AIMS Zone of Proximal Construction – Tools for Productive Struggle

The AIMS Center on the campus of Fresno Pacific University is an organization that I’ve admired for at least 15 years. I actually was an AIMS scholarship recipient as an undergraduate mathematics major and as a teacher I used AIMS activities whenever I could.

AIMS has changed a bit in the past few years from producing print curriculum – they realized there was a bunch of OER stuff out there that it was getting hard to compete – and focusing on professional development and research.

My wife Meagan and I were invited to speak last week on behalf of their Colloquium Series – research-minded talks to math and science teachers going through an AIMS-related cohort.

We spoke about ways to help teachers let their students experience productive struggle. Emphasis on productive. Meagan, a special education teacher and adjunct professor, focused on environmental factors and things that teachers can do specifically for the learner. I focused on making the math connections when possible, as well as technology tools for formative assessment and PD to give the best information to both students and teachers. Formative assessment should really be called feedback in my opinion, and I made the comment either on the podcast or during the talk that if our formative assessment tools aren’t giving the STUDENTS feedback we’re missing the point.

Universal Design for Learning – designing learning not to accommodate learners with special needs, but thinking about those accommodations and then designing the lesson to reach ALL Learners with those methods – was the biggest takeaway so much so that Meagan and I are planning on attending the CAST conference in Boston this summer. It was a great night with feedback from the, ‘students’, conversations with Dr Brownell for the podcast, and overall feeling like being home and fun to be in front of people with great questions. My main points:

  1. Make formative assessment something the students themselves can learn about  – don’t keep the ‘dashboard’ confined to the teacher!
  2. Give students multiple ways to express their learning and struggles
  3. Design your classroom in a way that there are no questions for what is expected of students – the less outside stressors, the more they can focus on the academic tasks at hand.


Video Archive of the Presentation


Thanks Lori Hamada, Dr Chris Brownell and the entire AIMS staff for making our visit welcoming and great! We can’t wait to be back!

Rekindling the discussion of better, cohesive OER discovery for math

Back in June, there were a flurry of posts related to Github for education (‘Curric-hub’) and possible variations of how it could work and why it wouldn’t etc.
Mike Caulfield
Max Ray
Chris Lusto (link is to his followup post)
Dylan Kane
Dan Meyer
Matt Lane
Obligatory Self-Post

and more!

There was discussion about cohesiveness, about how the resources would or wouldn’t be used, and the ‘grain size’ people were talking about – whole years’ worth, units, lessons…

But we didn’t talk about where or how they would reside as learning objects, in an LMS, printed out, or stand-alone. We didn’t talk about what tags we could add to them to help increase cohesiveness and usability for teachers across the country/world.

One of the million and one things I’ve learned at my job at OpenEd is how far behind education technology is behind the, “regular,” tech industry.

Sample API
Sample API

Many ed-tech companies don’t play well with API’s for accessing their systems or have crappy LTI implementations.  And while Metadata is so important to sites look google, but educational resources often are missing important pieces that make sharing resources difficult.

Google Yahoo and Bing banded together a few years ago to create  It’s a clearinghouse for metadata tags essentially, and many but not all of them are even used by . (which by itself is terrible in terms of usefulness as well, but that’s a different post). It used to be called LRMI but that functionality was absorbed into existing objects for Creative Works mostly.

There needs to be more efficient ways for this metadata information to get out to the world. To get out and be used by content creators and consumer sites such as Geogebra, Desmos, edpuzzle, Youtube etc. My idea is that these tags could be used to more gracefully piece together in a more cohesive way than creating Frankenstein’s Monster with OER.  For example, if tags ( ) indicated that the unit I published was for second grade, english language learners, included a three act math task and was accessible to deaf students, as well as had a spanish translation – that would be much better than searching for “barbie drop second grade” which is what we’d see now. Ideally a search engine would be able to piece together that needed information automatically.

But I’d want more that probably wouldn’t be in . I’d want to see how to best teach this lesson or lessons or Unit or Course in context of Literacy integration and STEM units to connect with. Classroom strategies for effective learning (online tools too!) and multiple ways to do assessment for it – not always just fill in the blank and multiple choice, but authentic assessment choices too. needs better ways to add metadata to everyday learning objects. We should shift towards an open set of decomposed learning objectives to be more clear about where a particular lesson fits into teaching a specific set of standards (Practices included!).

