Big Ideas

This afternoon I continued with the math course. The topic: Some big ideas -from five people for whom math is central in their lives.

This is the mind map – my interpretation of the big ideas offered by each (if you click on the mind map, you’ll be able to read it!):

 

Put this all together and you have “An Inquiry Relationship with Math“.

The question:

Now you have watched all 5 I would like you to choose one of them and write about a cool lesson idea (teachers) or/discussion topic (parents) that their words and ideas makes you think of. One of them, at least, I am sure inspired you to do something interesting in a class, or in a conversation with your child. I will give you space for that now to write a paragraph about that.

Big Idea: We’re using math all the time without realizing it!

I’ll start with myself:

• Today at the post office the postage on a small package came to $1.54 – I handed the clerk a toony (a Canadian $2 coin) and a nickel and said you owe me $.50 (we no longer have 1¢ coins in Canada) – because I wanted 2 quarters in change;
• This morning I was finishing up a swimsuit I’m making and had to measure out the elastic needed to finish around the neckline and the leg openings;
• I loaded the dishwasher – I group the plates by size, the cutlery by kind, the bowls get nested by size;
• I filled up the gas tank in my car – recorded amount of gas and calculated the number of kms since the last fillup;
• I’m a big tennis fan – they’re playing in Washington this week – I keep a close watch on the scores (tennis scores are unusual: love, fifteen, thirty, forty, deuce –> game; six games = a set; best of three sets = win).
• I set the timer on my video recorder to capture something I want to watch later….

Lots of “math” in my daily life.

For years, I’ve engaged teachers in an activity where they list all the reading/writing they’ve done in the past day or two, what’s the purpose, who’s the audience, and compare that to the reading/writing going on in their classroom; what’s obvious is the diversity of purpose and audience in the reading/writing going on in people’s everyday lives but reading/writing in school is largely limited to reading textbooks for the teacher as examiner.

Having people explore the diverse ways “math” comes into their daily lives would be interesting; doing that collaboratively would generate very rich lists. Taking that one step further, have them think about the “math” in their classrooms would show the same limited purposes/audience we see for literacy, I’m sure, and spark discussion about how can we change that.

I’d want teachers to make it possible for their students to develop an inquiry relationship with math. A relationship that lets students feel comfortable working at something they don’t know how to solve yet, that allows them to begin by inventing their own language for talking about the new math situation, where they know they’ll have some time to work at the problems, and a couple of other people to explore with; they’ll be able to experiment; where they can feel comfortable exploring their intuitive understanding, and they’re not limited by right answers, fixed procedures.

‘Number Sense’?

Back to EDUC115N.

What precisely is “number sense”? It’s not an easily answerable question according to Frank Smith (The Glass Wall; Teachers College Press, 2002). He points out

“numbers commonly don’t make any sense at all outside mathematics, and it is unclear what a ‘number sense’ might be within mathematics. A person may achieve a general understanding of how numbers behave, or relate to each other, within mathematics, but that is hardly a ‘sense’ in the way in which we can have a sense of smell or touch. Numbers don’t derive their meaning from anything in the physical world, but from something in our mind, and from the world of mathematics that minds have created( p.35).” He goes on to argue “numbers are relationships. Relationships with what? With other numbers. Isn’t that circular and completely self-referential? Yes – that characterizes numbers exactly. Numbers don’t get their meaning from anything except each other (p. 37).”

So where does this leave us with regard to learning mathematics? It is likely the first intuition a child needs to have is that each number is a magnitude within a set of ordered magnitudes – the rest is all relationships: one is one, two is one more than one, three is one more than two. They also need to have the insight that “one”, “two”, “three”, “four” are the arbitrary names we put on these “concepts.” The next big leap is that four is not only one more than three, it is two more than two, it is also two times two, or two less than six, or half of eight, or the square root of sixteen, or any other relationship we can think of for four. At this point, we’re beyond language, discussing numbers, not words. “The moment we start building numbers on numbers and examining their mutual relationships, we have left natural language and entered the world of mathematics (Smith, p. 37).”

