the standards (within grade and vertical). My huge take-away was how important this process is for any educator who is implementing the CCSSM. This in depth look is something I think is missing in much of the PD that surrounds these standards, which might explain a lot of the confusion about how to teach and help students understand the standards and practices, which in turn leads to the confusion among parents, the media and politicians.

My plan here is to write a couple posts about some of the ways looking deeply at the standards can bring clarity and understanding to how they work. More importantly, it emphasizes the importance of seeing the Math Content Standards and the Standards of Mathematical Practice as equally important in helping students learn, understand, and apply mathematics. You cannot look at isolated standards or practices or problems and hope to know how the standards and practices work to BUILD mathematical competence over time - (which, unfortunately, seems to be what is happening - looking at isolated components versus the overall picture).

This post is going to focus on the structure of the Standards of Mathematical Practice. Pretty simple really - there are 8 practices, and their structure contains the title and then the narrative description. The problem here being the title  is what most educators focus on, and the title DOES NOT give you enough information to really know what students need to do and what educators need to do to support students.  You really need to read the description to get a better sense of what students, as mathematicians SHOULD BE SAYING AND DOING, which then informs what resources the teacher needs to provide and what the teacher should be saying and doing.

I am guilty of assuming the title of the practice was enough. I will use Standard of Mathematical Practice #4 as my example. The title of this practice states that students should be able to:
Model with mathematics.
Seems pretty simple. On its surface it seems to imply using models - i.e. manipulatives or physical objects to help students understand math - which is wrong, or rather very incomplete thinking. Here's the narrative description of this practice:
Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.
I've highlighted just a few key phrases that show what this standard really means is students can USE mathematics to model and explain problems, which is NOT the same as using manipulatives to learn mathematics, which in my experience is what many educators think this practice means. What the practice says is students can approach a real-world problem and APPLY and use some type of mathematical model (i.e. equation, function, geometric figure) as a way to explain the problem and and come up with a solution. If you continue reading the narrative, you will see that it references students ability to make assumptions, approximate solutions, interpret their results and create mathematical models that help them solve the problem.  It's a lot more descriptive than just the titl, and something all educators need study and understand so that they are not misinterpreting what the titles alone might be saying.

The pictures to the right are of some of the work I did with teachers that shows their end-result of studying structure of the practices. These t-charts show an understanding of how the narrative informs both students and teacher actions (sorry, they are a bit out of focus!)

The point I am making here is that the title is not enough - the narrative is more important than the title because it gives expectations for what students should be saying and doing, which in turn informs what the teacher should be saying and doing. The structure of the practices, if looked at as a whole, provides information to help teachers create a classroom that supports student learning - you need to go beyond the title.