Greenhaus Education and Other "Effects"

Equitable Teaching Practices - What Does That Mean?

noreply@blogger.com (Dr. Karen Greenhaus) — Thu, 15 Apr 2021 15:50:00 +0000

I have been applying to jobs lately for a couple of reasons. Mainly, being an independent education consultant is really difficult in a pandemic, since most of that work involves travel to schools, conferences, or different states to support teacher professional learning initiatives and basically, most of that has dried up. Another reason, which almost seems the opposite reason, is remote positions are more prevalent now, so previous full-time jobs that required specific locations and hours are now going remote. I've been working remotely since 2007, and it's such a consistent part of my life, and important to my lifestyle, especially having to support and help care for my mother with dementia, and now there are more opportunities.

What I am finding as I fill out applications (all online) is the following:

It's hard to have a 1-2 page resume/CV when you have been working for over 31 years in education, so I've given up on that!
I am 'perfect' for quite a few jobs out there BUT I think my 31 years is working against me....i.e. I might be beyond the salary range the market is willing to pay due to my experience
Not everyone wants a cover letter, where you can 'outline' the specific qualities you have that match their job description. It's often hard to 'sell' yourself - they just want a few quick questions and a resume/CV upload
You don't hear back.

Sometimes there's a confirmation email, but most of them say "we will only contact you after we have reviewed your information and you pass the screening". Basically, if you don't hear from us, we don't want you, game over. So how long do you wait before you give up?
It's sort of demoralizing, especially when line-by-line they are describing you in the job, so you have to wonder what's wrong with me? Which brings me back to #2 - i.e. too old, too expensive, too much experience, too female.....(it may be none of these or all...this is the unknown!!)

It's hard to describe why you are good for a job in 200 words or less. Online applications seem to have a lot of limitations - "Write 2 - 3 sentences about what experience you've had with teaching online", for example.
Your social media presence is DEFINITELY being looked at!! From LinkedIn, to Facebook, to Google/Gmail, to Twitter, Instagram, etc. - they are all fair game, and sometimes links to your social media and blogs/YouTube channels, etc. are asked to be provided.
The questions asked are super interesting and reflect the politics of our time. These two have come up a couple times, in different formats /wording, which leads me to my posts main topic:

How does your experience show commitment to racial equity?
Explain how you use equitable teaching practices to make sure anti-racism is centered in your work

These are NOT easy questions to address in a short amount of space. They are not easy questions to answer being I am a white woman educator, so I can't guarantee my experiences and teaching practices have in fact always demonstrated racial equity and that anti-racism has been centered in my work. I think it has, and definitely has more and more over the past 20 years as I learned and grew in my profession....but it comes down to, what does it mean to have equitable teaching practices and anti-racism centered in your work? This is what I want to explore in this post, via finding and sharing resources, which is what I have been doing to really think about these questions.

Equity in Education - A Definition

There are a lot of definitions out there and explanations for what it means to have educational equity. I am going to share what I interpret this to mean, since this then connects to how I respond to the questions asked on the application. The image below is one I am sure many of you have seen in some form or another, but it's a nice visualization that shows how the same thing for all students in education (same teaching methods, same resources, etc.) is NOT equity, but equality, and does NOT serve all students - it still leaves many students lacking and behind, many of which are students of poverty, of color, and/or of special needs. Hopefully the definition of equity that I am using below clarifies this a bit more.

Equity in education means that the all students are getting what they NEED to learn and achieve their learning potential. It's not about the SAME resources for every student (which would be 'education equality'), but instead they have the supports, funding, resources, standards, teachers, modifications, that will help them reach the SAME high standards set for all students. They are NOT hampered by lack of resources, not restricted in what they learn (standards) or resources because of their background, race, gender, etc. What one student might need to succeed is different than what another might need, and both students are receiving those supports so they can both reach the academic standards. It's about getting all students to succeed, and some need different resources and supports that, if education is equitable, they are receiving.

Equitable Teaching Practices

Now I come to answer the questions that are posed in the applications, where I am asked to describe how my experiences and practices demonstrate equity in education and anti-racism teaching. Using the definition from above, here is what I consider equitable teaching practices:

Providing multiple tools and approaches when teaching topics. As a math teacher, this means to me showing and teaching with concrete tools, visual tools, text, etc.
Connecting the content I am teaching to something students are familiar with - i.e. bringing in their community, their culture, their world experiences. If I am teaching about slope and talk about mountains to a group of students who have never been to the mountains, I am not being equitable. Instead, talk about slope and connect it to hilly streets, or skateboard ramps, or railings on the steps, etc.
Asking for and listening to student voice. Creating a classroom where talking is expected and respected. Where collaboration is a daily part of learning - we learn better together. Where their experiences, understandings, questions are valued, shared, and more importantly, listened to and used.
Not assuming they know or don't know something, but instead providing opportunities for them to share what they know and using that knowledge to provide the supports and tools each student might need to get everyone to the same place. So often as teachers, following the prescribed curriculum and pacing, we just plug on through, assuming students are at a certain level of understanding, which means we are leaving those who are not there even further behind, and those that are past that stagnating.
Allowing students to explain, show, and share their understanding in multiple ways and accepting multiple approaches and solutions to problems. A student who is verbal might on paper look like they don't know something, but if you ask them to explain in words, you realize, wow....they do know a lot. So being willing and able to allow for students to learn and explain their learning in ways that make sense to them. Which is really different in our education system that expects 'one right answer in one specific format (think standardized tests). So this is a tough one as you often gets you bucking up against an educational system that is NOT equitable.

So - if I look at what I believe and have tried to practice and promote in my own teaching and in the professional learning experiences I provide to teachers, I think my practices strive to be equitable and anti-racist. I am attempting to allow all students a voice in the learning. I am attempting to provide different supports/resources for the differing students/teachers I interact with. And I am attempting to create a collaborative, safe environment for all learners (whether students or teachers). Was I always doing this? I like to think so, but I know I have continually improved over my career, especially after getting my masters and doctorates, that focused on these types of equitable practices. I know I still have a LONG way to go and am still improving. And learning from others. I will share some amazing webinars and podcasts episodes that have opened my own eyes to additional strategies and resources and perspectives on how to be equitable and anti-racist.

