## Friday, December 20, 2013

DEVELOPING AND FOLLOWING A ROAD MAP FOR PROBLEM SOLVING IN ALGEBRA

A friend of mine has a daughter who is struggling with the algebraic side of geometry, and, as my friend is beginning to grapple with understanding more about a Charlotte Mason education, she asked my take on helping her daughter with her math.

While the following is not pure CM in any way, it doesn't seem out of line with CM either.

The skill being addressed in this post has to do with developing a workable pathway to follow when trying to solve problems in algebra/geometry.  (The young lady in question is enrolled in geometry, but is working with linear equations, which, in my day, was considered part of the algebra curriculum.)

In short, I compared the basic process of working with algebra to the making of bread.

Since the young lady in question seems to have some weak visual skills, it is hoped that the allegory provides a road map for her to follow.  That way, she might be less likely to feel overwhelmed by the various steps involved, the number of picky details which must be attended to, and she might even feel more confident by knowing exactly where she is in the process.

Ideally, she will make a chart with two columns:  one on the left for the steps in bread baking as delineated below, and one on the right, with corresponding steps for working with algebraic equations.

And now, on to the allegory itself.

...............

1)  To start with, most people these days use a standard recipe for specific types of bread.  They typically set out their recipe of choice in a place where they can quickly refer to it while making the bread.  Unless they have memorized the recipe so solidly that they know they won't forget anything, they continue to set out the recipe each time they make that type of bread. This act of setting out a recipe:

IS LIKE:  knowing the basic formula in geometry or algebra the relates to the type of problem that needs to be solved  (e.g. - the 'recipe' in question might be a basic formula for a quadratic equation, the slope of a line, the formula used to find the perimeter of a rectangle, the area of a triangle, etc., etc., etc., etc.,

ON YOUR PAPER THAT MIGHT LOOK LIKE THIS

(Here we are using a specific problem as an example.  That problem goes like this:  What is the length and width of a rectangle if the perimeter is 36 ft and the length is three feet longer than twice the width'.  So on one's paper, one might write the 'recipe'/formula first, right above the workspace, before working the problem):

P = 2l + 2w

(which is the basic recipe   -or-    the formula for the example problem listed above)

...............

2)  After deciding what type of bread recipe is needed, and a recipe is set out and handy, many people recommend setting out the ingredients for that bread...., which.....

IS LIKE:  reading your problem in your textbook and writing down what is 'given':  e.g. - you might simply write down the equation the text asked you to solve, and/or you may need to write down information from a word problem......

ON YOUR PAPER THAT MIGHT LOOK LIKE THIS;

GIVEN:

P = 36 ft
l = 2w + 3'

...............all of which might be compared with the ingredients for the bread in question.......

3)  Other things might also need to be set out.  In particular, specialized equipment might be required for a specific outcome, such as a casserole pan for cinnamon rolls, or a cookie sheet for a free form loaf, or a tube pan for monkey bread, or etc.  ............... which.............

IS LIKE:  realizing what type of answer you are supposed to produce with your mathwork, such as 'finding the perimeter', 'finding the slope of the line', or etc.

ON YOUR PAPER THAT MIGHT LOOK SOMETHING LIKE THE FOLLOWING:

w = __
l  =  __

(the purpose of doing this is to help one ask, determine, and even remember where one is headed even when one is dealing with pesky little details............)

4)  Making sure you understand the basic process of making bread (proof yeast in warm water, add any 'extra' ingredients such as eggs or butter - yum!....), mix in dry ingredients, kneading the dough while adding more flour gradually until you get the 'right consistency' to the dough, allowing to rise, punching down the dough, kneading a little more, forming the dough, placing on a pan, slashing and/or adding any washes and/or garnishes, allowing to rise again, baking, perhaps brushing with oil or topping with icing, cooling - - - with those last two steps being somewhat interchangeable, depending on the recipe)....................... all of which.............

IS LIKE:  the order of operations in math  (think about it, if you knead the flour before you add the water, it won't do much good....., if you bake the yeast before it is even mixed into the dough, it will even kill the yeast.......... the 'order of operations' is really important.........!!!!)

