Aren't you fulfilling a stereotype?

Warning: this post takes this blog an overdue, brief-but-important detour into the topics of race, inequality, and identity.

During Christmas, my wife and I attended a family gathering.  There was food, there was alcohol, there was family.  There was also other extended family - wife's cousin's cousins family.  And as you will soon read, perhaps not enough alcohol.

My wife's family gatherings are typically pretty big.  Usually there are several big tables simply because there are too many people to fit with one table.  I happened to sit with these extended family - cousins of cousins.  I tend to eat a lot (people are often surprised by how much I eat since I don't look the size), so I remained eating at the table while most people had left to watch an exciting Heat vs Lakers game on TV.  Perhaps out of politeness, these related strangers attempted to strike some casual conversation with the inevitable questions pop up -- although the sequence was a bit strange.  They began by asking me what I was studying.  I suppose it has to do with how young I looked.

"I am pursuing a masters degree in math education."  I said, while stuffing my face with more turkey, and highly distracted since the Lakers were surprising me with these bench players coming on the court and throwing up big numbers.

"Oh" they said.

Yeah, I'm pretty terrible at small talk.  I think on some sort of philosophical ideological level, I just dislike small talk for the shallowness that it carries.  I am aware that deep conversations often necessitates initial small talk -- but I am usually just bored by them.  At least, that's the excuse I am giving for never fully developing a comfort level with small talk.

Completely awkwardly, and probably a full 5 minutes later, I offered some similar questions for them.  One of them just finished her masters degree in Asian studies.  The conversation drifted slowly to work and future work, and I mentioned that I am a teacher.

"In what?"

"Mathematics, at a high school here in town"

"Aren't you fulfilling a stereotype?" She referenced the fact that I am Asian.

I paused.  I didn't expect this conversation.  Didn't she say she completed a degree in math education?  Is she saying that Asian individuals should all avoid mathematics in order to not fulfill stereotypes?  How could someone who should be steeped in literature pose a question that narrows racial and cultural identities?  Tons of questions flash through my mind.

"Not the teaching part, I guess." Way too distracted by the questions surfacing in my mind, I threw out a useless response and kept eating.  And continued thinking.

Some people might have used this opportunity to elaborate, to flip the table, to argue, to discuss.

Not me.  Or at least not me at the time.  Looking back, I probably should have entered into that conversation.  But I didn't -- I am not sure why.  Maybe it was the setting - casual conversations at a holiday dinner table shouldn't evoke deep conversations.  Maybe it was the company - my lack of knowledge of who they are and where they stand.  Or maybe it was just me.  

I am not sure why.

I've always been interested in discussions of racial identity and equity.  Besides my personal experiences of racial profiling (e.g. my detour away from mathematics and my own interests), I began to read more about these related issues after discovering Angry Asian Man a few years ago.  Occasionally, besides reading literature on math education, I would get sidetracked by literature on racism, identity, and injustice.

So I certainly did not lack the interest or the background.  But I simply didn't engage in that conversation.

I am not sure why.

I think one of my weaknesses is that I tend to want to talk about the issues that I have not yet resolved for myself.  Issues that I have resolved to a point that I am satisfied with, I tend to lose interest (at least in the moment) in wanting to discuss them.  I am not sure what I just described makes that much sense, so let me give an example: 

Group work or no group work for students?

Yes to group work, I'd say.  There are benefits all around.  Mathematics is a discursive subject.  Talking and discussing mathematics is far more effective with respect to understanding than memorizing and regurgitating.  Beyond an understanding of the content, students are also developing an appreciation of broad transferable skills such as collaboration, teamwork, communication... and much more.

I hold some beliefs to most topics that I've thought through.  I don't mind discussions around these beliefs, as long as these conversations do happen.  Firmly held beliefs without reason or challenge often bugs me.  

It's not that I believe that these issues are completely resolved -- that there are no more conversations to be had.  It's just that these issues have been resolved for me to a certain point - and I am interested in moving forward from that point.  

And I am not really interested in convincing someone else of my point of view, I am more interested in considering their point of view.  Maybe this is another one of my weaknesses.  In a discussion about group work, for example, if someone challenges the notion of mathematics being a discursive subject, I would get sucked into thinking more deeply about: discursive nature of mathematics, perception of mathematics, purpose of mathematics...etc.  

All the while not really making an attempt to convince or defend.

If the person was posing the question in order to promote their own point: one of those "ah ha, you can't give an answer, so you must be wrong," then they would press their victory and end the conversation.

I often worry that this is a skill that I should be interested in - but not interested in.  This is why I am extremely glad that there are like-minded people who are promoting the values that I believe in.  The charismatic Dan Meyer on the values of engagement and inquiry, this talk on men's responsibility in gender inequity and domestic violence, Jay Smooth's brief discussion on marriage equality, Jason and Grace's discussion on supporting teachers of colour, Kamau Bell on transgender issues... and much more.

But this is really the easy way out.  I am basically taking a backseat while letting others create a path through ignorance, immaturity, and inequity.  I need to somehow find a balance between a) being completely confrontational and misrepresenting my desire to have conversations and b) being completely passive and not engage in important conversations.

What I should have done during that dinner conversation was to challenge this notion that in order to overcome stereotypes, we need to somehow control people's identities.  Yes, it is wrong with assuming proficiency in mathematics from students of Asian descent.  But it is even more wrong to prevent a pursuit of interest and curiosity due to a desire to oppose racial stereotypes.  By reacting to stigmatization of a "model minority," we are allowing the stereotypes to ultimately decide our lives.  How is this rejection of identity suppose to move us forward in the continued discussion of race as a culturally-constructed, historically-significant, often-misunderstood, even-more-often-miscommunicated, and often almost-always-misrepresented issue?