Some Resources Around Equitable Teaching Practices

There are obviously many more resources out there....so search, read, listen.

Here are a few webinars that TODOS Math for All and Benjamin Banneker did in partnership with Casio specifically around equity in math education. I was facilitating all of these webinars and learned so much from each educator that I know as informed my own practice even more. And while these are math-content focused, the strategies and frameworks shared could apply to any content area.

Equity in Education: Black, Indigenous and Latinex Parents as Intellectual Resources
Equity in Education: Rehumanizing Schools - Rights of the Learners (highly recommend this one)
Equity in Education : ICUCARE: Making Equity a Reality in Your Math Classroom
Equity in Education: Examining the Trajectories of Black Mathematicians
Equity in Education: Do Mathematical Practices Put Black Learners At Risk?
Equity in Education: Free Your Math and Justice Will Follow
Equity in Education: Creating Equitable Classrooms via Culturally Inclusive Teaching
There are more - so to see the entire list of on-demand webinars (many around equity, access, distance learning) go here.

As noted in previous postings, I host a webinar with my friend Tim Pope. We have interviewed a few folks so far that talk about equity and equitable teaching practices, so thought I'd share those here as well:

I hope some of this has enlightened you on what everyone means by equitable teaching practices. There is much to learn, much to practice, and it is a slow, continuous process to make change and practice equitable teaching. I will continue learning, as I hope you all will too.

And if you know of a remote educational/professional learning job, let me know!!

Education Podcast Listeners - What Topics are of Interest? Who Do You Want to Hear From?

noreply@blogger.com (Dr. Karen Greenhaus) — Wed, 07 Apr 2021 20:44:00 +0000

As mentioned in my most recent post, my friend Tim Pope and I have restarted our education podcast, 180days.education. So far, under the direction of our producer/editor, Quinn Greenhaus, we have come up with a consistent schedule, which includes releasing our episodes on a regular basis (the 2nd and 4th Thursday's of every month), along with releasing a monthly Newsletter (the 2nd Thursday of each month) that gives some insight into what the episodes for the month hold. We already had show-notes, but we have now made them more of a blog-type post, where you can listen to the episode and also read more about the episode and find any resources or links that are mentioned in each episode. We also are using some better tools for our recordings that make it easier for Quinn to edit each episode.

We use to use Audacity for recording, which is a perfectly fine tool, however we had to send our interviewees directions for downloading and saving the audio files and it had more potential for error. Now we use Zencastr, which is a simple web-link that everyone logs on with, and everyone's audio is recorded separately (and now also video if you want....that may be down the road!) Another lesson learned is it's worth it to pay for the subscriptions/services instead of only working with the 'free' options. As I am an independent contractor, I can use these as business expenses. Long story short, we've made a lot of changes - new website, new recording software, new equipment (high-quality microphone, boom arm, noise cancelling headset, newer, faster computer, etc.), paying for some social media 'ads' to try and garner some attention and followers. I would say this revamped podcast has us much more dedicated and excited.

One major change is every episode has a guest (or more than one), chosen because of a specific interest or question we were curious about. Tim and I are now just asking questions, and listening to others experiences and expertise, with of course some relevant commentary thrown in as it comes up. Personally, I enjoy this approach so much more than our previous approach, where we would pick a topic, each do research, and then try to share and disseminate the information we found. That type of format was a lot more stressful, since we were trying to be 'experts' on topics that we might have only had a passing understanding of previously. It is so much more exciting and interesting to let true experts do the talking - I feel so much more relaxed and involved in the episodes because I am learning as I listen. While I do research before each interview so that I have some questions at the ready, I do find that most of my questions come up organically, as I am listening to what they are saying. It feels much more like a conversation instead of an interview, and what I also love is that while we may have a specific 'topic' or interest we focus on, the interviews always end up being more or go off in unexpected directions.

Here are the guests and topics we have had so far in our 2nd Season (by Episode):

This month our focus is on instructional tools and curriculum materials. Our two guests talk about their experiences with developing quality curriculum versus open education resources and share their journeys. Dr. Steve Sullivan discusses higher education and Anatomy & Physiology using digital tools, and Patrick Vennebush discusses math curriculum and resources. We have upcoming episodes in May, June and July that discuss systemic change, teaching abroad, being Black (student, coach, teacher, mentor) in America and in education, and running for school board, to name a few.

To say that our revamped podcast is more fun for Tim and I is an understatement. It is such a pleasure to talk with all this folks who are so dedicated to whatever area of education they are speaking about, and to get insight into areas that we may know nothing about or that we know something, but are hearing a different perspective. We would LOVE to have even more variety in guests and topics (we tend to be a bit math centric, since this is both Tim and my backgrounds), so I wanted to put this out there to any of you in the education field - whether that be teacher, administrator, education resource provider, assessment provider, parent, student - what are topics of interest? And who do you want us to interview? I am not afraid to reach out to education leaders/pioneers I don't know and ask them to be guests - the worse they can do is say no, right?

So - please take some time to fill out this quick survey: https://www.surveymonkey.com/r/8PFYMS9

Be sure to visit the 180days.education website - leave us a voice message, check out all our episodes from Season 1 and Season 2, and subscribe!!

Back In Action - Coming Out of the Covid & DeVos Era Educational Haze

noreply@blogger.com (Dr. Karen Greenhaus) — Wed, 24 Mar 2021 16:33:00 +0000

I am embarrassed to see that the last time I posted on this personal blog was back in February, 2017. That is quite the long hiatus. No excuses other than I think I was hiding out until the DeVos era ended. I would love to say I have been hanging out at the beach and enjoying life, but in addition to that I have actually been working hard these last years. Teaching, supporting teachers, doing hundreds of presentations around the country and world (Italy, Germany, Japan, England to name a few), and working on new technologies for math (ClassPad.net) and doing webinars and creating lessons and how-to's to support technology integration. Never a dull moment....with occasional beach days thrown in!

Anyway, back in action on the blog. Going to try to be a bit more present in this personal blog. I've been blogging non-stop on the Casio Education blog, but it's time to get back to my individual voice and messages as well. I can do both!!

