I think I figured out why spreadsheets are fun for kids... (Computer-Based Math)

[Maria Droujkova]

Have you ever noticed young kids mesmerized by spreadsheets? Even kids who don't normally love number patterns?

This has been so puzzling to me! Are we all accountants at heart, when we are young?

For the last year or so, I have been working on a taxonomy of game mechanics, indexed by math topics and cross-referenced by pedagogical metaphors. There are a number of areas no game developer wants to touch with a ten-foot line of code, it seems. Most of these areas are "DIY math" - where users define their own shapes, formulas, patterns, in ways specific and peculiar to each math topics. 

There is http://www.crayonphysics.com/ - where is crayon math?

Grid reasoning is the math area that corresponds to spreadsheets. What is the basic DIY step for having a grid emerge? It is making the value in the next cell out of the value in the previous cell. Say, adding one more. One fish, two fish, three fish; one apple, two apples, three oranges (you will see this sort of math behavior all the time, because beginners, especially young kids, don't feel like being consistent in their patterns).

When grown-ups think of a multiplication table, it's all about multiplying row and column labels. But kids usually don't reason in terms of labels for whole rows and columns - not at first. Kids just "jump" from one cell to the next. And THAT is how spreadsheets work. The formulas users define in a spreadsheet are recursive: A2=A1+1, B2=B1+2, etc. A user defines a step from one cell to the next, drags the formula along a row or a column, and SEES WHAT HAPPENS. This is a perfect match for what little kids naturally do when they play with picture or number grids on paper or with manipulatives. They define a rule for going from one cell to the next, and see what happens.

And that's why kids love spreadsheets. 

Would it not be nice to have spreadsheet-like soft for toddlers and young kids, based on finger-painting on touchscreens? Does it exist somewhere I have not looked?

As I said, I can't find many such "DIY math" tools, specific to particular topics and aimed at young beginners. There are large platforms (GeoGebra, Scratch) but they are not topic-specific. Here are a couple of examples for fractals,of all things:

http://www.geom-e-twee.com/

http://illuminations.nctm.org/ActivityDetail.aspx?ID=17

Colleen King's new equation builder qualifies:

http://itunes.apple.com/us/app/shuttle-mission-math/id498617241?ls=1&mt=8

This is an popular example, which I can't match to a math topic yet:

http://armorgames.com/play/2205/light-bot

Both Colleen's Shuttle Mission and Light Bot are games - in that they have closed-ended goals (win conditions) attached to free (sandbox) maker/DIY actions. Games based on making something are even harder to find, and to build, than "pure" maker software. Actually, I can't think of any more examples. But there got to be more examples, right?

Cheers,
Maria Droujkova
919-388-1721

Make math your own, to make your own math

 

[Maria Droujkova]

On Tue, May 22, 2012 at 8:19 AM, Gary Davis <urgeto...@gmail.com> wrote:

Maria, glad you've discovered this. It's been well known to the spreadshets in education folk for at least 20 years.

Gary

Excellent! Then there must be good references. I like John Mason's work on structured variation grids. What else would you recommend on the topic of "local reasoning" (between cells) and especially software that would support it in young kids?

[S. Ali Ghasempouri]

Interesting! I have noticed such things when 8years old child worked with Binary-Tree by Mathematica,

and Functions with GeoGebra. 

I think that you are trying to draw our attention to Cellular Automata and role of AI in math learning. 

Questions you addressed, it's my concerns as well. I'm thinking of perfect space of all possible problems with intuitive controllers in CAP (Curriculum As Platform) We should define it clearly someday! I'm doing some simple experiments by Mathematica. 

>>Would it not be nice to have spreadsheet-like soft for toddlers and young kids, based on finger-painting on touch-screens? 

Yes. It would be so constructive! I'm learning iPad programming to build apps like that. Do you have any ideas?

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[m...@ms.lt]

Maria,

I note with Ali the connection to cellular automata, such as Conway's Game
of Life:
http://en.wikipedia.org/wiki/Conway's_Game_of_Life
see also:
http://en.wikipedia.org/wiki/Cellular_automaton
and Stephen Wolfram's "A New Kind of Science"
http://en.wikipedia.org/wiki/A_New_Kind_of_Science

I think that Stephen Wolfram would be very excited by your observation
that "They define a rule for going from one cell to the next, and see what

happens. And that's why kids love spreadsheets."

It would be exciting to see how you could relate Conway's Game of Life
with math games, perhaps create something deep and fun.

