Mathematician's Lament

My husband knows I'm a maths geek.  It was he who told me about this article a few years ago, after hearing about it at work (what else would you talk about whilst building a train?)  It was the perfect gift and I eagerly wrote a blog post to share it ... then forgot to publish it.  Oops.  

I still think the Mathematician's Lament is very clever:

Lockhart's Lament

and Lockhart's Lament - The Sequel

Lockhart's Lament was published in a book in 2009, then in 2012 a second book by Lockhart, Measurement, was released ... perhaps an answer to the question of how to change the status quo.

What is it?  According to Wikipedia, "A Mathematician's Lament is a short book on the pedagogics and philosophy of mathematics by Paul Lockhart, originally a research mathematician but for many years a math teacher at a private school. Characterised as a strongly-worded opinion piece arguing for an intuitive and heuristic approach to teaching and the importance of mathematics teaching reforms, the book frames learning mathematics as an artistic and imaginative pursuit which is not reflected at all in the way the subject is taught in the American educational system."

Here's an enticing quote from page 3 of "Measurement":  

"I am going to assume that you love beautiful things and are curious to learn about them.  The only things you will need on this journey are common sense and simple human curiosity.  So relax.  Art is to be enjoyed, and this is an art book.  Math is not a race or a contest;  it's just you playing with your own imagination.  Have a wonderful time!"

I think mathematics is tremendous fun.  I thought when I aced tests in Year 7, and when I happily battled through 4 unit maths in Year 12 with six classmates who would come over to do problems that took several pages to solve.  At that stage it was mostly about 'the test', but our goofy teacher still made it seem fun.  I felt quite disillusioned a month after our final exam when I realised I had already forgotten most of those abstract concepts, but I am still glad I had that opportunity to stretch my brain.

I am revisiting the marvels of maths, thanks to my children ... with their iPads, Vi Hart videos, Escher posters and a stack of brain-tickling maths books, the sky is the limit.  I'll share a list of our favourite resources and book lists soon.

This link is worth a look ... it presents Lockhart's Lament with a cool video, 'metamorphosis of a cube'.

I will leave you with a quote from an article written by Keith Devlin (whose column featured Lockhart's Lament) ...

"In fact, in the context of this country, bedeviled by the incessant math wars and the intense politicization of mathematics education that drives them, my view is that debate about the curriculum and the educational theory that drives it is a distraction best avoided (at least for now). To me the real issue facing us is a starkly simple one: Teacher education. No matter what the curriculum, and regardless of the psychological and educational theory it is built upon, teaching comes down to one human being interacting with a number of (usually) younger, other human beings. If that teacher does not love what he or she is teaching, and does not understand it, deeply and profoundly, then the results are simply not going to come. The solution? Attract the best and the brightest to become mathematics teachers, teach them well, pay them at a level commensurate with their training, skills, and responsibilities, and provide them with opportunities for continuous professional development. Just what we do in (for example) the medical or engineering professions. It's that simple."

Book Club fun!

A favourite game at our place is 'libraries' ... the children set up a borrowing desk, library cards and book displays.  After buying over a hundred new books yesterday, I was not at all surprised when a book club sprang up this morning!

Bookclub Guide (according to Elijah, 9)
-  Entry costs 5 cents, or buy a 132 day season pass for $1.00 ... bargain!
-  Customers receive a membership card showing their first name, ID number and a tally of attendance.
-  Ages 5 plus are welcome ... children under 5 can come but must be accompanied by someone over 5.
-  You can bring books to read or share.
-  The shop is open when the music is playing (currently playing Jack Johnson Curious George soundtrack)
-  Pram parking and comfortable seating is available.

iPad apps for 8 and 9 year olds

We have an iPad and an iPad mini ... when the children started saving a couple of years ago they both intended to buy laptop computers.  We agreed to pay half.   After months of counting and calculating, the time came and both chose iPads.  We are really glad they did!