I am working on a couple things of late that I am pretty excited about and hoping turn into some fun and informative things for others. This includes a podcast, possibly creating some PD courses (online) to support math teachers, and exploring some new avenues and partnerships.

I am still teaching online for Drexel University, and as we are all still under the COVID restrictions, I know how many educators, parents, students and people in general have really struggled with this remote-learning/working environment. As it has been something I have done for over 12 years now, I was not stressed, but I have seen and hear the struggles from friends and neighbors and colleagues, so I know transitioning to an online environment is difficult. This is one of the exciting things I am thinking of focusing on - i.e. creating some courses focused around how to teach mathematics online, since it's something I have done, with a particular focus on using technology appropriately in an online environment. Developing a PD resource for teachers/schools. It's what my whole dissertation was based on - how to provide hybrid PD around using technology to teach math - so, just creating my own courses to share with others.

https://www.180days.education/

A couple years ago, my friend and I, Tim Pope, started a podcast, 180days.Education. Tim and I had the pleasure to work together at Key Curriculum and traveled the country doing lots of work with educators on inquiry teaching with mathematics and technology integration. We started the podcast to share teaching with others, as both being former K-12 teachers and administrators, and then working in the publishing and training part of education, we thought we could provide some insight and a voice to others about education. What we found in our first podcast go-around was that the episodes where we had guests, talking about their expertise, were much more informative and fun, compared to those where he and I did all the research and tried to be the experts. Then life happened - career changes (for Tim mostly) and a couple moves, and so we took a 2-year hiatus. But - we have come back, and have just started our 2nd Season this past January with a different approach, much better equipment and recording capabilities, the same editor/producer (but with much more expertise under her belt as well (Quinn Greenhaus, my younger daughter)), and having a lot more fun with it. Our approach now is to rely on experts in the educational field to share their experiences, resources, etc. and Tim and I are just the facilitators of the conversation. We also are not limiting ourselves to math (though we do tend to be a bit math centric at times, as these are often the experts we know best), but education topics in general. We have talked to an expert on environmental literacy, someone on curriculum development, experts on professional development and systemic change, teachers in the classroom sharing their experiences with Covid and teaching, an astronomy and physiology professor.....it's exciting for us to be able to open up different avenues of interest to explore and share with others. So - hopefully you will give it a listen!! You can find us wherever you get your podcasts and also at our website, 180days.Education, where you will find more detailed show notes and links to resources mentioned in each episode.

Finally, I am exploring the idea of branching out and trying something new. I admit to perusing the job offerings at various edtech and math education companies to just see if there is something that sparks my interest. I know my strengths, of which professional development and online training/teaching are a couple, so I am looking in those areas. To be honest, being an independent consultant, while very freeing, is NOT very lucrative, so it would be nice to maybe bring in a bit more money, especially after a pretty tough year of COVID confinement. But that freedom of schedule is something I am not quite sure I want to give up, so looking for flexibility and more 'project-focused' timelines versus a required 9-5 or 8-4 work day. I have my aging mother with dementia that takes a lot of my time, so flexibility and remote work are not things I am willing to sacrifice. But - I am looking, exploring, putting out feelers....if anyone knows of anything!

Anyway - I am back and hoping to post a little more frequently - once a week is a goal at this point. Might be more, might be less...depends on the news and the mood!! Meanwhile, enjoy this sunset view from my home office....not a bad way to end each day.

Student Engagement & Learning - Connected to Teaching Intentionality

noreply@blogger.com (Unknown) — Sat, 23 Jul 2016 16:57:00 +0000

I read an article this morning from Nira Dale entitled "Why Instructional Design Must Focus on Learning Outcomes, Not Learning Activities". While the article was focused on edtech tools and making sure the games, devices and activities used to engage students also focused on learning goals, the message actually pertains to all education activities, whether technology is involved or not.

It reminded me of when I was working in Virginia Beach, where the whole district was working with creating lessons using the Understanding By Design framework from Jay McTighe and Grant Wiggins. The key message, that I have continued to utilize in all my work, is that engaging activities for the sake of having a 'fun' thing to do with students does NOT enhance student learning unless the activity chosen connects directly to the learning goals and desired student outcomes. Seems obvious of course - but, you would be surprised at how many teachers create wonderful, fun, engaging experiences for their students, with and without the use of technology, that don't connect to a learning goal.

The key point is intentionality in your choice of activities/resources/tools. Which of course requires deliberate planning and knowing your standards and desired learning outcomes.

I've been working as an International Fellow for the Charles A. Dana Center at UT Austin for the past two summers and as part of this work, we really focus on this idea of Alignment and Intentionality. Basically, understanding your standards for the grade/content you are teaching, really looking at what students should have learned and been able to do prior (previous grade), what the learning goals/expectations are for what you are teaching, and what they should be able to know and be able to do for the next grade. Aligning the standards vertically so that you can see the learning progression, which then informs your choice of activities.

Understanding the learning progression and standards informs what you as a teacher need to do to help students learn and reach those learning goals/objectives. This clear understanding of the learning goals forces you to be very deliberate and intentional in how you teach - what you do to review skills from previous grades, how you introduce new skills and concepts, what engaging activities you choose or resources/tools you use to support students learning of the concepts. Activities chosen for students to engage in should not be because it would be 'fun' and 'engaging', but because they would directly enhance and support the learning goals.

To sum up:
1) Understand the standards and the learning progression (vertical alignment)

Know and understand the standards you are going to teach
Look at the related standards from grade/course before and after so you understand what students should already know and be able to do and where they are going after they leave you

2) Based on the learning objectives you have defined and aligned from step 1:

Choose activities that will support the content and skills you have determined need some review - be specific, make sure activities align directly to the content and skills you are focusing on
Choose learning activities that will introduce, teach, and enhance student understanding of your current grade/content standards. Look for engaging activities that support your specific learning objectives directly. Be deliberate in the activities you choose - if you can't directly align them to the learning goals, then don't choose.

3) Build in deliberate formative assessment so you can assess students progress and keep learning objectives at the forefront

Activities you choose should allow for formative assessment - questioning, student discussion, student collaboration - so that you can continually assess student progress
Activities chosen should provide evidence of student understanding related directly to the learning objectives.

