Marsbarn Designs...

Radical thoughts

Some days, when washing one's hair in the shower, you think about radicals. 

No, not the kind of radicals that have counter culture beliefs (like carob is the same as chocolate), but the square root kind of radicals.  The kind of radicals that make you ask the question what number times itself equals 9. 

Well, that idea's a pretty straightforward one, 3 times 3 equals 9.. and well, -3 times -3 equals 9.  But what about the square root of 12? 

I work as a part time tutor at the local community college, and this week one of the sections of math has started work on radicals, so I've been fielding a lot of queries about how they work.  

And, I found a couple of reoccurring questions, which is why I found myself wondering what was tripping people up as I shampooed my hair. 

So I've been doing a lot of work like this:

We're thinking about the square root of 12, which we can think of the product of the square root of 4 and the square root of 3.  This is handy, because we know the square root of 4 is 2.  We (meaning math-y people) read the simplified form of the square root of 12 as 2 root 3.  

But there's more.  After we know how to simplify radicals, we start adding, subtracting, multiplying, and dividing them.  

And this is where the students I was working with this week started to derail. 
"What am I doing and why am I doing it?" 

There was a collective feeling of being thrown in the deep end of the pool after being told that water is just hydrogen and oxygen. 

I found myself throwing around words like "common terms" and stressing that relationship between the radicand (the number under the square root symbol) and the number in front of it is one of multiplication.

2 is being multiplied by the square root of 3.  I can also think of this has having 2 root 3's.  

Back to my shower, I'm thinking about all the furrowed brows I've seen this week, and it hits me. Mixed numbers. Whole numbers with fractions. Most of these students, when they've seen a number next to another "special" number, with no symbol between them, it's been a whole number next to a fraction. THAT relationship is one of addition. 

This isn't really something I've ever thought about before.  One of the great joys I experience with tutoring is learning from the students I work with.  Working one on one with someone, or several someones, makes you examine your math knowledge with a fine tooth comb until you hit a snarl.  

Consider this comparison:

When multiplying 2 root 3 by 4, we just multiply the coefficient (the number in front).  I explain it to students that they can think of this as being similar to multiplying variables.  2X times 4 equals 8X. 

But, when we think of mixed numbers and multiplication, we can't do this.

And I think this is one of the things that trips people up.  We have taken a familiar form, created new rules, and not really acknowledged that we've changed the landscape.   

What to do?  Be frank.  "Hi, we are throwing a bunch of new rules at you. This might feel overwhelming."

Also, use our language to our advantage.  Say it out loud. 

"2 root 3" sounds different than "1 and one third."  Hear that 'and', that 'and' means addition.  That is the difference here.  It is all in the way we read it.  "2 root 3" means "two times root three" the same way that "2X" means "2 times X".  "1 and one third" means 1 plus one third.  This is a great place to begin to untangle the snarl. 

 

 

Imaginary numbers, a basic primer with some non-mathematical imagery and pretty photos

Imaginary numbers. 
What is the deal?  What are they? 
Why are they freaky… or, wait, are they freaky? 

Some basics to get out the way:

Say we have a number. 
    Like 4.  
We can square that number (or multiply it by itself):
    4x4 and get 16. 

We can also take a negative 4, multiply it by itself
    -4 x-4 and get… 16.  

Two different problems giving the same answer because a negative number multiplied by a negative number gives us a positive number. 

Stay calm. This is just like in English.  An even multiple of negatives gives a positive outcome.  

“I don’t not like you.” means, “I like you”.

Yes? 
(Okay, so there might be some passive aggressive angst going on there, but you get my essential meaning.)

Now, let’s reverse our thinking.  (Mathematics likes to work like that, backward and forwards):

If 4x4=16 and -4x-4=16, then there has to be a reverse button that says what number times itself equals 16. 

This is our “square root”.  The square root of 16 is either 4 or -4. 

Now, what if we wanted to know what number times itself equals a -16?