Montessori is likely the first educator to use manipulatable materials to help children develop some of the intuitions about numbers as relationships (although she didn’t ever discuss it that way). Cuisinaire popularized the use of rods to discover relationships among numbers (again, he didn’t discuss number as relationship, although the activities using the rods allow intuitions about the many relationships to be developed). What’s clear to me is that the abstractions underlying mathematics are less likely to come from rote learning of “rules” and “procedures” dealing with numbers, than from the actual manipulation of concrete materials. Although some children make the leap without concrete supports, using manipulatives in a collaborative learning context (which opens children to discussing what they’re understanding) can make it possible for more children to reach the insights about number being an abstract, mental construct.

In this section of the course I was introduced to another vehicle for building some of the intuitions necessary for understanding math that I hadn’t encountered before: Dot Cards and Math Talks.

A Dot Card is a card with some number of “dots” on it – the point is to find out how students/people see them and the different ways of establishing the total number.

I saw the total number of dots as a series of vertical 2s – 2,4,6,8,10. But the dots could also seen horizontally as 3 + 2 + 3 + 2, or even on the diagonal as 2 + 4 + 3+ 1, or in clusters 5 + 5. The conversation the teachers had about what the card represented was a Math Talk. The point being to see number as a relational system, not as something fixed.

Another math talk arose from the problem

How can you solve 25 x 29 without paper and pencil?

Again, several solutions – using the associate property of a number changing 29 into 30-1 and multiplying:  (3 x 25 = 75; 75 x 10 = 750; 750 – 25 = 725).

But there are several other ways of getting there:

It could be seen as: 25=10+10+5; 29 x 10 = 290; 290 x 2 = 580; 5 x 29 = 290 ÷ 2 = 145; 580 + 145 = 725;
or it could be seen as:  25 x 2 = 50; 50 x 10 = 500; 25 x 9 = 25 x (4 + 4 + 1) = 100 + 100 + 25 = 225; 500 + 225 = 725.
As you can see, there are multiple ways of thinking about the problem.

We were asked which important ‘teacher moves’ did we see the teacher enact in the above  number talk? I made note of the following:

1. She presented a question/problem;
2.  She asked for answers followed by explanations/clarifications from several different people trying to get as many different ways of looking at the problem as she could;
3. She interjected some interpretation as she pushed for clarification;
4. She wrote a mathematical interpretation of the explanations/clarifications for each explanation as it was being given making sure she understood the explanation;
5. She offered a diagrammatic interpretation of one of the solutions and asked for an algebraic expression that described the diagram;
   
6. She asked them to diagram another of the solutions;
7. She asked for an algebraic expression to satisfy that second diagram.

Mental math becomes easy when you can be flexible about how to see the number relationships.

In a further video some third grade boys who were learning math in a classroom filled with Math Talk were asked about how they felt about math.  Their responses showed the boys have definitely begun developing intuitions about the openness of “number” – they understand problems can be solved in different ways, they have a couple of strategies they can name: decomposing numbers, using friendly numbers… and they mentioned that talking about math in connection with the problems gives them access to how others think and allows them understand there is more than one way to think about each problem.

Now add in materials you can manipulate to this mix and it would be very interesting to see how rich the students’ math understanding could become.

Working with a Growth Mindset

I was just going through some of the other students’ responses to the open ended task – came across this one:

64 X 25 =

How many ways can you work this out? Of course the underlying assumption is that the students understand place value and are comfortable converting units to tens, tens to hundreds, hundreds to thousands.

