If fractions make you pale, then scale, scale, scale! OR When kids invent algorithms

My goofy title is in response to the annoying rhyme you may have learned "Don't ask why, just invert and multiply".

In the course of writing a Professional Development module for teachers, and I've spent hours downloading papers about algorithms that kids invent for division of fractions.

Here's a summary of a not-so-common, but very useful method for dividing fractions and the representation that facilitates its discovery. I'll call this the scaling algorithm because it is born out of considering areas of rectangles and how the ratio of area to length of a side is preserved under scaling.  The algorithm is:

For example:

What representation would lead to this algorithm's discovery?  Ask the question in the context of area:

"What is the width of a rectangle whose area is 2 sevenths and whose length is 3 fourths?"

To answer this, let's make seven copies of the rectangle -- Why? Because integers are easier to deal with and we all know that 7x(2/7)= 2.  The question has now been reduced to "What is the width of a rectangle whose area is 2 and whose length is 7x 3 fourths?" 

Since 7x3=21, we have 21 fourths as one side of the rectangle.   We have exchanged ? x (3/4) = 2/7 for ?x (21/4) =2.  We may also notice that scaling each factor by the same number preserves the answer!

Let's scale again!!!  Take four copies of the big rectangle, which gives you a total area of 8, and a total side length of 21.
We have now replaced all fractions with integers and traded the original question in for the simpler sounding:
"What is the width of a rectangle whose area is 8 and whose length is 21?" At this point, we see that the answer is 8 divided by 21 or 8/21!  Woohoo!!!

Do kids invent this scaling algorithm?  Indeed they do if they are given the area context and their brains are not stuffed with "this is how you do it" algorithms.  Want to read a paper that talks about this?  Jaehoon Yim from South Korea has a 2009 paper that's fascinating -- Children’s strategies for division by fractions in the context of the area of a rectangle -- I'm not sure if you'll be able to access it for free.  But, the general gist is that ten and eleven year old students were able to "make the width equal to 1, make the area equal to 1, and change both area and width to natural numbers".  The strategy outlined above is of the last variety.  She even researched whether children could formalize their pictorial drawings to create a numeric algorithm -- and surprise, surprise -- they could!  Granted, these were students who were picked because they had a "positive attitude towards mathematics".

Incidentally, if the numbers are nice like 6/20 divided by 3/4, then they can use the area model to see that the answer is 2/5 since 6 divided 3 is 2 and 20 divided by 4 is 5.  In other words,

Here's a picture that shows this.  Can you see how?  The strategy is a little different from the one above as it starts from a unit square.

Six pointed stars -- just in time for the holidays!

So I'm on vacation in Maine where it was snowing today, and we took my little one for a walk in the snow.  A perfectly formed little snowflake fell on his eyelash, and we all just stood around and stared at him for a few minutes while it melted.  It was so beautiful!  I'd never been able to see that symmetry so well with my naked eye.  We took a picture, but it just didn't do it justice.
   Anyway, it reminded me of a book I'd seen once that would make a great gift.  The book contains snowflake photos taken and explanations by Caltech physicist Kenneth Libbrecht.  Apparently photographing snow is an expensive hobby, costing over a thousand dollars, not to mention you need to keep your camera warm!
   Then I just happened to see this wonderful video made by George Hart, Vi Hart's father and also one of the designers of many of MoMath's new exhibits.  The key in understanding what is obtained when you slice the Menger sponge in half along its diagonal is that we can view a star of David as being the union of three Rhombi, not just two equilateral triangles.  This different way of constructing the Star of David was exploited by an artist I recently saw at Tucson's 4th Avenue Street Fair.  Unfortunately, I can't find his site!  But the general idea can be seen in this basic picture I made.
It's just amazing to me how the things I see in everyday life sometimes link together in these nice ways... Anyway, there's a lot of six fold symmetry going on just in time for the holidays.

Learning to Teach by Teaching Teachers to Learn

I am currently helping twenty-seven K-8 teachers brush up on their math skillz.  The teachers I am working with are honestly a JOY to work with.  The first few days we had comments in the daily feedback that revealed discomfort and frustration with math and with the idea that we were asking the participants to NOT FOCUS ON THEIR TEACHING but to FOCUS ON THEIR OWN UNDERSTANDING.  Boy have they come around!!!!

Teacher participant (after seeing a colleague at the board):  "Are you going to ask for more people to share whether they did the problem differently?"

Me:  "Why yes, I was now that you mention it."

Teacher participant (in the process of standing up to come to the board):  "Oh, good because I did it a totally different way, and I want to show what I did!"

Another paraphrase of a student talking from her seat:
"Can I share something I thought of about the area model?  It's kind of like when we add 0 a bunch of times so that we could subtract negative numbers....because you can add 0 as many times as you want and still get 0.  It's like that, except... really I should just show you."