Our Favourite Apps

We learn plenty through games, and we have fun playing with education tools.  The lines between learning and fun are delightfully blurry!
These are all pretty good apps, but those marked * are frequently used and ** are current favourites

-  Squeebles Maths (KeyStageFun)     **

-  Dress Up Princess (Hooda Math)

-  Targeting Maths 3 and 5 (Blake)    *
-  Fun Soccer Math (Nghi Lee)   **

-  Tell the Time (Playerthree)
-  Math Drills (Instant Interactive)   *

-  PopMath Maths (AppBlit)   *

-  Ken Ken Classic (maths logic game)     **

-  Conceptis Hashi (Conceptis Ltd)

-  Speed!  (Jeffrey Norkoli)      **

-  Maths Trainer (Shalk)

-  Maths Songs Times Tables (1x-6x and 7x-12x)

-  Spelling 1-2 (School Zone)   **

-  Spellosaur   *

-  This Week's Words (Simulant)
-  Word Jigsaw (Megafauna software)
-  Phonics Genius      **
-  SpeakEasy French (Pocketglow)
-  Kids Apps (13 in 1)

-  Where in the World (UConnect)   **

-  Bridge Construction (Headup Games)   *

-  Bright Puzzles HD (Bright Idea Software)   *
-  Learn the World (Merge Mobile)   **

-  JamPad Plus (H2indie)   *
-  Sketchbook Pro (Autodesk)

-  iLuv Drawing People (MyVijan)
-  Drawing Pad

-  Read and Listen Now Bible KJV HD (Handech)   **
-  SimpleMind+ (xpt)
-  Notability (GingerLabs)

IF you want role play games, my children vote for:

-  My Horse (Natural Motion) ... they get a pet horse without the mess!    **
-  Survival Craft     **
-  Townsmen
-  Tiny Sheep  

Other favourite games include:

-  Harbor Master HD (Imangi)

-  Flow Free (Big Duck)   **

-  Banana Kong (FDG)
-  Angry Birds HD

-  Motorbike HD (baKno Games)   **

-  Mahjong (Byterun)   *
-  Mouse Maze

-  Montezuma Puzzle (ECO Pawel)

-  Square-Off Game (SchoolZone)
-  Memory (Irena Schens)   **
-  Cut the Rope
-  Jump Birdy Jump (Ovogame)

New arrivals, untested:

-  Star Walk HD (Vito) - astronomy
-  Planets (Q Continuum) - astronomy
-  Strip Designer (Vivid)  - cartoon making
-  BrainPOP

I haven't had to set limits for use ... except for myself!  We don't have a TV (only DVD and online viewing) so screen time isn't too much of an issue, and we have too many other interests to spend too much time staring at little screens.  We usually leave the WiFi turned OFF, but sometimes connect to use our Weebly site (a collection of all our online tools), for which I have created a desktop bookmark for easy access.

I would like a better writing app than our current Notes app ... preferably one with formatting options and somehow easy to share with the desktop computer.  If you have a suggestion please leave me a comment.

These cute little tablets have paid for themselves educationally and they add to the family fun!

-------------------&---------------------

A little extra waffle in case anyone is deciding what to buy, or wonders
why we choose iPads ...

*  I already share my lovely iMac and little ASUS netbook.

*  Lightweight, handheld tablets are easy to use anywhere (usually on the lounge).

*  The iPad combines loads of functionality in a tiny machine ... games, writing, art, movies, music, camera and video.  Games for fun and education were a priority.

*  It's super easy and inexpensive (often free) to add new apps.

*  Both children are keen writers.  For portable word processing the onscreen keyboards are fine.  Keyboard cases create a virtual laptop when desired.

*  We were slightly concerned about the smaller screen size of the mini for games and movies, but I changed my mind after trying it in the shop.

*  It's easy to turn WiFi on and off to control internet access.

Living Maths - Part One

Anyone who knows me knows I LOVE maths!

Here are a few more confessions to set the scene:

*  I'm interested in metacognition, patterns, and the links between maths, science and art.
*  I like order but am often disorganised, with many ideas and projects buzzing around at once.
*  I love imagining creative ways to do things.
*  My favourite part of teaching is seeing someone's eyes light up when they are inspired and their mind is humming with connections and understandings.

Steps and Stages ...

Since we started home ed., I've often wondered which maths curriculum would suit us now and into the future.  My kids are quite maths-literate, and accelerated themselves through much of K-1 maths before school age. There are gaps of course ... not many people learn neatly according to a schedule!  And we have had great chunks of time spent on Natural Learning, which helps keep things fresh and student-focused.