What I should have done was ignored the fact that it was casual conversation at a dinner table - because these conversations should never be casual.  It's like the "casual conversation" of why teachers are not important, and that "those who can, do; and those who can't, teach."  What I should have done was Taylor Mali that casual conversation and transform it.

What I should have done was become an angry Asian man.

What I should have done something was anything.  Anything at all.  

And that's something that I did not do.  

And my failure haunts me.

Capturing student learning - using technology

Assessment isn't just about what students write on tests.

I am not claiming to be an expert on this.  Far from it.  What I am, is someone who enjoys conversations around assessments, especially assessment of inquiry-based learning.  If anything, my journey through this M.A. has been an humbling experience that constantly exposes me to wonderful practices, thoughtful philosophies and a spectrum of ideologies.  But from experience and from literature, I can pretty confidently say that the more evidence we have of student learning – the better.  In addition, the more types of evidence we have – the better.

This is especially the case when we are engineering our classrooms through inquiry-based approaches.  Here I am talking about approaches that gets students to do and play with math, and not just memorize and regurgitate.  You can see many many many examples around the math twitter blogosphere right now (Dan, Bryan, Fawn, Al, to name a few...).

e.g. something like this where the students create their own questions and bring in their own plushies to do some bungee jumping

Or something like this where students ask how many marshmallows fit in the palm of their hands

So we’re doing all these amazing things to encourage and foster questioning, thinking and learning… but then most of us resort to writing paper tests as the sole source of evidence of their learning. 

I felt something was missing.

This missing piece is the focus of my thesis.

Up until recently I have been using ThreeRing as a primary way for me to store and interpret student evidence in picture and video form.

But a few weeks ago I went to edcampOttawa, and I bumped into these guys:

And they have developed a program that seems similar to ThreeRing, but different.

They still need to pump out that iOS version for me to fully invest everything in there (currently they have a browser and android version), but it looks ambitious and promising.  

Let me indicate the few things I like about it:

 1. Embedded Ontario curriculum expectations (and the US common core standards):

As well as details of the overall expectations of each course

2. Rubrics that can be custom created:

3. it doesn't tell you how to teach or how to assess.  It presents itself as an useful tool for teachers to use.  Like Desmos, they seem to welcome feedback (for now).

@PaiMath Thanks for the shout out, and all of your great feedback this afternoon!
— Sesame IO (@sesameio) November 23, 2013

4. the fact that programs and apps like ThreeRing and SesameIO are up and coming, is good indication that these type of evidence of student learning is catching on.

5. It's a Canadian company, and it actually has relations to our curriculum!  No offense to all my US friends, but having to read through your common core documents in addition to our own... is interesting... but time consuming.

A few things that I'd like to see them add are things like:

1. Currently we can only add notes (on the website) when we go into a specific student (unless I am wrong about this…)  It would be nice if the “add note” option is just right on the main page.  And then we could select from a drop down menu as to where each picture will end up.

2. There needs to be a common sharing ground of some sort between students.  This would allow room for peer feedback as well as self reflection.

3. It would be nice for simultaneous uploads to allow for simultaneous inputs for tags of students – instead of having multiple uploads for one student.


What programs have you been using to this end?  I'd love to hear about it!

Negotiating comfort levels with open questions

I am a huge fan of open questions.

I have held the set of beliefs that reinforces the benefits of asking a good question (and getting students to ask them as well).  Ultimately, the thinking piece is what it's all about.  The "math" (or what is falsely perceived as math) doesn't matter.  And I believe that open questions do this.  I've written about open questions a bit a year ago, emphasizing the importance of open ended questions.

But all beliefs stand to be challenged.

I naturally ask a lot of questions.  I think too much.  I think about something, and question myself, and those questions spawn more questions, and even in discussions those questions turn into potential answers which churn out even more questions.  I like doing this.  I enjoy the complexity of topics due to questioning.  (which sparked this post when I didn't get to have that conversation with someone).

But I am now questioning about this process of questioning.

This school year has been a strange ride for me.  Besides having a bunch of additional responsibilities that I've taken on this year, I picked up the teaching course one at a time.  I didn't start out with a full teaching load - they came individually later on.  In any case, one of the classes I am currently teaching is a grade 12 college preparation course.  For any non-Canadian readers, these are students in the "non-academic" (such a terrible stigmatic description, but I can't think of any other descriptors).

I have been juggling several different issues with respect to my teaching beliefs.  I am sure I will get to more of this later.  One of these beliefs relates to my belief that open questions encourage deeper thinking of mathematics.  Not only this, some students struggled with the project-based approaches to math education.

Let me elaborate.

A student has great difficulty with starting questions that have no definite answer.  Questions like "Why..." or "How might..."  He/She has a lot of problems knowing where to start, and prefer to have a "set method" of approaching mathematics.  i.e. give me a formula and let me follow it.  I have tried to emphasize the importance of developing an approach for solving a problem especially when you don't know what to do.  Afterall, that's life!  Life doesn't hand you nicely structured problems with numbers laid out for you. Life is all about messy problems where you are knee-deep in navigating through random shitstorm of information. Communications with the parents have not been extremely fruitful either.  Problem solving and critical thinking is the key in answering open questions.  I am out of strategies of negotiating this huge discomfort with open ended questions.  Of course, not all questions I give the students are open-ended questions.  I do include questions that are closed in order for students to develop confidence (even though I am a bit iffy about developing confidence simply because my answer is the same as your answer)... but I just don't know.

