Three Kinds of Math?

Money, Philosophical and Artisanal Math

There was a wonderful piece on National Public Radio this week that told the story of Harvard researcher Houman Harouni (, who had done historical research on why we learn mathematics the way we do.

(His full dissertation is here:

In the dissertation, he gives a very specific example of how this kind of approach to mathematics can be viewed through three different approaches.

Money Math
He makes the case that all of western mathematics has ended up looking like problems of this type:

Susan has 12 oranges. Her mother gives her 15 more. How many oranges
does she have now?

or: 12+15 = ?

This kind of mathematics came out of the economics of the time – money counters and accountants, business people needed to know this kind of math in order to balance the books. He makes the compelling case that the economies drove the need for this kind of math to be necessary, and it became the predominant way of thinking of mathematics since the Renaissance.

Philosophical Math
He offers two other types of mathematical approaches. What if the problem was worded this way:

27 = ?

This approach is a more philosophical approach about the nature of the number, and the ways that it might come to be and what it represents.what could go into the right side of that equation? 9×3? Three cubed? Log base three of 27? It invites a very different kind of mathematical thinking and exploration.

Artisanal Math
Another approach would embed the math in the professional work during apprenticeships with craftsmen. This was very reminiscent of the work of Jean Lave ( and her excellent work on situated learning. In studying the traditions of apprenticeship for Tailor’s in countries like Tunisia, it was clear that mathematical learning was built into the apprenticeship, but it is not anything like what we would call traditional teaching and learning of mathematics. Moreover, these tailors had a high functional ability to work with mathematics that were specific to their craft.

Tying it all together
Over the past six years in working in our MPX program I have been delighted and challenged to try and build all three kinds of mathematical approaches into the work we do with our projects. We have developed mathematical models in our scientific community to understand and categorize physical phenomena, we’ve looked at form and function in ways that they express themselves in artistic work in design and engineering, and we’ve practiced traditional math as a means to understand some of the ways that procedural knowledge in mathematics can help us unpack what we see behind certain expressions. I think the real challenge of the evolution of mathematics education needs to be in rethinking how do we approach these sometimes complementary but more often than not this connected or even underutilized approaches to building mathematical understanding in all of our students. In some ways, they fit the three legs of mathematical understanding that are part of common core: no (money math), do (artisan old math), understand (philosophical math).

Here is the NPR story that explains some of the research:

Laying the Foundation

Well here we are already more than two weeks into the start of the new year. Something new for me this year – I am teaching two different classes: 10th grade STEM for our integrated Mid-Pacific eXploratory (MPX) and a new course that is just starting this year which is Innovative Design Technology – a one year course focusing on the design process and wrapping in rigorous science content. I am co-teaching this class with my friend and colleague Lori Nishiguchi – we each have one section of this course.

As a result, I’m going to blog in this space for both courses – sometimes separately, sometimes, like today, a general summary of what both classes have been up to. As I said before the intent of this blog is to both inform about what we are doing in class, but I also use it as a space for me to share the work we do, the decision-making and rationale behind the kinds of efforts we make, and broader questions about education and the future of learning. Hopefully if you read this, something here will resonate for you.

So here are a few highlights from the past couple of weeks.

In the design class,

the most engaging work we’ve done so far was running the Stanford crash design course. Designed to take no more than an hour and a half, it is a quick iterative process that walks the participants through the design process beginning to end in the midst of trying to redesign a gift giving experience. Lori and I ran this on the second full day of class and it gave us both a chance to see how pacing, tempo and group dynamics were in our class, as well as a means to introduce the underlying foundation of good design principles that we plan to rely on for the next eight months. If you’re not familiar with their work, the homepage for their is here. I’ve embedded the design course video below in case it would be interesting to look at. Below that are a few pictures from the activity in our class which I think does a nice job showing the engagement and effort in this very quick process. What was most interesting was the debrief with the students revealed both their interest in understanding the importance of getting to the empathy and the value of understanding the end-user experience, as well as the high pressure nature of quick prototyping and iterative design.

Students doing the design process - going for empathy

Students doing the design process – going for empathy

Students doing the design process - going for empathy

Students doing the design process – going for empathy

Students doing the design process - going for empathy

Students doing the design process – going for empathy

Students doing the design process - going for empathy

Students doing the design process – going for empathy

Students doing the design process - ideating

Students doing the design process – ideating

Students doing the design process - pro typing

Students doing the design process – pro typing

I’ve created a Flickr page that will host all of the pictures as we take some during the school year – that is located here.

We’ve now launched into our first full design project which is redesigning all or some feature of a standard school student chair. Students have already gone out and interviewed their “clients” and have come up with some issues that they are looking to try and address and are now asked to design some sketches of what might solve the problem for students. The full design brief is pasted below.