Andrius

Andrius Kulikauskas
m...@ms.lt
(773) 306-3807
http://www.selflearners.net

> Interesting! I have noticed such things when 8years old child worked with
> Binary-Tree by Mathematica,
> and Functions with GeoGebra.
>
> I think that you are trying to draw our attention to Cellular Automata and
> role of AI in math learning.
>
> Questions you addressed, it's my concerns as well. I'm thinking of

> *perfect

> space of all possible problems with intuitive controllers in CAP

> (Curriculum As Platform)* We should define it clearly someday! I'm doing

> some simple experiments by Mathematica.
>

> *>>Would it not be nice to have spreadsheet-like soft for toddlers and
> young kids, based on finger-painting on touch-screens? *
> *
> *

> Yes. It would be so constructive! I'm learning iPad programming to build
> apps like that. Do you have any ideas?
>
>
>
>
>
>
>
> On Tue, May 22, 2012 at 7:58 AM, Maria Droujkova
> <drou...@gmail.com>wrote:
>

[Dor Abrahamson]

I'm enjoying playing a mechanical game with my young children -- good ol' "Avalanche" from Parker Brothers (I bought a used one on e-Bay and downloaded the instructions).

What is difficult for my kids is anticipating a succession of chain reactions, as marbles roll down and knock gates that release marbles that knock other gates that release marbles... One needs to follow an exponentially ramifying inferential scheme.

The trick here is that you cannot experiment -- it's a one-shot turn -- and, unless you copy the darn thing onto paper and go through tedious modeling steps (as Conway had to do for his pre-computer Game of Life), you must visualize the turn in anticipation of your actions.

My point is that, as Wolfram writes in NKS, humans' un-aided cognitive capacity is limited to a small class of phenomena. It is only with artifacts -- and the computer is the jewel artifact for him, as for Papert -- that we can augment on our cognitive frailty so as to perceive patterns in the world.

I just wonder what an iPad Avalanche would look like that did enable the simulation of various tactics. In a sense, it's a combination of a Galton Box and Chess. For educational design, this could create for children an incentive for developing insight into the mathematical notions we could embed as the game mechanics.

- Dor.

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"Let's build a better math game."

Dor Abrahamson

Associate Professor

Cognition & Development

4649 Tolman Hall, MC #1670

University of California, Berkeley

Berkeley, CA 94720-1670

USA

http://edrl.berkeley.edu/content/dor-abrahamson

http://gse.berkeley.edu/faculty/DAbrahamson/DAbrahamson.html

More about me: www.edutopia.org/blog/applied-mathematics-video-dor-abrahamson

[Maria Droujkova]

On Tue, May 22, 2012 at 4:12 PM, Dor Abrahamson <d...@berkeley.edu> wrote:

I'm enjoying playing a mechanical game with my young children -- good ol' "Avalanche" from Parker Brothers (I bought a used one on e-Bay and downloaded the instructions).

What is difficult for my kids is anticipating a succession of chain reactions, as marbles roll down and knock gates that release marbles that knock other gates that release marbles... One needs to follow an exponentially ramifying inferential scheme.

The trick here is that you cannot experiment -- it's a one-shot turn -- and, unless you copy the darn thing onto paper and go through tedious modeling steps (as Conway had to do for his pre-computer Game of Life), you must visualize the turn in anticipation of your actions.

What computer-based math can do that analog often can't:

- Save states

- Rerun from states

- Save, reuse and iterate modules (for example, the first third of your marble run)

- Do operations on modules (for example, inverse or rotate)

I think Papert had a similar list.

I just wonder what an iPad Avalanche would look like that did enable the simulation of various tactics. In a sense, it's a combination of a Galton Box and Chess. For educational design, this could create for children an incentive for developing insight into the mathematical notions we could embed as the game mechanics.

A good iPad Avalanche will have a level maker, right?

I think all math ed games must come with construction kits for levels (or other entities they have). Or maker mechanics inside the gameplay.

Cheers,

MariaD

 

[Chris Hazard]

I think I'd add one more to the list of "what computer-based math can
do that analog often can't": teach practical systems thinking:

>> The trick here is that you cannot experiment -- it's a one-shot turn --
>> and, unless you copy the darn thing onto paper and go through tedious
>> modeling steps (as Conway had to do for his pre-computer Game of Life), you
>> must visualize the turn in anticipation of your actions.

