X-rated Math: So inappropriate for their cognitive stage it's obscene, and what you can do.

No, I'm not talking about math with explicit language or subject matter.  The subject of this post is math concepts that are presented in a way that does not match the student's cognitive abilities.   If you want to skip down to some practical ideas to help your student, scroll down till you see the pink elephant.   The next bit is just background explanation.

It is happening more and more these days, due to the reality that the people making curriculum decisions are not knowledgeable about the cognitive development of children.  They assume, incorrectly, that the brain is like a container that you can simply pour knowledge into, and the sooner you get started pouring, the more time you'll have to pour more stuff.  They don't realize that the brain develops in specific stages.  In some ways, the early brain is like a colander whose holes will gradually close;  in the beginning, there is plenty of room to fill it with all kinds of critical information, but if you try to put in certain things too early they will simply leak through the holes.

The best plan is to put in the big "rocks" first-- the big ideas about the world and physics and love and math that children learn through play.  Taking turns, falling down, sharing, and quantities.  As the holes begin to shrink, we put in smaller, but still concrete rocks: more detailed ideas like dealing with anger, letter sounds, operations with numbers, and measurement.  These smaller rocks fit tinto the spaces between the bigger rocks.  As the student matures, the "holes" in the brain get smaller and smaller, allowing ever finer, more abstract ideas to settle into the spaces between the bigger concepts. Finally, the holes are completely gone, and even concepts as fluid as sand and water-- calculus, philosophy-- will be retained.

At least, that's the Holt model, based on what I've been taught about cognitive development and what I have observed in my years of teaching infants through adults.   The trouble is, when curriculum does not follow this model, abstract concepts are presented too early and the students simply cannot grasp them.  Or they may be able to hang on to an idea long enough to pass a test, but nothing is retained.  The students' time and the teacher's time is wasted, to say nothing of the potential for the student to give up on the subject entirely.

Case in point:
After a lesson on finding the constant of proportionality, a sixth-grader was recently given a problem like the following to solve (edgenuity.com; granted, it was actually from the website's seventh-grade series, but we're all about advancement these days, so the 6th grader was expected to learn it.)  Just a side note, I am not convinced that giving 6th graders terms like "constant of proportionality" to use even makes sense; something about the abundance of syllables tends to make the brain shut down.

Jenna is developing an equation that will represent the same proportional relationship as the graph:

How will the ordered pair help her when she finds the equation?

a) The product of the coordinates will be a constant term in the equation.
b) The quotient of the coordinates will be a constant term in the equation.
c) The product of the coordinates will be a coefficient in the equation.
d) The quotient of the coordinates will be a coefficient in the equation.


Those of you who have 6th graders don't have to imagine the deer-in-the-headlights look that accompanies this problem.  You've seen it.

So here's the concern:  the 6th-grade brain, generally speaking, is just beginning to be able to think in abstract concepts.  The most basic level of abstraction happened in preschool, learning that numbers can represent quantities: 7 can stand for seven apples.  The next level was learning that symbols can represent operations: 7 - 3 means taking away three from the original seven.  But all of elementary-level math is based on concepts that are no further removed than reality than that-- multiplication of decimals and fractions, long division, early geometry-- all of these ideas can be conceptualized just two degrees away from reality.  The student does not have to think abstractly.  Around the age of twelve, a child's brain matures and will generally grasp that x can stand for a variable quantity, and he may begin to solve for x using simple procedures. But this particular problem asks the student to think through concepts that are not just three degrees removed from reality, but several degrees:

1st degree. The numbers represent actual quantities.
2nd degree. A "product" is a result of multiplication, a "quotient" is a result of division; "constant term" means a number without a variable in an expression, and "coefficient" refers to the number in a term that precedes a variable.  A fraction can be called a quotient because it is essentially division.
3rd degree. The ordered pair (4,6) means that every time x = 4, y = 6.
4th degree. The graph represents a relationship between the quantities.
5th degree. The relationship shown in the graph can be expressed as an equation.
6th degree. The ordered pair can be used to create the equation.

So, to answer the question, the student must think at least six degrees beyond the concrete level.

Here are the thought processes needed to solve this problem:

1. The equation will be in the form y = __ x.
The student needs to know that this means "what must I do to x to make it equal to y?"

2. The student needs to know that the coordinate pairs in the graph represent the related values for x and y.  When x is 4, y is 6.  When x is 0, y is 0.