Basically - make your classroom an engaging learning environment but choose those engaging activities deliberately and intentionally so that your students are engaged in productive learning that helps them reach the learning goals. Learning goals should guide all your instructional decisions - from the tools you use to the activities your students engage in.

Lifelong Learning - It's About Finding Pathways and New Adventures

noreply@blogger.com (Dr. Karen Greenhaus) — Mon, 27 Jun 2016 20:14:00 +0000

Wow - can't believe I haven't posted since May. Crazy. Of course, I do post weekly on my other blog, so I am going to plead busy!!

It has been busy these last few months. I have been doing some very diverse things, all related to math education. It's one of the benefits of going out on my own and being "independent" - I get to try new things and do a variety of math and technology related activities which keeps me interested and on my toes. Here's just a few things I have been doing since May:

1) Working for Casio - so blogging, Tweeting, Facebooking and doing research on technology, BYOD, testing etc.
2) Working for the Dana Center at UT, Texas and DoDEA (Department of Defense Education Activities) as an International Fellow supporting their adoption of College and Career Ready Standards. I went to Austin, Tx for some training and just got back from England, doing training at LakenHeath Air Force Base. Such a marvelous experience! Heading to Austin again next week and then to SC for training in August.
3) Creating storyboards/video tutorials for OER Commons, so brushing up on my video-making skills and voice-over. That's been a blast.
4) Teaching an online Sketchpad course for Drexel University - we are in week 2 and I love it! Sketchpad is my all-time favorite edtech dynamic math tool so any chance I get to work with this is a bonus.
5) Heading to Hamburg, Germany end of July to support Casio R&D and attend ICME (International Congress on Mathematical Education) that happens only once every 4 years - THAT is super exciting!!

My personal experience with being able to work for many different folks is that I am constantly

learning, relearning or refining skills and knowledge. The work with DoDEA is really helping me focus on the Common Core Standards, which is then going to serve me well when I work with New Jersey teachers in August. Making videos/tutorials has forced me to dive deep into sound mixing and video clip editing and mixing, which is so helpful for making Sketchpad videos for the online course I am teaching or my blog, as well as the calculator videos I make to support Casio. All these different experiences force me to be on top of the latest education, edtech information/research and hone my math and technology skills. This "life-long" learning approach is exciting and helps me consider other avenues and possibilities for my career.

Looking back at my summers when I was in the K-12 public teaching arena, I realize that I've always been engaged in life-long learning and searching for new pathways & adventures. I took classes or looked for new strategies, lessons and projects ideas to use in my classroom. Every learning experience led me to that next adventure - i.e. teaching to administration to running the PD for a publishing company to getting my doctorate to starting my own business to now.

I often get asked how I ended up becoming an independent consultant - and I would have to say I just kept looking for opportunities, learning whenever I could, researching and reaching out. Make connections with folks at conferences or representatives from publishing companies, take some courses and explore new pathways. Challenge yourself. Don't just sit back and let the summer go by - be sure to learn one new thing because that one new thing could open a door to the next adventure.

Poor Math Questions and Misunderstandings

noreply@blogger.com (Unknown) — Thu, 12 May 2016 21:35:00 +0000

My daughter sent me this article, Parents are Freaking Out Because They Can't Answer a 7-year Olds Exam Question, the other day concerning a Twitter posting about a math problem that was frustrating students, parents, and lots of people on Twitter obviously. Here's the problem:

The issue is there are two answers that could be correct: 65, if you use the 'work backwards' method, subtracting out the 17 that just got on from the 63 and then adding back the 19 that got off. The other answer is 46, which results from simply subtracting out the 17 that just got on from the 63 currently on the train. The teachers exam rubric/'test key" stated the correct answer as 46. Hence - all the confusion, because that would mean the 19 was an unneeded number.

The real problem here, and it is NOT an isolated situation, is that this question is very poorly worded, and thus open to both interpretations. The work-backwards approach assumes that the question is asking for how many people were on the train at the beginning of the trip BEFORE the 19 people got off the train. The second interpretation assumes the question is asking for how many people were on the train before the 17 got on but AFTER the 19 got off. Both interpretations are right because the question is so vague - but, as the student obviously found out, the 'test key' accepts only one interpretation.

Poorly worded questions or problems on math tests or in math books are very common and a cause of much frustration and angst on the part of students, parents, and teachers. As a teacher I experienced it often in the textbooks and assessments my students had to use/take. Having worked in the math publishing arena for several years, when creating math lessons, assessments, etc. it is VERY easy to word things in a confusing way, making them subject to multiple interpretations. Without "vetting" and piloting questions, this can become a huge issue, thus where the problem lies - the creators of today's assessments and textbooks - i.e. the publishing companies - do not always invest in proper editing and piloting of problems to ensure that there is no confusion, misunderstandings, multiple interpretations, etc. That is time and cost prohibitive.

Poorly worded questions wouldn't be that big a deal if in fact, as this problem asks, the students worked actually counted for something. If their showing of work included their method and interpretation, and based on their method, the teacher was able to say that yes, in fact, your answer is correct based on your interpretation of the question asked, regardless of the key, then the poorly worded question would just be a minor issue. But - as this is more than likely a standardized exam with only one acceptable answer, the students' explaining and justifying of their interpretation and work has no bearing on the actual expected outcome. A true disservice to our students.

Ideally - students won't be exposed to these ambiguous types of questions. But - in a perfect world, even if these types of questions slip through the cracks, a students' explanation of what they interpreted the question to be asking, and the proof of their method to support their interpretation, would be counted as correct - i.e. 65 is a correct solution, even if the "key" says it's 46 because the student was able to explain and justify their answer to support their interpretation. That's a true assessment of student understanding - not an "set answer" to a crap question. So - first, let's push for better assessments, but more importantly, let's accept logical, well-reasoned explanations to poorly asked questions because there-in lies true learning.

By the way - a better worded question, if the answer was to be 46, would have been: "How many people were on the train AFTER the 19 left and BEFORE the 17 people got on? Or - even, "How many people were on the train RIGHT BEFORE the 17 people got on?