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My Montessori mind kicked into gear here – I don’t have any Montessori number bead sets (too bad, really because if a student doesn’t understand place value, the bead bars and squares represent the actual quantity, no abstraction, so when you lay out the problem it’s a direct representation of the problem). What I do have is Cuisinaire rods. So I go down to my basement to locate them. Come back upstairs and begin by laying out the problem. I set up a multiplication board with unit/tens along the right side and units/tens along the bottom. I lay out my first number 64, and show it as a 6 and a 4 placing the 4 beside the units and the 6 beside the tens. Next I lay the 25 along the bottom, the 5 beneath the units, and the 2 beneath the tens. (In a real Montessori board, the units block would be completely green, the two tens block would be completely blue, the hundreds block would be completely red, no confusing what the blocks represent.) (Oh, and the board actually has a layout up to ten thousand on each side so that you have a block for the millions – it’s a big board – the units of thousands are green, the tens of thousands are blue, the hundreds of thousands are red, and the units of millions is green – again strengthening the place value distinctions – units, tens, hundreds, units of thousands, tens of thousands….)

Now I lay out the array, I put 5 of the 4 bars in the units block – value = 20 units; I put 5 of the 6 bars in the upper 10’s block – value = 30 tens; I now put two of the 4 bars in the lower tens block – value = 8 tens, and two of the 6 bars in the hundreds block – value = 12 hundreds. For the purposes of this explanation I provide the value of the contents of the blocks: I have 20 units + 30 tens + 8 tens + 12 hundreds.

Next, I begin amalgamating the blocks: I’m only allowed from 1-9 units in the units block, but I have 20 so I convert them to 10s – leaving no units but adding a 2 bar (the small red one) to the tens pile (remember I have two 4 bars = 8 + five 6 bars = 30 ) already in the top half of my second sheet (I’ve drawn this second sheet so I could lay out the way I amalgamate the blocks – but were I using the Montessori multiplication chart, I’d do it directly on the board.) So I now have in the 10’s block – 30 + 8 + 2 = 40. Again, I’m only allowed from 1-9 tens in the 10s column so I convert the tens to hundreds = 4 which I put in the bottom hundreds column; leaving 0 in the bottom tens column. I take the 12 hundreds at the top and convert that to a 2 bar which I put in the bottom hundreds column and place a 1 cube in the thousands column. (As I make the swaps, I’d actually remove the pieces from the board leaving only pieces in the lower set of columns – I’ve left them on the sheet so you can see them – that may be confusing.)

So I’ve actually done the multiplication.

As you can see, after counting out the pieces, amalgamating them, I’m left with a single thousand and a 4 bar and a 2 bar in the hundreds, and nothing in the tens or units. My solution is 1600 – one thousand, six hundreds, no tens, no units. That’s one way of figuring this out. What I may do later is show how this translates into the multiplication of a binomial by a binomial – the Montessori materials for doing that are really elegant!

Teaching with a Growth Mindset

We watched a video of a teacher – she set a problem for her students to explore: what is 1 divided by 2/3? (Division by fractions is complicated to get you mind wrapped around.)

Cathy did a bunch of things to support her students’ learning.

• She encouraged students to draw, to visualize, to show their ideas in different ways, to value and explain different results.
• She has set up a culture for collaboration and shared work.
• She made it clear from the outset that she didn’t care about “the rule” right now.
• The students came up with two answers – she asked a series of questions:
  • “Why does each answer make sense?”
  • “Can you challenge this explanation?”
  • “Who has another way of explaining this?” The more ways we can think of this the better.
  • “What do we call the method used in this explanation?”
  • “Is this a problem that has more than one answer?”
• “The big danger in math is following the rules – we need to understand the ideas and relationships.”
• She affirms the students’ responses: “I like your reasoning about that.”
• And finality she introduces the notion of the Traditional Algorithm, at the same time making clear there is more than one way to think about this kind of problem.

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Here is a second example:

Faun Nguyen took a short, closed, one right answer textbook problem and turned it into a complex, open, many answer problem situation that challenged her students. It changed from being an exercise focused on performance and performing, getting right answers, to one that focused on learning and growth and thinking collaboratively – she made it possible for her students to own the task.

She invited the students to set up a problem – she offered some elements – she cast the problem in language they understood – a point (a golf hole), a blob (a water hazard), and another point on the other side of the blob (a golf ball) – at some location where a single direct shot won’t work (this is really more a snooker problem than a real golf situation but that didn’t seem to be problematic for the students). The challenge is to get the ball into the hole, but you can’t putt directly – you need to make a single “bank” shot.