3.  The question then is, what can you multiply x by to make it equal to y?  This requires a knowledge of fractions, as well as how to use multiplicative inverse to solve for x: if 6 = 4x;  6 = 4(6/4).  So, the equation for the graph is  y = 6/4 x.

4. The 6 and 4 from the coordinate pair are now recognized as the numerator and denominator in the fractional coefficient.  Fractions can be thought of as another way of writing a division problem; which means they are also quotients.  So, the answer to the original problem is d) The quotient of the coordinates will be a coefficient in the equation.

But remember, we are talking about a sixth-grader here.  If he has practiced the procedures enough to be able to find an equation for the graph from the ordered pair, he still may not be able to understand what it is that he has done.  If he understands what he has done, he may not be able to explain it.  And if he can explain the concepts behind the process, he still may be unable to wade through the terminology of products, quotients, coefficients, and constant terms in order to answer the question being asked.  Normal result?  Tears, screams of frustration, total meltdown.  And with enough repetition, unrelieved frustration produces defeat and apathy.

So, if your child were faced with a problem that was beyond her comprehension, how could you help her?  In an ideal world, you would be able to modify the curriculum to fit the student.  If you are homeschooling, by all means choose something that is student-friendly, such as Math-U-See, Mammoth Math, Singapore Math, Key to, Teaching Textbooks-- and every math curriculum worth its ink will offer a placement test so that you can select the appropriate level of its series; never buy a math curriculum simply because it is listed for a certain "grade level."  If your child is in a classroom, however, you aren't likely to have a say in the curriculum or the pace of instruction.  More often than not, even the teacher's hands are tied on those issues.

However, there are still a few ways to help.  In a homework session, for example, if she is already frustrated, crying, or in any way approaching meltdown, have her take a break.  No one thinks best when emotions are high.  She needs to go do something completely different-- walk the dog, play a musical instrument, do a chore, read a book.  Preferably, she will be doing something she enjoys and/or does well.  This is not the time to ask her to practice spelling if she is a poor speller!  But it's not necessarily the time to go play video games or eat a piece of cake, either-- that feels more like a reward for giving up than a temporary switch of brain focus.

When she is calm again, have her look at the problem and identify everything that she recognizes or understands-- whether it is a single word, or what a number represents, or what kind of answer the question is looking for.  For example, in the problem above, the student might know that in the coordinate pair, the first number is x and the second is y.

Next, look at what she doesn't understand.  Is it because of the "mathy" jargon used? Sometimes just knowing the vocabulary isn't enough if there is too much of it in the text.  Help your student translate anything in the problem or answer choices from math jargon into easily understood words.  In the example above, it might help to look at the answer choices first and rewrite the sentences:

If it makes more sense to the student, she should feel free to use symbols instead of words; "x" instead of "multiplication" and "÷" instead of division would be fine.


The final step is to identify what the student still needs to know to solve the problem.  The student may or may not be able to explain what it is that she doesn't understand!  Help her use whatever resources she has available to find answers.  This could be as simple as using the student's textbook to find examples of the given problem.  Or perhaps there is no textbook available, but the student has internet access.  Have her type in as many of the important terms from the problem as she thinks would be helpful.  For example, a quick internet search for "proportional relationship graph equation" brings up several helpful videos on Learnzillion, Khan Academy, and YouTube.  Have her choose the ones that seem most appropriate (some include grade level indicators), most understandable (with graphics), and/or easiest to digest (less than 5 minutes).  After watching a video, have the student decide whether her question was answered, or if she has learned anything helpful towards solving the equation.

If the student returns again and again to the point of frustration, please don't push.  Her brain may just not be ready yet.  Given a few more weeks, months, or in the case of very abstract concepts, a few more years, she can get the concepts.  Try to communicate the same sense of confidence you had when she was learning to ride a bike-- after the first few falls, you might say, "Let's try again tomorrow," but that certainly did not mean that riding a bike was too hard for her.  It just meant that she hadn't had enough practice to develop the skills yet.  It's the same with math.  Back off, stay focused, keep working, and she'll get there.

Meanwhile, make sure she's got the foundational skills of knowing her math facts and has good reading comprehension in other areas.  If she is weak in these areas, help her build them up and the rest will come as her brain matures.  And if she is developing gaps in her learning because the curriculum insists upon pouring in "sand and water" concepts before the holes in the colander are closed, please be ready to help her refill when her brain is ready to contain them.