Algorithms and Losing Control

noreply@blogger.com (Unknown) — Wed, 30 Mar 2016 20:02:00 +0000

On my way to the gym this morning I listened to yet another story on NPR about the FBI/Apple controversy surrounding accessing the iPhone of the San Bernardino terrorists. Quick synopsis - Apple refused, FBI found a 3rd party who was able to create their own algorithm to hack into the phone, FBI dropped case against Apple. But - everyone is worried. Will the FBI share this code-breaking algorithm with other law enforcement? Will we all be vulnerable now to some outside person accessing the privacy of our smartphones?!

Legal issues aside, what I find interesting is the fact that an algorithm unlocked the information. Or algorithms created the 'locked phone' in the first place. As a math person, it's always fascinating to think about how algorithms create so much of what is around us, and while I don't understand it myself, there are people out there who do. People who can create a code to break into a locked phone. Or an algorithm to pick stocks or predict weather. It's really incredible. A little mind boggling.

With this on my mind, I was excited to actually find a Ted Talk specifically on algorithms and the world around us. Seemed sort of ironic - though, it was probably a Google algorithm that matched my search of the NPR story to other stories and resources that contain algorithms. Spooky!! In fact - the talk, by Kevin Slavin, addresses this very fact - that algorithms and coding are responsible for some of our decisions (or many). So - are we in control or are the algorithms controlling us?

Watch the Ted Talk and freak yourself out too!

Common Core is NOT the same as Standardized Testing

noreply@blogger.com (Unknown) — Tue, 22 Mar 2016 18:18:00 +0000

I have read a couple articles of late that have kind of pissed me off. Mainly because they perpetuate the ideas that 1) The Common Core State Standards are handed down from the Federal Government, and 2) that the CCSS are the driving force behind Standardized Testing. While there are connections, both claims are a complete disservice to the purpose behind and the reason for The Common Core State Standards. And they completely ignore the fact that Standardized Testing is actually controlled by testing companies and publishing companies.

I've already posted on this before - "What People Think Is Common Core ISN'T" and "Common Core, It's NOT the Devil" so I apologize now for being repetitive. One article I read last night, entitled "What's the largest number you can represent with 3 digits? Nope. It's not 999." is a great article. I agree with everything the author said and applaud his sticking by his daughter, the second teacher, and the fact that he forced the standardized testing company to change their answers. Awesome. Then I saw the last line: "don't let common core stand in the way of your own children's education". Sigh. IT"S NOT THE COMMON CORE that's doing this - it's publishing companies, it's testing companies, it's misunderstanding and lack of training of teachers.

As I've written before, the Common Core actually represents problem solving, critical thinking, perseverance, multiple solutions (as this author was demonstrating his daughter was showing), communication, multiple pathways. It's all about unique minds and coming to mathematical understanding from different perspectives. Read the standards. Especially the Standards for Mathematical Practice. What the above article was demonstrating was the fact that the teacher was NOT following Common Core Practices, whether through ignorance or more likely, through the pressure of textbooks publishers and standardized testing companies to "standardized" the Common Core, and thus in fact become something completely different than what the Common Core represents. Let's go back to my two previous postings....what people think is Common Core ISN'T - it comes back to our testing culture and a need to standardize so we can "assess" students quickly, and in the process, take out all the problem-solving, uniqueness, critical thinking that is the essence of the common core.

Here are some facts about the Common Core:

State education chiefs and governors in 48 states came together to develop the Common Core, a set of clear college- and career-ready standards for kindergarten through 12th grade in English language arts/literacy and mathematics. Today, 42 states and the District of Columbia have voluntarily adopted and are working to implement the standards, which are designed to ensure that students graduating from high school are prepared to take credit bearing introductory courses in two- or four-year college programs or enter the workforce.
States led the development of the Common Core State Standards. In 2009, state leaders, including governors and state commissioners of education from 48 states, two territories and the District of Columbia, came together and decided to develop common, college- and career-ready standards in mathematics and English language arts. They worked through their membership organizations – the National Governors Association Center for Best Practices (NGA Center) and the Council of Chief State School Officers (CCSSO) – to accomplish this. The development process included defining expectations for what every child should know and be able to do when they graduate from high school and then creating content standards for grades K-12 aligned with these expectations. States relied on workgroups of educators, representatives of higher education and other experts to write the standards with significant input from the public in 2009 and 2010. States then appointed a validation committee to review the final standards. The federal government was not involved in the development of the standards. The final standards were published in June 2010 and available for each state to review, consider and voluntarily adopt.
Adoption of the standards is voluntary. It is up to each state and territory to decide if they choose to adopt the Common Core State Standards as their state educational standards in English language arts and mathematics. States can tailor the standards to address their needs. Here is a map showing the states that have adopted the standards.

The standards are voluntary, with the idea that if all states had the same standards it would be easier for students to be compared state-to-state and to ensure that all students, no matter where they live, were getting an equal and rigorous education. Notice - nothing about the Federal Government. The Federal Government did encourage adoption and funding with Race-to-the-Top money if states adopted the Common Core, but not required. And NCLB did require standardized testing of all students, which is the 'standardization' push of the Common Core Standards, which are difficult to assess in a standardized test. Standardized tests are NOT Common Core.

Standardized testing - states realized they needed to assess and that they needed some help with

standardized testing aligned to The Common Core. States got together to create consortiums, such as PARCC and Smarter Balance Assessments. These were led by Pearson (PARCC) and a "public agency" (Smarter Balance Assessment) and state leaders who opted to join the consortiums. These assessments are voluntary and there have been issues with both. There are currently only 15 states still in Smarter Balance and only 11 left in PARCC. Many states have dropped out.

With the adoption of ESSA, where states have more control over assessments, there is going to be a change in which standardized tests are used to assess students. But they are not "Common Core". ACT, SAT, NAEP - the most commonly used now, are definitely NOT Common Core. In fact - true assessment of Common Core should not be standardized, especially when keeping the mathematical practices in check. There-in lies the problem. We are trying to standardized standards that are meant to help students become problem-solvers, critical thinkers, and communicators. You can't demonstrate that on multiple-choice tests where one correct answer is the norm. So - rather than blaming Common Core, blame the 'standardized' way of thinking that forces teachers to teach to a standardized test, accept only "one" answer, or force ways of thinking on students when they should be encouraging uniqueness and multiple pathways. In order to change the way students learn, we need to change the way we assess or we are always going to be forcing students into one or two ways of thinking so they can 'pass the test'.