I am probably not explaining this situation well enough, but this has been plaguing my mind for the last few weeks.  I am not about to change my philosophy towards using good questions in order to elicit thinking... but I need to build a better cushion for students that feel overwhelmed with this.  Or... perhaps put a few flowery ribbons on these open questions?

Lots to think about.

Harnessing events that happen around you

Been swamped.

My life is pretty much unbalanced right now, and I am unable to fully commit to each of my responsibilities.

Basketball 5 times a week,

Teaching,

Grad school work,

 and sprinkle in all the different wonderful conferences, PD day organization, workshops...  Not to mention a new puppy at home

Wife's getting pretty angry at me at times because I'm never at home.  And when I am home, I'm not carrying the same load around the house - laundry, dishes, cleaning...etc.

Bah...  I need to re-evaluate how much I am taking on from now.

In any case, that wasn't the point of my first post back.

Things are happening -- That's really the point of this post.

Things are happening not only in my life, but also around the school as well.  For example, there is an addition to the school about to be constructed.

So things are happening - it's just about how we choose to harness it.  I mentioned before about harnessing a current event about a sinkhole.  So it's no surprise that I'd try to do something with something as delicious as the school addition.

Since I don't have much time, and I don't want to keep 100 drafts in my blogger...  I will just outline what's going on, and maybe go into details in a later post.

Topics covered:
- cost of materials (linear relationships)
- spaces, surface area, and volume (Geometry)
- measurement of inaccessible heights (trigonometry)
- budgeting and researching for suppliers and costs (budgeting, research, life skills)

Flow of things
1. beginning with a smaller box of figuring out prices, costs...etc
2. then carrying onto an existing section of the school, figuring out prices of glass, drywall, paint, tiles, bricks... more details the better
3. and finally an estimation of the school addition.  students submit their estimates, plans.

Some pictures (recently my iPad's doing some funky stuff, so I am unable to access the pictures on there... so just iPhone pictures for now)

More pictures to come... once the iPad is fixed.

What would you do with current events?  What have you done?  Share it with me!

Losing my shapeshifting powers

Another title would have been something like "Becoming a grumpy old man."

I will explain what I mean in a bit.  Let me begin by defining amorphous, courtesy of dictionary.com.

a·mor·phous

  [uh-mawr-fuhs]  Show IPA

adjective

1.

lacking definite form; having no specific shape; formless: the amorphous clouds.

2.

of no particular kind or character; indeterminate; having no pattern or structure; unorganized: anamorphous style; an amorphous personality.

3.

Petrography, Mineralogy . occurring in a mass, as without stratification or crystalline structure.

4.

Chemistry . not crystalline.

5.

Biology . having structural components that are not clearly differentiated, as the nuclear material incertain bacteria.

A few individuals in my lifetime have commented on me and my personality being "amorphous."  What they meant was this: I get along with everyone. I am able to adapt to any sides of an argument, and be perfectly happy doing this.  I can adapt different points of views, and my personality can change based on people that I am with.

I was like this because I didn't care.  My personality changed depending on who I was with because I didn't care.  I had no strong opinions about anything because I didn't care.  To some degree, I probably enjoyed being able to take different sides or being able to be different people.

I know, I know -- if I put it this way, it doesn't sound like a superpower.  Everyone, to some degree, has this ability.  I suppose mine was just extreme.  I was (and to some degree still am) able to basically be whoever you needed me to be.  I didn't care.

This trait of mine didn't come from nuclear waste or being bitten by a spider.  It probably came out of ashes of my childhood (or lack of one) of not really having parents around, and needing to make connections with others.  Obsessively so.  I noticed that I was doing this before some friends began to comment on it.  It was fine though, it was the way that I got by.  If anything, that was typically a likable trait.  For getting along with everyone, this was a useful superpower.

But I am losing this ability.  I can feel it drain away.

If what I am describing as a personality trait is something like "flexible," then I am slowly losing it.

Let me explain.

I am beginning to form rather strong opinions about issues and topics.  I am no longer able to fully adapt a different side.  Fully -- that's really the keyword here.  I am sure I am still able to present reasons for believing a different side, but I am no longer able to confidently defend multiple positions.

Sounds pretty crazy, so thanks for reading this far.

I began to notice my shapeshifting powers fading during the summer when I worked with someone who approached mathematics teaching and learning in a different way.  Someone who believes that math is important because marks are important.  Someone who disliked discussing changes, modifications, and explorations of different approaches.  Someone who believes in the importance of procedural skills.

I couldn't adapt.

I still attempted to be pleasant, but I was visibly (at least to me) unable to agree with these opinions.

I thought I'd take this opportunity to list some of the things that I am no longer flexible about.  i.e. a list of things that I have come to firmly believe (firmly is a strong word.  I suppose I just mean that I am unable to whole-heartedly support other sides)

• Collaboration and math talk is essential to learning mathematics
• Classroom management needs to be done through engagement as opposed to punishment
• Learning mathematics is about individuals constructing their own understandings and creating their own mathematical identity
• Procedural knowledge can only be a sideeffect of learning mathematics
• Pen-and-pencil tests are inadequate as the only method to assess student learning
• "Grades" on a test does not determine student quality, but merely the quality of that specific student product.
• Everything is a conversation, not a statement.