Well, that’s tricky, since we’ve already established that a negative times a negative equals a positive. 
We can’t pull a ‘real-life’ example that works.  There is a temptation to say,
“That doesn’t work. Full stop. Walk away. Let’s go think about a lovely holiday.”

Mmm… a holiday… wouldn’t it be nice to be able to just appear in Paris, with that internet buddy you’ve never met in real life, and to be able to eat ALL THE PASTRIES.  Yes. That would be nice.  I’m going to suspend my reality and just daydream a bit.

Oh!  What if I did that with math?
What if I just pretended that there was a number times itself that equaled a negative number. 
I could call it i, or my imaginary number. It will be the square root of a negative 1. 

Why -1? Because 1 is special.  I can multiply 1 by any number and end up with that number.  This means I can multiply any number by my special i and it can be imaginary too!  (The magic wand of maths land!)

Why is this a good thing? 

Well, if I suspend my disbelief (just like with movies or TV or a good juicy novel), I can think about things (here mathematical things) that may not exist is ‘real life’.  Just because something doesn’t exist in the touchable world doesn’t mean I can’t learn from it.  Being able to manipulate equations by using my imaginary number i allows me to hold onto information that would otherwise be lost.  Imaginary numbers are a tool for the "what if's" of the mathematical world.  

There is one more thing I should mention. 
If I take my i and multiply it by itself ( i x i ) or i squared, I will get my -1 back.  
It’s like closing a book, and being back in the ‘real’ world.   Or, it’s like getting a cleaning deposit back after leaving a vacation rental (I hear that can actually happen).   

Are imaginary numbers scary? 
Well, is it scary to think about things that don’t exist in a way we can touch? 

I guess the honest answer is ‘sometimes’.  We tend to think of math (and science) as being serious things made of “Real” concepts, but sometimes both fields are grounded not in proven fact or touchable reality, but in purely mental constructs.  There is a blurring of the lines between Real and not-Real. 

This happens in all facets of our lives.  The internet is filled with Real and not-Real things.  One of the advantages of imaginary numbers is that they come with a label.  That wee italic “i” states, 'Imaginary'.  Wouldn’t it be nice if all things from the not-Real realm came with a handy Imaginary tag? 

Multiplication work arounds

There are a lot of methods of double digit multiplication.  Personally, I don't feel that they are all created equal.  Some are cumbersome, some are confusing, and many don't carry forward.  Consider the following three methods: the Familiar, the Frustrating, and the Fabulous.

The Familiar. Yes, there is the traditional algorithm that  most of us learned in school.  This method is still taught, but it's one of several methods.

And yes. When done correctly, it works.  But the honest truth is that it isn't always done correctly.  Things go wrong (often it's that pesky zero in the second partial product that represents our multiplying by a factor of 10). So, having alternatives is a good thing. 

But then there are alternatives like this: 
The Frustrating:

Yes, it looks vaguely familiar, but upon closer inspection, it's just confusing.  Why am I starting with my 10's? How am I supposed to proceed?  What is that cross hatch thing in the middle supposed to be?  Can I expand this to more digits?

Sure, I can create an explanation for this method, but why?  Does it follow into any other methods?  Will my student see this again?  And the answer to both of those last questions is "No."  As far as I can tell, this is a stand alone method. It doesn't help with understanding the traditional algorithm.  It isn't useful in algebra or other higher math. And, trying to expand it to three digit multiplication is migraine inducing.  In my opinion, it is a dead end which confuses both parents and students.  

I vastly prefer this method:

The Fabulous:

This is another evolution of our array method.

The box on the right has the two numbers we are multiplying on the exterior of the box.  They have been separated into 10's and ones for ease of figuring.  Each interior box has a set of partial products, which added together (on the left) give us our total.  (As the student advances, the factors being multiplied in each little box don't need to be written.  This significantly speeds up this method.)