What is the "Math Path" We Should Encourage Students to Take?

noreply@blogger.com (Unknown) — Wed, 02 Mar 2016 14:48:00 +0000

I read this article the other day by Dana Goldstein, "Down with Algebra II!", which describes professor Andrew Hackers views on mathematics, and how the push for STEM, higher math like Algebra II, is actually creating a the failure and dropout rates we are seeing because it is pushing students into mathematics that is not necessary for their future endeavors and "destroying a tremendous amount of talent". The math requirements in high school and college are "highly irrational".

And then my friend sent me the same exact article via Facebook and asked my opinion. (This is what happens when your friends know you are a math person!). So - what is my opinion? I have to say, I completely agree with Andrew Hacker. In fact - I have written about it before a few times: Math Curriculum - What should we be teaching?; Financial Literacy - Real-world math, REALLY; Let's teach probability & Statistics - We need it!

Don't get me wrong - I think Algebra II and higher level math is important. I believe in the Common Core Standards, which recommend higher mathematical concepts. However - NOT for everyone. My friend who sent me the FB link, has a student with a learning disability who completely struggles in math. She was concerned because if Algebra II is a requirement, she knows her son will struggle, probably not get the grade needed, which will then hurt his chances of getting into a college of his choice. He is NOT planning to go into a field where he would need Algebra II, or Calculus. Which is true for many students. But - in our traditional curriculum, even with the Common Core, we push students along the following path: Algebra, Geometry, Algebra II, Pre-Calculus, Calculus. Maybe some Trig. Rarely do we push them into Probability & Statistics, and since Computer Science is usually NOT allowed as a math credit, we don't push them there either. And Personal Finance? Not even an offering in most places.

I will use my daughters as examples. The oldest is an art person - lives, eats, breaths art. If she could

Example Oldest Daughters Art - Acrylic

draw all day (which she does), she would be happy and satisfied, with the occasional break to eat. The younger - well, she is still a conundrum, but she is brilliant with math and science. She just gets it. She thinks different. When it came to what to take after Algebra II (required), it was a no brainer for the younger one to go into precalculus and calculus. She needed the challenge. However I told my older daughter she should take statistics. It would be more relevant to her more artsy/business direction (advdertising). Let me tell you - her counselor was NOT happy. Kept trying to push her towards Pre-calculus instead so she would get into "college". NO. This daughter would have done horrible in Pre-calculus - she was not interested in it, was not planning to use this mathematics in her future endeavors, so why should she be forced down this path? Well - we went against counselors wishes and took statistics, which she absolutely loved (great teacher) and totally excelled in. It was relevant, real-world, hands-on and - pertinent to what she does now. Perfect match. But - it was NOT on the recommended path so we had to fight for her right to take it. My point - two very different daughters, two very different personalities and interests, and therefore they should have two very different pathways.

I completely agree we need more students interested in STEM careers, however, not every student needs to be pushed here, especially if their interests lie in arts, history, business, computer gaming, etc. All students need math - and I am a big believer of math every year through grade 12. But - just different math. What I think, especially in this age of "personalized learning", is there needs to be choice in mathematics, and not the constant push to force all students down the same path, a path that for many is unnecessary and a road to failure. I definitely want students to learn math - and I truly believe in the Common Core State Standards - which emphasize problem solving, real-world application, critical thinking and conceptual understanding. All of these goals can be accomplished with other math choices. If we had other pathways for mathematics, we might actually find students developing an interest in math and pursuing higher level courses of their own free will.

A lot of districts will say they don't have the funding to offer more math choices. With online

Andres Marti Teaching Statistics w/TinkerPlots

learning that is not an excuse, as long as districts are willing to accept credits students take online from other places. Additionally, districts/states need to rethink the math requirements and what courses count towards a math credit. Thankfully, this seems to be coming up more and more, but courses like computer science should count as math credit (to replace Algebra II or Calculus), Personal Finance courses should be offered first of all, and should also count as a math credit. How does this work with Common Core State Standards and other standards? It shouldn't impact those at all - if students do take Algebra, Algebra II , Statistics, then the content standards should be followed. The Mathematical Practices should be followed no matter what courses you take. But - what is 'required' for math credit - THAT's what should be rethought. Let's make learning about what is going to help a student acquire the skills they need to pursue the career path they want and be productive members of society. Personal Finance would go a lot further towards that goal for all students than Algebra II, don't you think?

What is my ideal math menu?

1) Students should have at least 4 math credits to graduate high school. Math is important!
2) What are some required courses?

Basic Algebra (abstract thinking is important and useful to everyone)
Geometry (this helps with logic, spatial reasoning, etc.)
Personal Finance (yes - EVERYONE)

3) What are some optional courses that count as MATH credit? (not exhaustive)

Computer Science - all kinds (coding, robotics, gaming, etc.)
Probability & Statistics (voting, sampling, etc...important for being a functioning person in todays' society)
Advanced Algebra (Algebra II?)
PreCalculus
Calculus (A&B)
Trigonometry
Math for Medicine (i.e. for those interested in nursing or doctors)
Accounting

There are more that could be added to the optional courses list obviously - this is just a smattering. But - what matters here is there is choice. And based on a students interests, they should be given the choice.

Sketchpad & TinkerPlots - Still Out There, Still Awesome!

noreply@blogger.com (Unknown) — Wed, 24 Feb 2016 22:06:00 +0000

NCTM is coming up in April and is in San Francisco, which has me very excited because I get to have a Key Curriculum reunion (i.e. my colleagues from my years working for Key Curriculum mostly live in the SF area). It's hard to describe the amazing connection those of us who worked with Key, (Keysters, as we fondly refer to ourselves) have, and I have yet to find another place or another group of people that I so deeply connect to on both a professional and personal level.

