One by one (Regina Bittencourt)

Group members: Alina, Emily K, Keith, Tara

Regina Bittencourt: One by one (The original version)

Our remake of the original

In remaking the artwork, we chose to use acrylic paints on paper. During this process, we wondered about why the artist stopped at a maximum of 9 rows (before descending). Exploring this informed our group activity of attempting to make a 10th row, observing that we no longer have the same palindromic pattern, and discussing why that happens. We tried adding a row ourselves and soon found that the “place holders” get in the way and the pattern breaks. Inspired by the lesson on Babylonian place holders, this prompted us to wonder if there was a maximum number of rows for other bases as well. After trying out a few it became clear that this maximum is one minus the base you’re working in. For example, base 10 has a maximum 9 rows while base 2 has only 1 row before the pattern breaks!

Our extension

The extension that multiplies 9x9, 99x99, 999x999, … does not follow quite the same pattern as with the 1s, and so we worked to find out why this one has a pattern of its own. Of course the painting does not have 90-degree symmetry like the original, so we decided to adapt it slightly, “flipping” the bottom half to produce an image with 180-degree symmetry to give it more visual appeal. In making the extension, we had to make decisions regarding whether the color representative of each number should be the same as the original. We ultimately decided to keep the colors consistent between both works, in the hopes that it would make it easier for viewers to easily switch back and forth between viewing. Since 0 does not appear in the original, we chose the color to represent 0 (burgundy) based on what we thought would look best with the other colors in the piece.

In designing a class activity, we wanted to offer a way to further explore the idea of the One by one piece. At first we considered using sticky-notes or tiles, but decided this would be difficult to manage in the short time for presentation. We decided to create a visual handout with the images of our artworks on one side and an activity on the reverse. Providing this visual handout means the class can see the detail of the painting (the paintings are on 12” x 12” paper, so it would be difficult to see the colours from the back of the room).

For the activity, we ask the class to consider what comes next. That is, what would happen if we attempt to add a tenth row to the painting? We’ll work this out together on the paper and on the whiteboard. Then, we ask the class to consider what would happen if the painting were done in a number system other than base 10.

No action shots available of our painting because we were too in the moment. 🙂 

To set the vibe we played the September 17th NTS Breakfast Show w/ Flo while painting and learned a lot more about each other's interests and hobbies.

What Should Math Class Be Achieving?

It was illuminating to read about the highly politicized history of the math curriculum. I was surprised by a few points in the reading. First, I was struck by the timeline of Dewey's inquiry-based approach to teaching math (pre-1940s) described on page 396. If I heard about this approach out of context and was asked to guess from when it originated, I would have assumed it to be a quite modern idea. This made me feel sad for myself and other students who might have thrived in classrooms with this approach but didn't get a chance to due to the changing perspectives that favoured more traditional methods.

I found the shift to the Bourbaki group's "New Math" very interesting. It was truly ironic that a movement to create more skilled mathematicians seemed to completely backfire--alienating people further away from math rather than drawing skilled mathematicians in. How many brilliant mathematicians missed out on the opportunity to explore math further because the rigidness of the subject shut them out. I wonder if we still feel the impact of this today in the attitude many people have that mathematics isn't for them?

Jumping forward to the 90s, I was also struck by the desire to cling to traditionalist methods after the US didn't rank as high as they'd like in international mathematics standings, despite evidence that deeper understanding led to higher scores for the top-rated countries (described on page 400). This phenomenon or gut reaction to "dig your heels in" even when faced with evidence contrary to your actions is interesting to me, especially when it's done at the government level. How can we be more adaptable and open to change? In particular when faced with evidence that the change would be beneficial, and in particular on a governmental level (which can often feel quite out of reach as an individual).