 I was slightly worried, as in terms of number facts, Elijah has remained at a Year 2 level for 2 years.  But I can see some important connections happening now, and am seeing evidence that he's in another phase of rapid development.  For example, Elijah doesn't have quick recall of all addition facts to 20 and can't be bothered with learning to tell the time properly, beyond the hour and half hour.  His writing is way behind his comprehension, so I've often acted as his scribe.  But he knows how to work out simple percentages (eg. 20% of 1000 or 50% of 40), has a good grasp of some fractions, can count by 2's, 5's, 10's, and quickly understands any new concept we think to introduce.  I have been glad to slow down anyway, not wishing to hasten his acceleration unless it's driven by his own interest and readiness.

The Resources Dilemma ... 

My main problem is that we're spoiled for choice!  I've examined endless reviews, samples, placement tests and articles about all kinds of maths resources.  Some programs which partly suit my preferences include Singapore, MEP, Saxon, Targeting Maths and Go Maths.  They all seem excellent.  But I am never content, so haven't ordered a complete program.  Part of me wanted to find an online solution, but I wasn't fussed with the samples of Teaching Textbooks.  Would Elijah have to prove competence to gain entry to EPGY Grade 3-4, and would it be overly expensive?

Thus far in Maths ... 

I've been using Singapore Maths Textbook 2A with Elijah (no workbooks, just using it as a reader and doing the exercises orally).  I also printed and had bound MEP Maths 2A for him.  Both children have several Yr 1 or 2 textbooks to choose from when they are inclined to do bookwork (for Jasmine, this urge occurs several times a week, for Elijah it's only when I insist) ... Rigby, Maths Plus, Easy Learn Maths, Boomerang and Step Ahead.  Elijah likes the interaction of doing MEP lessons with me, but we don't formally 'do' maths often enough to use MEP properly ... for days we'll play with numbers and ideas in other ways, so that each time we go back to bookwork, he's progressed beyond the lesson we're up to.  I've never stuck rigidly to any textbooks, skipping pages which seem boring, redundant or repetitious.  So perhaps we've been too light on with number drills.  Instead we do sums at the dinner table, counting rhymes in the car.

Whilst some maths programs incorporate a lot of hands on activities, real life applications and motivations, I still find them a bit stifling.  Jasmine likes to do bookwork, but she enjoys the other things we do too.  Elijah definitely prefers learning through stories and conversations, shows, computer games, board games and puzzles.  We've used such a diverse range of resources, it's always a challenge to record them and make sense of the path we've taken.

Ideas ...

I've been toying with the idea of several maths lessons each week being devoted to fun and inspiring ideas, such as looking at M.C. Escher, magic squares, The Golden Section, Bible numerics.  And games.  And hands on, real life problem solving.  Doesn't leave much room for drills and paperwork though.  Do we really need them?  Do I need a 'spine' at all or can we be trusted to cover the 'basics' in unconventional ways?  Will my children be able to fit in if they ever need to go to school?

I'm coming to terms with all these questions, and the answer seems to be:

Living Maths ...

Our best 'lessons' over the years have been experiences such as making a 3D graph of matchbox cars, constructing polyhedra using straws and pipecleaners, counting money as Elijah saved for an electric drum kit, taking measurements for furniture designs, adjusting recipe quantities.

I'm going to TRY to stop looking at Maths Curriculums, and focus more on just 3 factors:
-  being more creative and diligent in RECORDING the incidental maths the kids do in life and play
-  identifying WHAT my children are ready to learn or do next
-  finding interesting printables, stories and manipulatives to ensure they have FUN experiences to gain that knowledge.

I want our maths 'lessons' from now on to be memorable, inspiring, lively, exciting, challenging and real.    I intend to read more on Living Maths and Charlotte Mason and Unschooling websites.  I am excited about diving deeper into Living Maths.  I'm keen to list the many ways MATHS happens in my home, and I think articulating what we unconsciously do will help me let go of my obsession with comparing curriculums!

I already have some great resources, and know of a few more I'd like ... I'm bursting to write about them all, but will have to save that challenge for another day.  I have bookmarked some helpful and inspiring websites, and will also list these and share some links and ideas in Part Two.

If you have any tips to share about studying, doing or recording LIVING MATHS, please share them!

Infinity Elephants

Sue often shares links to exciting mathematical wonders on her blog, Math Mama Writes.

Learning between the lines

If you asked me during the day yesterday what we were doing, I would probably have answered 'not much' or 'just the usual'.  As usual, I started with a vague idea of how the day might go ... an easy day at home, a bit of school work, reading together, lots of play (inside as it's raining), and as always going with the flow and having a lot of fun.