If you are having trouble understanding this method, think of it as an area problem.  You have a rectangle that is divided into smaller rectangles.  By calculating the areas of the interior shapes, you can find the total area. 

Why do I like this method?  Well, it builds on earlier visual methods.  The individual computations are easy. You can easily expand it into more digits (you just add more cells to your box).  And, most importantly for me, this method is built upon in algebra.  It replaces (or acts as a partner to) foiling in algebra. 

Ah yes, our frenemy, foiling.  The old "First, outer, inner, last" method that plagued my high school brain.  

If someone had shown me the box method back in the day,  I would have been a much happier camper.

The box method?  So much sweeter.  Notice how the x factors that can be combined line up on a diagonal?  So nice.  My high school students do not know the panic foiling used to give me, because they use this method instead.  I love a multi-use tool.  It slices, it dices, it reduces mistakes in multiplying and combining. 

So hang in there.  Some of the methods being taught will work for your student, and some won't,  but there is light in the spectrum! 

 

 

eggs & legos & clocks: fractions for the fourth grade set

I have an ongoing fraction-positive crusade with my older pupils.  Embrace
the fraction!  Learn to love it, and it will be your bosom friend.  I'm not sure
what it is about fractions that frighten and intimidate, but they do, and while
calculator created decimals might seem nicer, they aren't.  Decimals can be
messy, tricky, inexact creatures that can get a person in trouble. 

At it's base, a fraction is simple:  The bottom (denominator) number tells you
how many pieces you've cut your item (cookie, pie, cake) into, and the
top number (numerator) tells you how many of those pieces you have.  
Sure, there are rules about adding/subtracting, multiplication, and division,
but there is just so much you can do with a fraction!

In the hands-on, visually driven world of Common Core,  fractions are
introduced using egg cartons and clocks. Why? Well, they are comfortable
items most children have seen, they are cheap, and they contain units of twelve.
Twelve is a very friendly number when it comes to fractions and
equivalent fractions (different looking fractions which have the same value). 

(This blog post is part of an on-going effort to given parents an understanding
of how mathematical elements are being taught currently.)

To start, we'll practice dividing a dozen eggs into equal parts.
Here they've been divided into two equal parts (or halves): 

Three equal parts, or thirds:

Four equal parts, or fourths:

Six equal parts, or sixths:

And twelve equal parts, or twelfths:

Now, you'll notice that I've skipped, 5ths, 7ths, 8ths, etc. 
That's because 12 can't be evenly divided into those parts. 

Once students can recognize the shape of the different fractions,
we can start playing with equivalent fractions.

The fraction 1/2...

is the same as 2/4. 

Or 3/6.

When I work with my students, I often use legos, instead of
eggs.  I have more freedom to use fractions not included 
in the family of twelve (Hello, 8ths!). Also, they take up less
space, and all of my students like playing with legos. And, I have 
bricks that are set units, so it is easier to pick up 1/2 and hold it in
my hand and manipulate it.  It's also easier to show that 1/3 and 
1/4 are different sizes, so I can't directly add them.

Here I'm using the same basic set up of twelve as the 
egg carton, but I've expanded how I've represented the fractions.

Halves:

Thirds:

Fourths:

Sixths:

and twelfths:

Now, what else can I do with this model?
I can add fractions.  I can take a 1/4 piece and show that
if I'm going to add it to a 1/3 piece, I need to break both
sections into twelfths to do it.  Once I do that, I would 
have 7/12 for an answer. Similarly,  subtraction can be 
introduced this way.  

We can also explore the idea that 1/4 of 12 is 3.  Or that 
5/6 of 12 is 10.   To do this, I count the eggs or the pips (the
circles on top of the legos) contained in each fraction.

Improper fractions (where the numerator is larger than the
denominator) and their relationship with mixed numbers 
(whole numbers with fractions) are also easy to manipulate
with this modeling.

This method can also be used with clock faces, as they are
conveniently divided into twelfths.  That pathway opens up
pie shaped graphs, geometric elements, and time work.