Marie's "1 and 3/4 times 1 and 2/3"

She proceeded to place her drawing under the document camera, which looked something like the figure to the right.  She said

 "we had 1 and 2/3 going on to the right forever and 1 and 3/4 going down forever, and the product is just the overlap...So we don't need to think ahead of time of how big a rectangle we need".  So since you can see 35 squares, each of which is 1/12, you know that the product of 1+3/4 and 1+2/3 is 35/12.

I thought that her invention was really slick.  She noticed that adding extraneous area actually made the concept shine through.  We had just finished talking about how "adding 0" and "multiplying by 1" could be useful.....She really internalized that..

Marie asks LOTS of questions.  And as the workshop has progressed, the number of excellent questions that are brought up AND answered by the participants themselves just keeps growing.  Granted, these teacher participants are being paid to attend, but I don't think that's all that is motivating them to ask such wonderful questions.  I think one reason is that they are not being graded.


Many mathematicians (including myself) are appalled at the low level of understanding and lack of interest in understanding shown by many elementary school teachers as well as those who aspire to be elementary school teachers.   But being appalled doesn't accomplish anything.

LEARNERS' RIGHTS and MY RESPONSIBILITIES AS AN INSTRUCTOR:


Many people talk about the "Rights of the Learner".  But we (my teaching partner and I) REALLY took that seriously.  Clearly it's important to create a classroom environment that is comfortable, but that doesn't mean "feel-good" or "dumbed-down".  I now attempt to do the following:


1) meet anger about and frustration with the material with a sense of humor as opposed to dismay or disapproval.   


2) acknowledge that learners may have previous experiences that we can learn from.  So we should solicit anonymous feedback, show that we have read it, and publicly address or discuss it.


3) encourage learners to reconcile old ideas (whether correct or incorrect) with new ones   Asking them to "just start from scratch" or "just forget about the algorithm" is unfair. 


4) keep in mind that learners want to please the teacher (especially if a grade is involved).  When there is no grade involved, students are more likely to express their frustrations honestly, but still hesitate for fear of "upsetting" the instructor.


5) be consistent in "walking the walk".  In other words, if I talk about the philosophy of "why something should be discovered or learned independently", I have to try to avoid things like writing on students' papers or directing them on what to write.  I have to invest in their thought process whole-heartedly regardless of my concern for "covering material".


 For instance, after actually discussing and debating about the abhorred-by-mathematicians "FOIL method" of multiplying binomials and how it was related to the distributive property, one student jokingly said "Well I prefer the SARAN WRAP method."  After we had discussed the method (its advantages and disadvantages) heatedly AND with humor, even though they knew I "didn't approve", students felt comfortable saying that they "used FOIL" to solve the problem.   To me, this is much more preferable because they are being HONEST about how they thought about the problem rather than trying to make me feel good.

That's all for now...So many thoughts are running through my head.  But here's a link about Finland's lack of grading which has means that in Finland The differences between weakest and strongest students are the smallest in the world!!!

Let's just say a lot's happened since my last entry.
For instance, I had a baby!  So what's it like teaching and doing mathematics with a baby?
Let me just tell you that my 8-month-old does not think that he should have to share me with my computer...

   After a particularly productive bit of time spent typing on my computer, feeling proud of myself, and ready to email my work out, I realized that I had been ignoring the little firefly who was screaming and wailing for me to pay attention to him.  So I picked him up, and as I turned around, he spit up ever-so-adeptly on my keyboard.  Of all the directions he could have chosen, the spit-up landed just so, and tt took me a second to realize that my computer was probably fried.  After turning it off, waiting a while, and turning it on only to find out that the "k" key now controlled the volume of the speakers, I went to try and get it repaired!  "Water-damaged?" they asked.  "Why, whatever do you mean?" I replied.

   Another morning, after staying up late to grade a katrillion exams, my kiddo woke up with a fever!  Teething had commenced, and he didn't really care if I still had the last day of classes, reviews, extra office hours, and more grading to contend with.  What good are exams unless you can chew on them?

   But really he's a pleasure--- in office hours, he plays in the pack 'n play quietly while my students do math at the board, and they are more likely to try and engage him then he is to distract them.  That's right, I take him to school with me about once a week.  And no, I don't leave him in daycare.  Either my husband, one of my friends, or myself takes care of our sweet baby at all times....

   But I'd like to hear if anyone else in mathematics has stories about raising children while working.  I'd especially like to hear about women who are doing that...  I don't meet too many, but maybe that's just because we're all too busy to be bumming around at coffee shops or chit-chatting in the hallways....or reading/writing obscure blogs :).....