I want to focus on the function topics in these write-ups, for the book I am writing. So my notes detail these parts of clubs, and I also include descriptions from parents about other aspects, at the end. We still do Show and Tell and Apple Math activities.

Where club descriptions are

• All photos – thank you Annette and Ali! 9/29/2011 http://www.flickr.com/photos/26208371@N06/tags/naturalmathclub09292011/ and http://www.flickr.com/photos/26208371@N06/tags/naturalmathclub10062011/
• Maria Droujkova’s blog http://www.naturalmath.com/blog/functions-3-4
• Natural Math wiki http://naturalmath.wikispaces.com/MathClubs
• Natural Math email discussion group http://groups.google.com/group/naturalmath; send email to naturalmath+subscribe@googlegroups.com to subscribe
• Google Doc draft of a book about math clubs – let me know if you’d like to help, and I will add you

Introducing equivalent functions and graphs (9/29)

When kids guess one another’s function machines, they frequently get into arguments about correctness of guessing. “No, this machine does not add one! The correct rule is – first add two, then subtract one!” – was the example from this club. I wanted to focus on the idea. I put LEGO blocks on the table and used large blocks to build a graph of my function machine for inputs of 1, 2, 3, 4, which I also drew on the whiteboard. My machine added two, so the results were 3, 4, 5 and 6. I asked kids to build the machine that would produce towers just like mine, but use a different rule.

Exegesis: why graph?

Graphing allows to see the global behavior of a function, at a glance. Is the function linear? Is it increasing or decreasing? Is it periodic? You can also manipulate function, as a whole, using the graph representation, for example, double the function or invert it.

Goals and hopes for introducing graphing through equivalent functions

Equivalent functions may have wildly different formulas or word descriptions. This is exploited in “magic math tricks” – for example, it is not obvious to kids that “Take a number, double it, subtract five, triple the result, add fifteen, divide by six, subtract your original number” will always result in zero. I bet it was not obvious to you without computing as you read, either! But looking at graphs of two functions on the same coordinate plane, it’s always obvious if they are equivalent or not.

So, I thought the task of creating different function machines that give the same output will work well.

Equivalent functions and graphs: the unexpected

This activity totally did not work, so I am not even writing the “expected” part.  I think only a couple of kids understood what it was about – the boy who created x+2-1 for his guessing game, and his guessing partner! When I worked with kids one-on-one, in the past, we always moved into equivalent functions after it came up in a guessing game, as a disagreement. Without the playful frustration and arguing of whose guess was right, the task made no sense! Kids interpreted it as making graphs just like mine, which left them baffled because the task had no creativity to it whatsoever – very unlike our usual tasks.

I abandoned the attempt and asked kids, instead, to make a LEGO graph of any function machine they want – their old ones, or a new one, or my machine. That worked much better. Some interesting functions were made, such as the constant function (y=2)

LEGO graph of y=x:

Observation: Don’t answer questions kids did not ask. The issue of equivalent functions wasn’t problematic for most of the group and I could not make it interesting, from scratch.

Introducing linearity (10/6)

For that day’s task, I asked kids to make up function machines and then to create LEGO towers of outputs from inputting 1, 2, 3, 4 and 5 into their machines. After kids were done, I asked them if the tops of their towers made “straight stairs” or a line – we checked with rulers or markers.

Exegesis: why linearity?

Unfortunately, many people hold misconceptions that “everything in life is linear.” For example, math students think that the square of the sum gives you the same result as when you square numbers individually and then add them. Even more dangerously, people often believe that doubling the efforts will lead to doubling of outcomes – in policy, education or the environment. Without early childhood experiences, it’s very hard to appreciate how fast exponential functions grow – such as population explosions, or the growth of debt.  That’s why exploring linear and non-linear functions, early and often, is crucial.

Goals and hopes for the introduction of linearity

I hoped that by building LEGO graphs, kids and parent will see that all of them only produced linear functions. It would be nice if the question, “But does it have to be so?” became available to them!

Linearity: the expected

When I asked kids to create some function machines and their LEGO graphs, I fully expected all machines to be linear. This is what happened. I am working toward a future where this is not so, but at this point in history, young kids just don’t get exposed to nonlinear operations enough to start designing with them.

The task itself was engaging, partially because everybody loves LEGO, partially because they like to make function machines with friends. We assembled on the stairs to show off our functions.

