## Wednesday, January 9, 2013

### Followup on exponential activity

So happy to be writing about math for a change! If only so that next year I have something to start with that is a little further along the effective-f2f-time continuum than this year.

So here's what happened:

I showed the first slide to all, had random volunteers find a few simple equivalent rules, to warm up before putting them in pairs to do the rest. After about 10 minutes, we talked.

Here are the screen shots of the different groups' sorting of the "exponential things":
 amje

 dafr
 jali

 mibo

 pama

 saca

 tyelel

Observations during their work time:
• almost no team was able to finish this first slide, and most started with the algebraic rules.
• some gravitated to the graphs and sentences later. I think that for a lot of them it is a revelation that a graph represents the same thing as an equation. Or a sentence.
• lots used geogebra to see if rules were equivalent, instead of working through the algebra
• only two groups looked at the rule with e in it, and only one found its correct equivalent (at the left)
• saw lots of discussing and questioning of each other, just a little of "miss, we're totally lost"
What I did:
• I brought everyone back and went through the answers, pausing to let them either high-five or ask questions. They are really weak overall on evaluating things like $27^{-\frac{1}{3}}$.
• Since we ran out of time, I decided to put the whole set of slides, answers and all, on this voicethread: and told them to look at it and leave their questions there, or answer other peoples' questions. Another opportunity to get their reflections, so that makes 3 ways.
• In only one class were we able to get to the next activity, which was all english situations. I gave the hint that 3 of the 5 were equivalent, and immediately regretted it. Won't do that next time.
Next time: start with a slide that has only rules, then the next has rules and english, then the next has all three types of representation. Actually, it might be better to leave out graphs, not sure they helped at all.

It's only after doing this type of thing that I realize what I was trying to do. I made this activity up on a hunch, it seems, and now I what it could have done if I had done it better:
• I want them to not only know THAT two things are equivalent but WHY they are - why they are algebraically and why they are realistically. It's not enough to get the algebra behind
• $2^{3x} = 8^{x}$, I want them to see that if something octuples every hour, it also doubles every third of an hour. I want them to be able to read that kind of info off the rule, like if the nbr of bacteria after x hours is $y = 3^{2x}$, then they're tripling every half hour.
• I pushed geogebra so much that now they rely on it too much. I have to find a way to balance things. Maybe I need to model using it more, as a way to verify their algebra, rather than to guess and check.
• The idea that the base can be ANYTHING as long as you make the necessary adjustment to the coefficient of the exponent - is that too much for them at this age? Or will this help them out when we get to logs?
• I think they need to sleep on things sometimes, so it will be interesting to see if anyone gains any insight tomorrow. Including me.
But the point is, I never even tried getting this idea across before, so even if it doesn't work, it has to have stretched some minds a bit. I haven't received any of their reflections yet. So looking forward to that, almost more than the math!

## Tuesday, January 8, 2013

### Activity for exponential functions

The continuing story of my flipped classroom, in which I try to make the best possible use of that f2f time:

I spent yesterday working on this activity for tomorrow's Science Math gr 11 class. It's another one that was inspired by Malcom Swan's Improving Learning in Mathematics, and I think it would fall under the category of  "Interpreting Multiple Representations", on page 19.

What I'm really looking forward to is the part where I ask them to reflect on what they learned doing this. I am giving them the option to do so either in private, via this week's checklist: (4th item):
)

or on their blogs. I'm hoping to get more blog action this way, but I do plan to share the private reflections, and keep them anonymous. I just want them to know what each other is thinking!

Here's the activity: We're just finishing up with the exponential function, and I wanted to tie up a few loose ends, and give them time to chew on things. For instance, that when something gets doubled every hour, it gets root2'd every half hour. You didn't know that "to root2" was a verb, did you? It means to multiply something by the square root of 2. (In my world it does, anyway.)

I'm going to give them this jumbled up slide of exponential things (I just couldn't think of another noun....) and ask them to sort them by which ones are equivalent. Lots of  "things" here - graphs, function rules, sentences. I plan to put the kids into small groups, then gather them back every few minutes. I'll ask them to use the nifty new highlighter pens in our virtual classroom.

But this is what I'm really getting at. Equivalent situations, ie. that you aren't locked into a base of 2 just because the problem says something doubles. The next sequence of slides is all about getting them to be able to move easily between these equivalent representations, not only from algebra to english, but from algebra to algebra, and english to english.

Here's the full set of slides, with answers. I'm thinking that I will go over the answers to the first jumble, but then let them sleep on the next set. We'll see how it goes. The thing about these types of activities is that they sometimes go in a direction you hadn't thought of. And they as a result end up taking either way less or way more time than you had anticipated. It certainly keeps me on my teaching toes!