I find that my students really like this method of fraction 
manipulation and exploration.  It works for most of them.
They have a deep understanding of why you need common
denominators to add, and how a single fraction can have
multiple representations.  There is often a stumbling point about
why 1/4 is 3 eggs, and 1/3 is 4 eggs (out of twelve), but we sort
that out.

Down sides?
My  main complaint with this model (and with many other
visual based methods) is that I wish the mathematical 
part of fractions was more integrated.   Most of my 4th and 5th
grade students memorize the equivalent fractions in an egg carton.  
They don't actually know  how to mathematically reduce fractions.  
Their knowledge of fractions is tied directly to the fraction families
you can make in a base of 12. If you give them 8ths, they don't
know what to do. Do they have a "deep understanding"? Yes.
But without the numerically driven (non-model) base, it isn't
as flexible as it needs to be. In my experience, this void isn't 
addressed effectively in 5th grade, and then in 6th grade I have 
students who want to draw fraction out in a model.   None of 
my high school students (those who fear fractions, and turn 
to decimals as their (faulty) saviors) had the complete Common
Core base.  It will be interesting to see if the students who 
grow up completely immersed in Common Core, and who
have that model based knowledge at their heart, are more 
comfortable with fractions.  I hope so. 

Dyslexia and Understanding, by the numbers.

I have dyslexia.  I reverse letters, shapes, numbers,
number sequences, and directions. I have tools I constantly
use to coach myself through left and right.  I have good 
days. I have bad days.  And no, this doesn't stop me 
from tutoring math. In fact, I think it makes me better 
at it. 

One of my students has a math teacher who looks 
blankly at her when she says she doesn't understand 
where an element in an equation came from. He cuts
corners in his Algebra 2 class.  He skips steps in his 
equations, because he doesn't need them.  He can see the
sequence, so it's obvious.  We've talked, he and I, about
showing his work, but because these steps are all so 
clear cut to him, he doesn't know when he's doing it. 

There are two kinds of passionate practitioners in a 
craft. There are those who excel naturally, for whom
all the elements come easily. Yes, they work hard,
but they are assembling a kit where the pieces are
preformed.  And then there are those who love
something, and strive, and struggle, and move forward
without preexisting skill.  They create the elements 
they need from scratch. 

Personally, I think all math teachers should have 
had the experience of sitting, ill prepared, for 
a math test.  They should know that pounding of the 
heart, the shaking of the hand, the dry mouth,
the swimming numbers, the cold knowledge that 
there is a pattern, and method, and you haven't 
been able to absorb it.  They should have had 
that bitter taste at the back of the pallet when
they turn in an unfinished test that they know is 
a fail.  

Why?  Because, guaranteed, they will have students
who feel that way.  

Back to the teacher for whom math is always easy,
if you've never had to struggle to learn something,
how can you explain it to others?  If the pattern is
always obvious, how can you describe it?  It is much
harder to put yourself in a place of deficiency, if you've
never been there.  

If you had asked me, twenty five years ago, if I'd ever
be tutoring math, I would have laughed and laughed.
I would have said, "I'm not good at math."

There is a confusion, I think, about proficiency and passion.
I talk to people who respond to the question, "Do you like..?"
by saying "I'm not good at that."  I think that being "good"
at a task isn't the partner of enjoying something.  I don't 
think people should be limited by their innate skills. 
"I cannot do that, because I am not good at it." is one of 
the saddest statements around. 

What happened to turn me around?  
Calculus and physics.  In college, I had a passionate fling 
with physics.  Which meant retaking all of my math,
from the beginning.  And then I met calculus. 
My (eventual) husband and I went to the Devil's Cauldron
for a weekend beach getaway*, and as we watched 
the waves bound and rebound against the rock walls
I thought, "Math can describe all this."  and that 
was the start.  I was not suddenly "Good" at math,
but I was smitten with it. 