CC BY License

Short idea about NCTM Elections

This is a short response to:

Every wanna-be elected official is talking about creating community, NCTM leadership etc. What I’d really like to see is this:

  • People are better than, “resources”.
  • We know there are a ton of talented math teachers.
  • Many NCTM teachers have deep knowledge in things like equity, Desmos, Geogebra, OER etc
  • Create a badging system (mozilla Open Badges) and have members identify their interests etc – make it searchable for members-only and have that link to their social media profile of choice… twitter, Linkedin etc. Most teachers that are really into certain topics have their own professional websites as well for other teachers. Let’s link everyone up much like the current MTBOS search engine.
  • Having another repository of content wouldn’t do any good because it’s already out on the web just in disparate places. Being a driving force of curated, assessment to instruction to leadership is needed and NCTM’s job as part of supporting new and experienced educators.

Developing Growth Mindset in Teachers


My favorite college professor always said in intro-level math classes, “You’ll learn about that more in Number Theory,” without further explanation. By the time we took number theory of course we were CURIOUS about topics such as, “for all real numbers…” and somehow dealing with the abstract made all of the concrete calculations we’d done over the years make much more sense. We started thinking not about how to multiply radical numbers but how multiplication itself worked and even, when it  didn’t work. We went from learning in the relatively concrete to seeing in the abstract.

First Classroom

Teaching Math is itself a bit of the same. The first year of teaching – even with a great student teaching program – is often an eye opening thing. I still remember the first time the door shut to MY CLASSROOM at McLane High School in 2006 (old photo gallery)  and the feeling of, “oh crap,” there was! There were about 200 students that I was to teach math to that year. Of course there were some classic first-year teacher stories later told with glee to newbies. My next door neighbor teacher wasn’t hired yet, so there was a sub for 30 days while paperwork was finished. I should say a series of subs. I often had to open the door in between classrooms to get the other kids to quiet down  – which in retrospect actually improved my level of respect from my own kids because they thought I was a little bit of a badass. But teaching-wise, I didn’t have quite the arsenal that I would today. I went through the book for the most part, coming up with cool ideas when I could. Famously I took a picture of my Toyota Corolla’s windshield because I thought it’d be cool to have kids find out the area their family car’s windshields measured as part of a unit on circles – finding sector areas to be exact. Of course, in a school with 94% poverty (average income for a family of four was about 2200 a month, which although that was about 400 more a month than what I brought in, not much…).

But that experimenting made me stronger. I shared my ideas with colleagues at my school (twitter had just been invented so wasn’t yet an option), and that first summer had the great experience to attend a two week institute put on by the San Joaquin Valley Math Project. It was like a two week summer camp of math mentoring and even as a first year teacher I felt respected and challenged to think about better ways to challenge kids – and myself – to teach math differently. It was liberating to be around others as passionate about math and kids as I was – when I would make a mistake, I was ASKED about it, not just told no… which made it a high-growth camp for everyone! (Thanks Lori Hamada, now Exec Director of AIMS!)

I taught mostly Geometry that first year – and in the years following did Algebra 2 for a couple of years, Alg/Geo III, CAHSEE, Algebra I, Independent Study which I turned into History of Math sometimes… and later Pre-Algebra when I changed schools to help start a middle school water polo program. Every class influenced the others and that deep exposure across grades 7-12 definitely helped me deliver professional development and now at OpenEd. And when opportunities to grow as an educator – always at my own expense except for the SJMP training – I went!

Not all teachers have the time or motivation; that I know. The best PD experiences I had were hands-on MATH experiences with supportive and mentoring peers(Pre/Algebra University in FUSD, 2011-12?). I remember once going to a lesson where we were to trace Functions from 7th grade through high school. There was a group of middle school teachers who proclaimed – “Why do we need to know this? We don’t teach high school…” I wanted to shout something about the Progressions or, as a former high school teacher, how tricks like FOIL and even PEMDAS don’t hurt but actually hurt students mathematical understanding if that’s all they’re taught. Yet as we did the hands-on math, some teachers didn’t know the rationale behind the tricks either! This isn’t their fault – they probably took the CSET or got an emergency credential or – just forgot after years of teaching lower mathematics.

