Monday, February 27, 2012

Math - Learning the Four Processes!

Today was such a big day for us - Yoav learned the names and symbols of the Four Processes (and equals).  He didn't know any of these until today.  Like the letters, I've been able to shield him from any discussions of these.  He has been doing math problems for quite some time, but we always simply say "and" for plus and 3 4s for times, but never the words plus or times.  I doubt it makes much of a difference other than staying in a slightly less intellectual/symbolic world, but it's interesting to me that it's possible to have a child who recently did 57+57 in his head but doesn't know the word plus :)

First we folded a piece of paper into fourths and I drew five sections (middle for equal).  We started with equal and I told him about how he and Elie love to go on the teeter-totter and what does it look like if they got it balanced so it didn't go up or down.  Then I drew a picture and he copied and then I said, ah in the picture, you're both in the same place - you're Equal!  And then I drew the equal sign and said it's like the top and bottom of the teeter totter when it's perfectly balanced.

We did a similar format for each of the Processes.  The picture ideas are from Jarman's "Teaching Mathematics in Rudolf Steiner Schools" except the plus which is from Schuberth's "Teaching Mathematics in First and Second Grades in Waldorf Schools".

So Plus is a man holding out his two hands to collect things (we drew flowers); Minus is a man giving something (we drew a tack based on our example from last week - also note that I spoke of division as giving versus taking away - Waldorf stresses the moral benefit thinking this way, which I agree with); Divide is a man with one hand on top and bottom of a plate of muffins to share.  Multiplication is a sower of corn with arms and legs outstretched - we drew the seeds on the left and the fully grown corn on the right.


Next we went back to our work from last week to add the symbols to our equations:


Yoav had so much fun with this.  He kept saying how much fun it was.  I'm curious to see how well the ideas will stick.  I think we'll do one more day with the pictorial computations and then move to straight equations.

For many Waldorf people, this might seem quite contrary to the "standard" Waldorf treatment of the Four Processes.  Jarman and Schuberth are both against the overly pictorial introduction of the Four Processes.  I can see the benefits both ways but Jarman's explanations in particular really made me decide to go with this particular method.

About the different methodologies, Jarman says (in his book, "Teaching Mathematics in Rudolf Steiner Schools for Classes 1-8":
Whist it is of great value to introduce any educational subject to children by means of pictorial presentation - and each topic within that subject, too - there is a danger in adopting a similar attitude to the development of the subject or topic irrespective of its nature.  When teaching reading and writing, or history, or art, the pictorial element always needs to be present.  Mathematics teaching, though , is quite different in this respect.  Human feelings and thought pictures nurture the essence of literature and history, but mathematics is essentially a will subject.  The will has to be brought into thinking.  While the introduction to mathematical topics does require the pictorial element - only in contemplating the pictorial is a growing human being left free, for pictures do not compel - this needs to give way to a musical element in the development of such topics.  To use Rudolf Steiner's terminology, whilst literature and history rely on Imagination, mathematics relies on Inspiration.  Mathematical progress depends upon overcoming and freeing oneself from the pictorial and living in sense-free concepts.  This is why all pictures used in introducing mathematical topics need to be precise.  Teachers who like to bring fairies and gnomes into Class I arithmetic need to be aware of the dangers indicated above.

Wednesday, February 22, 2012

Math - The Four Processes

We started our Four Processes math block this week.

I'm getting ideas for our Four Processes math block from the books:
Teaching Mathematics in Rudolf Steiner Schools for Classes 1-8, by Ron Jarmon
Teaching Mathematics for First and Second Grades in Waldorf Schools, by Ernst Schuberth

This is our first week of the block.  We did some practice writing the numbers because I noticed that Yoav was writing some of them backwards.  I wrote out the numbers 1-20 for him to copy and, as you can see, it was too much fun for him and he kept going :)

Yoav's numbers up to 70 (stars between to help keep space between numbers - idea from Donna Simmons)

We are also doing word problems based on ideas in Ron Jarmon's book.  The first day, I used an example straight from his book.  He includes a word problem for addition, subtraction, division and multiplication all based on the same story line.  So I sat next to Yoav and we each folded our papers into quarters and we did the problems in one quadrant each.  Next week, when I introduce the symbols, we'll go back to these papers and add the equations below the pictures.

photo (1)
Top page is exercise from Jarmon's book; bottom one is my example below.

I'll give my story line (the bottom piece of paper):
1) Mama Bird found this many (I drew 8 lines in the first box) twigs to make her nest, but she wants this many (I wrote 17).  How many more does she need?  He'd answer and we'd both write the answer on the right.
2) Mama collected this many (drew 12 pieces of string in the second box) pieces of string for her nest, but only used this many (wrote a 7).  How many will she put back in the pile?
3) Mama is feeding her babies.  She has this many worms (draw 12 worms). Each baby needs this many (write 3) worms.  How many babies does she have?
4) Mama wanted this many (draw 16 bugs) bugs.  Each time she flies down to collect the bugs, she can only fit this many (write 4) in her mouth.  How many trips does she have to do?

Yoav LOVED these.  He wanted to do more each time we finished, which we didn't, mostly b/c I wouldn't be able to make up the story on the spot.

Yoav counting by 5s.
We also did skip counting on the balance beam - counting by 2s, 3s, 5s, 10s, (and then by 100s, 1000s, millions by Yoav's addition) as we walked forward and backward on the balance beam.  The 2s, 3s to 30, 5s, and 10s Yoav can do very fast.  By his choice, he did 3s up to 90 or so and beyond about 30, I could see him counting in his head.  This is to help memorize the multiplication tables "by rote", as Steiner says to do.  Ideally by third grade, a child should be able to quickly say the answer to any random multiplication question up to 12x12.

I need to spend more time on these exercises.  Schuberth has some fun exercises, like the +2 one, where I say a number and the child says what number you have to add 2 to in order to get my number.

Next week I'll introduce the symbols.