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The assignment has become a standard one for grade one students in Ontario: gather a collection of 100 items to bring in and explain to the class on the hundredth day of school, usually in mid-February. It filled my six-year-old with glee. She knew exactly what she wanted to bring in. I was less thrilled, because the collection she chose did not belong to her, but to me: my dice.

Nevertheless, we spent an hour one Sunday afternoon gamely dividing dice into baggies, categorizing them by number of sides, then by colours. Along the way, we explored concepts relating to the base-ten system and the absolute basics of multiplication.

My daughter has plenty of experience with dice. They’ve been a part of her life since birth. Plush dice, foam dice, and the vast array of dice used for roleplaying games by both her parents and all their friends, were her first introduction to numbers that weren’t on her fingers. For her, dice represent fun times with friends, groups of people laughing and telling stories around the dining room table, the adults who take an interest in her life even though they aren’t related to her – and math. So it was natural that when she needed a real-life collection to bring in for Hundreds Day, her first thought was dice. Dice are math as it is in her life.

There is a great deal of emphasis in mathematics education on making math real, on finding the ways to make the numbers concrete, tactile, visual. This emphasis is a vast improvement over numbers that never left the page, because it does facilitate a deeper understanding of mathematics, and that is the ultimate goal of mathematics education. (1) But my daughter’s experience with dice is evidence that it doesn’t go far enough. For her, dice are not something a teacher brings out to show how numbers work; dice are real life that we describe using numbers. The educational establishment has been getting it backwards. The goal is not to make math real. Math is already real. The goal is to teach how reality can be described using math.

Paul Lockhart, in his article, “A Mathematician’s Lament,” discusses how mathematics is the art of pure idea. (2) When we teach it procedurally, we strip from it the inherent creativity and beauty of it; but when we use it to describe our ideas, and engage students in describing progressively more complex ideas with mathematics, we find that everything is math. There’s no need to make it real because it already is. As teachers, our vision for our students should be to bring their mathematical understandings into the classroom. Where is the math in their lives? What forms of art exist in their cultures, and in the culture to which we’re introducing them, that can further their understanding of number and pattern and relationships? It is when we follow students’ mathematical understandings and extend them that we get the deep understanding of mathematical ideas that creates lifelong learners and problem-solvers.
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Finding connections from math to the real world seems backwards to me, somehow. It also negates the value of the other connections students could be getting out of the same material. If you use arrays to model multiplication, you've modeled multiplication and nothing else; but if you use something built as an array (like a quilt block) and look for the math in it, you can get multiplication, but also geometry, symmetry, co-ordinate grids, fractions, growing and shrinking patterns, division, principles of design, and measurement. You might easily miss them if you start with the math. From our perspective, that makes sense; we see the job as making math real. But kids are seeing it from the other side; they need to describe reality using math. Disparate topics taught with a variety of representations don't do that as well as a single representation that is deeply explored.
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On Friday, I showed the kids number lines that went from 0 to 1, divided into tenths.

Today, they had to use a number line from 3 to 5, and show 4.25 on it.

There were some kids who made a number line, divided the spaces between the whole numbers into quarters, and could justify 4.25 as the first quarter after four - perfectly acceptable. Others did it the way I'd been expecting, drawing a number line with ten divisions between each whole number and putting 4.25 halfway between 4.2 and 4.3. Also perfectly acceptable, of course.

The problem was with the kids who drew a number line with ten divisions and put 3 at one end, 4 in the middle, and 5 at the other end, leaving them with five divisions between each whole number. I should have gotten those kids together and asked questions to get them to clarify - maybe have them label the divisions so I could see if they actually labelled them correctly (0.2, 0.4, etc.) or some other way, incorrectly. If they labelled them incorrectly, I could then have gotten the kids to think through the concept of the divisions and compare them to the other number lines they've done, until they realized for themselves that it didn't look right. If they labelled them correctly and figured out that 4.25 had to be one quarter of the way between 4.2 and 4.4, then there would be no problem - that's another perfectly adequate solution.

Instead, I handled it in precisely the wrong way - I told them they were wrong and needed to think it through again, which didn't give them any way of doing that and didn't really establish what they understood or misunderstood of the question.

Um, oops. I'm chalking it up to post-long-weekend brain fuzz.

Thursday's lesson will be better. The problem: As a class, we're going to send a parcel to a school in Afghanistan with school supplies on it. Problem is that Canada Post charges a lot more to ship parcels over 50 kg, so we need to figure out how best to organize our school supplies to come as close to the weight limit as possible without going over. (There follows a list of simple school supplies, in sets of ten or twelve each, which we will weigh as a class using a scale borrowed from the science room, accurate to hundredths of a kilogram.) Given these weights, make a proposal for what we should include in our care package to the school in Afghanistan.

If they're really interested, there is a program like that through Plan International and another through the Canadian Military which I'm quite happy to look up - though without the weight limit, which is artificial to give them an adding problem. We'll get half a dozen problem-solving strategies out of it and get to discuss a world issues topic as well. It's win-win.

Got it!

Jul. 13th, 2009 09:49 am
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I just came up with the most fabulous project EVAR to combine social studies and math.  Get a load of this:

    Project:

     

    With your partners (groups of 3) you will research number systems in at least three ancient civilizations.  One of those three must be the Hindu-Arabic system; the others are up to you and will depend on what resources you can find to learn about the number systems.  I've included a list of links, available on our class's First Class page, to get you started.