Finally, as a whole the reading reminded me of a time in my masters degree when I heard a CBC radio show discussing narrative assessment. I'm paraphrasing, but one of the interviewees essentially said "Narrative assessment might be possible in 'soft' subjects but could never be used in a subject like math". I was a bit outrage by this use of math in their example. What did this person know about teaching math? What did they know about math at all? In my experience as a student and as a TA, narrative assessment was where the most influential learning happened. When we shifted the focus from the grade, we could instead discuss the actual content.

Sometimes things don't work out the way you planned

(Here's a parked car I saw squashed by a tree on a not even windy day, no one was injured)

Curriculum: Explicit, Implicit, Null

Prior to reading the article, I defined curriculum as the set of standardized outcomes required to meet the learning objectives of a particular subject at a particular level. My first thought on this is that it is a baseline, a minimum.

I found the article enlightening as it highlighted that my gut definition of curriculum focused on what it "is" and not what it "isn't", both in the "implicit" and "null" sense that Eisner talks about in depth in this excerpt. The full picture of the learning happening for a student is not complete without these other elements.

What struck me about the "implicit curriculum" was the frequent nuance behind the points Eisner was making. For example, on page 95 he discusses how the timetable could teach adaptability and skills for context switching, yet on the other hand it could also demotivate students when they feel they don't have the time to be fully engaged with what they're working on. So, these contexts that are often taken for granted or invisible to us are still providing a lesson. It is extra important then to be mindful as to whether the socialization we are teaching is what we actually want to (or should) be teaching.

The "null curriculum" is a powerful influence to be aware of because it is telling students what is deemed worthy of study versus unworthy. This sends a message about what our society values. Looking at the BC curriculum it is encouraging to me to see there is required indigenous-focused content. Yet this requirement is much smaller than many other subjects. It is notable also that the vast majority of requirements focus on language, math, science, and even career education (a topic which did not exist when I went to high school), with less emphasize on physical education and the arts. This is remarkably similar to what Eisner lamented in this excerpt which was originally published in 1979. It is both surprising and not that almost half a century later his words ring true. 

I chose this picture of a river near Whistler because there is a lot more river

hidden behind the rock formations including a partially visible waterfall

Favourite and least favourite teachers

My least favourite teacher was a math professor during my undergraduate degree. It was clear from the first class that they expected us to already know the material we were meant to learn. They would call on us during class to answer questions and then shame us in front of the class when we struggled to answer. There was no rubric, no guidelines, and no clarity of expectations. This professor did not seem to have a plan for us: even during exams they would simply write questions on the board and then leave the room, telling us to hand in our answers to their office (thank goodness we brought our own paper and pencils). The feedback on these exams consisted only of red exes. The two biggest struggles learning from this professor were the culture of shame they created as well as the complete lack of preparation.

My favourite teacher was also a math professor during my undergraduate degree. This professor expected a lot from their students, and really pushed us to struggle with complex ideas and problems. They opened my world to ways in which math disciplines can overlap, encouraged me to persue a masters degree in math, and surfaced opportunities like math conferences and government funding. They were organized, clear, and expected a lot from us. The biggest thing that made this teacher great was their confidence in me: their confidence in me gave me confidence in myself. 

A puzzle solved in subsections

The Locker Puzzle

 I started out the locker problem the same way I start most math problems: writing out some examples. 

The leftmost column in the block below enumerates the students from 1 to 20 (that is the highest value I went to for my example, figuring it may be enough to illuminate a pattern). The top column enumerates the lockers starting from 1.

Beside that I have 20 "lockers" represented by - if the locker is closed and + if it is open. To the right of this I have some commentary of my thoughts for each row.  It is important to note that because the subsequent students won't change the state of the lockers in line 20 because their values exceed 20.