This list is not exhaustive.  The last one probably looks a bit out-of-place, since it is basically saying "I am not flexible with someone who is not flexible."  However, that's really the only one that sparked all of these.  I was unable to accept, and unable to work with, someone who did not want to discuss differences and changes.  Even all of the points above -- although I have strong opinions about those topics, I am willing and able to chat about them.  But if you are someone who isn't even willing to talk about these points -- then I really can't work with you.

So basically, like my alternate title for this post, this is about me becoming a grumpy old man who is unable to be flexible about certain topics.

[AnE] Ordered Pair Assessment - formative, summative, and evaluative

I've talked a lot about assessments (Thoughts and practice on Assessment for inquiry-based approaches (Standards Based Grading) Part 1, Part 2), so I thought I'd share some examples.

Here is an assessment method that I like to use from time to time.  This particular method primarily functions formatively, but I also use video evidence for evaluative purposes.

The task: Ordered Pair Assessment

The idea is fairly simple.

The room is set up like this - where the students face off each other

They have to find their ordered pair and then sit down.  I personally arranged all the pairing so that good conversations could ensue between the pairs.  On their table they get the following sheet:

The assessment takes over two days.  The first day is where they get the opportunity to explore all the questions with their partner.  They then get to take it home to review the ideas that they've connected together.

On day 2, each group gets assigned a number where they focus on preparing to present their discussions.  

If a pair has been assigned #2, for example, student x would tackle:

"what is the difference between direct and partial variation in the context of a table, a graph, and an equation?" 

and student y would tackle:

"what would be a linear function that goes through the first and third quadrant? are there more than one?"

Throughout day 1 and day 2 they have the use of a white board where they can record whatever they like on there.

Then on day 2 they would present their explanations to their classmates.  Most of them use the blackboard to describe their findings.  Some of them decided to pass their own white boards around.  Other students decided to just speak to their question.

What worked well

• The questions were open and required explanation (which then requires understanding)
• students used this opportunity the refresh their conceptual understanding
• the task had multiple entry points for different levels of understanding.  I was happy with most of the questions.
• this effectively incorporated assessment as learning (where students were reinforcing and extending their understanding through the task), assessment for learning (since both the students and I were getting a good sense of what they understand), and assessment of learning (since I was able to use the video evidence of their understand to inform my evaluations)
• students were engaged! All of them were talking math.  Some were more effective than others with certain questions, but all of them had something tangible to work with, and they were allowed to review their notes to inform their understanding.

What needs improvement

• Time was a huge issue. There was not enough time for presentation. May need to reconsider this task for multiple days (more than 2).  Perhaps embedding new materials if I am concerned about course time.  In my opinion, this was too valuable to be cut short.
• The instructions had to be changed multiple times as we went along.  Sometimes because of time restraints, others because I change my mind... a lot.
• While I am happy with most of the questions, some of them dug too deep.  It was good that I was able to choose numbers for pairs that were more suited to answer the question, but day 1 was definitely challenging for some of them.  I am unsure how I should address this.  On the one hand, I like the fact that all of the students had all the questions.  On the other hand, I am wary of students who were stuck on questions that demanded too much depth.

A Fountain of Data

I have been neglecting the blog for a while... Mainly because of this new addition to our family:

Being a puppy, she's been sucking up most of my time.  The rest of my time was, and is, filled with: reading journal articles, working on Jo Boaler's rich How To Learn Math course, creating some project based plans for a new course I am teaching next semester, furnishing the new apartment that my wife and I have moved to...etc.

That's enough for excuses.  I thought I'd organize my time a little better and post about what I did in the last semester.

I am going through some of the pictures in my phone, and making some brief descriptions of what I did... Hopefully it will be coherent.

There are these water fountains that have been installed in every school in our school board.  There are two at our school.

We began by asking if they've seen the water fountains before.  Then I ask them if they saw numbers on the top right corner of the water fountain.  Some of them said yes, and proceeded to share what they thought it said.  Here's an example of what that looks like:

Two students then receive the top secret mission of obtaining these numbers from the fountains.

This is where we talk about being green, saving the environment, and how mathematics can help us do this (beyond just recording).

Once the two students return with the numbers, I announce that we will be retrieving these numbers over the next few weeks.

As you can tell, we also built other estimations that was related to tweets (students chose who we were "following").  Every day after this I get them to estimate what the numbers are for that day.  I get them to guess numbers that were close.  At the same time, I would send students to retrieve the numbers while the rest of them estimate, justify, and defend their choices.

The water bottle count, in particular, lead to some nice real life data.  The gaps caused by the weekends gave opportunities for the students to think about how to deal with data and how to use data.  Interpolation fit nicely into that discussion.  They come up with the ideas/concepts.  I help them build the details and supply them with terminology as necessary.

Besides using this for data management and opportunities to talk about linear and non-linear relations, I also noticed the difference between the two water fountains.  One was used a lot more than the other.  We discussed the reasons for this, and then I showed them this video that I made.

note: So apparently the video I uploaded onto youtube didn't work... let's try this instead

Even with my poor video editing skills, it's clear that the rate of the water flow was not the same.  We proceeded to uncover the mysteries there.

It was fun.  Including data collection, this took about 2 weeks - and it was well worth it.  I think I have some data and graphs somewhere...  maybe I will find it later.  Although based on my last post, I would strongly encourage you to find something similar in your school!  What can you track and record daily? What kind of relationship would it be?

e-mail me or leave me a comment if you think of something cool!