ON YOUR PAPER, THAT MIGHT ACTUALLY MEAN WRITING DOWN THE ORDER OF OPERATIONS FOR A WHILE (perhaps in the margin or on the back of your paper), AT LEAST UNTIL THEY BECOME ABSOLUTELY AUTOMATIC EVEN AFTER TAKING A BREAK FROM MATH............. (a simplified version of the order of operations might be useful, though the specific steps are still vital; that simplified version with bread might be:  combine, rise, form, rise, bake.... whereas in math it might be something like:  simplify - isolate - solve..... those simplified headings might be able to slip into portions of your order of operations depending on which order of operations your math class is using - -- if not, it is probably important for you to identify which of those 'subheadings' would apply to each step in your class's order of operations)

Your written list might include something much like this (though I didn't look up any specific order of operations, because that depends on your curriculum/teacher);

FOIL method
simplify
combine
isolate your selected variable using opposite functions (but always doing the same thing to both sides of the equation to keep it 'equal and balanced')
simplifying between each 'opposite function' step as needed

5)  Make the dough by combining the ingredients for the bread, following not only the basic process, but also any particulars for a given 'recipe' ..... which...............

IS LIKE:  trying to solve the math problem (setting up an equation and solving it, or just solving an equation that was given to you)

ON YOUR PAPER, YOU SHOULD EITHER SHOW THE EQUATION THAT YOU WERE GIVEN OR THE EQUATION THAT YOU SET UP, THEN SHOW EACH LITTLE STEP ON A SEPARATE LINE, REWRITING AT LEAST HALF OF YOUR EQUATION FOR EACH STEP  (this not only helps you know what step you are on, but which steps you have left to do, keeps you from skipping steps, and allows you to assess any mistakes you made down the road........, people who are struggling with math but who try to do more than one step at a time often miss little picky details - - - so only do one step at a time..........., at least until this is something that is utterly solid for you................, oh!  stick with showing each tiny step  until the order of operations has been utterly solid for quite a while..............

.                         36' = 2(2w + 3') + 2w
.                              =  4w + 6' + 2w
.                              =  6w + 6'
(flip)              6w + 6' = 36'
.               6w + 6 - 6 = 36' - 6'
.                          6w = 30'
.                   6w(1/6) = 30'/6
.                            w = 5'

IS LIKE:  using the results from solving your equation to produce any other data that is needed (in this case, having solved the initial equation for 'w' allows you to solve for any other unknowns, in this case, solving for 'l')

ON YOUR PAPER IT MIGHT LOOK SOMETHING LIKE THIS:

l = 2w + 3'
= 2(5') + 3'
= 10' + 3'
=  13'

7)  Presenting your bread; might include allowing it to cool, and/or topping it with icing or butter/oil, or fresh herbs/pepper, including a side of balsamic vinegar and oil with or without pepper, etc., possibly slicing it, arranging it nicely on a platter, or etc., which....

IS LIKE:  neatly presenting your answer in full........., which sometimes is simply listing the answer(s) ***again*** at the end of your work, or which might require a chart or graph of some sort

ON YOUR PAPER, EACH PORTION OF YOUR ANSWER MUST BE REASONABLY NEAT, LABELED APPROPRIATELY IF NEED BE, CIRCLED IF REQUESTED BY THE TEACHER, AND/OR etc.,   ....  In our example, all that is required is a relatively basic-answer-type-format example, which would look something like this:

w = 5'
l = 13'

..................

So, that is the end of the analogy.

...................

I realize that there are people who can simply look at a math problem and do all of the thinking necessary to solve the problem in their head without writing anything down.

I'm not one of those people who can do math in my head, and, my my friend's daughter, it appears, can't do that either.

So, different people need to approach math somewhat differently - some just thinking through the whole process easily in their head, and some having to find a way to keep track of all of the picky details along the way.

I'm hoping that this little analogy can help some struggling students in their efforts to keep track of those seemingly pesky details, and ?maybe? even help them become solid with math.

(That said, learning to visualize the ideas in a mathematical equation helps these types of students to break down 'math walls'.  Perhaps I'll post on that at some point.  Feel free to let me know if anyone would like that done here on this blog.)