The students made drawings fulfilling the conditions (most of them it seems were able to set up the problem so a single shot would actually work).

Now comes the real open-endedness of the situation:
She said: You need to convince me and your classmates that the ball hitting the edge at the spot you’ve chosen will bounce out and travel straight into the hole.

The students had rulers and protractors at their disposal. Some of the students played around with rules and constructed similar triangles that way, others used the protractors to find the bounce point (so that the adjacent interior angles were equal), some used both tools.

The students’ explanations were well thought out and illustrated several different ways to solve the problem all of which worked.

(Take a look at Faun Nguyen’s blog)

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A Third Situation:

My original thought process:

We have a triangle growing – the total number of cubes is the progression of the squares of the natural numbers – triangle 1 = 4 (1+1=2 squared), #2 = 9 (2+1=3 squared), #3=16 (3+1=4 squared), #4 = 25 (4+1=5 squared), #9 = 100 (9+1=10 squared). In other words, case n = (n+1) squared Case 100 = would have 10201 cubes (101 squared)

After watching the class video further:

Growth Mindset Task Framework

1. Openness
2. Different ways of seeing
3. Multiple entry points
4. Multiple paths / strategies
5. Clear learning goals and opportunities for feedback.

Look at the task framework and write about the dimensions that the two versions of the task (“How do you see the shapes growing?” and “How may cubes are in the 100th case?”) differ on.

How do you see the shapes growing?
1. Openness – got to figure out that the “triangle” grows either by adding bottom rows, or by adding one to each column and adding outside columns. (I hadn’t considered the columns idea at all; I’d just seen the problem as adding rows)

2. Different ways of seeing – two very different ways of seeing the problem – either as growth by adding bottom rows, or by adding columns of increasing height

3. Multiple entry points – can start by actually counting the cubes, and seeing if you can figure out the next in the series and work from there; could try to abstract it from the series of three…

4. Multiple paths / strategies – the adding rows is the simpler conceptually since it goes more directly to the (n+1) idea for the rows and # of cubes (as the square of (n+1); whereas the columns has the complication of the columns being abstracted to (2n+1) in addition to the sum of the cubes being (n+1) squared. I imagine students would come up with other ways of seeing this – I’m bound by recognizing the series being the squares of the natural numbers.

5. Clear learning goals and opportunities for feedback – here the learning goals involve seeing problems in multiple ways, finding more than one way to start on a problem, getting to answers more than a single way, before finally getting to the algebraic expression of the problem.

How many cubes are there in the 100th case?
1. Openness – there is one right answer
2. Different ways of seeing – there is only one way of seeing the problem implied in the way the problem is presented
3. Multiple entry points – there is a single entry point for most people here – through the numbers, not the conceptualizing of the problem
4. Multiple paths / strategies – a single path – through the squares of the numbers
5. Clear learning goals and opportunities for feedback – learning goal is singular – to know how to set up an algebraic expression for the nth case.

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Overall, I found this stuff interesting to think about.

An Open Learning Task

This comes directly from the course.

Question:
Find an example of your own – a task that starts off as closed and short and encouraging fixed-mindset thinking to one that becomes a growth-mindset task.

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Here’s a simple example of an open activity to start – to discover the sum of the interior angles of any triangle = 180°
1. Ask the students to construct several different triangles (scalene, equilateral, isosceles), in different sizes (using a ruler and pencil)
2. Ask them to number each point and colour the points of each triangle (the points of each different triangle probably should be in a different colour so they don’t get them confused!)

3.Cut out a triangle, then cut it into three pieces (each piece now has a coloured point from the original triangle)
4. How many different ways can you fit the coloured points together so they are along side each other?

5. Do this with each of your triangles, and stick the results on a piece of paper.
6. Describe what you observe.

The goal of this part of the activity is to discover the sum of the interior angles of any triangle is 180° (forms a straight line or straight angle) – feedback is directed at helping the students in aligning the coloured-in points of the triangles in as many ways as they can.