I tutor math because I want people to be able to
find the joy and the beauty in it.  I want to take away
the fear and the frustration that comes from not being
"good".  I know that intestine knotting feeling of 
failing a test, or not understanding, or needing another
viewpoint.  I know what it is to have a teacher look blankly
at me, when I say I don't understand. I know how much
harder it is to understand if you are scared.  One of my
math sessions isn't a good math session if there isn't
laughter in there somewhere.

And, if you asked me today, "Are you good at math?"
I would say, "No, but I love it and it brings me joy."
I work at math. I study. I practice. I think about it.
Sometimes, I make mistakes.  Somedays, my Math
Brain suffers, or my dyslexia is stronger.  Sometimes
my students point out my mistakes, and when that
happens, I say, "Good job!  That was awesome! 
What do I need to do?"  

Perfection is overrated. Compassion, isn't. 

 

* Not my video, but it illustrates my point

Evolution of arrays, or why they aren't just one trick ponies

I have been a slow (okay, glacial) convert to arrays.
They appear early in the Common Core curriculum. 
This is what they look like in third grade, where they
are used for basic multiplication skills.

Then, they move onto this in fourth grade:

They start out on grid paper, and, in the beginning,
students are encouraged to manually count squares.
The numbers being multiplied get larger, and are 
broken down into tens and ones.  Students are shown
that larger numbers can be broken down into components,
which can be multiplied and then added together.
Often, these components can be easier to work with
mentally. It's easier to think about 10x8 and 4x8 and 
adding them (80+32) than it is to mentally multiply
14x8.

Then they move away from arrays done to scale, 
where you can count grid squares, to skeleton arrays
that require multiplication work.  Again, larger numbers
are broken down into units of tens and ones, and those 
pieces are added together. Is this shorter than doing 
an algorithm?  No. BUT, this does demonstrate the 
varied way that we can pull apart and reassemble 
multiplication problems, which is HUGE for algebra.

Then, we start to move away from the visual, to
the number driven math:

(I wrote in little number number sentences
to show where the product in each cell came from.)

Here's a purely algebraic version, to illustrate where
the values in each square come from.

Oh, look, distributive property and algebra.
Hey, we're in middle school now!  8th grade, perhaps?

We've got our factored quadratic, and we're going
into standard form.  And, voila, it's work we've been
doing since 4th grade.  Isn't that nifty??
This is where we would foil in old school math.

If foiling is your jam, then you don't need this, but
if you are a student who looses track of what's been
multiplied and what hasn't, this is an easier to follow
method. (And yes, it works in reverse. You can use this 
method to factor a quadratic, but that's a different post.)

I'm not quite done.  This method continues to 
evolve.  Take the following:

Oh yes, imaginary numbers!  This is big kid, high school
math, with things I can't visualize! Three factors being 
multiplied by three factors! But, our array can handle 
it. And look at how pretty it is! I can see my pairs 
that cancel so clearly!

The symmetry and cleanness of this method makes
my little dyslexic heart SO happy.  SO HAPPY.
Now, look below at the method I learned in school.

You tell me this foiling method is better.  That mess 
above.. that is how mistakes are made. That is how
elements are lost and math students cry.  This is where
the new method is faster, neater, kinder, and sweeter.

Do I always love arrays? No.  Sometimes they are slow,
clunky square wheels that make me say, "Put more 
math in your math!" to my students, BUT, it has a 
purpose and it is a great, powerful tool to have. 

Arrays, or one of those mysteries parents don't understand

My main, number one, burning hot issue with common
core, is the alienation of parents in the curriculum.  The
"you can't possibly understand the intricacy of what we're
doing with your child" tone of the math roll out. But here's
the thing, if my fourth grader can understand this and
embrace it, why can't I?  

Take the array. An array is simply a grid which is 
colored in to represent mathematical situations.