I’ve spent several years since that time doing as much as I can to help  Math teachers. Leading Math Mindset Book Studies, starting a Facebook Math Teacher group, Math and Beer Nights, attending and speaking at California Math Council conferences, leading the content arm of an outstanding mathematics formative assessment and resources company, being an adjunct professor of technology at Fresno Pacific, and even helping with the social media arm of CMC.

All of this because as a high school and college student, I struggled with math because it was presented to me as something to memorize, not something to think creatively about. That failure meant I had to stop- not regroup and ask questions about my failure to learn through it. Long before growth mindset was popular, I had a sign in my classroom – Celebrate Mistakes.

That first year of teaching, I went off on a rant once at the end of class how if, (paraphrased)”You were that kid who played the game of school just good enough to get an A, who asked for extra credit from your teacher and that’s what helped you pass the class, you know that those teachers actually failed you. Because in this class there aren’t games to play and I don’t give extra credit. You will work hard for and earn your grade, and you will pass this class and be ready for Algebra 2 and college.”

A young man taller than myself came up to me after class and said that I must have been talking to him. He admitted he didn’t really know his times tables and other basic math, so for a couple of months would come in around 7:30am a few days a week (which turned into almost every day) to just practice. We just kept it about math and he started feeling more confident, even asking questions and answering others in class.

As educators in and now out of education we need to find ways to help teachers express their unknowns in math. We need to encourage and mentor them, let them ask WHY they need to know where functions are going unlike judgmental old me. Like my mentors at McLane and the SJMP, teachers like students need support in going from a fixed to a growth mindset! Of course, once they all read Math Mindsets, with Math They Can!

Math Tasks and OER – More Than Rhetoric

The past couple of weeks have seen engaging blog posts from two math powerhouses – Dan Meyer and Matt Larson.

Screen Shot 2016-09-07 at 10.14.51 AM

The basic premise of the discussion back and forth is that teachers using open educational resources can’t always be trusted due to the fact that many math tasks on the internet are standalone – that is, not connected to the previously covered standards, tasks and curricular sequence.

I do partly agree with this – back in 2013 I was tasked with assembling (among other teachers) so-called example curriculum unit exemplars for my school district. While I loved the work of Geoff Krall’s Problem-Based Curriculum Maps, it was true that using work from many different authors often required finessing and sometimes modification if the particular task covered topics that hadn’t been adequately covered.

Both Dan and Matt make the point that teachers – with proper curriculum training – should be able to take the disconnected tasks found on the web and adapt them to their classrooms for coherent instruction. I am concerned however about a few things and will address those concerns here:

  1. Reuse : While from what I can tell there is little in say, Classroom Chef and Hyperdocs that isn’t already online in some (perhaps less refined) form, there is still a question about better formats to deliver instructional materials. Books are great and tangible, but most online math tasks etc live on webpages and blogs. Definitely not the most dynamic of content, but useful and simple. However, one way I’ve often felt Dan Meyer’s tasks for example could be made better was ‘student versions’ of the pages for teachers to be able to send their students from an LMS/Google Classroom – currently there is nothing to stop someone from linking/setting up just that since he lists the open license in his sheet. 
  2. Redistribute: Many authors of tasks do not properly license their work. Even a mention of a CC0 license would do well to ensure fair use by others for derivative tasks, etc. In addition, as more publishers in the future want to incorporate tasks by online teacher-authors, protecting their work and having the freedom to specify usage is important.
  3. Remixing: While John Stevens 3 Act Search Engine is useful, it essentially is a cobbling together of what shouldn’t have to be such a hard thing. I also strongly feel that when teachers take someone’s task and improve upon it, it should be easier to find those derivative tasks and see what/why they made those modifications. The community is thriving already
  4. Revision: When the original authors revise their works, it may not be immediately clear (although most post revision statements; Dan Meyer has all of his tasks on a spreadsheet that simply updates the source tasks, etc). A format to get more eyeballs on tasks before they are published/disseminated before revisions may or may not need to be made would be helpful. While posting anything to #MTBOS is sure to get you at least some views and comments, I almost wish there was something like “#MTBOS_CHECK” for an author wanting to release something to the world but asking for revisions or commentary first. Sometimes I’ll look at a task and not quite like it, send feedback to the author etc… then forget about it.