Now do the same thing with shapes that have more than three sides – try 4, 5, 6, 7, 8, …
Describe what you observe.

a quadrilateral:

a pentagon:

a hexagon:

When they’re done, the question is can we summarize this in some way?
My hunch is students will come up with more than one way to bring the data together but in the end, what I’d want them to understand is the number of straight angles formed from the interior angles of  any polygon is two less than the number of sides (or the sum of the interior angles is the number of sides minus 2 times 180°)

To Summarize:

1. Openness – the activity is open because the students are working from their own triangles/polygons – they’re all going to be different in size and shape – hence the possibility of the generalization
2. Different ways of seeing – the triangles/polygons are all different and it isn’t immediately obvious how to align the coloured-in points together so they are touching – but playing around with them makes it possible to discover that the sum of the interior angles of a triangle = 180°, of a quadrilateral = 360°…
3. Multiple entry points – while I suggested starting with triangles, it’s possible for the students to start with any of the polygons – a triangles makes it possible to bring up the nomenclature of “straight angle”
4. Multiple paths / strategies – the important thing here is the discussion the students have as they try to align the points of their triangles/polygons – trying to figure out how to align the so the coloured points all touch one another
5. Clear learning goals and opportunities for feedback – The goal of the activity is to discover the sum of the interior angles of any triangle is 180° (forms a straight line) – and that as you increase the number of sides of the polygon, the number of straight angles is two less than the number of sides.

Not-Learning

http://www.lupinworks.com/snippets/notLearning.php

[ Journal Entry ]

Herbert Kohl’s notion of “not-learning” is a very powerful one. It shifts some of the responsibility for student learning to us—students may make the decision to “not-learn” but we have the option of changing the learning situation so that they might choose to learn instead.

I can think of several not-learning decisions I have made in my own life. I chose, for example, to not-learn to use the phone system at UM. It did stuff like transfer calls, allow group calls, etc. But I just couldn’t force myself to bother with that stuff.

I also chose to not learn the Collective Agreement. I figured there were lawyers on staff to help me out when problem situations arose and they would alert me to contractual necessities. So I just learned, instead, to check in with Alan (the university lawyer) before making a decision I thought might get me into trouble with the collective agreement.

We make all kinds of ‘not-learning’ decisions in our lives. So do kids. I interpret your (my students’) hesitation to push full steam ahead with fast reading as having some not-learning roots. Is that a fair interpretation?

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I found myself exploring the “Backstreet Boys” recently, because I’m working with a thirteen year-old who loves them. I bought her a book about the guys and last evening didn’t I run into a program on Much Music which I actually watched! They’re not bad, particularly when they’re singing a cappella. I learned a bit more about how they got together and got their break, what they’re working on at the moment—and I choose to learn more because it allows me to engage in conversation with Maggie (I never listen to Rock, R&B—I’m a classical music person!).

The reason I’m working with Maggie is that she’s on the edge of being a not-learner. She’s just finished seventh grade. She’s quite bright, but definitely turned off—a classic not-learner—she procrastinates, avoids doing assignments, does them hurriedly at the last minute or late—she’s not far from doing nothing at all. What I’m trying to do is show her that she doesn’t have to avoid learning—that she’s capable of making sense and finding ways of negotiating assignments that she finds satisfying.

Kohl’s description of not-learning is one of the most powerful notions I’ve come across recently. What I find fascinating is that we’re all not learners in some situation or other. You might find it useful to think about those situations where you choose to be a not-learner. because it’s a sure bet that every student in your class will at some time or other decide to engage in not-learning.

Establishing A Mistakes Friendly Culture

Speed Is The Enemy!