 

The array is used in many different configurations in 
this "new" math world. It can be used for multiplication,
division,fractions, square numbers, square roots, decimals,
and factors. Oh, there are more ways to use it than that,
but these are just from the top of my  head.  In short,
arrays are a simple but versatile visual math tool.

They can be expanded to larger multiplication problems to 
demonstrate the commutative property of multiplication.

The purpose of the array above it to try and demonstrate
how more difficult multiplication problems can be broken
down to easier products, which are added together.
Is it successful?  Well, that depends on how you learn. 
No one-sized method fits all.

Here is an example of arrays being used to show
equivalent fractions. Again, a visual method to teach
what, in the past, was taught with numbers.

I primarily see arrays in elementary school, but they do 
pop up in middle school.  They can be used to demonstrate
the concept of square numbers and square roots, for 
instance.

(There is a rather beautiful introduction to the Pythagorean
theorem which uses them.  That particular example needs
a post of its own.)

Is it really the bomb? Meh. I have personal issues with it.  
I like the visual element of teaching math, but I think the
curriculum spends too long on it, and not enough time
transitioning to  number based math.  I see kiddos in 6th
grade who want to draw arrays for fractions or
multiplication, instead of using numbers, and honestly
there is a time when the array holds you back.  It is also
slow, and that can be discouraging to kids who are ready
to move forward.

Arrays are a tool.  I'm a woman who likes tools.  They
don't have to be fix-alls, they just have to have a purpose,
and they shouldn't be mysterious.  

Keeping multiplication facts light and fast

How do you build speed with multiplication,
teach place value, build confidence, reinforce
the joy of being challenged, and listen to a
fourth grader giggle?
Welcome to math tutoring! 

Multiplication facts, times tables, whatever
you want to call them, you need to know them.

Well, okay, you can not know them, and I guess
the world wouldn't end, but your ability to enjoy
the beauty of more complex math would be stunted. 
And if you don't care about that, well, all that repeated
addition is going to get old. 

Our math curriculum, the one that my local school system
uses, which is part of the Common Core system,
doesn't subscribe to multiplication memorization.  
Memorization doesn't denote deep understanding,
which is the goal of Common Core math, so the time 
isn't set aside for refining that convention.

But, as much as arrays and unit blocks might create a
visual for 4x6 which can be expanded to other subjects,
my students are not absorbing the quick answer of 24. 
My fourth grade students, all 5 of them, retreat to
calculation for their multiplication facts.  
Some of them want to draw the array, some use the
mental prompts (like 4 tables are double/double), 
and while having emergency fall back methods
are awesome, having to recompute constantly slows
down the fun of math. We don't want to recreate the
wheel every day, we want to explore on our hot rims. 

As an alternative to doing sheets of timed multiplication
facts (gag me!), we've been doing a series of multiplication
games. My current favorite is a simple dice game using
10 sided dice.  

The set up for this game requires trust, because I'm asking
the students to go quickly and to not worry about making
mistakes.  "I want you to go as fast as you can, and to just
call out the first number that comes to mind." is a
surprisingly difficult thing to ask. (I think it's easy to forget
how vulnerable it leaves us to move quickly in uncertain
areas.)  

I start with a pair of ten sided dice. (For those who don't
game, dice come in a range of sizes and shapes, including,
but not limited to 4, 6, 8, 10, 12 20, and (the rare) 30
sided die.  You can also find a 100 sided dice, but they are
very grumpy to use.)

I roll the dice, and say the two numbers aloud.  They shout
out the product. I encourage speed. I don't want them to
have time to recalculate. If they are mistaken with their
product, I gently correct. If the product they call out is
the answer to another, nearby product,I give the the
correct factors. If they say, "I don't know", I encourage
them to guess. I don't push. I never scold.  This has to feel
safe.  