Chair design Challenge

In The MPX 10 STEM Class,

we have started with initial activities to lay the foundation for our year-long project in investigating urban transportation. The ultimate goal is really four steps: students being able to build and take apart bikes, students being able to design and build an electric bike from a standard bike frame, students designing a payload system for their electric bike to accomplish a course challenge, and the culminating set of activities around designing a more fully thought out urban system for transportation.
Although this set of challenges will be ongoing during the year, we will be doing breakaway work in a variety of other smaller projects and activities – both to create some other learning opportunities as well as to make sure our core content is covered in physics and algebra two. We spent the last week going back to some mathematics work around linear equations, and investigating constant motion which lends itself nicely to the beginning conversation around moving bikes as well as exploring linear functions and their modeling capabilities in the real world. A few pictures below show students white boarding, conducting experiments, and actually taking apart the battery-powered cars to investigate the way by which the batteries stored energy turns into motion.

Students Whiteboarding

Students Whiteboarding

Running a motion experiment

Running a motion experiment

analyzing data

analyzing data

Sketching out the inside of a battery powered car

Sketching out the inside of a battery powered car

Just like the design class, I’ll be keeping a Flickr account with a full set of photos here. The next blog posts will be more philosophical, but this was a good way to get started with year and what we’ve been working on. Last attachments for this post: the syllabi for the two courses which outline the main content areas, pedagogical practices, and assessment that we will be using in the courses.

MPX10 Stem 2014-15 Syllabus
Hines Design Technology Fall2014 Syllabus
E Kūlia Kākou
Let’s strive and aspire together

Words lead to culture which leads to learning

A reflection on the book “Choice Words” by Peter Johnston and a commitment to move this work in my classroom forward this year.

Reflecting on the book “Choice Words”, there are powerful examples here about how to build a classroom culture that supports and nurtures learning, builds a culture of agency and makes a real community of learners happen.

One of my goals for this upcoming school year is to not just apply these ideas into my practice but to have my learners become fully aware of them as well – both to help me be more accountable in supporting and building a classroom of kindness, reflection and growth mindsets.

I am powerfully struck by the link between the goals of our program, the IB learner profile and the comprehensive tenants of the book.
One of ways that Johnston expresses this is by referring to the ideas in the book as an epistomology – Which I agree with. In a sense, how we interact and nurture conversation say a lot about our theory of knowing and learning.
As a science teacher, I think I do many of these things because of my philosophical (epistemological) underpinnings of being a constructivist. Questions liked “what would happen if?” And my favorite “why do you think/say that?” align well with my preferred ways to engage a class in discovery ad reflection.

What I feel I need to do more fully and work on are the aspects of building a community of respect and reflection in my learners. Let me give a few specific examples from the book. After some exploratory and explanatory chapters at the beginning, Johnston give specific types of situations in the language that can be engendered for deeper more reflective and powerful learning. These include chapter titles like “noticing and naming”, “identity”, “An Evolutionary Democratic Learning community”.

Most of the chapters start with an introduction to the idea, with some exploration of teacher exchanges, but then zero in on specific kinds of open-ended questions that engage students to think deeper about their learning.

So for example, in the chapter on evolutionary, democratic learning community, he includes examples like

“any complements?” – a means to create focus and listening strategies as well as building a culture of feedback

“are there any other ways to think about that? Any other opinions?” – He makes the point that research shows this is rarely done in classrooms, but the opportunity to create places where students see differing viewpoints is critical – Johnston mentions that research shows that disagreement in ideas is a more powerful and effective way to advance thinking that agreement.

In the chapter on identity, another beautiful quote is:

“what have you learned most recently as a reader?” – What a simple crafted idea that is wonderful because it honors students thinking, engages them to think about their own process in comprehension, and builds a sense that we are learning this together.

In the final chapter titled “who do you think you’re talking to?” He summarizes and dives into the application of these ideas and the outcomes from a culture of purposeful use of language. In doing that, he contrasts some case studies of teachers. He compares two – one whose name is Pam who was well-liked by her students, but does not engender this sense of agency with her students.

In one passage he states:
“It is fairly easy to hear the way Pam thinks of her students. They are people who cannot be trusted to read independently or make productive choices, children who are incapable of having a conversation.”

To move from a teacher-centric class to a community of learners, teachers need to change their beliefs on learners and learning. Johnston doesn’t come out and say or promise we can change this internal belief about learning by just changing our language. He does encourage the possibility that by starting to adopt the language we can start to evolve our beliefs about the transactional exchanges that happen and the value they can bring to improve classroom discourse.

In my role as an instructional coach I need to embrace and model this more fully. As a teacher/facilitator, I need to continually improve my craft and ask my learners and peers to help me be mindful of and shape my language.