You may notice that I specifically pick out the so-called four R’s of OER – Remix, Reuse, Revise, Redistribute. 95% of all online math curriculum I’ve seen at least posted through say the #MTBOS on twitter adhere to these principles. My main point here is that more needs to be done within the Math community for education about what makes their task/game/lesson OER or not and if so – how to leverage that for maximum, even crowd-sourced potential. I am keenly aware of my own lack of contribution to several projects I’d love to devote more time too – chiefly among them – but it doesn’t mean I wouldn’t if I had great ideas to share. Sometimes I feel I’m so busy exploring what’s already there!

I’ll also come back again and call out the importance of dynamic curriculum maps importance to ensure that students DOK levels are being seen and adequately addressed – as well as coverage of both the standards and mathematical practices. Side note: Dan Meyer’s spreadsheet already lists the MP’s, CCSS, and License.

Defining Open, blog by David Wiley
Dan Meyer's Blog
Geoff Krall's Emergent Math
Matt Larson 

Github for lessons… another post

For the best background start here:

(TL,DR: Teachers want a searchable archive but don’t use what’s out there because they either can’t find it or it takes too long to modify to their needs.)

First read:


As someone who has created and shared quite a bit of curriculum, I’m always curious when someone says at a workshop, “I used your xyz activity in my class, thanks!”. As in, did they use it how I thought they would? Did they modify it and make it better? How did they adapt it to their students?

I brought this up on Twitter a few months ago in regards to Google Docs. There are a lot of great resources and links and lessons on Google Docs, but if someone “Makes a Copy” there is not a record of that copy being made. Github on the other hand is a system that would record such differences, but is very difficult to use for the average teacher and doesn’t allow concurrent collaboration on documents, which is important for many teachers working in teams…

There are a few ideas in play here:

  • People want lesson ideas and instructional tags (ie, visual learners, ELL, growth mindset etc) not just lesson plans
  •  Videos of the teachers teaching the specific lessons would be cool
    • Let’s add student work perhaps “Math Mistakes” style too!
    • And blog posts about certain lessons from teachers so teachers can know what to expect before they give it? Is this too, “meta”?
  •  Version control and the ability to show different variants of the same types of lessons that may be useful: Ok, this might be only something I want,
  • A ranking system for the best lessons for a given
  • Topic/Lesson/Unit/Standard/Style of Learning (capitalizing for emphasis) so that lessons can be taught in continuity
  • The best lessons from the so-called MTBOS websites to be updated automatically as well…
  • Not tied to any one LMS or system or format but yet cohesive enough to be cloned and edited from this one interface

Again, as Dan and others mention often teachers are just going to search the night before they do a lesson – goes back to how curriculum mapping is incredibly important in or out of an LMS or public system as well in the long-run.

While my very own company site is a great source for student-facing assessment and learning content that fits within any LMS (through LTI etc) that you choose and thus the content is not tied to one site or system – but doesn’t have the “ideas” such as three act math tasks, Desmos lessons etc. mentioned in the other blog posts.  However there are enough open tools out there (Drupal? Wiki’s?) I feel like it could be built but is beyond my code knowledge and time scope – and unless it’s easy to use and pretty, teachers won’t use it.

Side note: We’re about to start the Mathematical Mindsets Summer Book Study! Come join us.  

The Polygon War

I wrote this as a comment on a friend’s post and wanted to share it here.

The post:

Screen Shot 2016-06-02 at 7.40.11 AM



My response:

