Issue #9

We don't face any major high stakes testing for 2 more years. This is the quiet before the storm. If you're just joining the readership of Competitive Parent Magazine, you're joining at a good time.
Issue #9

A month ago, high school enrollment letters were issued and we met our goal. We sluffed off 5th grade (step 1) and then started in earnest with test prep at the end of 5th grade. While my current 5th grader refused my offer to blow off 5th grade, he's starting the program today whether he wants to or not.

From the Editor
Issue #9 is getting back to the basics.

Today we kicked off the long slow ramp up to the high school selection process. There is a lot of work to do now (end of 5th grade) until game time (middle of 7th grade). Yes, I know how to do high stakes testing on demand with little to no preparation. However, I can't type that fast. It will take me at least 2 years. In addition, growing the core cognitive skills that are needed in high school is much more important than passing a test, and growing cognitive skills takes longer than faking them on a test.

In this issue, I'm closing out 8th grade (data point number one) and getting organized the soft topics like projects, activities and chores, while I gear up for 6th and 7th grade (data point number two). I'm older and wiser now by 3 years. So it's not just about test prep anymore. In the next two years I also want to answer the question, What will it take now so that my future high school child excels academically and socially, in a high pressure environment, and has enough other material on his high school CV to prove that he didn't have to spend all of his time on the AP course load?

Activity Update

It's a bit early to be worrying about college applications. Instead, I'm worrying about laying the groundwork...

Activity update

Activities have unfolded nicely in this house, almost naturally. Like everything else, behind the scenes with a concerted effort on a daily basis to arrange our values, activities, priorities and sanity toward a desirable end goal. The end goal is the exact same thing college administrators are looking for on a college application: fortitude, sociability, hard work, giving back.

I'm going to recommend one mandatory activity and then demonstrate how we pursue two others that have a very high payout. Your chosen activities will vary. I'm make a strong case for the two we picked based on numerous conversations with parents who have little to go on except for a few sensational outcomes taken out of context.

Start with chores

Chores are the foundation of all the qualities that you want your child to take into their clubs, sports, and organizations. A lot has been written about chores. Think 'opposite of the spoiled, privileged child'. Think hard worker. The principles are similar to everything else that's worthwhile.

  • Lead the way. If you hire someone to clean the house, you're sending the wrong signal to your child.
  • Slowly transition all chores to your child, like over a period of two years.
  • Do not assign chorse that involve with hazardous chemicals. I'm not worried about an accident, but I prefer that my lungs bear the brunt of noxious fumes and not my child's.
  • Allow plenty of room for ineffectiveness and sloppiness. Be proud of a vacuumed room even though you can still see dirt on the rug. There will be time later to point this out, but chores are discouraging enough in the beginning without you being a jerk about it.

My favorite benefit of chores is watching the child learn to get things done. The vacuum backpack was their idea. My second favorite benefit of chores is free appliance repair. We've ratcheted up the bar on chores;this is a competitive gifteness blog, after all. Vacuuming is just a start.


Music and math success go well together. This is well known. There are two routes to go with music after that, and I'm going to argue that band is the better of the two.

If you've read the Tiger Mom book, you've seen the first option. Long hours of practice, lessons, stellar performances. I think this route applies to a few people who like this sort of thing and is oversold to the rest of us.

Option B has a much easier entry point. To be a contributing band member, one simply has to practice about 15 minutes a day. It's more of a social activity than reaching excellence. While option A is teaching kid not to quit in the face of relentless rigor, option B is teaching the kid not to quit in the face of bordem.

Or so one would think. We discovered contests (memorize the last and hardest piece in the book), honor band (a weekend of fun, as it turns out), and city band. We found about about city jazz band too late. Lesson learned. In other words, you can work as hard as you want with option B. Based on sitting at math competitions, 90% of high performers in math are perfectly qualified to sit alone for hours working on things, but need more practice in the social skills department. Thus option B.

My proudest moment as a parent happened a few weeks ago at contest. My son had an hour to kill before his performance, so he decided he would memorize the piece. He spent an hour screwing it up, but went into the contest room and nailed it. I'm not proud because he succeeded, I'm proud because I saw him fall short many times before, including the previous year. Option B has a low bar and kids naturally step up on their own without nagging.


Some of the benefits of obtaining Eagle Scout are well known. It's a rare and highly respected little addition to a college application. It represents 7 or 8 years of training in all sorts of things, teamwork, discipline, giving back.

My personal selling point for Scouting is this: on Monday night, most kids are sitting at home in front of screen. My kids are at a Scout meeting learning CPR or planning a trip or teaching the other scouts how to tie rope. We've done enough math. I need my kids to review the achievement list of the younger scouts, grill the newby's, and then approve the list. It's all social and leadership skills and no screens. Weekend activities are even better. When I said 'discipline' above I was thinking about the time we showed up to a campsite at 9:30 pm in freezing rain and the Scouts had to set up the camp as fast an organized as possible while I sat in the car admiring their work. I personally slept in the back of my SUV, but I was thinking proud thoughts while I did it.

Here's the best part of Scouting. It's our little secret. Scouting before 5th grade is nice (aka no screens etc.) but doesn't matter. If you join Scouts any time between the end of 4th grade and 5th grade, your child is starting fresh as the other scouts start from the beginning. In other words, 'Cub Scouting' doesn't count. In my opinion, it just burns out the kids before the real fun begins.

There are thousands of activities to choose from. This summer my older Scout is going to spend a week on a research vessel off the coast of Florida with one of the scout leaders (my wife). Cheap cruise vacation and learning experience all rolled into one. I'm looking forward to herding cattle with the little brother in a few years on a large ranch in New Mexico but some of the other scout leaders are thinking about mountain climbing or the Appalachian trail.

Even better, Scouts accept girls. I am personally cursed with only boys, but if I had a daughter, she'd be a Scout.

If anyone wants to ask questions about scouting, I will answer them. In the Midwest, the scouts own 4 sailing sailboats and 25,000 adventure camping site. When we go to a nuclear lab or an airfield or to launch rockets, 60 year old Eagle Scouts show up to lend their expertise in that field. It's like we're in some secret society that rules the world secretly. The only caveat is that anyone can join. Here's the website for our troop (in addition to hiding in my car during campouts, I'm the web master).

Advanced Projects

All those years of projects and crafts and science experiments with vinegar and baking soda is finally paying off. Today while doing dishes...

Advanced projects

We have an old set of rules that govern getting out of math. I've never spoken these rules, but I think my kids know them.

