Infinity

An acquaintance recommended a site full of "math in daily life" videos.  Lately, math around here has become a little dull.  SpiderGirl is practising multiplication tables; BatBoy hasn't had anything new for awhile; so the inspiration was welcome.

"The Infinite Life of Pi" caught BatBoy's imagination.

He calls these his "infinite drawings."  He made one that looked just like a visual representation of Fibonacci's sequence too.  (It was a series of adjacent squares.)  From there, we went on to explore ways infinity shows up.  His drawings reminded me of fractals, so I found some videos to show him.

And of course, the Ted-Ed had a video on fractals too.

Finally, we looked at what happens when we allow long division to go into decimals instead writing a remainder or changing to a mixed fraction.

Take Me to the Moon

Daily math doesn't have to mean practical situations that everybody gets into all the time.  Sometimes, daily math just means math that happens to come up. 

 

Today, A. made a comment that she could jump six times as high on the moon.  When I challenged her idea that she was jumping as high from the floor as the top of her head at the peak of her jump, we decided to measure.

 

The very high tech measurement device:  a piece of sidewalk chalk in a chalk holder, taped onto a headband.

 

First, A. marked the chalk's height against the wall while she was standing.

 

 

Then she jumped while applying enough pressure for the chalk to draw on the wall.  She did this several times to achieve the highest jump she could.

 

 

She measured the distance between her initial marking and the highest point on the chalk line:  20 cm.

 

 

Multiplying 20 cm by 6 gives 120 cm, or 1 m 20.  How long is that?  She rolled out the measuring tape, and we also held it up vertically to get a sense of how high the bottom of her feet would get on the moon.  A. was impressed.

 

 

 

 

Baby Steps

On the topic of supporting the development of mathematical communication, today I want to address patience.  One of the greatest obstacles to learning is the fear of making mistakes.  In other areas of study, we know that mistakes are the path to learning.  In Math class, we get points taken away for every mistake we make, and if the teacher doesn't understand the steps you made to get to the wrong answer, we get no points at all.  But in fact, we don't want children to know what steps to show; we want them to understand how to show enough so that their audience can follow them. We, as guiding adults, need to exhibit patience with our children as they learn to communicate their mathematical ideas.

When children are learning to write in the early years, we encourage them to get their ideas on paper without fussing about perfect spelling and punctuation.  When they are learning to express themselves as toddlers, we give them the words we think they are trying to use.  We need to do the same with "showing steps."  Before we ask children to show all of their work on paper, we need to talk out their reasoning with them.  We have the advantage in that we can make educated guesses about how they came up with a solution to a problem or what steps they might take.  The back and forth of a conversation allows the child to make baby steps in learning to communicate by both watching and hearing the adult model, and making small attempts themselves and getting immediate feedback.  As we converse, we might write the parts on paper that we think would go into "showing work."  Eventually, the child will make their own written attempts.  

The process of learning what steps to show takes time and is unevenly paced.  A child not only needs to learn the language and vocabulary of math, but also takes years to understand that different audiences require different steps to be shown or different ways to show them.  We need to embrace that what goes on inside the head of a child is usually more than they can express well.  Rather than the patience of waiting -- waiting for the child to mature when one day they will magically "get it" --  what is required is the patience of building.  Celebrate each small success.  Laugh over misunderstandings.  Learn from mistakes.  It's all valuable.

Supporting Communication Through Understanding A Child's Process

Earlier I wrote about the importance of supporting the development of mathematical communication in a child.  I thought break my thoughts up into chunks.  Today, I'd like to talk about listening to understand a visual thinker.

One of the difficulties in communicating with a visual-spatial person, especially a child, about how a solution is reached is that they often understand things in pictures.  For one thing, there isn't really a sequence of steps they took to arrive at "an answer."  Rather, everything is there all at once in the picture.  For another, since we converse using language, ask children to explain themselves using language, and often model using language to explain, children expect to explain themselves using language.  As children get on in Math, this might evolve into a habit of trying to explain steps in a process using numbers or equations.  We need to break out of these two boxes. 

Communication II

Continued from an earlier post...

Now where was I?  (That's the trouble with taking quiet time in the morning, rather than at night:  there's no way to steal from your sleep to get a blog post finished.)  Right, communication.