Within cellular automata and other multiagent systems work, there are
all sorts of interesting puzzles that can be difficult to reason
through without simulation. Many times it is very difficult to
visualize how things change at different scales of abstraction. Some
problems can be fun to reason through, such as "boids"-like algorithms
(take a group of people, have each of them randomly pick another
person, run half-way to that person, then pick another person and
repeat - what is the behavior of the group?), but others can be
downright impossible. I'm sure some people on this list have seen
some of Bret Victor's work/essays, this in particular:
http://worrydream.com/KillMath/ . Search for the section titled "A
Possibly Embarrassing Personal Anecdote" in particular.
(Most of the rest of Bret Victor's website is worth exploring too,
especially this: http://worrydream.com/LadderOfAbstraction/ - he's a
mathemetician turned lead UI designer at Apple for a long time and
designed many of their leading products.)

Programming and process implementation is often seen as a difficult
thing. Mathematics-oriented computer scientists typically prefer
stateless functional programming, but it hasn't been adopted by the
larger public because of some initial ease-of-use issues. One thing
my company is working on, and I've had many discussions with Maria
about, is a way to allow users to more expressively and easily create
program or process -like objects.

-Chris

> What computer-based math can do that analog often can't:
>
> - Save states
> - Rerun from states
> - Save, reuse and iterate modules (for example, the first third of your
> marble run)
> - Do operations on modules (for example, inverse or rotate)
>
> I think Papert had a similar list.
>
>>
>> I just wonder what an iPad Avalanche would look like that did enable the
>> simulation of various tactics. In a sense, it's a combination of a Galton
>> Box and Chess. For educational design, this could create for children an
>> incentive for developing insight into the mathematical notions we could
>> embed as the game mechanics.
>
>
> A good iPad Avalanche will have a level maker, right?
>
> I think all math ed games must come with construction kits for levels (or
> other entities they have). Or maker mechanics inside the gameplay.
>
> Cheers,
> MariaD
>
>

[Dor Abrahamson]

Having children learn complexity is an exciting endeavor. 

Indeed, a very powerful way of going about this is by having the kids build the models themselves.

But they can also learn by working with ready-made models.  I was a post-doc on a project that did just that in Chicago Public Schools (Uri Wilensky, PI).  We used the NetLogo extension "HubNet" for conducting Participatory Simulation Activities (PSA) in a networked classroom, where each child used either a TI calc or a laptop as the client. The actions were aggregated on a server and projected onto the wall as a "giant video game."

Here's one of the closing paragraphs from a paper we presented in AERA about 6 years ago:

Wilensky, U., & Abrahamson, D. (2006, April). Is a disease like a lottery?: Classroom networked technology that enables student reasoning about complexity.Paper presented at the annual meeting of the American Educational Research Association, San Francisco, CA.

***

4.1 Effectiveness

Analysis of classroom discussion during the PSA revealed several dimensions of complex phenomena that initially triggered incorrect agent-to-aggregate explanations. These included spatial–dynamic cues inherent in the simulations, such as the individual agents’ velocity and group density, as well as more conceptual or mental-simulation reasoning that interacts with the spatial–dynamic cues, such as: (a) failing to anticipate emergence inherent in agents’ rule-based interactions; (b) proportional and linear reasoning; (c) randomness–determinism confusions; and (d) ignoring the effect of feedback loops. The data suggest that “complex-system heuristics” are cognitively challenging because they often run counter to the automatized “simple-system heuristics” that students typically employ.

Nevertheless, analysis of the pre/post data revealed a patterned shift in student reasoning. We interpret the progress in students’ reasoning as indicating that students leveraged heuristics embedded in the participatory simulation activities as reasoning tools that they applied when engaging in problem-solving complex-systems situations. In particular, it appears as though students learned to inhibit “simple” responses and, instead, to complexify their models of systemic phenomena to build and defend their assertions. That students successfully navigated between agent-based and aggregate descriptive models supports Wilensky and Stroup (2000). That students could ‘storyize’ an emergent process as a confluence of parallel local interactions raises questions for Chi (2005), who posits that these are ontologically distinct and, thus, unbridgeable. 

***

[Maria Droujkova]

Wow Stephen,

Nice rapid prototyping there! 

So this construction kit can be used to make mini-games, right? What are some interesting setups, and what is the game mechanic there - drop all the given balls all the way through? Hit some targets on the bottom without running out of balls? Maybe these are questions to Dor, since he has the physical game!

The questions come, in part, from a conversation I had with ThinkFun president Bill Ritchie about their upcoming curriculum for helping kids design levels for their puzzles, such as Chocolate Fix:

http://www.thinkfun.com/chocolatefix

These questions came up and I think they apply to Avalanche as well.