  • Rule #1 - You can get out of doing math if you do a project of any kind.
  • Rule #2 - If you stall or defer math by reading a book, I won't bother you.
  • Rule #3 - Neither of these rules in any way impact the 'No math, no computer' rule.

Recently my son and I found ourselves staring at each other from opposite ends of the hall way, me rubbing my fingers above a math book, and him slowly reaching a hand into the project closet. He won the duel. I added a corollary to rules #1. If you're going to do a project in the kitchen while I'm trying to eat breakfast, I'm going to up the ante.

How to make candles

The first step in making candles is getting out the candle making kit that we got him when he was the inappropriate age of 6 or 7. Our candle making kit is in the closet with two dozen other age inappropriate kits of all kinds. You can get directions and ingredients on the web and under your kitchen sink for projects, if you prefer, saving you hundreds of dollars.

A bit of science

The next step in making candles is for me to remove as much fun as I possibly can from the activity by turning it into a hard core science project. The only thing I could come up with on such short notice is emulsifiers, since we're combing wax and peppermint oil (don't know why), which probably don't combine. Unfortunately, there was only enough wax left for 3 little candles and we don't stock emulsifiers, with the possible exception of the spray I use to loosen up rusty screws. Instead of charting the results of 30 different recipes, we had to think through not screwing up our one recipe. I call this the NASA approach.

What is the melting point of wax? According to our constant companion wikipedia, parafin wax melts at 320 degrees or so. According to our thermometer, our wax melted at about 160 degrees. What explains the difference? Our wax is obviously not parafin wax. Thinking in action.

Mr. Science wanted dye, but all we had was food coloring. We thought about emulsification, and decided not to risk food coloring. Peppermint oil was risky enough.

Once again, the magic of vocabulary saves the day. If I would have said 'wax melts at 320 degrees', we probably would have boiled the wax and ruined the candles. Knowing the term 'emulsifier' made us skeptical of adding anything to the wax. In 18 months, during the summer before 7th grade, I'll demonstrate taking the Vocabulary Approach to it's insane conclusion. For now, we start everything with vocab.

The difference 7 years make

We've been doing crafts and projects for many years. If I could sum up Postulate One of my pedagogy, it's that learning takes place doing projects and it doesn't take place reading a book, for most subjects, most kids, most of the time. Our school program is all projects all the time. I once asked my oldest how he learns science concepts if all they do are science labs. 'Once a month my teacher hands out a few definitions or makes us read something for about 15 minutes. Then it's more lab work.'

A few years ago, I would have watched my child spend 15 minutes doing nothing in response to being asked to put away the dishes. Putting away the dishes is a daunting task, what with all those individual dishes. Which one to select first? Now, I've got a dish putting away machine. Same with candle production sans directions (couldn't find them). Instead of being lost in the steps as would have been the case a few years ago, he was brainstorming how to upsell his candles. Here is the result.

A note on problem solving skills

When I teach higher order problem solving skills, the lower order problem solving skills like algorithms and solution strategies tend to emerge on their own, with a few exceptions. When you do projects at home, the higher order problem solving skills emerge on their own. Think about that. Projects provide the best learning environment for higher order problem solving skills. Really hard new math problems are a close second.

The High School IQ Leap

With high school four months away, it's time to outline academic goals. I only have three important ones so far....

The high school IQ leap

Years ago when I was studying cognitive skill growth for ages 3 to 5, I came across research that demonstrated a mystery leap in intelligence at the macro level. Roughly 15% of teenagers gained 16% intelligence for no apparent reason. Since then professors have retired, websites have been removed, and I've lost the studies. In 2011, a study was published in the journal Nature that once again showed this effect with a smaller sample size.

I've found in my own research that positive effects on the population as a whole can produce a stunning impact on a population of one. In other words, there is a brass ring worth grabbing and we plan to grab it. I surmise that there is a magical time period where brain development intersects with academic work, hormones, and socio-emotional development to produce the leap. It could be that some academic sleepers wake up to their potential at this age. It could be that IQ tests are flawed. But I'm taking no chances. I'm on the quest for another holy grail.

Where to begin

What is it that produces such a big change in high school sophomores? Is it finally seeing academic material that is worthy of effort? An inspiring teacher or a really good book? Is it sports or other job that teaches hard work and the reward for effort?

I'm quietly organizing a list of things that might help. I've got to keep this super quiet. If word gets out that a PARENT is trying to motivate their CHILD, it will backfire into a big fireball of lameness.

Currently I'm evaluating adult sized books that might be inspiring and lead to endless hours of reading. That's all I got so far. I've managed to keep my son from quitting the trumpet for 5 years, and he's going to high school with a jazz band in a school districts with a deficit of trumpet players. In high school, I became very interested in sports, but I didn't attend a high school of 4,000 with world class athletes. The school doesn't have a Video Game Esports team, which will be primo college app material for the kids who start it.

Progress so far

To pave the way for this breakthrough in my child in a few years, I've done two things. First, I wrote a PC virus that shuts down video game programs every 2 minutes except weeknights from 8pm to 9pm and weekends from 1pm to 4pm. I'm really proud of this. I wrote it in powershell.

Next I set up a guest network and changed the regular network password. There are 2 phones and one computer in the house on the regular network (the parents' devices), and every day I change the guest network password. This not only regulates internet usage, but it makes PS4 playing impossible because the poor PS4 can't keep video games up to date in off hours like it used to.


No, this isn't competitive. This is way beyond competitive, unless you have no internet in the house at all, in which case it is competitive, somewhat.

Begging Summer Math

We begin 5th grade summer math today. We're two months early as usual so I can prep my readers for this activity...

Begging summer math

We're beginning a 2 year math program. This is a repeat of teaching 2nd grade math to a 5 year old in many ways, including the complaining, the pain, the constant 'I can't do this' and the string of incorrect answers. As you can see, we're using a high school SAT test prep book.

In order to gear up for this, I've spent the last 2 months getting mad at internet usage and video games. This is really hard for me to pull off. I started a new job 6 months ago and I've been in front of the computer day and night all weekend fixing problems at my new job. Today I announced no internet usage until 10 hours of chores, music, jogging, and other activities are completed (2 hours of actual activity and 8 hours of complaining and wasting time doing things like reading, the Kindle being the only screen available).

Did you notice that the article's title says 'begging' and not 'beginning'? I didn't notice this until I published it.

The program

Welcome to the SAT program. Here is a little primer on how it works. Between now and next summer, we're going to do about 5 math problems a day from whichever test and problem we finished the last time. This time next year, we'll start in on the reading comp tests and repeat the whole painful ordeal.