Interestingly enough, young kids don’t have the misconception that simple arithmetic operations, such as “add two” or “multiply by three” are not really functions.  Michel Paul explains it well in a recent discussion of procedural vs. functional programming languages at Math Future:

However, trying to implement Scheme in a traditional Algebra setting was extremely difficult.  One issue is operator prefix notation.  In Scheme, if you want to express 2 + 3, you have to express it as (+ 2 3).  Both kids and math colleagues regard this as exotic and strange and a diversion from ‘real’ math.  However, it really isn’t.  In fact, the opposite is true – it points directly to the core of much mathematical misunderstanding on the parts of both students and teachers.  Operator prefix notation emphasizes the fact that even ‘ordinary’ arithmetic operators are functions!  Kids, and probably even most math teachers, regard arithmetic operations and functions as two separate topics – they would say we ‘use’ arithmetic operators in constructing functions, but no, even simple arithmetic operators are themselves functions.

However, kids don’t necessarily see addition as a binary function, that is, a function with two inputs. They see adding two, adding three, adding 100 as different unary (one input) functions! But when they start playing with wild big numbers – “Mine adds 100!” and “Mine adds 1000,000!” they are ready for two-input function machines.

Observation: For a kid, 100 or 1000 is a variable. It plays the same role X does for those who do formal algebra. When kids start to jump around the number line freely, they are ready for variables in this context.

Linearity: the unexpected

I asked kids to wave if their function machine was linear – either flat or “straight stairs” at the top. Everybody waved. I then asked kids to work with LEGO blocks, and create function machines that are not like this!

I have never done that before and therefore, wasn’t sure what would happen. By now, we introduced three math representations of functions: verbal descriptions, input-output number tables, and LEGO graphs, as well as “portraits” of functions as machines.

Observation and a Russian fairy tale: The portraits of function machines don’t represent math within functions, but they play a key role in children’s thinking! Portraits turn a function from a set of separate, different actions (add two to one, add two to five, add two to ten) into a single object, with some universal properties. For example, each function kids created had the property of linearity. The fact that very different functions may have the same property is a powerful idea, a high-level generalization not everybody appreciates in Algebra I and II. Yet our seven-year-olds do it.

As I am writing it, we finished the fifth club meeting and all kids still love to draw portraits of their function. A portrait takes time and it carries no mathematical information. Should we encourage kids to switch to formulas, graphs, and tables of values exclusively? I would not! They still need the strong roleplay reminders that a function is a single object.

“The Frog Princess” is a Russian fairy tale. To start families, three prince brothers shoot arrows, and marry women living where their arrows land. The youngest, Ivan, shoots so far his arrow lands in a swamp, by a frog. The czar sets some quests for the brides, and the frog princess proves to be creative and smart. Unknown to Ivan, she turns into Vasilisa the Wise (the fairy tale archetype somewhat like Sonya Kovalevsky) every night, to finish the quests. The final quest is a ball dance, where Ivan finally sees Vasilisa in her beautiful, smart and magic human image. He sneaks away, finds her frog skin and burns it to keep his bride in the human form forever. But this act invokes a powerful curse, transporting Vasilisa to an evil castle beyond the edge of the world (“beyond the three-of-nines domains, in the three-of-tensth kingdom”). Ivan has to quest for a long time and level up in intelligence and kindness before he can rescue his bride.

The moral of the tale: don’t destroy children’s roleplay prematurely, or their emergent formal math will disappear into faraway lands and it will take a lot of effort to rescue it!

http://youtu.be/LPSsT4lARAY

It was hard for kids to create nonlinear functions, even when they work with LEGO graphs directly. I would not even be able to pose the question of linearity without some hands-on or computer-based graphing representation. Some attempted to make constant functions – after all, they did not look like stairs. Some tried multiplication by larger numbers, like y=9x, to see if it won’t break the linearity. We celebrated all ideas, but we checked with a ruler and did find out if the result was linear or not.

However, we got a nice variety of nonlinear functions as well:

• Colson made a periodic function that was equal to one for odd numbers and two for even numbers.
• Ava made a piecewise linear function, with three different rules for numbers under five, over five and five
• Kaya made a piecewise linear function with infinitely many pieces: it added zero for the first three numbers, subtracted one for the next three, added zero for the next three and so on
• Moreghan made something I want to call a random function, and I think a couple of other kids tried this idea too – we will need to chat about it more
• Noah made an exponential function that started with 10 and doubled – 10, 20, 40… The formula is y=10*2^x for those who want to know. It’s interesting what complexity kids can reach with appropriate tools.

Parent stories

~*~*~*~*~*
Ali:
Very good day yesterday!  It impresses me that no matter how the group changes (in terms of size, new kids, etc.) it just keeps flowing seamlessly.  Kids are really able (and comfortable) to work at whatever level of complexity (or simplicity) that they are comfortable and / or capable of.   Kids who have not been at every meeting blend right in with the ones who have.  I just love the obvious feeling of confidence, comfort and ease that everyone has.  I think that has a lot to do with your style of allowing the kids to freely share their thoughts, ideas, questions & feelings; as well as rarely saying that anything is ‘wrong’… rather things are ‘interesting’, a ‘good thought’ or a ‘good try’.   That is a major confidence booster & I think plays a big role in why the groups is so lively.  :0)  No one, even shy kids, seem afraid to speak up.  Which sadly, I think is the case often times in a traditional school setting.  I just love the feel of the group.