Reflecting on the implementation of 3 acts

So I suppose I'm doing a bit of spring summer cleaning (yes, even this post was a draft from several months ago way back), and trying to turn some of these drafts on blogger into actual posts.  I threw out some introduction on assessment, then again with a bit more details (which I will continue when I have some time), and now here's another topic that's been on my mind.

Maybe we shouldn't share act 2's.

Well not exactly.  Let me unpack my thoughts a bit.

There are many different versions of the 3 act idea.  I am talking about a fairly specific one.

Of course, act 2 is extremely important.  It's the meat and potatoes of what teachers are going to teach their students.  So what exactly am I saying? Ditch act 2?  Well not quite.

Here's the way I see it.

It is important for teachers to develop their own approaches and mold ideas into ones that will work for them.  In other words, just like how we want our students to construct their thoughts and understanding - why not do the same with teachers?  Makes sense to be constructivist through and through.  Emphasizing act 1 and act 3 may be the best way to approach this.

I've heard people say that "the three act idea doesn't work!" "Students get tired of always seeing these pictures and videos!" "I can't do this every day!"... so on and so forth.  Here are some discussions from a while back several months ago on when it just doesn't work out:

• @mathhombre's post on what could go wrong with Barbie bungee without the setup for intellectual need
• @MrPicc112's mention on how "even... great ideas can [fail] or be used improperly, or rushed to the point that no meaning is taken from them."

There are certainly many more examples than this (and definitely many more since I wrote this draft...).

I choose to believe that these criticisms and problems are actually problems for some teachers (surprise).  I'm going to take a crack at the reasons:

1. Teachers are all different! Teachers have different teaching styles, different personalities, and varying comfort levels with the materials/students/time of day/weather/life.

2. Students are all different!  Students also have complex learning preferences, different personalities, varying  comfort levels with teachers/prior knowledge/time of day/weather/peers/life

Broadening our consideration to the learning environment as a whole, we have dynamic classrooms that are unique to every moment/class/lesson.

Why am I even writing about all this (this is the same question I ask about many of my own posts)?  Recently I've encountered several teachers who have gotten interested in implementing the 3 act ideas.  This is exciting.  However, they are looking for exact lessons to implement.  Painstaking, step-by-step instructions for how the class will go.  This worries me.

I don't often confront teachers in how they should facilitate learning. Discussions, yes.  Confrontations, no.  In fact, I get so caught up in thinking about multiple sides of the argument of any most situation that it is pretty much impossible for me to get confrontational (unless it's something fundamental that I disagree with, like saying math is just a collection of procedural skills... ARGH).

I think the solution is the following:

• Developing an abundance of act 1's
• Not all act 1's will work due to the reasons listed above about the unique classroom environment.  Luckily, several excellent websites have been developed for giving teachers and students lots of options: e.g. 101qs, Estimation 180, Visual Patterns...etc.  The key idea is about generating student questions that is interesting to them.  Act 1 is our bridge of connection to student engagement.  It is important to build this bridge authentically.
• Encourage discussions surrounding implementation
• I am a big believer in learning from everything: mistakes, successes, distractions... Talking about one way of doing things is never the way to go.  Students and teachers need the freedom.  Just having the option of discussion is valuable.
• Draw emphasis on student engagement
• The main selling point of the 3 acts is engagement.  We should never lose sight of that.  The main consideration for any teacher should be "how engaged were my students during that implementation?" and never "how well did I follow Dan Meyer's method (substitute Dan's name for your favourite teacher) of instruction?"  We are not perfect, nor should we expect to be.  But regardless of perfection, our focus needs to be on how many students we managed to engage and inspire during the process.
• Sharing personal stories but discouraging copying
• Not to say that "copying" won't work.  Imitating best practices has its values, but I think developing and exploring your own best practices is even more valuable.

(It's like @ddmeyer read my mind.  Since the initial draft, he's expanded and encouraged conversations on act 1, act 2, and act 3)

And... drum roll please...

Maybe we shouldn't share act 2's.

It's not to say that act 2 is not important.  In fact, it's an important part of student learning.  I often recommend the following structure of problem solving for students:

The process is not linear, it's complex and messy - just like our lives.  So if we add in arrows, it would probably look lke this:

If act 1 is the door that invites students in, act 2 is the essence of what they discover inside the room.  Act 2 is where students "overcome obstacles, looks for resources, and develops new tools" (according to @ddmeyer).

So act 2 is important, I've stressed that one to death.  What am I suggesting then?  This is basically what I am proposing: that students and teachers discover their own need for act 2's and find content in the act 2's.

What I am suggesting is to not have act 2 be predetermined.  It may sound like chaos to not have the necessary information immediately ready for students -- like on a card or something -- but I think it is important for students to explore and collect information on their own.  There are several benefits to this:

1. Students learn that data in reality is complex, and mathematics is our way of modelling the world.

2. Students take ownership of not only the questions, but also active collection of necessary information.

3. Students become responsible for the accuracy of their results.

4. Students are forced to actively reflect on their strategies and conclusions.

What I am saying also effectively promotes a specific kind of three act lesson.  As I mentioned here, I am distinguishing between 

A) a three act lesson that is open to a wide variety of questions, and

B) a three act lesson that is relatively closed and geared towards a specific objective.

I prefer type A.  Even though currently I am not able to explore type A fully.  At least not unless I am the only teacher teaching the course type, or I am able to convince all the other teachers to go with the same approach.  We'll see what happens next year.

Just thinking out loud, perhaps having a predetermined act 2 may also have some interesting dynamic that may not be necessary.  Students may feel good or bad about whether they did end up asking and exploring a question to which you had some additional information for.  It takes a bit of the ownership away.