The work is both impossibly hard and really easy at the same time. The easy part is our pace of about 5 problems a day. I don't care if my child wants to do the problems and then redo them all a few times, or if he wants to talk through each problem with me as he goes and ask for help, which is "what do I do?" followed by my answer of "the problem", or "how do I do this" followed by "by reading it and doing it". If we get to a problem that requires some knowledge, like exponential operations, we may take the day off and do exponential problems instead. I'm looking for about 30 minutes of effort, not necessarily any achievement.

The hard part is having to do new, unknown, hard, thinking problems on a Saturday morning when other kids are playing video games. But, we're in a gifted program, and the secret of a gifted program is that the top 2 or 3% of academic performers do a lot more than just school. That's why we're here. We're like the travelling team of academics.

Right now my son is trying out various forms of whining and moaning looking for a response from me. My response to him and to you, the reader, is that we will muddle through for the next year, moaning and all, and magically my son will be doing high school level SAT problems with some skill and success. He might even finish the SAT some day and get a decent score.

Why am I doing this insanity?

My goal is that my son is challenged with some insanely hard and new work each day in order to make him think. There isn't a course in school called 'Do new insane work' so we fill this gap at home. When he goes to math competition, he will be blown away by kids who practice routine calculation problems like 1/x or probabilities so that they speed through the timed competition test. I envy these kids; I love competition and if we had the time and effort, we would do both long slow problems and practice on timed tests. But I won't let my son go the speed route because it will have negative consequences to his future academic work.

During our SAT work, week by week, my son is going to pick up a complete set of formidable academic skills along the way. There are some really big ones like jumping into the unknown with courage and fortitude, knowing that the effort will be rewarded with success. There are lots of little skills like reading a problem and knowing whether the easiest way to address it is just to guess or check each problem. (This is a subset of time management skills as well). Since my child doesn't know the correct way to do things, he will solve a few problems the wrong way, and I'll be proud. He'll need this skill in graduate school. There are so many more. I'll point these out when I see them.

Here is a short list of skills that I don't want my son ever to learn. The skill of being able to do something because I just told you how to do it. The skill of being able to do something fast because this is the exact same problem you just did 72 times previously. The skill of doing a problem quickly because it only requires the application of some stupid math calculation or concept. The skill of doing a one-step straightforward problem. The skill of doing math quickly because you practiced your math facts and in doing it quickly without thinking, you don't appreciate the actual math thing on the page because you're too busy calculating the correct answer.

The Big Five are back

My main skills goal right now is just to refresh the Big Five Problem Solving Skills that we learned starting with COGAT test prep at age 4. When you put a 10 year old in front of an SAT test, each problem is going to have a light bulb experience, and it may take 10 minutes and 3 question readings to get there. These are all multi-step problems with unfamiliar math concepts. A high school sophomore would have a completely different experience. When you do grade level work, you don't get need to use the Big Five skills.

Here is an example from today. If mxm7 = m28 and (m5)y = m15, what is the value of x + y? When we tried this type of problem 6 months ago, we ended up backtracking on exponents and algebra for about 3 months. Now we're back. After getting yelled at, I asked him to explain exponential operations to me. This shut him up, because the last time I asked it took him 3 weeks to work out exponents. If anyone cares, let me know. There is an easy way for little kids to work these things out without knowing anything about exponents and never see or memorize a formula again. Nonetheless, this was a hard problem.

Status Update

Round One of my research for middle school is complete. It was high risk, but it ended well. Round Two will refine our methods...

Status update

In this article, I'm going to reveal my current research with a bit more clarity.

We concluded the quest for high school. My approach was based on best practices, extensive Q&A with older parents, an eye toward identifying our goals and lining up the fundamentals under these. Before 6th grade, we set up a plan to prepare for 7th grade, and during seventh executed the plan. Round 2 removes the risk element, now that we know the methods work, so we'll set our objectives a bit higher.

My only misstep was outlining the bare minimum that my child had to accomplish - and then I watched in frustration and horror as he shot for the minimum.

The end goal

Of course, my end goal is graduate school, but with that way off we set our sights on getting into the right high school. There are 5 top nationally ranked high schools in Chicago, believe it or not. Each has a different flavor. There are two high schools that bill themselves as the most competitive and admit a small number of candidates. There is one high school that bills itself as the best but it isn't. And one high school for slackers who will nonetheless go on to great things.

The fifth high school is the biggest, with 1,000 freshman entering each year. It is not as highly ranked simply because it has to lower the cutoff to get to the 1,000 mark. When we looked closer, I noticed two things. First, it has a group of perfect scorers, maybe 300, that is the size of the entire freshman class at the two most competitive schools. Second, with a student body of 4,000 kids, there is an endless array of music, art, theater, clubs to fit any student's emerging agenda. Plus, I calculated using trend data that this school would eventually rise to the top. It's on it's way there already.

I determined that we could get a B, forgo 8 out of 99 points on the big test and still get in. I informed my son. I'll never make this mistake again.

I shouldn't have been shocked to see a B on the fall quarter report card. I'm paraphrasing to keep this article family oriented, but I asked 'Why did you get a B?' My son responded, 'Because I can.' On the big test, all 15 hours of it, we got a stellar math score on Tuesday. On Thursday, my son informed me that he finished the reading section early, as in he knew by the nature of the questions he was already a bit ahead of the cutoff, so he quit early, as in started guessing, as in QUIT. Again, paraphrasing, I said, "You realize you have to get a perfect score on the last exam to get into high school." He responded, "OK". In the end, he was 2.6 points above the cutoff, and I was a nervous wreck. He did get his perfect score on the last exam.

Next steps

I've already got really great plans for high school. I haven't been allowed to see or hear or know anything about school work for 3 years. By court order I had to give my son power of attorney to sign my name to anything from school so I'm in the dark. He takes train, bus or bike to school orientation and similar events. In this context, my 'high school plans' consist of showing him how he can get much better results doing much less work and doing what I can to steer him into the most challenging parts of the program so he'll expend some effort in high school.

I should point out that we're not talking gifted here. We had a dinner with various friends after the high school letters came out; the table was oozing with super intelligence. So I'm not being humble but honest. Edison, Einstein, Childs, Bojaxhiu, Kowalska and many others weren't really gifted. So whatever that thing is that defines a substantial contribution to humanity, that's what I'm aiming for. I'm going to check the box that states [ ] Knows and communicates well with gifted students.

Big plans for 2020 and beyond

Like my previous website, I'm going to lay out my methods and research as we go with Child #2. Right now we're in the quiet before the storm, so you haven't missed anything yet. He's finishing up fifth grade. The fun will begin in a few months. You'll see step-by-step how to get from point A - where we are, to point B - perfect scores for high school entrance.

