Saturday, November 25, 2017

Math Over 20 Minutes a Day

I'm back in the daily math business thanks to the break.  It beats listening to 2 boys make up video games and verbally describe the action because they are not allowed to play real video games unless they do some really serious math, read, practice their instruments (together without fighting) and clean the kitchen and vacuum the place.

Each child began their official math career with 6 months of math at a level of current plus 2.  (This was Every Day Math, hard but not too hard.)  It's a right of passage and a way to impart senior executive functioning skills, problem solving skills, core learning skills, and grit.  The exercise below is exactly the exact identical same situation, just with different math.  Since then, I haven't worried about an organized program.

When the video game talk became pushing and shoving, I upped the ante.  The fourth grader would get a 6th grade math test from Virginia and the 7th grader got a rigorous high school Algebra II final.

For math at an inappropriate level, we follow these rules.  First, the kids do it.  Since they are missing vocabulary and concepts, because I haven't done current+2 in a long time, we do it again together the second time.  (We've been skimming current+4 and current+6).

I'm going to describe how we do some of these problems so that you'll see that it really isn't about math at all.  It's about a high degree of analysis and problem solving because a) the MAP is a big deal this year for us and b) multi-digit multiplication is 100% useless and distracting from important skills.

Here are the rules for doing math at home:
  1. Let the child do the work on their own first with no help.
  2. When you go through it, pause to take the material on it's own terms because the child hasn't seen it yet and he started with a final and not the actual text book.
  3. Figure out a way to cheat.
#3 is the key to everything including academic and life success.  It's why I expect my kids to open the book the first time the night before the Chemical Bio Organic Genetic Engineering Chemistry final and ace it the next morning.

Here's how it works in action.  As you can see from the picture, we got off to a slow start on the very first question.


The way to cheat on this question is to note a) the answer is in the vicinity of 25 or 26, and since no addend ends with 9 in the 10,000th place, the answer has to be C.  Much more gratifying than multi-digit addition which is used no where in life or in any other class or in college.

Question 3:  6x + 3 = 3(2x + 3)
Here are my comments:  Every time you see x in an equation, be prepared to rearrange and transform the equation.  This is a good place to learn how parentheses work without the spoon feeding and repetition of Pre-Algebra. "x" requires a few long discussions under the heading of Power Bucketing (aka setting up future math) but we already had those discussions.   The equation becomes 3 = 9.  Now pick the answer.  On the SAT, we'll switch to looking for a subset relationship to find the answer and other cheatiness, but for now they need to learn transformations.

Question 5 was even more fun.


It took a while, but we settled on putting 6.23 x 9.3 within the bounds of 6 x 9 and 7 x 10.  You can see this work in the middle of the page, and you can see on the right the framework my son used to actually calculate 6.23 x 9.3 using successive digits.  The cheaty-est way requires the most work, higher order problem solving skills, more creativity and more time.  I love it when a student is excited that cheating turned a 5 minute problem into a 20 second problem, never stopping to think that it took us 20 minutes to get there.  

By the way, one reason to let the child do the work first is because children will most likely resort to calculations and they all need practice in arithmetic.   The main reason is that whatever they answer is expected to be wrong, and Math House loves mistakes.  

Question 6 in the picture is awesome.  It involved a wiki definition of "statistical" followed by an evaluation of each statement on those terms.  'Statistical' is a summary or characterization of the data, and 3 of those answers ask for a single number.  The last one is seeking an average.  Also, statistics usually follows the rest of the material in a math course and in terms of timing usually ends up being taught after the you-know-what test.  I'm not sure how the little one ended up with the correct answer the first time.

The Algebra II final was loaded with solution strategies, most of which were not discussed yet and ended up being more boring to re-work than the 6th grade test.  The breadth of the Algebra II test was good so it was worth it.  On both tests there was this little gem:

Alegra II
Evaluate 3n/(n + 3) + 5/(n - 4) 
Pre-Algebra
Order 1/4, 1/5, and 11/40 



In both cases, you can't compare apples and oranges, or 1/5 and 11/40.  And you can't directly add things with different denominators.  What happens if you multiple something by 1?  Does 100/100 qualify as 1?  What about (n + 4)/(n + 4)?  These are my comments as we meandered to comprehension.

Many parents complain that they don't feel qualified to understand math in a child's terms, let alone teach it.  Here is my response.  What they are really saying is that they don't want to take the time to understand and solve the problems of a) the math on it's own simple terms (the 'simple' takes time to get there) and b) the problem of working slowly with the child until the child has learned some skill.  And yet they expect their child to magically acquire patience in analysis and problem solving skills?

You've got a third problem, which is that at school the child is learning boring spoon fed repetitive work that values memorization, speed, and 100% accuracy.  So if they don't learn the important skills at home from you, they won't learn it.

Learning starts with unknown, slow, and 0% correct.  I'm happy if you feel like you're at this point too.

First, acknowledge that your baffled.  You can be baffled on how to teach this while the child is baffled on what to do.  This sets the right tone and makes everyone comfortable.

Next, look at what you know about the problem.  a) Your child stinks at parentheses, isn't 100% conversant with variables that span R, doesn't see the need for common factors and b) you don't know how to teach them.

Third, come up with a strategy.  It will have the following components:  a) it will take a long time, b) it will be step-by-step, c) you have to back track on something simpler, like having the older child do the fraction problem first and having the younger child compare 1/2 and 1/3 first, d) you will have to reevaluate and try again.

The preceding paragraph is going to be the child's take away.  Someday they'll understand "n", common denominators, how to use parentheses, algebraic transformations and the rest of it.  At least they'll be comfortable when they get to it in school or on a test.  But those problem solving skills starting with 'Baffled' and ending with 'Try Again' are so powerful both of you are on a path to 99%.

I'm going to conclude with a warning about accelerated math in school or after school.  It is now common to teach Algebra II in 7th grade and Geometry in 8th grade, or to skip to Geometry in 8th grade, despite the compelling evidence that the result is many kids quit math early in high school and reporting hating it.  At this young age, speeding along, the kids memorize and learn to use the concepts.  There is little if any time devoted to the key skills that are taught in high school, especially in Geometry, learning adult level problem solving skills doing Geometry proofs.  I would be surprised if any 8th grader could prove to me that a straight line is straight, let alone prove each building block of Euclidean Geometry all the way up to trig, but that's what a rigorous high school class teaches, because it is essential to Calculus.  It's essential to thinking.

We've been doing high school Geometry proofs for the last 6 months with both kids.  Algebra II is a belated detour.  You might imagine that the approach is dramatically different than an accelerated school math program.   In 14 months I'll tell you why this is so important.






3 comments:

  1. I'm struggling with 4th grade math materials. What's the best way to teach my second grader how to solve these questions?

    There were three times as many jelly beans in Jar A as in Jar B. After 2685 jelly beans in Jar A were sold, Jar B had twice as many jelly beans as Jar A. How many more jelly beans were there in Jar A than in Jar B at first?

    Aileen and Barry had an equal number of postcards. After Barry had given Aileen 20 postcards, Aileen had five times as many postcards as Barry. Find their total number of postcards.

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    Replies
    1. This is an awesome question and I'll need at least a post to answer it, maybe a book. For now, it will have to be a post.

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    2. Shortly after I wrote my post, which is chock full of useful advice, I sat down with my 4th grader to do these 2 problems. Please see my comment at the end of that post.

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