The General had been fighting the Polygon vs Circles War (historical note: 0 to 360 ANGLE or approximately 4000-6000 AD) for what seemed like forever. Skirmishes on the Great Plane had taken their toll on the forces, often forced to use the Z axis plan to escape the battlefield (and they were never seen from again, although some claimed to see ghost shapes moving in and out of the (x,y) axes. . Over time one group had emerged as the dominant rebel force – the parallelograms. Infighting had occurred between squares and rectangles – squares insisted they were not a splinter group but the rectangle contingent ended up crushing those rebels. At the battle of Rhombus, some squares took so much damage as to lose the 90 degree corners that had so defined them and a new shape type was born – named after the battle where they were first seen. Now there was a rumor of another new shape formed from rectangles or squares as well that was threatening the very definition of a Parallelogram. It only had two sides parallel and was often irregular – or so the stories went. The General didn’t’ believe it – probably just a pentagon with a short side he reckoned (and pentagons themselves were just a variant born from of the triangle revolt of 180 ANGLE). Still, it was something to be considered dangerous as those could be nasty competitors when encountered on the battlefield. Finally as he moved between two parallel lines he noted a strange intersection point ahead. It was blurry at first but then he saw something – it looked like a rectangle as he faced it but as it turned he saw one corner had been ripped off and now repaired. There was one side where its rectangle roots were clearly visible – 90 degree angles and all – but there was something different. “How can it be part of the parallelogram family yet only one side parallel?” he muttered. As the shape got closer he got his triangle angle sum Formulator ready to fire. The strange shape however stood up so that both of it’s bases were now perpendicular to the earth and exclaimed: “If you Formulate me, you’ll see I am still a quadrilateral! The only thing different is my area formula will not compute as a rectangle – but I can prove my lineage. I am a new shape in town – a trapezoid.”


Knowledge of the Common Core Math Standards and Teaching

I was privileged to go on a hike Sunday morning with amazing math teachers from the CUE Rockstar Math Edition camp in Los Gatos.


One of several conversations I had on the five mile morning hike was about knowledge of the standards (specifically, right as Matt Vaudrey walked up). I commented how teaching high school for 5 years before going to middle school was a great thing for me professionally due to better knowing where the students were going. In my school district at least, we had a lot of training before Common Core were implemented but not as much afterwards. Moreover, there was a lot of resistance from middle school teachers to learn about the actual mathematics involved and where it was going instead of more traditional PD about methods and activities.

Now that we’re a few years in – and I’m not in the classroom – I was curious about what’s happening out there still. I put up two polls on Facebook and Twitter asking if teachers still need more training on the standards. Thus far (although I’ll give it a couple more days) almost 3:1 on Twitter and 19:1 on Facebook teachers say they still need more information. For math specifically, the Progressions Documents are incredibly rich mathematical learning documents, but often underutilized in training due to their density. Likewise, the Coherence Map is great for big-picture thinking as long as concrete examples are also shared for teaching and to make them come alive. I also of course highly recommend OpenEd’s resource library filters to make the standards content come alive.

In my job as Lead Content Curator at OpenEd, I get to work with standards from across the nation and even around the world. One thing I have really enjoyed is seeing how other bodies have both structured their standards and modified/adapted them. In states such as Indiana that have modified/rebranded their implementation of the CCSS, they’ve often done things like break up certain standards in high school to more their individual component parts. Others have simply renamed the standards due to politics. I wish all teachers had the time and the impetus to dive deep into the shifts, strands and knowledge that the standards contain on a purely academic basis. While one may not find hidden amazing math activities, they will find a coherence that is real and be able to be better, “learning guides,” for their students as well.

I hope to do some thinking and posting on this space about specific learnings I’ve had and give opportunity for teachers to go on their own journeys of learning about the CCSS as well.

Comments on Coding

When I was 14, I learned basic HTML from tutorials I could find on the internet. After a couple years I was making tables, animations, and formatting, but once things got more WYSIWG, I stopped. I tried picking up Java because it was the hot programming language of the late 90’s, but made it through one or two chapters of a book on it before giving up.

Flash forward 15 years, and coding has become an increasingly mainstream topic and something that we should teach our students. In an age where kids won’t be able to take apart their computers and add things as much like I was, learning the internals is just as good.

At my job a month or so ago the boss suggested that I learn the Go Language, by Google. It’s been slow going as there is a ton of syntax and even command-line stuff that I just forgot, not having had to use the command line in about 10 years and never programming a real language like this before. The hardest part has been not knowing where to start – like what a package is, or where to put a function etc. Some resources I’ve found helpful that were suggested by colleagues:

I was talking to a colleague the other day and we noted that the cool thing about learning is that almost every time we sit down to learn something new, we incrementally get better. It is frustrating not to be able to just sit and learn and do it all at once, but it’s a process I’m trying to be patient with.

Coding should definitely be taught in schools instead of say, keyboarding for sure… and incorporated into math classes when applicable!