The goal is to spend as little money and time as possible. We don't just want to get into high school, but to prepare for a strong showing. That's the topic where the money and time will go (again, as frugal as possible on both accounts because I'm cheap and hate to drive places). The academic work doesn't take much time to explain, leaving me plenty of room for articles that deal with the softer topics. In all humility, if there's one thing I know how to do, it's coach a kid to a perfect score on a test. As we go, I'll also delve into additional topics like secrets of an awesome high school experience and how to outdo people who go to Yale without actually having to go to Yale; all I need is a lot of research and experimentation. That's why I have an oldest child.

Issue #8

To say that this issue presents the best achievement in my career of At Home Schooling is such a gross understatement that I can't even find click baity enough click bait to introduce this month's topic. How do you combine math (great pay out of college) with writing (CEO earnings by 40)? The answer is philosophy.
Issue #8
To say that this issue presents the best achievement in my career of At Home Schooling is such a gross understatement that I can't even find click batey enough click bate to introduce this month's topic. How do you combine math (great pay out of college) with writing (CEO earnings by 40)? The answer is philosophy. This field combines the rigors of math problem solving with thinking and writing skills, and throws in a heavy dose of arguing.
From the Editor
Issue #8 is a leap in thinking.

The very first issue of Competitive Parent Magazine was published in My goal with CPM was to refine problem solving skills using Algebra and make some headway into writing skills, with some chores thrown in. That and my kids were getting older. It's one thing to teach the cognitive skills needed to pass the GAT test, but quite another to prepare for long term success. Specifically it takes more writing on my part. The core skill set is still the same whether the child is 4 or 14.

This issue transitions from math to philosophy. There are few high paying careers in STEM, but every large company has a CEO. Can you take a child's mastery of problem solving skills and turn them lose on everything else? I first approached philosophy as a bridge between math and writing. I now think of it as a bridge between thinking about math and thinking about everything else.

I laid the groundwork for philosophy by 6 months of writing exercises. Writing got pretty boring after the first month, so we switched to reading a page or five of something interesting followed by 30 minutes of discussion. I turned to philosophy when I ran out of other topics to discuss.

Fun with philosophy

Most philosophers throughout history were arogant, drunk morons. There is no other way to explain how they could come up with such stupid ideas...

Fun with philosophy

If you have gifted children, defined as average children who are consistently exposed to gifted level exercises, then you probably have argumentative children. This is the hallmark of a gifted child. You may not notice this if you have a single child and you don't hang out at the playground listening to their conversations. If you have multiple children, then every dinner conversation topic (e.g, 'pass me the ketchup') devolves into a vigorous debate over who is right. Mom's hate it. Dad's are too busy arguing to notice, (e.g, the ketchup bottle is in reach, stop moving it, you already have enough ketchup, no that's not the exact amount that belongs on a fry).

Lately, my favorite At Home Schooling exercise is to state the thesis of a philosopher and then invite my children to evaluate how totally right this person is on this particular topic. It's hard not to smile. The first reaction I get is a 10 minute look of disbelief at how wrong I must be while my child unpacks the logic. With a really good argument, it might take a few days.

In fairness to long dead philosophers, we have much going for us today that was not available in the 18th or 19th centuries. Kant and Descartes provided some really powerful tools for thinking that might not have been accepted as widely back then as they are today. The IQ of the average 13 year old today is much higher than the average IQ of an 18th or 19th century philosopher. Eighth graders don't drink wine during breakfast or lunch, and so have a greater capacity to think through a complicated problem after dinner. The only commonality between an 18th or 19th century philosopher and the modern teenager is that both are convinced that they are right and everyone else is stupid.

As a side note, I never understood why The Left decried the narrow focus on studying the Eurocentric view of the world. After all, math is math, and math historians are careful to point out where other cultures discovered Algebra, Trig, Calculus and number theory before it was rediscovered and codified by European academics. Granted, it's well known that historical research is usually wrong; this reputation is well earned by European and American historians who are blind to the fact that there may be something they don't know. Since I started studying philosophy a few weeks ago, my eyes are opened to just how arrogant and clueless the average European philosophers is, and this field would benefit enormously from input of the other continents.

An example of stupidity

The whole field of ontology is based on the same faulty premise as alchemy. According to Anselm's famous logical fallacy, God is the best thing you can think of, but this thing would be better if it existed, therefore God must exist. What?

We thoroughly debated a few of these arguments realizing that the definition of God, no matter how carefully defined, will not logically prove his existence.

On the other hand, Karl Popper's description of what is science versus what is not science is as profound as it is simple. (Thanks Crash Course in Philosophy on youtube). For every 10 philosophy arguments that I read, one of them is really good. The other 9 are the reason I think 8th graders are smarter than 18th and 19th century philosophers.

The never ending debate

Every philosophical concept and argument is fair game for learning. Some of these are not only good for learning, but age appropriate as well.

How do you define the self? I put this question to 3 thirteen year old boys. Here is the list I received from the first one, we'll call him Arthur: "I'm humorous, play video games, and have friends." So if you're travelling abroad and in a bad mood and all you have is a book for company, then you're not you? The next friend, Stuart, proposed "International, sports minded, sarcastic, and not Arthur." Well played. You're still you on a trip.

We didn't even get into the philosophical arguments on self since this discussion took so long. Are you your cells, your brain, a soul, etc. The next morning Arthur announced "I have 4 moods. Humorous, serious, bored, and before a cross country meet I go like this a lot (flailing about)". So he is still himself on a trip abroad. But that's not the point.

Why don't you want to die? "Because I can do fun stuff and death is boring". I had to help here - we don't know death is boring but we don't know that it isn't, but we know for a fact that today's fun stuff is in fact fun. I'm not 100% sure of this, but I don't see this argument in the body of material on the death of philosophy. A fun discussion, but again, not the point.

What is the point?

The point is that the kids are learning to 'see'. I develop this skill in pre-math kids who learn to 'see' shapes and shape attributes in a complicated diagram during COGAT test prep, or 'see' the missing equation in a complicated math word problem at age 8. Now they're learning to 'see' relationships and characteristics and logic that they didn't know existed behind a simple word like 'self'. This is not just a skill, it's a super power. When paired with the super power of vocabulary (ontological, abductive, etc) I've got a kid who's going to do well in AP English. This assumes that their grade school teachers are doing the same thing with literature, which they are, or I would be doing it at home.