The bottom line to me is the title of this post. If I want to get to better, deeper learning in my deeper learning class (MPX), I need to develop a culture that exemplifies the language of an effective community of practice. In order to get to that place I have to craft and shape all conversations and interactions in my room with a mindful approach towards the ideas of “Choice Words”.
My words WILL lead to development of a classroom culture that will ensure that everyone learns with respect and depth.

Addendum: After writing this blog post on the plane back to Honolulu from ISTE, I flipped through Johnson’s more recent follow up book “Opening Minds” and was struck by his close work with the research on mindsets by Carole Dweck amongst others. Not surprisingly, the work between language and student agency is powerful and very closely aligned. but that is another blog post for another day…

My Deeper Reflection about Deeper Learning 2014

Deeper Learning 2014 Reflection

I suppose I should start my reflection by reminding myself what the purpose of the conference was based on their own language:

Connect – I have made several new connections and have deepened my relationship with several old connections.
Innovate – I have new ideas for how to better support deeper learning in my organization, school, or classroom.
Experience – I will experience deeper learning for myself, which will help me to apply deeper learning principles to my own work.

so measured against their own questions, here is some of my thoughts

***It is worth pointing out that ALL the conference materials for the sessions are free and available at : **********

Wednesday: Deep Dive: Teaching Science with Robots with Dr Karl Wendt

After walking us through a series of robots that he had built – from underwater robots to programmable motorized robots – Karl introduced us to the project of the day – building, and testing a small entry-level robot that cost less than $10 to build called spout.

resources for all of his work are located here:

One of the strengths of what he’s doing, is he made a series of short instructional videos that show how to build the robot. My partner was Ben Daley, and we agreed that by doing it as a series of self directed videos, there was a sense of pacing and authenticity in constructing that would’ve been very different if it’d been synchronously teacher led. We found his work on these videos to be excellent, and felt that we could initiate this project right away.

My Take Aways: The thing that struck me about this project/deep dive was that there was a good principle at play here: the fun part begins with the students building the robot. The hook is that in order for the students to solve the challenges, there needs to be enough core content knowledge (electricity, torque, light, friction, switches, etc.). Depending upon the way the activities of the challenges are set up, it’s entirely possible to cover different branches of physics and physical science. That to me was the strength of this activity – in the kind of work that we do with STEM education, the way to get kids engaged in by bringing some authentic design challenge or construction. Careful selection conversation and building an adequate resource library to support learning will help shape the learning that needs to be part of the academic rigor. Karl’s webpage for his resources is located here:
Nice. Design.

After dinner, there was movie night and a panel discussion. The teaching channel has built a resource area that includes 50 videos that are meant to be short explorations of specific practices of schools in the network (looking at student work, advisories, comity partners, etc). this is a great source that I plan to use extensively in my work both on campus and outside of campus. the video series is located here:

Keynote with Ron Berger.

At some point, they will put the video of this keynote online. My guess it will be linked somewhere from the main page here:
Ron talked a bit about beautiful work, but the highlight of his presentation was four 8th grade students from Polaris Charter Academy in inner city Chicago. Their presentation was engaging, compelling, executed flawlessly, and a powerful story. Their work started with the investigation of the preamble of the Constitution but evolved into a study of gun violence in their neighborhood and what that told them about their freedoms and liberties and protection for the common good. 
If this video does get put online, I highly recommend you watch it!

Looking at student work
attendees were put into groups of about 10 and given a piece student work. Facilitation of the session was led by a high-tech high student – our young lady was Killian and she did an excellent job leading and guiding our conversation. We looked at a compiled magazine that was created by sixth grade students to answer their big question of inquiry about different cultural and scientific theories about the end of the world. Our group had an excellent conversation about the things we saw that indicated deeper learning was evident, and places where we think the work could have been made stronger.
My take away from this one was a recognition that we need to more regularly look at our own student work as a team to help us decide how we are doing in looking at the rigor of our work, as well as how well it meets the other deeper learning characteristics like collaboration, developing academic mindsets, and problem solving.
Hour. Well. Spent.

Session 1: From Scratch with Scratch with Don Mackay
In this session, we were led through a series of design, Inquiry, and challenges about our understandings of electricity and magnetism. The documents for the workshop were here:

After investigating , we were challenged with building a spinning motor from the equipment that he gave us. One of the things that made this interesting was the play between designing and exploring what we understood about electromagnetism. You can see that in the document linked above.
What were my big takeaways? I really feel that the pedagogical approach Don took was excellent – in our modeling terms from ASU, he began his studies with a discrepant event which challenged our notion of what we understood about the phenomena that underlie a particular field of study – in this case of electromagnetism. I like the bite-size nature of what he designed to explore this naïve understanding that we might have. I think this would suit well as ways to build knowledge in projects we do that a more extensive so that we’re creating place for rigorous conversation about what we understand. I’m going to try that with our light project will be doing in May.