When they relax and I can see them starting to get the
groove (or grow bored), I swap out one of the ten siders
for a ten/ten sider (Okay, I made that word up. It's a ten
sided dice that has 10, 20, 30, etc on it, not 1, 2, 3).
Now they are multiplying 30x4.  If they balk, I remind
them that they know 3x4. Now they are combining a
factor of 10, or adding a zero to the end of their product.

When that gets easy, I switch out the ten/ten for a 100/ten
dice (100, 200, etc).  I also have a 1000/ten dice
(1000, 2000, etc).  If they are having a good time, I will
mix up the dice and have numbers like 400x7000. And then
it's "Hey, you just created 2,800,000, and that's pretty
damn cool." (For the higher numbers, I let them use a piece
of paper to track the place values created by the added 
zeros.  This can slow down the multiplication facts, but
now we're concentrating on other things, and the
multiplication facts are coming fairly naturally.)

(For clarity, all of my students have already done
multiplication work in 3rd grade.  They have "learned" their
math facts, but aren't fully fluent in their recall. Much of
what they need is confidence and refinement.)

We process the world individually

I was looking over a unit assessment test 
that one of my students had taken in school,
when I was struck by one of his answer sets. 
This student is in 5th grade, and we've been
working together this year to try to get him
to grade level before middle school.

The problem in question had a the outline 
of a shape, and the students needed to  
make at least three "distinct" observations
about the shape, as well as label the shape
or make their own drawings
to show each of their observations.

The shape looked like this:

Under the shape he'd written:
"6 faces
cube
quadrilateral"

His teacher had marked the first answer right,
and the second two as wrong. His answers puzzled 
me.  This is a part of math he is gifted in, and I
was trying to see why he'd answered what he had.
I looked closer at his shape and saw light pencil lines.

Me: "Do you see this shape as a cube? In three dimensions?"
Him:"Yaaaa." (in the tone of 'because that's what it is'). 
Me:  (Nodding. Thinking.)
Him: "I tried to draw in the other lines, but I couldn't get 
it to look right." (This was his trying to meet the second
part of the question.)
Me: "You see it like this." and I drew this:

Him: "Yes. It's a cube. That's what I wrote. But I got it wrong."
Me: "Your teacher was seeing the shape in two dimensions."
Him: "What?"  (Mystified.)
Me: "Two dimensions. Not three. Can you see the shape as
a flat shape?"
Him: (He tilts his head.) "Oh. Ya. It's a hexagon. Oh. That's
what he wanted."

And this is one of the reasons testing is so hard for this
student. He just isn't processing the problems on the test
in the manner the teacher is looking for.  Does he know
what a hexagon is?  Yes.  
Did he accurately describe the more complex shape his 
brain processed?  Yes.
Did he get full marks on the problem,  No.  
He received partial credit for the first answer, which 
is true for both the two dimensional and three
dimensional  shapes. 

So, my further quandary is this. How do I work with him
so that he knows what is being asked in his work?  How
do we train his brain to spot what he's to be looking for?
It's a brilliant quandary, but frustrating when you're the
student who isn't perceiving things like the rest of the class.  

 

Common Core: Educate the parents, Get to the mathy bit, and Communism. A layman's take.

Common Core.  Those are naughty words in a some circles. These are just a few thoughts about Common Core Math.  (Common Core English has totally different issues that are beyond my purview. )

I’m not a math teacher. I don't even play one on TV.  My experiences with the evolution of the “new” math of the Common program is gathered through working with my children and my tutorees.  This is my fifth year tutoring, and each year I’ve experienced the curriculum change through my students.  I primarily work with 5th through 9th graders, and the way my students are spaced, I have a nice cross-section of materials and the way they are evolving.  My thoughts about the math program are my own, and are based solely on my own observations. 