  0102030405060708091011121314151617181920
01 - - - - - - - - - - - - - - - - - - - - Student 1 touches all the lockers
02 - + - + - + - + - + - + - + - + - + - + Student 2 touches all even lockers
03 - + + + - - - + + + - - - + + + - - - + Student 3 touches all lockers that are a multiple of 3
04 - + + - - - - - + + - + - + + - - - - - Student 4 touches all lockers that are a multiple of 4
05 - + + - + - - - + - - + - + - - - - - + And so on...
06 - + + - + + - - + - - - - + - - - + - +
07 - + + - + + + - + - - - - - - - - + - + 
08 - + + - + + + + + - - - - - - + - + - + 
09 - + + - + + + + - - - - - - - + - - - +
10 - + + - + + + + - + - - - - - + - - - -
11 - + + - + + + + - + + - - - - - - - - -
12 - + + - + + + + - + + + - - - - - - - -
13 - + + - + + + + - + + + + - - - - - - -
14 - + + - + + + + - + + + + + - - - - - -
15 - + + - + + + + - + + + + + + + - - - -
16 - + + - + + + + - + + + + + + - - - - -
17 - + + - + + + + - + + + + + + - + - - -
18 - + + - + + + + - + + + + + + - + + - -
19 - + + - + + + + - + + + + + + - + + + -
20 - + + - + + + + - + + + + + + - + + + +

So each locker is "touched" if the value corresponding to the student is a factor of the value corresponding to the locker. 

Great! So we now know something about when lockers will be touched (i.e. when they're state will change from open to closed or closed to open). So what is it that influences whether a locker is closed, or in our picture above negative, or whether it is open, or positive?

Looking at the picture vertically now, we can see that a locker ends up closed (negative) if it has been touched an odd number of times, and open (positive) if it has been touched an even number of times. Given what we said earlier about factors, these number of "touches" are equivalent to the number of factors each enumerated locker has.

So which numbers have an odd number of factors and which have an even number of factors? If we need a hint we can look at the signs corresponding to line 20 above. The negatives correspond to lockers 1, 4, 9, 16, ... and voila! Perfect squares have an odd number of factors, while all other numbers have an even number of factors.

So after 1000 students (or any number of students for that matter) have had their turn, lockers corresponding to perfect squares will be the closed lockers and all others will be open. In the case of 1000 lockers 1, 4, 9, up to 961 will be closed.

Lockers at my previous workplace, giving wicked witch of the east

Sidebar: In my original exploration I used 0s and 1s to represent the state of each locker but changed this to - and + to hopefully provide better intuition around odd and even factors. I also made an early mistake to the state of one locker which greatly confused me along the way when I had locker 20 as closed. When I thought more about the patterns going on I knew something went wrong with my example.

Relational Understanding AND Instrumental Understanding

One thing that struck me was that I have felt the exact same reservations towards "instrumental math" that Skemp lists in this article. This was a sadness I felt to how (I thought) math was often taught, as if some of the beauty within was completely overlooked and ignored. In grad school I read "The Mathematician's Lament" by Paul Lockhart, an article that plays with the idea "what if we taught music like we teach math". I was struck to read the very similar analogy given by Skemp in this article, some 30 years prior to Lockhart. Back in grad school I felt validated in my feeling that "instrumental" is somehow missing the point, and I was annoyingly sanctimonious about it (despite never taking the "devil's advocate path").

Since that time, my perspective has gained some nuance. When I read Skemp's football analogy on page 3 I wondered to myself: are there not skills used by both sports that would transfer from one to the other? I finished the article disagreeing with Skemp that instrumental and relational math are so distinct, and my long-standing skepticism of instrumental math has softened. I see these different approaches as two sides of the same coin. Though the sides are certainly different, they are not necessarily opposing. And each offers a different way to teach, both of which carry their own merit and both of which I will try to use appropriately in my teaching. After all, the title of the article is "Relational Understanding AND Instrumental Understanding" not "Relational Understanding OR Instrumental Understanding".

Maybe we can have our cake and eat it too?

(A strawberry cake my friend made that is still to this day the best I've ever had)

Hello World!

Welcome to blogging

Temperamental Mount Rohr