I have one comment about the interface - can you make the balls fall differently, maybe faster? I am thinking of Drop Seven or the online version Chain Reaction. But maybe not that fast. Simulating gravity/acceleration would be hard, right? There is just something about the way balls fall that needs to change, because it draws attention to itself.

http://chainfactor.com/ 

Cheers,
Maria Droujkova
919-388-1721

Make math your own, to make your own math

 

On Sun, May 27, 2012 at 12:45 AM, Steve Thomas <stho...@gosargon.com> wrote:

I created a version of Avalanche where kids can construct their own games, dragging in gates and marbles.

It could also be used (need to make some minor modification to create your own binary adding machine.

Game and short video demo available here (its free and open source).  Feedback and suggestions appeciated.

Stephen

On Tue, May 22, 2012 at 4:12 PM, Dor Abrahamson <d...@berkeley.edu> wrote:

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[Dor Abrahamson]

Double wow, Stephen, I'm blown away.

It's like you're a genie in the bottle!

I love the combination of virtual and physical media. I think ideally kids will learn to use the virtual as a laboratory for creating the physical. LIke engineers!

BTW, here's a link to NetLogo Galton Box (built by Uri Wilensky). You can run it there on your browser w/o downloading NetLogo (just need Java). When I was at the CCL (Center for Connected Learning and Computer Based Modeling) I created a whole suite of interactive simulations of probability experiments, that I called "ProbLab." Most of those models are an integral part of the free NetLogo download. But there's also a (possibly clunky by now) site here from which you can launch them on your browser as applets. If you want some theory around these builds, there's a short 2006 paper on one of them, a more theoretical 2009 paper here , and so on.

The vision of NetLogo is that learners investigate phenomena by modeling and simulating them. Of course, as often happens ... <SIGH> ... it gets swallowed by the system, so that most activity out there might be to use the ready-made models that had been envisioned only as samples... In particular, there are researchers in my own graduate school of education who embed the NetLogo models in scripted environments. Anyway...

BTW, here's the Avalanche instructions pdf I downloaded from the web.

[Steve Thomas]

Thanks to all for the nice comments.  Its not that I am that good, its that Etoys provides a wonderful set of abstractions and first principles which you can use to rapid prototype and build things.  The actual prototype was built using just four scripts (ranging from 2 to 4 scripting tiles each).  The longest time was spent drawing the gates.

I like the idea of kids building models and simulations.  Agreed the challenge is how to get this to scale and be used in classrooms. 

Thanks for the NetLogo links, I am exploring as my limited free time allows.  I am especially interested in your ProLab, as I plan on building some materials and teaching probability and statistics in 2013.  I liked your basic idea of providing visuals and simulations kids can play with to get a "feel" for the subject.  I am also looking at having them program and create their own models.  Any suggestions on resources are welcome.

Building what Dor calls in his paper "interactive computer-based artifact designed to help students learn mathematics" is a good idea especially when combined with more kinesthetic (step away from the computer) activities as well.  I have built a number of these artifacts myself and think it provides a bridge for teachers who have very limited amounts of time to learn how to do this on their own.  The challenge is to introduce these to teachers and provide them resources and "training materials" so they can with limited time, understand and start using them.

In regards to Simulating Gravity, that is a good idea, and would not be hard "its just a simple matter of programming."  Now finding the time, that's hard.

Cheers,

Stephen

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"Let's build a better math game."

cheers,

- Dor

[Dor Abrahamson]

Stephen

As for stepping back from the computer, you'll see here or here that my designs are always mixed media. In fact, my current work is Wii/Kinect.

As for kids programming, I have found with NetLogo that what works is "Ready, Set, Go!" -- 

Ready -- kids work with ready-made models

Set -- they change value, tinker with superficial code (like changing some code values, even just the background color -- wow)

Go -- they build their own model from scratch. Well, actually, they borrow lots of code from existing models, bricolage style, and that's ok.

Personally, I've worked mostly with university kids -- mainly undergrad pre-service teachers as well as grads in a semester long programming-based seminar on agent-based modeling of probability simulations. 

But as for kids, let me paste below a recent exchange between member of ccl and edrl lab about kids programming. There might be some resources there for you.

I think a good outlet for your work would be Journal of Statistics Education, and in particular their recent "STEW", which are teacher-friendly to-go lesson plans.

cheers,

- Dor.

Begin forwarded message:

From: colleen lewis <collee...@gmail.com>

Date: May 23, 2012 12:13:01 PM PDT

To: Brian Meyer Waismeyer <squi...@berkeley.edu>

Cc: Dor Abrahamson <d...@berkeley.edu>, EDRL List <ed...@lists.berkeley.edu>, Michael Hoffman <archangel...@gmail.com>

Subject: Re: [EDRL] Fwd: [ccl] Programming for a six-year-old?