Apparently philosophers, despite their hang overs and limited IQ's, knew this all along. Philosophy is about asking questions and extending, changing, refuting, or proposing arguments. In other words, it's really great preparation for getting papers done quickly and not being challenged by college despite taking advanced courses.

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How to torture your child with philosophy

Our At Home Schooling math program got lost on the road to arguing and veered into philosophy. Next we added our normal rigorous methodology...

How to torture your child with philosophy

There are so many different fields within philosophy. The only two that I'm familiar with are logic (known as logic) and the structure of a good argument (known as arguing). There are so many fields. Where to begin?

I settled on this question. Did mathematicians discover the circle, or did they invent it?

The first step

This is a nice baby step. A child as young as 10 should immediately appreciate the need to define 'invent' and 'discover' before answering the question. My job as the team member is to follow up each assertion with a question doubting it's validity. Of course, there is no answer, and I don't care for one. I want to see thinking.

This is also a good exercise for 'going deeper', although ironically my 10 year old went much deeper than my 13 year old. Here's the potential of this problem:

  • Did Columbus discover America even though there were people living here already?
  • Or did he discover it for the queen of Spain who didn't know it existed?
  • Does anyone invent anything?
  • Does every invention exist within an infinite permutation of words, colors and atoms, or do they just pluck it out of the realm of possibilities that already exist? (That was my contribution.)
  • Does a painter create a painting or does he just make one that already exists conceptually?
  • Does he invent the concept or just put together things that already exist?
  • Then is an invention really new or just a combination of things that exist?
  • If I tape a flashlight to a broom, did I invent something?
  • What is a circle? Did they discover it by looking at the sun?
  • Or is it all points equidistant from a center point, in which case it doesn't exist in nature?

The second step

I presume the second step is a framework for tackling the problem. Something like a solution strategy. We're not making progress, just asking more questions. Maybe our questions are the formulation of a strategy. We're not that far along yet as philosophers.

We need baby steps here is well, like assuming the definition of invention and discovery and then applying it to the circle. This is a solution strategy from math. Or we look for examples of invention and discovery and categorize them looking for patterns. Again, math. Or take the simplest of all examples - use a stone to break a rock, and see what comes of it, working our way up from there (stone tied to a stick). Math. The only thing you can't do strategy-wise is try out every answer from a pick list because even though I have one I won't share it. That leaves adding the missing equation, working backwards, and problem decomposition, but these solution strategies naturally emerged from analyzing the problem.

In case you're interested, I just paraphrased the Big Five solution strategies from math plus The Solution Super Strategy (aka the stone). I wonder if the mechanics of philosophy presents an additional strategy or just redefines the usual suspects from math.

The torture

We're getting no where. Switching sides, changing our minds, tossing out steps of reasoning that lead no-where, and asking a lot of questions. My counter questions are frustrating because every answer has multiple follow up questions. I think this is the point of philosophy.

My philosophy partner decided that ideas can be invented, and an invention is the application of the idea or concept to a bunch of prediscovered objects. Each person can discover something or be told about it from the discoverer. The same thing can be discovered by many people. That's why Columbus was able to discover the new world, even though there were people already living there that obviously already knew about it. If you attach a flashlight to a broom, you invented the concept (The Night Sweeper 2000), but you didn't invent or discover its parts.

Good enough for now. Your answers may vary.

Philosophy and Writing

My original concept for an At Home School writing program was something that involved writing. It's morphed into the skills you need to ...

Philosophy and writing

My original concept for an At Home School writing program was something that involved writing. It's morphed into the skills you need to sit down and bang out a solid A paper without a lot of effort. I haven't formalized the list of writing skills. Here's where I am:

  • Writing skills have a little to do with math so we'll start there.
  • Writing skills have a lot to do with philosophy skills which is why we're focusing on philosophy now.
  • As a kid practices skills, he gets faster. After a lot of practice, he moves quickly, avoiding mistakes and dead ends from past experience, characterizing new versus old problems quickly, and applying the skills much more elegantly the next time.
  • There are no new skills in writing, but practicing them in the context of math doesn't help using them in the context of writing. Mostly.

In the last article, I summarized math solution strategy skills in the context of philosophy. In short, the broader list of problem solving skills includes reading the question, devising a strategy, getting the wrong answer and trying again, and checking the work. 'Seeing' and 'Vocabulary' are super powers and not skills but play a major role in all fields. Creating fancy algorithms to solve all problems in this class of problems is another Super Power.

I call these the Big Five Problem solving skills. The first 4 in the list above are the first 4 of the Big Five, and the last one is called bucketing - taking on work that you won't be able to solve until next year and then looking like a genius when you get to it in school because you're already familiar with the material. The 3 Super Powers of problem solving emerge with practice on the right types of problems (aka problems that need seeing, vocab, and algorithms).

Here is my view of the exact same skill set in the context of writing.

  • Understanding the rubric on a writing topic is a bit like reading the question in math and philosophy.
  • The 'question' phase in writing involves additional information gathering and organizing the thoughts.
  • The solution strategy also includes organizing the sub-topics and determining the best way to present them.
  • I feel like getting ready to write and actual writing are two phases that each need application of the Big Five problem solving skills.
  • Making mistakes, trying again, and proofreading are identical and already well known as the secret to good writing.

As I mentioned before, I see no benefit to writing from practicing the Big Five in the context of math. The reason is that the 3 Super Powers don't develop for writing without actual writing. Vocabulary might be an exception, but the seeing and algorithm skills require topical practice. An algorithm in writing is a way of presenting material in an interesting and compelling way. Organizational skills in math which we haven't seen since we tackled geometry proofs aren't going to help a young writer crank out best selling novels like James Patterson.

At one point, we spent time writing. Our records are 4 straight hours crafting improvements to phrasing and word choice (my son did this to spite me) and 90 minutes writing a summary of The Brotherhood of Steel until I cried uncle. Then our insane 7th grade writing teacher made the kids write 7 1/2 hours a day for the last two weeks of school in preparation for more work in 8th grade, so I started focusing on the other parts of writing that don't involve a keyboard. Mainly talking. My kids are learning to 'see' deeply into questions they never saw before, like what is the self, organize and argue. They are developing something to say and practicing it in the context of conversation.

At one point long ago, Plato complained that the introduction of scroll in the education system is ruining education. Similar complaints followed the introduction of the printing press, what with the masses getting a hold of the material reserved for the educated class. I used to think of that when people complain about people standing on the side walk reading mobile devises. Lately I think Plato might have been on to something.

Philopophy in Spain

Normal people show up to work on Monday and say things like 'How was your weekend' and 'Did you see the Bears lose again'. I show up and say things like 'We're studying philosophy - do you known anything about Kierkegaard?' This is approach, while social inept, helps a lot with my ongoing research.