What brought on these thoughts about helping children communicate their math?  It was a combination of things that grabbed my attention this year.  One contribution was the oft-repeated assertion that visual-spatial thinkers "just know" solutions to problems.  The other was the similarity in approaches used by teachers in a variety of subject areas:  In The Writer's Jungle, Julie Bogart emphasizes the importance of helping children express themselves by first listening to them and scribing for them, then offering your own words and structures to clarify or help them to better communicate their ideas.  Grammar, spelling, punctuation are all there to support communication; they aren't ends in themselves.  When my daughter brought her music composition to show her teacher, her teacher didn't critique her very unusual timing (In fact, she commented that she found it interesting.) but rather helped her by showing her how to add the bar lines and time signatures that would allow readers to understand the sounds she wanted to create.  Visual art, too, has been described as sharing with the world what you see the way you see it.  Learning to communicate in math needs the same type of support as learning to communicate in visual, linguistic, and musical mediums.  

In North American society today, we tend to think of Math and Sciences as going hand-in-hand, perhaps because the Sciences so regularly use Math to communicate and understand their own ideas.  However, Math is not itself a Science.  There are parts of Math that are irrefutable and reproducible; those are the parts that seek truth in the way Sciences do.  Math is also about beauty and creation; those parts are like the Arts.  To validate only the mathematical ideas of a child that agree with a textbook is akin to accepting only the drawings of a child that are exactly like the given sample.  To ignore or correct a child's own mathematical ideas is to dampen the mathematical spirit in him.

How then do we support the development of mathematical communication?

To be continued in... 

• Understanding a Child's Process
• Baby Steps

Communication

Lately, I've been hearing a lot about kids who "just know" an answer to a problem and don't know how they got it.  I must say, at first I felt a bit perplexed.  In all my years of teaching and tutoring, I have never come across a single student who "just knew" an answer consistently and couldn't show work.  Even if they began by saying that they didn't know how they knew, we could always tease out the line of thinking that led to their answer.  What I believe, then, is that it's not okay to become complacent with these very intuitive kids.  Showing work for marks is not necessarily a goal for everyone, but being able to communicate thoughts is an important step in learning.  Yes, communication is necessary for the sharing of ideas that leads to piggybacking and synergy.  More importantly, though, successful communication ensures that a child will understand that their mathematical ideas are valuable, that their logic is valid, that they are capable of "doing math."


Continued...

Math and Science: Measuring Angles

Up until now, our studies in Math and Science haven't really overlapped much.  There has been some reading of thermometers and volume measures; we haven't even graphed results.  Use of the protractor is yet another form of measurement, though one that isn't used in our daily lives, so I consider this experiment to have led to as much math learning as science.

Factor Trees

BatBoy and SpiderGirl have been playing with multiplication lately.  The Beast Academy workbook (3B) has many problems requiring the use of multiplication and BatBoy is learning a good portion of the lower tables just through use.  In the car one evening, he was talking to me about finding all the combinations that would give a product of 12 (I think) and so I offered to show him factor trees.  SpiderGirl immediately wanted to know all about it too.

When I was in school, factor trees looked like this:

Snowflake Symmetry

We got a new microscope in the summer and around November, we saw some beautiful photographs of snowflakes.  I've also been telling the kids that real snowflakes have six-point symmetry, which they, of course, wanted to see for themselves.  I planned (without saying anything to anyone else) that at the first snowfall, we should bring our microscope outside and take a look.



The first snowfall of the year happened a week or two before Christmas.  I was tired from running around to rehearsals and concerts and parties.  I figured, we can wait until the morning.  Then, I thought, "You know what?  It will probably snow again." 

It never did.  Now, I'm thinking about getting the garden ready for spring.

Look Mom!  I can SEE the six-point symmetry.  She has excellent sight.

Scaling Tangrams

BatBoy loves tangrams and tangram puzzles.  Over the years though, our sets have a few pieces.  When it was suggested that was play with tangrams as part of Chinese New Year celebrations, it seemed like a good excuse to make a new set.  SpiderGirl also wanted to make a set for herself.

Enchanted Learning has a page on tangrams and how to make them.  They do it by folding.  Since we have graph paper, though, it seemed like a good time to introduce scaling diagrams.  We scaled the template on the website by a factor of 4.  The kids did their own multiplication. 

Then they glued the graph paper onto cardstock and cut out their new tangrams!  They made some zodiac animals off the Enchanted Learning page and did some puzzles from Fun O Rama.

Place Value and Bases Other Than 10

Our family loves Penrose, the Mathematical Cat.  Here, one story about numbers in base 2 inspired explorations of number bases in general.  These snapshots show BatBoy using counters to translate numbers between base 10 and other bases, while SpiderGirl looks on and gives input as she is sewing.

10100 in base 2 is 20 in base 10.

Setting up to work in base 3.