It didn't seem like much, just an ordinary home Monday, as much as any day here is 'normal'!  Looking back I'm amazed by the scope of what went on between the lines, during our 'ordinary' day:

5:15am     Snuggly little ones in sneak in with me after Pete left for work, neither of them sleepy ... we talk and pray about what we're thankful for.  Jasmine starts a Sudoku puzzle from my bedside table - needs a lot of help, but she has the general idea.  They start a debate about pets ... Jasmine wants a white rabbit, Elijah wants a turtle ... though they know we are not likely to become pet owners any time soon!  I half-sleep, half listen as their thoughts bubble out about care, habitats, natural environments, food (I'll grow carrots, said J.), temperature regulation, cleaning etc ... the possibility of further sleep melts with the rising sun.  Around 7am we drift out to a sunny dining room.

Breakfasts are made to order ... toast with all the colours of the rainbow for Jasmine (a rectangle each of jam, vegemite, peanut butter and honey), a banana sandwich for Elijah, toast for me.  The kids eat and watch cartoons from 7:35 to 8:30am, while I shower, check emails, read blogs, wash dishes ... away from the screen, kids dress, make beds, brush teeth and play.

The kids play variously with dolls, books, bikes and scooters and a big cardboard box, which they wallpaper with paintings done last week.  This ends in tears later in the day, as some paintings rip during removal.  Meanwhile, I start a new blog, Ink Island , to write more about writing, and I write my first post.  It's fun playing with layouts, background pictures and words.  I also finish my previous, jumbled blog post for Chrysalis Island.  My mind is a bit clearer after getting these thoughts down!

We come together for morning tea (fairy floss and nuts) and read about Sarah and Abraham's baby and their praising, from The Little Girl's Bible .  I write Psalm 34:1 on the whiteboard.  The kids copy the verse into an exercise book and draw and label what they praise the Lord for.  Jasmine makes hers sparkle with glitter pens and glitter glue.  Elijah's picture spreads to the next page as he tells us a story as he draws.  He writes on the whiteboard - sunny (Jasmine wants an 'e' on the end, but reluctantly trusts me that it must end in 'y' like baby and puppy).  Elijah draws boxes for a checklist and labels the first one 'litrse' (literacy).

Jasmine does a page of a spelling textbook then starts colouring in and doing patterns inside bubble-writing.  I ask Elijah if he'd like to write about pets or knights, but he says chocolate!  He tells me a title, which he copies from the whiteboard (Chocolate Flavour Inventions).  He narrates a bit while I write (the new chocolates will be sold to raise money for poor people), then he draws and labels some new flavours, which somehow turn into icecreams along the way!

Around 1pm, Pete arrives home and we have lunch (he's home early as today's work was overtime on a rostered day off), toasted sandwiches.  He offers to clean up, so the kids and I sit on the lounge to continue our informal (make it up as we go) unit study learning about Poles and Polar Bears.  We've watched DVDs over the years, and read a few books recently, like Polar Bears Past Bedtime , by Mary Pope-Osborne, and parts of Horrible Geography: Perishing Poles .  We talk about (and I record) what we know and what we want to know .  It's fun and I'm impressed!  We look at the globe and wall map, to see Antarctica and the Arctic.  Then we re-read a picture book, Antarctic Dad , share a few pages from non-fiction Antarctica , and a new picture book Over in the Arctic .  We take a long time to read, as each page is fuel for discussion, dominated by Master E.!

The kids put away a stack of about 30 books they've been browsing.  Jasmine plays boxes/ dots with Pete while Elijah 'reads' Knights and Castles .  From 3-4 pm the kids watch ABC TV (playschool, cartoons).  Pete starts our tax returns, and I read in bed!

While Pete has a well earned rest I keep the kids quiet with a game of Rummikub .  It's their first time playing, and I'm surprised by Jasmine's patience and by how interested and eager they both are.  We play open hands, but by the end they can make their own moves if I give a small clue.  At 5pm the kids call Pete out and we all watch Survive This! on ABC ... an adventurous Canadian survival show for kids.