I found out this week (on Monday) that philosophy is taught in high school in Europe. Furthermore, there is a comedy team that ends every set with 'Que va, que va, Yo leo Kierkegaard'. This is very amusing to Spaniards, but I'm not the slightest bit amused that US schools don't teach philosophy. How is the US going to maintain it's superiority complex if our education system is sub par?

Resources for Middle School

Here is my list of resources to help you get started with philosophy. Think of it as the new math or Math 2.0. Same brain part, bigger challenges...

Philosophy resources for middle school

An internet search for 'middle school philosophy' turns up websites that are not helpful at all. You'll find either questions or a directive to create a syllabus. One exception is the Plato website which is loaded with canned lessons (thanks plato website, just what I need). You may note that I stole their picture of bored middle school students having a discussion. We don't find the topics boring at all, such as this topic which involves an explitive.

I'm just beginning to experiment with the material on the Plato webstie. My approach is to peruse philosophy for my own enjoyment, making note of questions or themes that are age appropriate. Some of these are duds, like anything truth or beauty, and some of these are gold like the concept of the self. My go to website to prepare material is Stanford Encyclopedia of Philosophy if you're weird like me. If you're normal, PBS put together a series on youtube call Crash Course in philosophy.

Our syllabus for middle school philosophy is to view all 46 eight minute videos from the Crash Course. The challenge with these videos is that they are too fast for kids. One of my child's teachers uses Crash Course for history, but he slows down the video. My approach is to watch it and propose the questions to my child. Then he has to watch it or he doesn't get to play video games. A discussion argument follows, sometimes for days. Sometimes I wiki the topic to bolster my argument. The topic concludes when my student can articulate his position. It's like the Word Board all over again.

What I'd really like to do is make my kids watch Shelly Kagan's course at Yale. I'm afraid they would be as bored as I am fascinated. Would it help the kids to get a real live taste of a college course? I don't think so. It would just turn them off to philosophy. They can experience it when they're ready to appreciate it.

Jumping into the debate so late in the game, I've noticed that all of the arguments for God, the soul, the afterlife etc are now hopelessly out of date and invalid. I can't tell if the students at Yale came to the same conclusion because I don't grade their papers. For example, the concept of a soul credibly existed because science could not explain the mind. In fact, philosophy is the bucket of knowledge that can't be explained by science. The material in this bucket is shrinking. I think we need a brand new set of unexplained concepts to explain with philosophy. I'm working on it.

Issue #7

This issue explores the topic of math competition from the standpoint of general academic skills. Each article will break down the sub-skills of a state level question...
Issue #7
We kicked off At Home Schooling academic work at age 4. focusing on the core academic skills like reading the question, getting the wrong answer, being baffled and the like. In this issue, I'm going to show how these skills mature in the context of much more difficult problems. Spoiler alert! They don't mature at all. I took these skills from Poyla's seminal work on how to solve high school geometry proofs and Poyla took them from career mathematicians.
From the Editor
Issue #7 has three themes.

We're getting into competitive math. The first theme is roughly 'all competitive math all the time'. In this issue, I'm going to break down how we tackle these problems in the context of higher order problem solving techniques. Only one competitive math website mentions the skill set, and they only mention 4 of the skills. This issue covers foundational skills in anticipation of problem solving strategies.

In this year's Competitive Parent Magazine, I'm going to demonstrate foundational skills, applications to all subjects (not just math), and then delve into advanced solution strategies, not necessarily in any order. The second theme is problem solving strategies. The third theme is application to all subjects. Regardless of the topic, all themes show up in bits and pieces every time we do anything, math or otherwise.

A Time for Guessing

Sometimes the best way to do an problem is the easist. The easiest way is simply to guess and check, adjust and repeat until you have the right answer.

A Time for Guessing

There are three integer values of x that make the equation x3 + 6x2 + 11x + 6 = 0.

This is a difficult problem for any child in grade school. It's nearly impossible for a 10 year old. This class of problems shows up on all tests, including the SAT. Yet my kids have figured out a way to get these problems right and frustrate my ability to teach them math concepts. In this article, I'm going to describe the direct way of doing this problem (their way), the problem solving way (the right way), and the math way which I teach after they get the problem right quickly with no discernible effort (the penalty box way).

The right approach

If this problem shows up on the SAT, my lazy child will just try all 4 answer choices and circle the one that works. This is a real skill with a name and this is how it works.

  • I recognize the math concepts in the problem and go about solving the equation.
  • My son, meanwhile, notices that he doesn't recognize the math concepts and goes about plugging in random numbers until he gets one that works.
  • In the process of plugging in random numbers, sometimes the result is too high, sometimes too low, so he'll adjust accordingly. Sometimes he sees a pattern between 'too high' and 'too low' and adjusts with intent.
  • Just as I'm beginning my work, he announces the correct answer and makes me look stupid and slow.

The problem solving skill is called estimation, and the sub-skill is to recognize a problem that makes this technique the most expedient. Advanced concepts usually fit this category, and younger children have an advantage recognizing things they haven't learned versus an adult who has already learned the concept.

The second sub-skill is iterate and adjust. This takes some practice. A child will adjust the wrong way and make things worse and start to learn what adjusting is all about. In this case, starting with -1, 0, and 1 should point to an 'ah ha' moment because -1 works but the other's don't. A survey of the equation again (aka reread the question) shows why this is the case.

An alternate approach

The acceptable approach is to spend some time understanding the equation under the heading 'read the question'. To understand how this equation works, one plugs in a few numbers to build a mental graph. Since this is a timed test, there are 3 numbers that need to be plugged in; any other number is usually wrong. These numbers are -1, 0, and 1. The correct numbers are the easiest to work with - thus small or trivial ones.

I'm OK with 1,2 and 3, provide it's followed by -1, 0, and 1, but 4 is definitely the wrong approach and results in a lecture. The little one always says 4 to buy some thinking time while I lecture him. I always fall for it.

In every problem we've seen so far, the numbers we plug in are very close to the answers. I think the test makers intentionally put a time waster like this on the test to give the advantage to kids who stop, think, and go through the Big Five problem solving techniques. 'Estimation' and 'Try An Easier Problem' are somewhat related on these problems.

The penalty box

In the event that a child guesses correctly within seconds and ruins my plans for a good 20 minute math problem, I'll follow up the problem with a lecture in question form. They should know all of the answers, because they've heard them before, but they don't regurgitate terminology like '3rd order polynomial with complex roots' quickly no matter how many times they've heard it. Nonetheless, sitting through a lecture - in question form - is good practice for getting thrown into a math class beyond their comfort level.