With this math conference and potential to see so many of my former colleagues, I have been a bit nostalgic about some of the things I loved from Key - i.e. Sketchpad and TinkerPlots to name a couple. Sketchpad, now owned by McGraw-Hill, is still around thank goodness, but without the support and push in math education it deserves. Thankfully, some former Key folks are still out there making a difference with Sketchpad - check out Daniel Scher and Scott Steketee's most recent article in The Mathematics Teacher, Connecting Functions in Geometry and Algebra, where they use Web Sketchpad to create dynamic, in-article representations. Having worked with Web Sketchpad myself, it's amazing, and a great addition to desktop Sketchpad, and hopefully McGraw-Hill will at some point provide access to this tool for everyone. Meanwhile - Sketchpad is still out there, and of course still better than Geogebra for many aspects of math (I have done extensive work with both). (Shameless plug - if you need any Sketchpad training/support for your teachers, please reach out to me, as I do that as often as I get the opportunity!).

TinkerPlots, which McGraw-Hill gave back to it's creators, is available again as well. If you haven't ever explored it, I highly recommend you do so. TinkerPlots is dynamic statistical software that allows incredible visualization and exploration of data, quickly and easily just by dragging and dropping. It's so fun!! Better than anything out there, hands down. The website on TinkerPlots has great resources so be sure to check it out. I use to do workshops and webinars with TinkerPlots when I was with Key, and wish more folks would discover the power of it, as it is especially helpful when working with students just beginning to explore data and connections. Especially helpful for those of you working with the Common Core State Standards for Mathematics, which has a huge emphasis on data analysis. (Another Shameless Plug - if you need TinkerPlots training/support I am available!)

In my reminiscing about my favorite math software and Key, there is another group I should let you know about, a group that also fell in love with TinkerPlots, but wanted a web-based version for their data site. Tuva Labs - a data literacy site that has really expanded over the last couple of years, has created an online version of TinkerPlots and has lessons and data sets that can be used, for free, in classrooms. Here's a link that lets you see the data tool. The site has data sets, lessons in addition to the online dynamic data tool. It is exciting to me to see a web-version of TinkerPlots out there that teachers and students can access, as it makes data come alive and available for students to explore and analyze data. Data literacy is such a need in our world.

Anyway - it warms my heart to know that dynamic math tools, like Sketchpad and TinkerPlots, are still out there, still available, and still being used. And better yet, there are web-based versions out there, though not necessarily readily available for all (Web Sketchpad). They are both evolving to meet the demands of a web-based society, which is exciting. There are many other math tools and apps out there, but in my humble opinion, having worked with so many, these two are some of the best and easiest to access and use, with and by students.

OER Commons - A Good Place for Common Core Aligned Resources

noreply@blogger.com (Unknown) — Thu, 18 Feb 2016 17:09:00 +0000

There are a lot of Open Educational Resources out there, which is exciting if you are a teacher, as you can find myriad of lessons and ideas. The big issue with OER is of course the quality of what you find, does it really align to your instructional goals, and naturally, do you have the time to search through possibly hundreds of resources to find that perfect fit.

In my recent work, I have been exploring The OER Commons website, which is a digital library and network of Open Educational Resources. It allows you to search, with a search feature that really lets you refine down to exactly what you are looking for - whether that be full lesson plans, videos, full units of study, online courses, etc. You can also review and comment on resources, which is a nice feedback feature that allows for others to gain from those who have used a resource. You can create your own content to share or link to others and collaborate.

Fig 1

What I particularly like, as a certified Common Core advocate, is the Common Core Hub, where the OER Commons digital librarians have organized thousands of resources so that they are specifically aligned to the Common Core Standards, even the Instructional Shifts for ELA and Math. There are also some CC Online Courses and other supports for educators. The search within the Common Core Hub allows you to find resources down to a specific content standard in the Common Core.Including the Grade, Domain, and standard. In Fig 1, you can see where I have narrowed my math search down 5 resources (from over 3000).

Once you narrowed down the search, you can look at your results, click on them for more

Fig 2

information,where you will see the standards written out, be able to view the resource in full detail. If the resource has been rated by others (there are some that have, many that haven't yet, since this is still relatively new), you can see a rubric of how well the resource supports the standards (Fig 2). This will be a really powerful support for teachers, since finding OER resources that are of quality and actually support your instructional goals can be a challenge. It will of course require those who use these resources to take the time to write a review and provide feedback, but that is part of the collaborative environment the OER Commons is trying to foster.

Again, there are many places to go to find OER but I like what I have explored so far on the OER Commons website, in particular the ability to find Common Core Aligned lessons, resources and educator support. Worth exploring.

Parlez Vous HTML? Coding as a Foreign Language

noreply@blogger.com (Unknown) — Fri, 12 Feb 2016 17:08:00 +0000

Yes, there is some thinking-differently going on, at least in Florida. There is a bill in the Florida House of Representatives to make in-person and virtual computer-science classes count as Foreign Language credits towards graduation. THAT. IS. AWESOME!

I am all for changing the way we teach and what we teach and where we teach in this country. I would love for us to break out of our traditional classrooms of desks in rows and teachers as sages, and let students have more choice in subjects they, and have more options of courses. For example, not every student needs calculus, but EVERYONE needs Personal Finance - and yet, it is not an option in most school districts, or if it is, it is often an add on. (See my previous posts on Financial Literacy if interested....1) Math Curriculum - What Should We Be Teaching 2) Financial Literacy - Bring It Back to School! and 3) Financial Literacy -
Real-World Math, Really!)

Math itself is a foreign language, so the idea to make coding officially designated as a foreign language is just brilliant. Now, granted, it's just at the higher ed level - public colleges and universities, but it's a start. I think K-12 schools should adopt a similar approach. Computer-science is such a vital skill these days, and so few students actually take computer-science in large part because it doesn't fit in the path of required courses for graduation. I look at my own daughter, a recent graduate with a degree in Advertising, who has had a difficult job finding work because she was lacking coding skills, which were NOT part of her advertising/marketing curriculum. And yet -

every job out there for an entry-level marketing/graphic designer requires the ability to code for websites, emails, etc. She is thankfully employed now, and is learning to code on the company' time, so she will be gaining some needed skills for future endeavors. But the fact that she spent four years in college, and before that, four years in high school without the opportunity or the push/requirement to take computer science is ludicrous. Especially when she is in a field that REQUIRES this skill. How can it NOT BE TAUGHT in the very program she graduated from (and she was at a very prestigious school in a very prestigious program - Creative Advertising at UT in Austin).