I’ve never heard the term ‘piecewise linear’.  Gonna have to look that up. :0)  I think Kaya’s idea was boggling her own brain. :0)  She explained it to me but had a hard time remembering how it worked & correlating it consistently to the non-even angle of lego solutions.  I helped her write it out the beginning number; the function (which varied); and an arrow to the lego solution.   She LOVES to do math in her head… but she is a visual child.   She closes her eyes, puts her head in her hand and will sit silently doing a division or multiplication problem mentally for 5 minutes.   I often times have to remind her that it’s ok, and beneficial to use visuals – objects, written numbers etc. to simplify, clarify and ease all that tiring brain work! ;0)

Thanks so much as always.

~*~*~*~*~*
Alexia:

The poem Maxime brought for Show and Tell.

Monday, at the end, Girl Scouts.
Bees fly flower to flower.
Tuesday, Tae Kwan Do.
Fine for 4:15.
Wednesday, violin.
The sounds of kittens purring and the wind blowing.
Thursday, a game that’s not lame:
Tae Kwan Do and Math Club.
Friday, the sounds of the sweet steps of my dear brother.
Saturday, relax. No school.
Maybe go to the pool.
Sunday, church. And everything starts again.
–Maxime

~*~*~*~*~*
Laura:

We have been participating in Math Club since the beginning. In fact, Colson was a Math Club of one a few times. The group settled in somewhere around six to eight kids over the next couple of years with some participants constant and others rotating out. This year there are at least 14 kids. Holy Moly! Talk about a new Math Club dynamic! I have to admit that I had some trepidation at first – Club was very loud, very busy and a little chaotic, but the group is finding its stride now. My husband has taught me the stages of group development: forming, storming, norming and performing, and by week four we were coming out of the storming phase and entering the norming phase. The 14 kids present were attentive, the craziness level was down and the focus was good. Many kudos to Maria for her group management and leadership!

Now on to what we did.

1. Show and Tell – The keepers of the math terminology board were identified. Keepers post on the board the new math ideas that the group identifies during show and tell.

• Maria showed a folded up inflatable ball. Exercise balls are a big deal at Math Club because up until meeting four there was only one ball. Now there were five and another to be blown up. She described the folded up ball as a cuboid that is a ball, kind of like origami.
• Noah and Colson did an art show and tell. First Colson posed and Noah drew a gesture drawing of Colson in under 30 seconds. Then they reversed roles. Maria called the drawings “space filling curves.”
• Ava brought a 12-sided die and the kids talked about the sides and angles.
• Toriah and Alden brought a rubber band weaving board. The kids discussed the shapes/patterns and the meaning of parallel.
• Jessica brought a stuffed bunny she has had for eight years. The kids discussed the parallel and perpendicular stripes. She also brought a phone and they discussed the numbers on the phone.
• Marianne brought a Magic 8 ball. Discussion was of the diamond-shaped dice and the randomness of the message.
• Kaya brought a shell chime that spiraled and represented infinity. Maria noted that the shell threads of the chime get longer as the pattern swirls.
• Colson built a Lego pyramid – a square that gets bigger and bigger.

2. Function Machines

The kids worked in pairs to make function machines, with some kids working separately as they desired. Maria gave the example as:

1 2 3 4 5

2 4 6 ? ?

What is this machine doing? Noah responded that it multiplies by 2.

Maria directed the kids to make their own machines and model the outputs with Lego towers. Maria explains that these are linear functions such that kids can take a ruler and make a straight line to connect the towers.

Photo opportunity with the Lego tower outputs. All the kids sit on the sides of the stairs and place their Lego towers in the middle.

3. Apple Math – Prisms

• How many cuts to make a diamond/rhombus prism out of an apple? Guesses were 4, 5, 6, 7, 8, 9 and 100. Answer: 6
• How many cuts to make a trapezoid prism? Guesses were 5, 6, and 4. Answer: 6

Picture: modeling a prism with hands

4. Make a Non-linear Function Machine

• Make a machine where the outputs do not form a straight line.
• Colson develops a periodic machine of 3, 4, 3, 4.
• Noah develops and exponential machine of 10, 20, 40.
• Other machines were periodic.

5. Rachel, Ava’s mom, was periodically observing kid attention spans during this lesson because she was curious. I believe that she found that their attention was fairly high. She may want to chime in with her findings.

Thank you, Maria, for a highly productive and creative meeting!