I'll spare you the questions. Here are the answers.

  • This is a 3rd degree polynomial and therefore has 3 roots according to the fundamental theorem of algebra.
  • I'll draw a few examples. If it only crosses the x axis 2 times, this means that one root is complex
  • New York gives a test in high school with a famous name I forget that expects you to decompose the equation into it's roots, but we are not going to do that because it's tedious and boring.
  • In a 3rd degree polynomial, like a 2nd degree polynomial, once you get past the outside roots, y continues to grow in magnitude.
  • If I'm in a bad mood, we'll delve into the slope of the curve, a 2nd degree polynomial on it's side is not a function, or anything else I can think of until we get to the 20 minute mark.

There's a big difference between a kid sitting in class hearing '3rd degree polynomial with complex roots' for the first time and one who's heard it all before and finally gets to learn what it really means.

The answer

The answer to this question is the roots -1, -2, and -3. The purpose of this question is to test the child's ability to apply basic arithmetic to a complicated, unfamiliar topic and move on quickly to the next question.

I've outlined 3 approaches above that you can use as you're child's academic coach to impart learning skills. If you take the time to not teach math, your child will pick up a variety of really create cognitive and learning skills. The tree examples above involve cognitive sub-skills, resilience in the face of complexity, and becoming inured to tops that are age-inappropriate.

My todo list includes preparation for a future math competition. Based on past experience, we'll never get that far down on the priority list.

Final note - I 'borrowed' the picture above from I like this website. It's thought provoking.

Back to algebra

Until we get to algebra, returning to algebra is a consistent theme in our daily practice until we take the step from solving algebra problems to using algebra to solve them.

Back to algebra

Teaching algebra to a child who is not ready for algebra is like teaching arithmetic to a 2nd grader without letting the child memorize math facts. In the latter situation, the child builds number sense. In the former, the child is building algebra sense.

We started doing algebra problems about a year ago, and now can silently look at a problem that requires algebra and derive the equation not using algebra. All of the basic problem solving skills are required to get to this point, especially the ability to work diligently for an extended period in the face of frustration. The frustration in this case is me watching my child not use algebra. It's worth it.

Here is the problem for this article: Jack London wrote Moon Face in 68 days. If he would have written an extra page a day, it would have taken only 51 days. How many pages are in the book?

False start

A child steeped in problem solving might say 'n times 17 = 51'. Is this the correct answer? I don't know. I have to work it out, because I didn't have an academic coach with the forsight to not teach me algebra when algebra is so obviously needed.

I presume that the child split this problem into 2 sub problems on the way to 17n = 51 but can't untwist the jumble of thinking to explain how he did it.

Where genius is born

If you withhold prepackaged math concepts from your child like carry the one in double digit addition, long division, and in this case algebra, you're opening the door to advanced thinking. Once you provide pre-packaged solution methods, the child's brain will shut down while they follow the steps in order and reduce the complexity of advanced math to applying arithmetic steps.

The time is coming in their academic career when the child will be required to think. The whole purpose of an academic career is to get the child to the point of thinking. A child who applies formulas and mechanisms to solve problems year-after-year will get to the point where thinking is required and fall apart academically. It usually happens in pre-algebra, if you are lucky and there's time to rectify the problem, but may happen in high school where the only recourse left is a tutor or the school's psychologist.

Coaching genius

At some point the child will a) have the correct answer, b) have an incorrect answer, or c) be totally stuck.

What's the difference between coaching a genius and coaching a complete math moron? Nothing other than how many of these problems you've done. Every math moron is a genius waiting to happen. If you think you have a math genius, you're using problems that are too easy.

Working out the answer

When it's time to check the answer, I'll work out the answer verbally and with step-by-step algebraic equations. I ask a question for each step like 'How many pages is in the book if Jack wrote for 68 days?' 68p = n. Write the equation to describe writing an extra page a day for 51 days (not me, I'll wait 30 minutes for this) which is 51(p+1) = n. Now what do we do?

We do about 3 of these problems a week. At some point last year, I explained that n =n, and the fundamental theorem of algebra means that you can't solve 1 equation with 2 variables (does it?) so 68p = 51(p+1).

If it takes 10 minutes to answer this problem, and I don't see a step-by-step solution, it will take 20 minutes to do it properly with algebra. I'm inching toward the point where I can ask for the child to write down the step-by-step equations without my assistance. We're not there yet. I think my older one is there, because he's studying algebra in math class, but I'm out of the picture so I can't say for sure.

Getting to algebra

How long will it take for my little student to stop working out problems mentally and start using algebra? When these problems are no longer challenging, he'll start using algebra. It's as if his brain is trying to eek out perfect comprehension before substituting mental effort with abstract equations.

The bonus

A child with number sense knows he's wrong when he gets 15 x .3 = 45. A child with algebra sense needs no formulas. He will eventually need algebraic equations, but topics like profit and probability are just known without help. He'll have a dozen or so internal algorithms that he created himself. Occasionally I see one of these in action and I'm in awe.

It is impossible to use this pedagogy in a classroom setting. If I were to use this method in the average American 6th grade, parents would watch their children get most problems wrong between September and May, and these kids would probably fail a pre-algebra test on May 31. Then by June 15, they would ace anything on the SAT. I would be fired in October, so it's a mute point.

Solving all questions

I'm half finished writing this issue and I realize that I want to write an article on every question from every released test. Instead...

Solving all questions

Every time we practice a problem, I want to write an article on the process, because it's worth an article. Instead of a 30 article issue, I'm going to summarize the process in this article. In our case, we're tackling the team level questions from state level competition. Me and my 10 year old buddy. We both pitch in - it's a team sport, after all - and nail each question. Why schools don't teach this framework is the latest in a long series of shortcomings in math education.

Practicing rigorous math at home is a team effort between parent and child. In competition, the team is comprised of 5 students, and they can divvy up ownership of the 5 steps below. All 5 members contribute to each step, but one child is assigned the role of gate keeper. With the parent-child team, I like to let the child do all the work and be the gate-keeper for all 5 steps.

Here is my example question for this article. I'm paraphrasing the question. On an actual test, the question would be worded in a much more confusing, vague style with more sentences and clauses.

The math coach passes out 10 felt tip markers to each team member. Bill and Joe will always receive the same number of felt tip markers. How many different ways can the coach distribute markers to the team?
Step 1 - Read the question

I like to jump right into solving the problem and get it wrong a few times because I missed subtleties, implications, and missing parts of the directions. Don't we all. The first and most important task is to get the question right. That's more than half the battle and this step is the hard one.