• Value mistakes
• Talk about brain growth and why mistakes are important
• Give work that encourages mistakes – the key is to offer experiences that are at the edge of students’ understanding
• Don’t share your anxiety about math, your lack of confidence, your bad experiences (that just increases stress/anxiety and keeps the brain from working – or – it makes it OK to avoid math learning altogether)
• Grade differently or not at all
• Dissociate math learning from speed

Do I agree with all of the above – not the exact wording precisely:

“work” – that will be interpreted by most parents and teachers as drill – doing math exercises/problems from a textbook or worksheet; whereas “work” ought to be open-ended exploring which offers unexpected surprises and individual learners make different discoveries which they then share.

“talk about brain growth and why mistakes are important” – “talk” – that’s the problematic word for me – more important is having success at math activities – that success demonstrates to the learner that it’s possible to be successful – talking about why mistakes are necessary for learning doesn’t help a lot; being successful makes a huge difference.

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In 1981 Frank Smith published an article in Language Arts, 58 (1): 103-112) (reprinted in Essays Into LIteracy 1983, 95-106). I still think this is an extraordinary piece of writing that sums up the dilemma of teaching.

Smith discusses “the learning brain”:

Learning is not an occasional event, to be stimulated, provoked, or reinforced. Learning is what the brain does naturally, continually. It is only in artificially contrived experimental or instructional situations that the brain usually finds itself not learning – and tolerates not learning. (And even, then, learning is probably taking place, that such experimental or instructional situations are artificial, unproductive, and boring.)

This is the time bomb in the classroom – the fact that children’s brains are learning all the time. They may not learn what we want them to learn. They may not learn what we think we are teaching them. But they learn, if only that what we try to teach them is boring or that they are unlikely to learn what we think we are teaching. Learning is the brain continually updating is understanding of the world; we cannot stop the brain from doing this. The hazard of so much instruction is not that children do not learn, but what they learn.

The important question for researchers is not how we learn, in the sense of the underlying brain processes, but why a brain which normally learns so effortlessly, so continuously, should sometimes be defeated by tasks that are intrinsically not exceptionally difficult. 

Smith goes on to discuss three big ideas: Demonstrations, Engagement and Sensitivity.

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I have written about this elsewhere.

Demonstrations

[ Journal Entry ]

I’m going to start with “demonstrations”—let me add my 2 cents to Smith’s discussion. What I understand from his argument is that we teachers have no control over the demonstrations we offer. We do what we can to create a learning situation; HOWEVER, what we set up are POTENTIAL demonstrations. What makes something an actual demonstration is a learner’s engagement. In other words, learners are in control of demonstrations—they determine what aspects of a situation they will engage with. What decides whether a learner engages or not is what Smith calls ‘sensitivity’—the learner’s expectation that it’s worth engaging in an experience. And because learners determine what constitutes a demonstration, it means that what I think the situation might be demonstrating may have nothing to do with what catches any learner’s attention; different learners in the same situation are likely engaging in different demonstrations at the same time. Another thing I’ve come to believe is that there is no such thing as NO demonstration—there can be no absence of demonstrations because everything is a potential demonstration–that’s where Smith’s time bomb comes into play—the brain is learning all the time, even if what it’s learning is “not-learning”. (Herbert Kohl writes brilliantly on “not-Learning)

So, supposing that… if we created learning situations so they were like the best kindergarten classroom—helping learners feel confident that nothing, or very little, is beyond their capabilities, to have a reason to shoot at being able to do anything that seems worthwhile, able to explore freely and pursue tenaciously, explore with an open mind, to feel free to take intellectual and social risks—to ask outrageous questions, to make wild and improbable connections, to take on tasks that might require a long time to complete, and even to abandon some tasks unfinished—what might learning be like for learners of any age? Meier’s experience at CPESS showed that enterprises, a collaborative learning community is feasible.

It’s the engaging in tasks that might require a long time to complete—Conroy’s contention that education doesn’t end until life ends, because you never know when you’re going to understand something you didn’t know before—that’s compelling. Another point he makes which I found very powerful is that much of understanding is retrospective—as he puts it

“I remembered what I hadn’t understood…until my life caught up with the information and the lightbulb went on.”