Not knowing how things will turn out is a scary thing, but I think there is a lot of richness that would come from it.

There is also an important consequence of not having a closed act 2: a powerful act 3 may be difficult.  

This is a direct result of the fact that everything is left open after act 1.  Students all learn, but they may learn different things.  Act 3 looks different from class to class for me.  Typically I may have one resultant act 3 that most of them would have explored, but there are lots of other solutions for other questions that others have arrived at.  I am usually fine with this.  It ends up being time for consolidation and celebration.  What questions did the groups ask and how did they get there?  I would let the kids spend a little more time exploring what others have explored.  In a sense, some groups are creating their own act 3's for other groups to try out.

Going back to my analogy with act 1 being the door and act 2 being the wonders you find inside the room.  I am suggesting that students decide what is interesting inside the room.

I am often left with the conflicting thoughts of whether this is a good way to go about it.  But that's why I am throwing this out there.

School's out: Irrational emotions

Classes ended earlier last week.

Finally finished marking yesterday.  Not quite done the grading and reporting yet.  I will probably post about that process later on when I have time.  With all the exciting blooming interest in SBG right now, I want to share the things that I do (and try to do) when it comes to putting it all together.

I know I should be excited -- I will soon be able to get to my giant backlog of drafts and blogposts that I want to finish up.  I will soon be a part of the summer writing team for math assessments.  I will probably also have a lot more time to do work on my thesis.  Depending on how things turn out, I may end up helping out with a bunch of research initiatives as well. -- and I definitely am excited (although a bit sad that I can't join everyone at TMC), but I can't shake this irrational sense of sadness/helplessness that I am feeling.

The day class ended, I actually couldn't sleep that night.  I stayed up most of the night thinking about the kids from my two semesters this year.  Students I reached, the students I couldn't reach, and whether it was good enough.  I spent a lot more time worrying about what they would be like when they come back (and in the case for the 12s, if they come visit).  Stupid me, right?  I've always missed students when I don't see them, but ever since reading this from @approx_normal, I've started to worry about them more.  It's stupid and irrational.  I can't control the future, and I certainly don't control them.  It's completely irrational, and I don't know if I am alone in feeling like this.  I am also feeling an overwhelming sense of loss.  I understand that I may see the students again next year, but it won't be the same.  Even if I get some of them in my classes next year, it won't be the same.  The classroom dynamic would be different if not gone.  The culture that I've built would be gone. Maybe growing up without parents resulted in lots of attachment issues, but I have problems letting go.  My wife jokes and asks what it would be like when we have kids.

I support this idea of being a school teacher and not merely a classroom teacher, but I still can't shake feeling a little lost, or that something has been lost.  Like a huge part of my identity is missing and I am just replacing it with this irrational fear.

It's ridiculous and irrational.  Not sure what else to say or do.

I am hoping writing this out will help me overcome this... whatever this is...

Not good enough

This post is not so much about Math.  I am essentially treating this as a place to release some thoughts.

Just some disconnected and unorganized thoughts.

I had an interesting time growing up.  Long story short, no parents around, took care of my younger brother since I was 12 or 13... etc.

Maybe as a result of this, I have always cared a lot about the well-being of others.  It is especially the case for the students that I get to know, teach, and listen to.  I even try to build opportunities in the class to discuss the "important" and "heavy" aspects of life.

Not good enough.

I have always tried hard to reach out to students that are having a hard time.  I tried to build up their confidence within class, or try to establish groups that are beneficial for them.

Not good enough.

I've had students seek my advice about difficult situations.  I listen, I gather support for them, I help them where I can.

Not good enough.

Some students that I can't reach at all.  They get on the defensive whenever I approach for conversation.  Maybe it is because the environment I build is simply not good enough.  Or maybe because I am simply not good enough.

Even the students I do reach, that do feel comfortable with me, what I do is STILL not good enough.  I haven't solved their problems.  I haven't changed their lives and prevented their hardships.  I haven't given them enough reasons to stand up or to smile.  Besides being there to listen, support, and connect them to parties that can actually do something -- I've done nothing.

Or at least it feels like nothing to me.

Colleagues tell me "you can't save everyone," or "some just can't be saved."

But I refuse these words.

It is simply not good enough.

[AnE] SBG - where I am and where I want to go

Previously on AnE...

"Assessments are not just in written form.""It is what we do with [assessment] that determines the function."

This time I plan on tackling a different but related topic of standards based grading (SBG).  This is quite a monster of a post... so be prepared.  I will begin with some background first...

I am open to different definitions.  In fact, I love reading about slightly varying definitions because it helps frame the conversation.  But for the sake of what I am about to talk about, it's important to indicate what I take SBG to be.  Let's first establish the difference between norm-referenced vs criterion-referenced interpretations of assessment.

Norm-referenced is something that we math teachers don't -- and shouldn't -- use.  It's the idea of the bell-curve.  Criterion-referenced involves the idea of establishing scores according to how well the student matches up to established criteria.