My point here - lets rethink the requirements for students as they go through high school and college. What skills do they need in the careers they want to pursue? They should have choice to choose subjects that will benefit them in the future so they are, as the popular lingo these days goes, "college and career ready". Let them take computer-science as either one of their math requirements (instead of say Algebra II or PreCalculus) or as Florida proposes, a foreign language requirement. I know my daughter would have been a lot better off with a couple semesters of computer science instead of the required two-semesters of Spanish, for the career she has chosen. The same goes for Personal Finance - a very needed course for all students. It should be an option and count towards graduation. Not everyone needs the traditional path of Algebra, Geometry, Algebra II, Trig, PreCalculus, Calculus. If a student plans to be a business major, Personal Finance is going to be a much bigger benefit than Trig.

Let's rethink our curriculum. Let's making learning relevant, useful, and personal. Coding as a Foreign Language? Hell yes!

Libraries, BookBots and Algorithms

noreply@blogger.com (Unknown) — Fri, 15 Jan 2016 17:15:00 +0000

I was listening to NPR this morning, as usual, during my workout. The Pulse did their entire hour on the Rebirth of the Library. As an avid reader, this was of great interest to me. I love books. I have a crazy collection of books, all down in my basement now since I live in a tiny house with little room for my shelves and shelves of books. But - I don't have the heart to get rid of them, and I do reread them - it's like coming home to old friends.In fact, I just reread over the holidays The Lord of the Rings Trilogy (an annual tradition)...which can only be done correctly with a the actual book that allows me to flip back to the maps a million times.

NPR's many stories related to the library included many things about how libraries of today are still relevant, and how many libraries are changing. I would suggest reading/listening to the different stories. The one that really sparked my mathematical interest however was the one by Peter Crimmins entitled Will BookBots be the revolutions libraries are looking for? I'd never heard of a BookBot, but basically it's a robot that finds books that have been stored in metal bins in large warehouse spaces. The bins are arrayed in a matrix (math!) and a searchable database is used to identify a book and send an algorithm to the BookBot, which then finds and retrieves the bin and brings it to the librarian to get the ordered book. It's a space-saving method of storing thousands of books, allowing for libraries to provide more space for meeting rooms, makerspaces, wifi, social gathering areas, etc. But - what fascinates me is the math.

The BookBot works on algorithms to find the books. The books are not arranged in a typical Dewey-

decimal system, but based on an algorithm. When books are returned, they don't go back to the bin they came from, but go to the closest bin to the crane. Storage is based on efficiency and proximity, not what we typically associate libraries with - alphabetical, chronological, genre. Fascinating. And clearly an amazing amount of programming and mathematics goes into this system. Naturally, this just made me think about students and the never ending questions "when are we ever going to use this". Here is yet another example of math at work....and in the library!!

I love finding examples of math in places you don't really think of as "mathy". Apparently this BookBot technology comes from similar technology used in the automotive and textile industries. It's fascinating to consider the mathematics needed to store up to two-million books and be able to deliver a single item within five minutes of clicking the online catalog. Yet another example to provide students where math and technology are used on a regular basis.

I found this video of the BookBot in action at the Hunt Library in NCSU. Very cool.

Being Slow in Math is A Good Thing

noreply@blogger.com (Unknown) — Wed, 16 Dec 2015 17:05:00 +0000

"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):

Everyone can learn math at high levels
Believe in yourself (your beliefs about your abilities actually changes the way your brain learns)
Struggle and mistakes are really important in learning math
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): https://www.youcubed.org/four-boosting-math-messages-from-jo-and-her-students/

#HourofCode Sparking the Need for Computer Science Curriculum

noreply@blogger.com (Unknown) — Fri, 11 Dec 2015 17:22:00 +0000

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.

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: http://globaloria.com/ 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.
Code.org Has some great #hourofcode tutorials kids will love. Just some listed below:

Minecraft
Star Wars
Frozen

Tynker has grade-level games and coding and teaching support as well
Khan Academy - coding tutorials among the many free content videos available.
http://www.teachcs.org/

Technology without Training & Sustained Support Will NEVER Succeed

noreply@blogger.com (Unknown) — Mon, 30 Nov 2015 15:55:00 +0000

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:

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

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

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

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.

Purchasing Digital Resources - Things to Consider

noreply@blogger.com (Unknown) — Thu, 19 Nov 2015 19:43:00 +0000

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:

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

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.

#Edchat Discussion - Politics, Religions & Education

noreply@blogger.com (Unknown) — Thu, 12 Nov 2015 00:13:00 +0000

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:

"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.

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.....

What if? A simple question to engage students.

noreply@blogger.com (Unknown) — Wed, 04 Nov 2015 13:37:00 +0000

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!

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

noreply@blogger.com (Unknown) — Tue, 27 Oct 2015 20:38:00 +0000

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.

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

noreply@blogger.com (Unknown) — Wed, 21 Oct 2015 14:08:00 +0000

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.

Common Core - Final - What Do You Mean Rigorous?

noreply@blogger.com (Unknown) — Fri, 16 Oct 2015 14:12:00 +0000

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.

Focus
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.

Coherence
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	Grade 2	Grade 3	Grade 4
Domain	Operations and Algebraic Thinking	Operations and Algebraic Thinking	Operations and Algebraic Thinking
Cluster	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
Standard	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
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.

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

noreply@blogger.com (Unknown) — Thu, 08 Oct 2015 18:10:00 +0000

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.

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

noreply@blogger.com (Unknown) — Fri, 02 Oct 2015 19:46:00 +0000

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.

Introduction
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:

developing understanding of multiplication and division and strategies for multiplication and division within 100;

developing understanding of fractions, especially unit fractions (fractions with numerator 1;

developing understanding of the structure of rectangular arrays and of area;

and describing and analyzing two-dimensional shapes.

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) 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) 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) 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?

Domains

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

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)

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?

Summary
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).

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

noreply@blogger.com (Dr. Karen Greenhaus) — Fri, 25 Sep 2015 16:48:00 +0000

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 #1: Mathematically 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).
Notice, START WITH THE STUDENT!!

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.