First off,  Common Core took a harpoon gun and nailed both of it’s feet to the deck of the Titanic when they utterly failed to include parents in the how’s and why’s of Common Core Math.  I’ve seen some really interesting and thoughtful reinventions of how math is taught as the new lessons have been fleshed out, but I can’t name a single (non-teacher) parent who has any idea what their child is doing, or why.   I sincerely LOVE some of these new innovations, but by not educating parents with their children, the designers of this plan have lost their best supporters.  Are all parents interested in the details of how math is taught to their children? No.  But, those parents who distrust mysterious re-writing of techniques they were raised on, or who actually want to help when homework is frustrating?  Those are the parents who make up the icebergs that sink unsinkable education programs.  

Through my layman’s observations, the Common Core’s mission is to deepen the understanding of students in mathematical thinking.  They want students to understand the greater how’s and why’s of numbers and functions.  Instead of taking on faith or blind acceptance, they want young people to have a profound knowledge which is backed up by a framework of mathematical thinking.  For example, when I was in grade school we were taught to divide by fractions using the mystical axiom: “Invert and multiply, ours is not to wonder why.”  The new curriculum seeks to teach division with fractions by first teaching multiple ways of seeing fractions, then to teach the interrelated relationship of multiplication and division through a range of activities.  Only with this base do you encourage the students to formulate their own understanding of division with fractions.  All of this teaching is done through multiple methods, which might include physical objects, grids, arrays, coded shapes, words and, finally, numbers.

This brings me to my second serious issue with Common Core math. I love learning multiple ways to solve math problems. I love the ideal of deeper understanding. I am a person who can wake in the night thinking about how many different ways a sum can be calculated, BUT, as much as I appreciate the very thoughtful building of ideas that I see coming from this program, I am often frustrated that there is too little time left to consolidate ideas.   When the day ends, and the understanding has been built, the student must be able to do the simple conjuring of facts.  They must be able to numerically divide fractions. The algorithm of invert and multiply must be known.  It is not prudent, in higher math, to need to draw illustrations to divide fractions.   I find students who understand the Why, but they cannot just Do.   The Guess and Check process lingers way too long.  There isn’t enough emphasis on just Doing The Math.   And this lack of nitty gritty doing is another element of this curriculum that frustrates parents who were raised to simply solve the problem.

My third issue: testing.  Give me the latitude to use this comparison.  The ideals of Common Core Math are akin to the ideals of Communism. (Wait, take a deep breath.)  There is beauty in the idea of Common Core Math, just as there is beauty in the idea of true, pure Communism.  Communism, the idea of co-ownership, of communal living, of pure-hearted workers striving to live together with the single goal of doing their best, what is not to admire?   The rub is when Communism becomes a reality and all of that lack of freedom and poverty kicks in.  It doesn’t live on a large scale like it does in theory.  Common Core Math, with its goal of deep, fluid thinkers who fully grasp mathematical theory is utopian.  The dissolution point is testing.  How do you test thought processes?  If you encourage multiple paths to understanding, how do you demonstrate that you’ve all arrived at the same destination?  Because on the Common Core tests, you can't just given an answer and show your work, they want you to say "WHY".  I would say the majority of school children simply do not have the capacity to explain how they think with clarity.  Just as Communism strips it’s participants of rights and freedoms,  Common Core Testing undermines the tenants it tries to instill. It says, "Think in the these grand expansive ways, and then give me this specific wording as you jump through institutional hoops."  If you could remove the tests from the equation, perhaps Common Core could be as idyllic as it intends to be.  

I have other, wee beefs with individual elements of curriculum here and there, but these three issues are the big ones for me.  Educate and include the parents,  make sure the algorithm can be used after the base is built, and let go of the testing.  

 

My intent is to follow up this blog post with other posts hi-lighting techniques in the new Core that I LOVE.  I mean, seriously, love.  As in, I could build little shrines to.  The build up to quadratic equations?  I am filled with envy that I did not learn them this way.  It’s like finding out that you don’t have to worship a math God that mocks and requires flesh sacrifices.  Instead, there is a kind and benevolent Goddess that just wants you to understand.  (Of course, I firmly believe that quadratics should be able to be factored without divine intervention.  My point is, now that is possible.)