I've never graded with "marks" where a question is out of a certain number of "points." Okay, "never" is pushing it.  In the very beginning while I was at my first practicum I had to grade with marks, but ever since that point I haven't done percentages or marks.  I have always believed it's better and (with experience) easier to evaluate based on achieving a standard or not.  There are lots of examples we can take a look Math Mistakes, a beautiful website edited by @mpershan.  I will use one of my own students's example instead:

Nothing special.  It's a relatively closed question with one right answer.  Regardless, we can take a look.  The student knows a couple of things.  They know about the general concept of what an optimized shape with the biggest volume is - given a set surface area.  They are not efficient with establishing the formula, and somehow their solving step went to a 6 instead of dividing by 2.  But who's to say that the mistake of cube-rooting is worth more than others?  I believe it's more meaningful by looking at what the students understand and what they don't understand.  First and foremost, we're looking at how much of the standard they have achieved.  Instead, we begin with the expectation of what we want the student to achieve.  Which according to our curriculum, it's this:







Note the "investigation" part.  That's important.  The Ontario ministry of education also provides some "specific expectations," which can be helpful.  One of them reads like this:


















This problem is not quite an open question that will fulfill the expectation, but the next is more of an open question that will have opportunities for that.  That's really all it's about -- opportunities.  Opportunities for students to demonstrate their mastery of expectations, skills, or standards.

There's a lot of conversations that need to happen around the idea of SBG, which as I mentioned in the previous post, I am extremely excited about.  Daniel Schneider (@mathyMcMatherso) recently wrote several pieces on assessment, like this one.  He hits several important points in assessment, and the piece has gotten a lot of buzz - which is also exciting to me.

After attending several sessions at OAME today, I came to this realization:

Interesting, and exciting as well.  I want more though.  I need more.  I recently had several wonderful conversations with Jonathan Newman (@newmjh3) on Tiered assessments and Standard based assessments.  I quickly shared what I did last semester with him, and he came up with the wonderful idea of using Google spreadsheet to develop something similar.  We're still working on it.  There are a lot of awesome things that we'll talk more about too.  But in any case - let me finally frame the meat of this post.

SBG - Where I am

I begin with something like this in my spreadsheet:

and each assessment is recorded in the vertical columns.  But then I use a few formulas to make it automatically visual:

There are a few things here

1. Easy to use
Because of how I set it up, there is just 1 visual tab (and not one for each student).  I just have to change the student name, and everything will automatically change to reflect student achievement.

2. Proficiency levels
Instead of percentages and arbitrary points, I get an overall look at whether the student has achieved understanding.  In the example given, there are 2 main expectations shown.  Each assessment is chronological, so we can see a clear trend of improvement.  I also use different colours to indicate what forms and types of assessments they are (I will probably talk about forms and types of assessments later).  Students get a clear picture of what they need to work on.

3. Incompletes do not count against them
I separated the incomplete assessments as just missed opportunities.  They are not rigid "0's" that students received which transfers to some sort of percentage after mysterious averaging functions.  Incompletes are simply missed opportunities.  Lack of evidence does not imply that students do not understand the concepts.

4. Multiple opportunities
Students will always be able to come for reassessment.  It's a loaded statement, but I will unpack this in a later post.

In addition to this form of reporting student achievement, I have also started to integrate these elements into my Ninja Board.

It looks a bit like this on the back of my classroom wall:

As students gain understandings of masteries, they also gain these crests.  There are essentially 11 overall expectations in the course that I am referring to, and each correspond to a crest that I have created/found/edited.

There are clearly more that I need to explain about the various aspects that I am doing with assessment in class (construction of a sample assessment, for example), but I will get to those later.  I want to get to the next part - which is where I want to go with assessment.

SBG - where I want to go

This is more important to me.  This is the reason why I have been looking at how others have approached the same concepts.  This list is open to change, but I don't think the main points will alter very much.

1. Completely projects-based spiral curriculum
I alluded to this before here, and then here.  Essentially the idea is this.  Opposed to the U.S. curriculum, where topics of mathematics are separated to Algebra and Geometry...etc.  In Ontario, Canada, our curriculum revisits ideas in what is called a spiral curriculum.  Recently, I became interested in the idea of creating a tighter spiral within a classroom.  The main idea is this:

• Everything is activity driven.  No lectures.
• All concepts covered are done through rich activities.
• There are no units.  Just problems.

No Units.  that's really what I am trying to get at.  Rich problems for mathematics are everywhere.  it's up to us to harness them.  through this method, concepts of mathematics evolve naturally as students work through interesting problems throughout the year.  They learn to create their own problems, and learn to solve their own problems.

This approach is actually perfect for mediating the issues surrounding re-assessment.  If topics are revisited naturally through each problem and activity, then there is no problem with reassessment.  They are reassessed anyway!

2. Focus on Mathematical Processes

Within the Ontario curriculum mathematics document (grade 9-10 example), it describes 7 mathematical process expectations: Problem solving, communicating, connecting, reflecting, representing, selecting tools and computational strategies, and reasoning and proving.  These skills are common and are suppose to flow throughout every grade.  Students learn to employ these skills through the different "content expectations" described before.  I have been doing process portfolios to focus on these important transferable skills.  However, this is not a focus in all classrooms.  I want/need to bounce more ideas of how to assess these important skills.  Hopefully this blog post will garner some interested people who are doing similar things with these transferable skills so we can have good conversations around them.

3. Improving evaluative properties of alternative assessments

This is pretty much my M.A. direction.  What are valid methods of incorporating observational assessment evidences of student achievement?  What qualities do activities need to share in order for alternative assessments to thrive?  If students are demonstrating understanding, does it really make sense to make them sit down, and to write out their understanding?  Should the additional step of "writing it out" become the determining factor of student achievement?  ...etc.

Of course, there are lots of aspects that I am interested in as well.  My conversations with Jonathan has been focused on the feedback piece.  Implementing Google Spreadsheet is an excellent way to achieve this.  Hopefully we'll both get some off time soon so we can chat more about this.