There is one complexity in the problem above and one missing element that is critical to the answer. Step 1 continues through step 2. Reading this question should take some time.

Step 2 - Understand the question

In the previous article, understanding the question means trying a few values of x to see how the equation behaves. In this problem, understanding the question means brainstorming different distributions of markers or laying out at a high level groups of distributions. If a diagram were applicable (think geometry), the diagram would be mandatory and might solve the problem with no additional work.

During this process, my teammate said 'OK, Bill and Joe each get zero and the other 10 pens go to the other three kids.' After he said this, I realized that I read the problem wrong. Where does it say that you can't give zero pens to one of the kids? Is zero a valid distribution? Yes it is. Kids who read questions quickly might miss or forget that Bill and Joe get the same number of pens. Kids who haven't screwed up numerous problems in the past might miss the trivial solution, and the trivial solution is about 40% of the total count of permutations. Sometimes the trivial solution is referred to as the corner solution. (What's the maximum value the graph y = 4x + 3 can take if y is less than or equal to 30? Draw the graph and the constraint and you'll see the corner.)

We have a rule in this house. "Always start with the easiest possible answer, and the easiest possible answer is always zero." This rule doesn't apply to this problem. The rule that applies to this problem is 'Don't ignore trivial solutions' and trivial solutions often involve zero.

After half our team re-read the question again, we determined that we both understood the question.

Step 3 - Devise the strategy

This step will take many issues to elaborate because these strategies are fundamental to success in all types of endeavors. Success in math does not mean that the child can win math competitions. It means that the child does well in all subjects that depend on strategy, including doctoral level history and literature, not to mention science. Math is the raw application of logic and problem solving that will be used elsewhere.

In this case, the question is the last one on the state competition test. The kids probably have 4 minutes or less to solve it. The strategy is to group permutations in buckets, solve the buckets, and aggregate the permutations. Divide and conquer. We keyed off of Bill and Joe, with buckets of 0-0, 1-1, ... , 5-5. Order least-to-greatest or greatest-to-least is imperative. This was my contribution because I felt bad about missing the trivial solution and wanted to pull my weight.

How did we get to this strategy? Our overriding objective in math is to find the fastest easiest cheatiest way to the answer. Lots of counting is tedious and error prone. Grouping and order is required.

Step 4 - Get off track and adjust

Success doesn't result from perfection. Success results from fixing mistakes. Mistakes can only be fixed if the kid is not flustered by their own incompetence and dispassionately continues the struggle.

This problem can only be solved by an ordered list. We started with 0-0-10-0-0, 0-0-9-1-0, 0-0-9-0-1, 0-0-8-2-0, etc. At one point, we skipped 0-0-6-4-0 and went right to 0-0-6-2-2. It wasn't intentional, but we found groupings of 6 and groupings of 3 - a huge time saver - and in the excitement of overloaded working memory lost our disciplined order. In my role as strategy enforcer I raised the flag. Getting out of numerical order almost guarantees permutations will be skipped.

Most problems have 2 or 3 viable strategies, one of which will yield the answer and 2 which require a do-over. With different categories of problems, this step might be called 'get it wrong and try again' or 'get nowhere and go with plan B.' A child who works with problems that are time consuming, frustrating and error prone, instead of the spoon feeding from school text books, is a big leap closer to life success.

Check the work

On a math exam or a test like the SAT, this step can be worth 15% of the final score. It usually involves recalculating or calculating a different way. In our case, we were delaying a pizza dinner, and stopped as soon as we had our first version of the answer. The team member in charge of checking already had his coat and car keys. My older son, a veteran of math competition and the SAT, mentioned that no one ever finishes, let alone checks answers. Under these conditions, our easy-cheaty way of working is superior because easy-cheaty means less mistakes.

The solution

I don't remember our solution. The trivial distribution where Bill and Joe each get zero is 72 distributions and the total is around 170. This article provides a partial list of all the things the team learned doing the problem. That is why solutions are so pointless to education. If we had 172 and the actual answer was 184, would I diminish the achievements in learning by ending 20 minutes of effort with 'You got it wrong.'

To be more precise, the diameter of my child's brain grew 1/2 inch during this problem. He improved on at least 6 different sub skills and I didn't have to yell at him like I always do on certain elements of technique. Knowing the solution isn't going to add any additional benefits. If more parents took this approach, Stanford would have to put trailers in the parking lot to handle the overflow in math majors. Instead, they have 2 or 3 freshman sign up for the math department every year.

During 2nd and 3rd grade, while we were learning to make mistakes and check the work, I repeated "Wrong, try again" over and over again until neither of us cared. In this case, I said 'Good job. I think we either got the right answer or got close. Get in the car so we can go to dinner.'

Not Really Math

The first question uses only two skills, but demonstrates them in a powerful way...

Not Really Math

How many of the ways that 27 cents can be made using quarters, dimes, nickels and pennies use and odd number of coins?

I rate this question about a 5 out of 10 because it has a few things going on at once and it's poorly worded so that the child has to read it a few times and think. This is warm up questions for a nervous middle school team, but a great question for 5th grader. On a 40 minute, 8 question test, this question should take no more than a few minutes. We spent about 15 minutes on it.

The skill

There are two ways to do this problem. You can spend 15 minutes on the question and 1 minute getting the answer right, or one minute on the question and 15 minutes getting it wrong. We don't use solutions. If the child does the problem correctly, the answer will be obvious to parent and child or easily provable. If it's not, I consider the question wrong. The 'right way' is the payoff, the solutions never show more than a single number, and I'm interested in the learning process and not the final number.

When I'm not helping as the missing team member, I usually look at the answer and declare it wrong. My trainee does it again, usually getting a better answer or learns to verbalize his defense.

The right way

There is only one correct way to do this problem. That would be the easiest cheatiest way or organize the work so that the least amount of effort is expended finding the qualifying combination of coins. Solving this problem is like coming up with 'alphabetizing' the qualifying combinations.

The actual result

We started brainstorming combinations. This technique is a way to understand the problem and works in other cases. In this case, it led to an unverifiable mess of combinations.

The only correct answer - the answer that a 5th grader is unlikely to come up with on the first try, is to start with quarters (one), them move to dimes (2 and then 1, in that order), and on to nickles. Any other approach is too much work for me to grade.

The problem seems deceptively easy if you've never seen one of these before (which is the whole point). I asked 'How do you know you have the correct answer' and with a bit of staring, we found some missing combinations. The faster this problem goes, the worse the result.