What that means is that learning isn’t uni-directional—learning occurs in hindsight—when something happens that permits a “wonderful idea” and we see a connection that was impossible before. He also makes the point that understanding doesn’t mean resolution—many situations are very complex and although we understand some of the variables at play, we may not be able to resolve what’s problematic in them.

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Smith’s article on “demonstrations” is an important one. I keep highlighting several quotes:

Learning is not an occasional event, to be stimulated, provoked, or reinforced. Learning is what the brain does naturally, continually.

This is the time bomb in every classroom—the fact that children’s brains are learning all the time. They may not learn what we want them to learn. They may not learn what we think we are teaching them. But they learn, if only that what we try to teach them is boring or that they are unlikely to learn what we think we are teaching them….

The reason why collaboration is crucial is that we learn by seeing how things are done—demonstrations. That means we need to be in the company of other learners so that we have maximum exposure to the way people go about doing things we might be interested in doing ourselves. It’s that learning brain that’s learning all the time—coming up with those wonderful ideas.

Paul Lockhart

I tried the problem – and yes, you get parallelograms!

Paul Lockhart, gained a certain degree of fame for his “Mathematician’s Lament,” a stinging critique of the state of mathematics education. He is full of passion for the creative art of mathematics. He’s not so worried about how broken mathematics education deprives students of the development of their other talents; he’s worried about how it deprives them of math. Lockhart believes the solution to the current situation is to adopt an approach to mathematical education that frees students from memorization and exposes them to the wonders of what he calls “mathematical reality.”

(Adapted From: http://harvardpress.typepad.com/hup_publicity/2012/07/is-mathematics-not-beautiful.html)

Barriers To Learning Math

1. In the current testing culture success is doing well on tests, getting the right answers; just exploring isn’t valued (by teachers, by students, by parents)

2. Speed: Laurent Shwartz worried that because he wasn’t “fast” he might not be intelligent. His schooling conveyed the not so covert message that speed=intelligent/smart – so if he was slow at thinking about something he didn’t already know, he mustn’t be smart.

3. Persistence isn’t valued – where what matters isn’t how fast you get to understanding but how thoroughly you make sense of something complicated. But that’s a hard sell in schools because they’re geared for “one size fits all” – and don’t make room for people need time to figure things out for themselves.

Outside The Box

• to explore seemingly wild possibilities
• to persist with difficulties
• to be willing to be misunderstood

That describes me as a learner! Most of what I bother learning these days has to do with my interests – web development, quilting, knitting, cooking, gardening.

I’m constantly taking on new projects each of which pushes me in new directions, requiring me to learn something I don’t yet know. I drive my friends crazy because I can’t let a “problem” go – keep at it until I have figured out what I need in order to move ahead. Most of them will just give up, particularly with the computer stuff, whereas I’m in the discussion groups trying to find someone who can help me out, or googling for a way to accomplish something, until I’ve sorted out what I need to know.

What does this mean for learners of math?

First of all, here is a link to the Common Core: http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf. It’s going to be very interesting to see just what kind of learning experiences get written and what kind of standardized testing will go along with these “standards”.

Most important will what kind of investment will States make in teacher development – want my prediction? Give it five years, ten years – not much will change. There are too many teachers who are invested in teaching as they’ve always done, professional development is usually so poorly done, that while the textbooks will change to reflect these standards, they’ll be interpreted as current curriculum objectives.

Just look at the Grade K Overview:

• Counting and Cardinality
  • Know number names and the count sequence.
  • Count to tell the number of objects.
  • Compare numbers. Operations and Algebraic Thinking
• Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
• Number and Operations in Base Ten – Work with numbers 11–19 to gain foundations for place value.
• Measurement and Data
  • Describe and compare measurable attributes.
  • Classify objects and count the number of objects in categories.
• Geometry
  • Identify and describe shapes
  • Analyze, compare, create, and compose shapes.

I can imagine lots of ways of setting up open-ended experiences that would make this challenging for children. But I can also imagine lots of prescriptive teaching going on.