Session 2:  From Maker Space to Maker Campus with David Stephen
Wow. I had never met David until this day, but he was a teacher and the architectural designer of the interior spaces for the high-tech high buildings. Wow. He led an excellent, progressive session about the work that he does, and how he sees the importance of space as one of the ways to create and support student agency and 21st century teaching. Instead of summarizing what he did, here are my tweets I sent out for his session (there are in reverse order last to first):

#deeperlearning Session with David Stephen  Provocative, Progressive, Powerful ideas on Agency by Design

#deeperlearning ideas for transforming spaces on the cheap: Make Space  Great book. Easy to read. Low hanging fruit.

another great site on maker ideas – The Fab Lab  #deeperlearning

#deeperlearning the resource at @AgencybyDesign is tied to the Making Thinking Visible team as well. too cool!

Two useful resources on design:  and  Great ideas abound! #deeperlearning

#deeperlearning SO much of our conversation on Agency by Design is reminding me of the work by Pine and Gilmore on the “Experience Economy”

#deeperlearning Agency By Design is a project out of Harvard Project Zero …

#deeperlearning great turn of phrase from Davis Stephen: “airport grade” soft furniture. Love it! I’ve seen blown apart bean bag chairs!

#deeperlearning how to create inviting educational spaces – David Stephen uses the term ‘artifactorium’ to define exhibit space feel of HTH

#deeperlearning with David Stephen of New Vista Design talking about creating Student Agency By Design “From Maker Space to Maker Campus”

His three shared documents are on the materials page under session 2 :

My Take Aways: This session rekindled in the and interest and passion around the importance of facility design editing needs to both support and lead teachers needs and behaviors into the place they need to be and to not reinforce things that are keeping them from advancing their professional practice. This topic has been of high interest to me since we started designing the Weinberg technology Plaza on our campus back in 1998. I definitely want to spend more time going through his resources and the agency by design website mentioned above.

Session 3: Beautiful, Beautiful Math with Marcia Dejesus-Rueff

We had about 16 people in this session and we looked at ways to bring Art (poetry, dance, musics) into the mathematics that we do. I really liked the fact that her compelling case was mathematics is a search for patterns. These patterns can be found in poetry, dance, music, just about any art form we looked at a Dylan Thomas poem, and a Frank Lloyd Wright stained glass window as ways to create a linking between art and mathematics. Her documents can be found on the session 3 resources page here:

my takeaways: I am still professionally challenged by looking for rigorously appropriate and engaging art that can lead to deep understanding of algebra to mathematical principles. the session provided me with opportunities to look at some other examples of how that looks.

Unconference Sessions:
Polls were taken on Thursday to determine the topics that would be offered on Friday. They were a variety of sessions offered that looked at everything from how to start your own deeper learning school, to what assessment looks like, to professional development strategies for teachers.
I attended:
The Leader’s Guide to Deeper Learning with Ken Kay and James Gibson
Ken is now at EdLeader21, and the session focused on walking us through the deliberate planning process and leading a school through transformation to deeper learning. At our tables, we modeled the process of developing core competencies, identifying new student skills necessary, and finished by proposing strategies to move towards these goals. The pictures of the chart paper are located here:

the resources that they shared in the workshop are located in session 3 folder here:

Other Thoughts:
Larry Rosenstock mentioned in his address the importance of recess for our group as a means to connect with familiar and new colleagues. I can’t agree more with that impact – whether it was lunch time, or just corridor conversations, or the after hour social events, the group used the time outside of the sessions at least as powerfully if not more so as a means to explore, connect, challenge and extend their thinking about their work. Looking towards our conference in November ( there was a lot we could take away from what work here at deeper learning 2014.

Student work is everywhere in High Tech High. Every time I come I take pictures to add to my collection – but this time with different intent for me. Since our students are going to work on building projects for the rest of the year, I wanted to capture some examples of already displayed student work to challenge my students to think of how they could make something as good as or better than what they see in the images. I have included a few examples below.

photo 1-1

photo 1

photo 2-1

photo 2

photo 3-1

photo 3

photo 4-1

photo 4

photo 5
Last Word
One of the last things we were asked to do was fill out a post card with at least one idea we would implement. I wrote three:
for my MPX program, bring more student work protocols into our meetings
for our work with Kupu Hou, find ways to sustain a community of learner model that will more directly influence the attendees to stay connected over a longer period of time
for work with the Hawaii education leadership Summit, rethink what the topic might be by spending more time asking our intended audience what would suit their needs past so that we can align our work with their needs