The kids take turns doing dot to dots in a booklet of A-Z and 1-30 puzzles, easy but fun, and play 'memory' with a deck of picture cards.  Dinner (chicken schnitzel, mashed potato and veges) sparks talk of volcanoes (as Elijah's mash mountain erupts with sauce), and is followed by the final few chapters of our latest chapter book, The Littles and the Lost Children by John Peterson.  Together the kids quickly do a set of Maths Mentals from New Wave Mentals B , and Elijah asks to do more.  They find it fun because it's new today, but I'm thinking "it's too easy, even the next book is too easy, what's the point?" but I do want them to have quick recall and some 'schoolish' skills in case they ever decide to do maths tests, competitions or need to slot into a classroom at some stage.  I have always done maths just for fun, and my kids do seem to be wired the same way.  We'll see.

Time for pyjamas, tooth brushing, tucking in, chatting and praying.  Pete read a chapter of the next book, The Littles go to school .  Then the kids play in bed, read and colour in until about 8pm when they fall asleep.  I go online to renew library books and update the kids' book lists for the Premier's Reading Challenge (both are now finished, Jasmine's first and Elijah's second year).

In bed at 10pm, I'm too tired to read for once (aside from a quick scripture), but I can't sleep.  Several times I flick on the light to jot down notes to myself.  I finally do some maths (double 2, 4, 8, 16, 32 etc) to switch off my meandering mind, and drift to sleep trying to double 16384 ... finally .... zzzz

Patterns and Tessellation

I had a ton of fun preparing a Patterns and Tessellation workshop for FBI (Fun Brain Institute, our local home ed co-op) this week.  The kids who attend FBI usually range from 0-10 years old, and my voice is still a bit dodgy, so this presentation was very brief ... we did a whirlwind tour of tessellation, patterns in nature, maths and art.  After writing up the first few Fibonacci numbers (1, 1, 2, 3, 5, 8, 13) a few kids were able to continue the pattern.

I flicked through these library books (and some printed pictures) as we talked about bee hives, snow flakes, architecture, mosaics, M.C. Escher and cells in the human body.

We also did a craft round-robin (cutting snowflakes, icing biscuits and making pop-stick puzzles), table games, played tunnel ball and had a story and music session.  Many children coloured in a tessellating pattern at some point during the day.  Here are some of the games & manipulatives we used:

Tantrix is shown top left.  Other games we enjoyed playing included Tangram, KogWorks, Mastermind and Block by Block.

I promised to share some links to relevant websites with explanations, printable pages and best of all, interactive tools to play with ... so here they are!  Please look at them with your children, because there are links within links and I can't check that all of them are appropriate for little eyes.

The maths of tessellation:
http://www.coolmath.com/lesson-tessellations-1.htm
http://mathworld.wolfram.com/Tessellation.html
http://mathforum.org/sum95/suzanne/whattess.html

Galleries, information, DIY instructions and summary of software to create tessellations:
http://www.tessellations.org/tess-what.htm
http://www.tessellations.com/

Tessellation grids to view or print and colour in:

http://gwydir.demon.co.uk/jo/tess/index.htm 

M.C.Escher:
http://www.worldofescher.com/gallery/
http://www.mcescher.com/
http://britton.disted.camosun.bc.ca/jbaraki.htm

View Escher-style patterns, or buy a cool colouring book:

http://www.tabletoptelephone.com/~hopspage/HopsTiles.html 

AND NOW FOR THE REAL FUN...
Free Interactive games (play with tessellating shapes on your screen):
http://shmuzzles.com/shmuzzle/shmuzzle.htm
http://www.mathcats.com/explore/tessellationtown.html
http://nlvm.usu.edu/en/nav/frames_asid_163_g_3_t_3.html

Easy on-screen activity ... cut a snowflake:
http://snowflakes.barkleyus.com/make-a-flake.swf  

Read about snow, ideas for fun ways to MAKE snowflakes:
http://britton.disted.camosun.bc.ca/snow/snow.html

AND more ... lots of links ... tessellation, patterns, fractals, Fibonacci and other fascinating maths!
http://www.uen.org/themepark/patterns/tessellation.shtml
http://42explore.com/teslatn.htm
http://www.cap.nsw.edu.au/bb_site_intro/stage2_Modules/tesselations/tesselations.htm
http://members.cox.net/fathauerart/index.html
http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html
http://en.wikipedia.org/wiki/Fractal
http://www.youtube.com/watch?v=G_GBwuYuOOs

It's hard to stop ... but there you have it.  A glimpse into the glorious world of maths ... I hope you and your kids will enjoy some of these links!