## Tuesday, January 1, 2013

### Our Mother's Books

I love ebooks, but I also love the real ones, and every time I have to buy a new book, I am torn. This holiday, something happened that made me really appreciate the real ones. I and my four siblings were faced with a difficult task that left a mark on us all, and I don't think it would have done so if not for real books, with real paper, and dust, and handwritten notes, and of course, that old-book scent.

What happened was that in order to accommodate our mother's care team at the residence where she lives, we had to free up some space in her room. And that meant books had to go.

 Marion Rae McLaren
Our mother was a librarian. She loved books, history, antique furniture, and her 5 kids. Everywhere she has ever lived would most accurately be described as a library with nice furniture in it. Our mum had every kind of book imaginable - all the Time Life series', science references, World Wars I and II, The History of Flight, the history of probably the entire world, Birds of North America, the Royal Family, Bibles, Bible History, Films, Lady Diana, complete works of Shakespeare, Jane Austen, Joseph Conrad, Charles Dickens, Winston Churchill, you name it. Not to mention all the magazines to which she subscribed - Time, Newsweek, The Economist. We rarely, if ever, had to go to the library for research.

Over the years, we have had to move our mother several times, due to her declining health, and with each move, her world has shrunk a little more, both the physical living space and her memory. Each move has required that we remove more of her things. Some we have distributed among the five of us, but most we have had to give away. We've had to pack up quite a few boxes of her beloved books, her Franklin Mint collections, fine china, and beautiful linens, keeping only those things that we felt she would want to keep, and also those things that we identified most closely with her. It's been a kind of slow and painful filtering process, most of which she mercifully missed. The last move was three years ago, after which we were down to two bookcases (each way overstuffed, with sagging shelves), one antique chest of drawers, one antique mirror, a tv, a CD player, a lovely old armchair, and one side table that was made by her father, our grandfather, a country gentleman in the finest tradition.

This holiday, the physiotherapist needed us to remove, among other things, one bookcase, which in this case meant removing roughly the equivalent of two bookcases-worth of books. As has always happened when we have had to go through this, we found ourselves falling into a pattern. We began by industriously going through the books, sorting them into two piles: give-away or keep. But every few minutes, someone unearthed a book that made us all stop, gather, and remember. This slowed us down quite a bit, but even though we had a limited window of time, we allowed ourselves this luxury, knowing that this was probably the last time we would do this.

Sometimes we gathered over a book with a name and date handwritten in it:

 Mum's writing in The Complete Works of Shakespeare(note the price)

 Mum's writing in Pickwick Papers, a gift to her brother, Alex

Sometimes it was a book we remembered reading ourselves. I found the book on Stonehenge that I had used for a project in elementary school. My siblings paused over The Rise and Fall of the Third Reich, which they were all surprised to learn each had read at different times - and I mean this actual copy of it. My sister and I wondered how many other people had read The Double Helix after plucking it off their own living room bookshelf. We found comfort from the pamphlets and travel books she brought back from her trips to Europe and Asia, knowing that long before her world was reduced to this tiny room, she had realized a lifelong dream of travelling. After repeating this cycle of sorting and gathering many times, we eventually managed to finish.

By then, the overall effect on us was a sense of being steeped in a rich broth. By the end of the whole endeavour, we are all freshly imbued with our mother's intelligence, her passion for history, her love of reading, and her deep abiding devotion to us.

I'm not saying she is now reduced to her books in our minds. Of course, the books don't tell the whole story of our mum -  that as a girl she first saw The Wizard of Oz in a movie theatre, that she spent a few years being a career girl in New York City, that she raised the five of us virtually on her own, put us all through University, did her Master's degree at night after working all day, even took some math courses that tortured her but that she got through, that she was a news junkie, she survived an abusive mother and absentee husband, she was a snappy dresser, she loved M*A*S*H*, and she quit smoking just before retiring. Or that she was always enrolled in some continuing education course, or presiding over the St. James Literary Society, providing us all with an authentic model of a lifelong love of learning.

But the books do tell a part of her story, and our story. Sometimes, things, concrete things, do matter. I wonder how, or if, any of this could have happened if her entire collection had been in digital form. There would have been no need to go through anything, because digital books don't take up space. The five of us wouldn't have had to be there at the same time to mutually decide anything. There would have been no handwriting, or dusty old library scent, because there would be no pages to hold it. And I don't know how it works with borrowing books from someone's e-reader, but I bet it's not free, and it's not as cool as finding the actual book you and your sister both read, and that your mum read too.

In fact, I wonder what collection of mine would my own children have to go over so that they came away with a sense of what I loved in this life, what I made time for even when it was really hard to. And what about their kids? What are we leaving behind that tells our loved ones who we were, and does it even matter?

Well, I know it mattered to the five of us that because of our mum and her books, we were all there together, just the five of us, if only for a short time. And because of the memories those books held,  it was like Mum was there too.