  1. Write a paragraph explaining the main features of the three number systems you chose (each group member should write one paragraph; I will be looking for rough work during conferences.)
  2. Use number cubes or other manipulatives to model each number system.  (It may be easiest to pick one number and model it in all three number systems.)  Take pictures of your models.  Don't forget to include a group member in each photo so that we know who the pictures belong to when we download them off the cameras!
  3. Make a chart that explains the main features of each number system, and compares it to our base-ten system.  Some suggestions for headings on your chart:
  4. Use of zero

    Base number

    Strengths

    Limitations

  5. Write another piece in the format of your choice, explaining which parts of those ancient number systems are still in place today, giving examples for each.  (Use the cameras or images from the internet to back up your points!)  If any of those systems have been completely abandoned, explain why you think that happened.
  6. Each group will do a brief oral report on their findings for their classmates.  Every group member should be able to discuss any aspect of the project - even if somebody else worked on it - so be sure to teach each other what you learned!

I'll be making up a rubric and some lesson plans when I get back from my swim, and I've already got a thorough list of expectations that can be assessed using this project.  When I'm done I'll post the whole thing to Ontario_teacher.  Any suggestions appreciated; this is the first, very rough draft. 

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The question always comes up in debates about teaching mathematics: how come the Japanese do so well at math, when they stress rote memorization, if memorization is the wrong way to go about teaching math? (It's usually said in a tone of smug satisfaction, as though there's no possible comeback other than to admit that a constructivist approach to mathematics must clearly be wrong because the Japanese aren't doing it.)

The answer: a little from column A, and a little from column B.

The Japanese do indeed expect kids to memorize a lot of facts, but that doesn't mean they teach by modeling procedures and then assigning students practice questions to follow those procedures. No, they present a problem which the students do not yet have the skills to solve, then the kids work in small groups to figure out the problem. At the end of the lesson, the teacher summarizes what was learned, and students are assigned a small number of questions to practise on at home.

In other words, they use EXACTLY the method we're being told to use - with the exception that they do not allow calculator use.

The single biggest difference between North America and Japan is the number of school days - Japanese students have far more. The next biggest difference is in average expectations. When American moms are asked what mark is acceptable in math, they generally say a B or a C. Japanese moms expect an A.

So - high expectations, a constructivist and problem-solving approach to mathematics, high support in the form of parental help and extra tutoring - that's nine-tenths of the items we're expected to include in our mathematics programs.

Oh, and I should point out that the rumours about this kind of math instruction ignoring basic computation skills are false. We do drill math facts; we just make sure to drill them AFTER students have achieved comprehension, rather than before or instead of.

For future reference: http://www.gphillymath.org/ExempPaper/TeacherPresent/Mastrull/SMastrull.pdf

Doing Math

Jun. 28th, 2009 09:02 pm
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I haven't been posting all my discussion posts here, but here's another one for those who care.

Doing Math )
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This is not the first big assignment I've done, but the others have all been posted to the discussion forum and have resulted in immediate feedback there; I'm not going to repost everything, I promise. :)

The question )
My answers )
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My assignment: First Journal entry
Think About It
Take a few minutes and jot down your thoughts in your Journal. Do you think we teach children mathematics or do you think they learn mathematics? What images do you think of when you think of teachers teaching? What images do you see when children are learning? Is it possible that learning can happen on its own or from a peer, with the minimum involvement of a teacher?

Teachers play an important role in the process we call learning. They become the "guide on the side" not the "sage on the stage." Showing and telling are not teaching. Rather, we hope that children can be actively engaged in learning, questioning, analyzing, predicting and constructing knowledge from meaningful contexts and real-world experiences.

My answer:
I don't think this should be an either/or question; that is, I don't believe there is a great divide between teaching and learning. They are two parts of the same whole. Facilitating learning is called teaching, and it can take many forms. When I think of teaching, I generally picture direct instruction, the teacher modelling and then the students practising with guidance. Then I think about the activities I have students do that look nothing like that - jigsaw activities, where my whole role is to write questions on chart paper and let groups come to their own answers through exploration; whole-class discussions, where my role is to moderate who speaks and when, and possibly ask leading questions if the discussion stalls; critical literacy and exploration activities, where I provide materials and guide students to see them in light of certain key questions. I'm not doing much modelling or direct instruction in any of those situations - and together, they make up well over half of the classroom activities I plan.

I don't think the traditional teaching methods - imparting knowledge to students who lack it, in a top-down model - is really as traditional as recent research would have us believe. It seems to me that the method we're being told to consider, including the one hinted at in this journal entry with that very leading question, is just a variation on the age-old Socratic method, where we teach by asking questions that lead students towards more, deeper questions, and the knowledge they require to ask the next level of questions. Indeed, in the Socratic method, the students ask at least as many questions of each other as the teacher asks of them, resulting ideally in a depth of discussion that is totally lacking in so-called traditional educative methods. (I call it a variation because the Socratic method is mostly a thought exercise, with the students studiously avoiding getting their hands dirty, whereas the modern version of it requires kids to get up to their elbows in manipulatives of all kinds.) Most learning has been done like this since the beginning of time; it was the people we think of as traditional teachers who changed it, bringing Skinner's behavioural model of teaching into the forefront of pedagogy.

So, to bring this journal back around to its point, students learn mathematics in a variety of ways, including direct teaching, exploratory learning, peer interaction, and observation of the world around them. My job as a mathematics teacher is to highlight the connections between mathematical concepts, to ask the questions that will lead to students deepening their understanding of those concepts and their connections, and to facilitate their exploration of their world through the language of mathematics.
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I've discovered that Twitter is actually a good tool for summarizing notes, because it forces me to be very succinct; however it's not a good tool for storing those notes, even with tags attached, so I'm transferring them here starting now. Cut for those who don't care to read teaching summaries; also, cross-posted to Ontario_teacher. )



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