Leave a note if you are interested in talking more about this.  I am currently pretty sick... so it's possible that I am lacking in a lot of explanations.  But don't hesitate and leave a note for elaboration.

[AnE] Assessment for Inquiry-Based Approaches

I have been meaning to write about assessment for a while now.  Things kept getting more and more overwhelming in my life, so it had to sit on the backburner.  Another reason that this kept getting pushed back has to do with the content itself.

I have been interested in assessment ever since I stepped foot into education.  I have always grabbed onto the constructivist framework of education - specifically the inquiry-based approaches that has been promoted by people like Darling-Hammond, Boaler, and more recently from the blogosphere, people like Meyer (both of them [@doingmath]), Doran @nik_d_maths , Nguyen @fawnpnguyen, .... and so many more.  The question for me has always been - how do we appropriately assess these approaches? (this was in fact the question that I asked @ddmeyer when I met him a year ago)

Mathematics Assessment is in fact the focus of my M.A.  All the research I deal with are primarily built on the foundation of inquiry-based learning, and necessitates standards based grading.  With the emergence and acceptance of inquiry-based approaches, the challenges of assessments has taken a drastic turn.  How do we develop appropriate assessments for the way that we are facilitating discussions?  An activity like this where students are engaged in the process of learning, and are effective in demonstrating their understanding of various concepts... how exactly would we assess this? (my concerns are currently the summative and evaluative functions... which I will explain later) In my mind, there is a huge disconnect between the way that we are facilitating learning, and the way that we are presenting our evaluative assessments.  All this stuff is precisely what I am (have been, will be) working on.  It sounds like I gave a lot of detail on what my thesis entails... but my topic is actually a lot more specific.  I will probably write about it later.

Which is why the recent growing interest in assessment in the blogosphere is exciting news to me.  Daniel  @mathymcmatherso and Tina @crstn85 , for example, were one of the ones that have written about assessment recently.  I have yet to figure out a way to fully engage these people in the same room for conversation, but I am sure that'll happen down the road.  Another reason why I haven't approached them for these conversations is that our concerns are currently at different places.  They are concerned with written assessments and establishing the how-to's for those.  The development process is rich, and is probably not something I should impose on.  Swimming around in the sea of research, as well as various reform efforts for assessment around Ontario, I have developed a way of doing it.  I don't want to interfere with others, because I am interested in seeing what comes out first before we engage in conversation.  I am actually genuinely interested, and excited, to see the evolving product.

In any case, I thought I might start off what will likely be a series of posts about assessments, beginning with my general understanding of assessments (I won't say it is THE way to think about it, for obvious reasons!).

Assessments are not just in written form.  There, I said it.

It seems like a silly distinction to make, but it's actually a very useful one.  It's difficult to narrow down a one-liner definition for assessment, but we can begin to unpack this big idea by looking at the associated functions of assessment: formative, summative, and evaluative.

These three functions are huge ideas, but I will talk about them generally and separately.  I will just tackle the formative function first.

The formative function of assessment is a huge piece of student learning.  It's not just a buzz word that's being thrown around in interviews or teachers college.  Dylan Wiliam, in my mind, brought this idea of formative function out to the forefront back 2 decades ago.  The main concepts behind formative assessments have not changed.  However, the details have been altered throughout the years.  The most recent one that I am aware of states: "An assessment functions formatively to the extent that evidence about student achievement is elicited, interpreted, and used by teachers, learners, or their peers, to make decisions about the next steps in instruction that are likely to be better, or better founded, than the decisions they would have taken in the absence of that evidence."

Heavy sentence, I know.  Wiliam unpacks some features of this definition:

• it is based on the function served by the information yielded by the assessment, rather than a property of the assessment itself.
• the assessment can be carried out by the teacher, the learner, or her peers.
• the focus of the definition is on decisions regarding next steps in instruction, rather than intentions or outcomes.
• the definition is probabilistic
• the assessment need not change the direction of instruction (it might merely confirm that the planned subsequent actions were appropriate).

Now, you may be wondering why I wrote "formative function" instead of "formative assessment."  Isn't that easier to write?  Well, that's actually exactly what I've been trying to avoid.  I don't believe that assessments should solely be formative -- and since "formative assessment" implies that it has a singular purpose, that phrase is not as useful to me.  What do I mean?  Well an assessment is what it is.  It's an assessment.  It is what we do with it that determines the function.  Let's say it's an assessment in written form.  If you decide to have them write it, give it back, then have the kids discuss and generate answers together -- then that's formative, and probably summative as well.  I think a lot of teachers jump straight to the evaluative function of assessment, and they develop separate special "formative assessments."

Everything we do as teachers is for the sake of facilitating effective student learning.  Assessment is a huge part of this.  In a way, teachers are like researchers.  We are gathering information about our subjects so we can accurately construct a picture of their learning.  Any puzzle pieces that we obtain -- are assessments.

I plan on writing more on assessments and what I have learned about assessments in the future.  I will likely explore more of the conceptual structure of assessment.  If you are more interested in the pedagogical concerns, let me know.  I can share how standard-based grading functions in my classroom, or how I deal with open-questions and why my tests have never had marks on them?...etc.

See, this is another reason why I didn't get to this blog post... It's such a giant topic for me.  I have way too many things that I want to talk about that it's hard to break it down to manageable pieces.  Oh well, one step at a time :)

Hopefully this roused some interests and questions for you to explore too!

*Click here to continue to read about my assessment practices