A few other possible resources are

http://ruby4kids.com/ruby4kids 

Scratch curriculum (possibly more challenging than a student would take on individually)

colleenmlewis.com/scratch This has mainly been used with advanced 10-11 year old students.

The "Build your own blocks" version of Scratch byob.berkeley.edu also has functions and you can access the Berkeley curriculum for free http://sage.cs.berkeley.edu/ titled "The Beauty and Joy of Computing." This is for college students, but maybe if the 6-year-old is "done" with Scratch, this would be an option.

-colleen

On 23 May 2012 12:08, Brian "Meyer" Waismeyer <squi...@berkeley.edu> wrote:

Hmmm. I have too little experience with programming to suggest one language with any certainty (especially not for such a young fellow)... but I've had a lot of folks recommend Python to me as a nice blend of free, reasonably beginner friendly, and functional (actually used professionally). 

I randomly stumbled across these books last week. Though they may be a bit advanced for six years, they're something I think I would have loved to play with as a kid:
http://inventwithpython.com/

The books are free online or in PDF format and were written to accessible for 10 years plus (e.g., for a 27 year old like me who stumbled into programming late and likes random projects).

Brian

On Wed, May 23, 2012 at 11:45 AM, Dor Abrahamson <d...@berkeley.edu> wrote:

Michael -- fyi

d

Begin forwarded message:

From: Seth Tisue <se...@tisue.net>

Date: May 23, 2012 11:40:44 AM PDT

To: c...@ccl.northwestern.edu

Subject: [ccl] Programming for a six-year-old?

A Scala coder friend of mine in Boston wrote me and asked:

Hey Seth, My older son is soon gonna turn six and I am thinking

of starting to introduce him to programming.  He already did

some Scratch <http://scratch.mit.edu/> but I believe he's ready

for some code writing.

Do you maybe have any recommendations as to which language

would be most appropriate for him?  What age group is NetLogo

targeting?

I responded as follows:

Scratch has a new 2.0 alpha that's a lot more powerful than the old

Scratch, in that you can define your own blocks.  That brings it much

much closer to "real" programming, IMO.  I haven't tried the new version

myself yet, though.

A kid that can handle any textual programming language can definitely

handle basic NetLogo.  However, we're usually targeting junior and high

school so we don't have learning materials that are targeted for younger

kids.  For example,

<http://ccl.northwestern.edu/netlogo/docs/tutorial3.html> moves too fast

for almost any six-year-old, even one who's already done

<http://ccl.northwestern.edu/netlogo/docs/tutorial2.html> which

introduces light coding.

If you're willing to learn the language yourself and work with it with

your son, then NetLogo could be a good choice.  But if you want something

with lots of books and on-line tutorials targeted for six-year-olds, and

an online community where he can interact with other kids using the

language, then (sadly!) you'll need to look elsewhere.

A good guy to discuss this with is Dave Briccetti.  Maybe you already

asked him -- didn't you and I both sit with him at the Scala Days

banquet last year?  anyway, see e.g.

<http://briccetti.blogspot.com/2011/05/koja-scala-python-and-scratch-for.html>

There's a lot to be said for teaching your kid a language you use and

enjoy yourself.  So Scala+Kojo might be a good choice.  (Though I think

there's only the one guy working on it and I'm not sure how much online

community.)

A good local person to ask is Josh Cough; he has a son who's nine or so.

I've cc'ed him.

If anyone here in the CCL wants to amend any of this, or make additional
suggestions, I'd be happy to pass your thoughts along.

-- 
Seth Tisue | http://tisue.net

Dor Abrahamson

Associate Professor

Cognition & Development

4649 Tolman Hall, MC #1670

University of California, Berkeley

Berkeley, CA 94720-1670

USA

http://edrl.berkeley.edu/content/dor-abrahamson

http://gse.berkeley.edu/faculty/DAbrahamson/DAbrahamson.html

More about me: www.edutopia.org/blog/applied-mathematics-video-dor-abrahamson

-- 
_____________________________________

Doctoral Student and Researcher
Department of Psychology
University of California, Berkeley
_____________________________________

Dor Abrahamson

Associate Professor

Cognition & Development

4649 Tolman Hall, MC #1670

University of California, Berkeley

Berkeley, CA 94720-1670

USA

http://edrl.berkeley.edu/content/dor-abrahamson

http://gse.berkeley.edu/faculty/DAbrahamson/DAbrahamson.html

More about me: www.edutopia.org/blog/applied-mathematics-video-dor-abrahamson

[Steve Thomas]

Thanks Dor, I could tell you use mixed media when I read the first paper you sent.  I will check out the other resources as well.

Cheers,

Stephen