Fractions in 2nd Grade

Last month, a reader asked me how to teach fractions to a 2nd grader.  I'm assuming that the reader is asking me to provide the magic formula for a child of a specific skill set (which I haven't measured) paired with a parent who has a specific skill set (which I haven't measured).  This is a tough problem that requires the Force.  I retreated to the island of Skellig Michael off the coast of Ireland so I could meditate.

This article is going to describe how to teach a math topic (in this case fractions).  The approach for fractions shares a lot with other topics.  Fractions is especially important in math because it is slightly abstract and always multi-step, follows 3 straight years of spoon-feeding one step problems in school, and therefore befuddles students and parents alike for want of cognitive skills.

The Parent

Before we launch into this topic as a parent/coach/Jedi Master, we need to take a step back to appreciate the importance of this math topic.  It is fascinating, alluring, captivating.  Without this appreciation, it's hard to pass our love of math to our children.  If your child is ready to take on fractions at a young age, you will convince no one by saying "Fractions are totally boring and pointless to me but I'm going to make you do them anyway".

What are our goals for fractions?  We want the child to know fractions so that they can look at a super hard problem and see the answer right away.  We want the child to emerge from these fraction studies with a formidable skill set that can be applied to other quantitative areas.   We want to present this child with something like exponents or algebra someday and the child will say "Leave me alone.  I can do this all by myself".

There are some years from K to 8 where you want your child to do 45 minutes of math a day and be really, really good at it.  In our case, there were 3 of these years.  I wouldn't do it every year because the child will learn to hate math, and it's not necessary anyway.  If this happens to be one of these years for your child, in addition to what I describe in this article, get them a decent workbook that includes fractions and make them do every single problem in the book no matter how long it takes.  Otherwise, just do what I recommend here.

Fraction Foundation
The first thing we need is a high level definition of fractions.   When you divide 20 by 4, you end up with 5.  This means splitting 20 into 4 groups gives us 5 in each group.  If you have 20 skittles, but I'm only going to let you eat 1/4 of them, you're only going to get 5.  These are two different concepts, but the exact same mathematical operation, namely 20 ÷ 4 = 5.

What does it mean to divide 7 by 2?  What does it mean to divide 1 by 3?

There are two times when we have these discussions.  The first time is when I think we're going to be studying fractions in a few months or next year.  I call this Power Bucketing.  This discussion will create a brand new bucket in the child's brain called 'Fractions', and when the child sees fractions in school, while the other kids are trying to come to terms with fractions, my child will already have an empty filing cabinet in their brain for fractions and will have a permanent head start in this area.

When I teach fractions, we spend the first week just asking what fractions are.  I will give the child 10 to 20 minutes for them to think through these simple problems, like 7 ÷ 2 = 3 1/2.  After we've exhausted the mental capacities of the child, I'll ask for a picture or show them how to diagram fractions.

If you look through 2nd, 3rd, and 4th grade math curriculum on the topic of fractions, fractions are introduced slowly.  I'm not going to speed through this process.  Please view this Ted Talk on J.J. Abrams from 2007.  Look at that box with a question mark (in the video).  As long as the box is sealed, your child's imagination is in play.  As soon as you open it and describe its contents, you've ruined your child.  Let the child figure out what is in the box on their own.

Fraction Lifestyle
At this point, you can introduce fractions into your conversation.  Think about a really smart parent with multiple PhD's who just talks their child into Stanford.  We want to be like this parent, only not as nerdy.  The two most obvious uses of fractions are time and baking.  Get your child a brownie mix and make them do all of the work.  Put post it notes on the refrigerator reminding yourself to talk about time only in fractions, as in 'it's 1/3 past 5, what time is it?'  By the way, my older child has been in charge of making desert for years thanks to fractions.

Fraction Overdrive
This page from IXL describes the basic fraction related skills expected of 4th graders.  You can also look at grades 5 and 6 because fractions is going to appear every year from now on.  I didn't read any of it because it's too boring.

Instead, like all topics in math before calculus, with the exception of geometry, we simply have to state the obvious.  How to you add, subtract, multiple, and divide fractions?  Throw in 2 more operators (greater than and less than) and transformations (aka equals) and that's pretty much our goal.  This is exactly 8 things to learn (transformations are 2 things - equivalent fractions and transformation to and from mixed fractions).

This little exercise is going to be repeated with rational numbers, exponents, complex numbers, and other pre-pre-algebra topics.  When this child is doing algebra for the first time at age 9, and is stuck while trying to reduce a really complicated algebraic expression, I say 'Dude, you've only got 4 possible operations - addition, subtraction, multiplication and division, just try all 4 of them to see which one works.'

When your child sees decimals and percentages some day, we'll have 2 additional transformations involving fractions.

Where did I get all of this material?  I spent a month thinking about it.  Your child's teacher does not have a month to spend on fractions because there are typically 6 to 8 topics each day, plus statistics.  With some math topics, I also wiki and read about Egyptian or Babylonian history.  Your child's teacher won't have time to do this either.  She has 8 subjects and 30 kids of cognitive profiles to teach.  You have 1 child and fractions.

The Student

Children are natural learners.  Once the parent is prepared (95% of the battle), the rest is easy.  Just give your child as long as it takes and don't help at all.  Ask the questions and expect your child to work things out mentally, when your child doesn't succeed, ask them to draw the picture.  Help as needed, but only after the child has exhausted their mental faculties.  I generally observe mental exhaustion takes place at about 20 or 25 minutes (because I always choose really hard material), and I'm prepared to sit there, sometimes silently, for 20 or 25 minutes.

If you hand a 4th grade book to your child, there will be gaps hidden in 2nd and 3rd grade material.  The child will get stuck on a problem, and the way forward is material that they either never had or never mastered.   Be prepared at all times to go back to 2nd or 3rd grade material as needed.  Suppose they get stuck on a really hard problem, and you can see that it involves transforming from mixed fractions or comparisons.  Take a few days off and do some problems involving mixed fractions or comparisons.  IXL is good for this.

Step 1 - Comparisons and Transformations
I'm not sure why a book would be needed at all.  The most important fractions are 1/2, 1/3, 1/4 ... 1/10.  If the denominator is greater than 2, then you've got 2/3, 2/4, 2/5 ...,  3/4, 3/5, 3/6... and so on.  Then you can multiple any fraction by 2/2, 3/3, 4/4 and you've got a set of un-reduced equivalent fractions.

Pick any 2, and ask for >, <  or  =.

If your child was adept at division and had a really strong number sense, I would not create flash cards to drill my child on fraction comparisons.  If your child did not have a strong number sense because they never had really great curriculum at age 4 or 5 that built number sense, I would not only create flash cards, but I would create spread sheets with 100's of problems from the fraction list and drill the child until their number sense was invincible.  In our case, we did this at age 4 with SSCC and never looked back.  Except when we did this again.  And one other time.

You can search the internet for "comparing fractions worksheet" and see thousands of examples.  If I never met your child and you only gave me 30 minutes to address "comparing fractions", I would print 3 of these: One with pictures, one with simple fractions, and one for harder fractions (involving primes versus composite numbers, like 12/13 versus 10/12) and I would find out quickly where they are.

This exercise requires transformations - like comparing 5/6 and 10/12.   This is a two step problem.  5/6 and 9/12 would be a more obvious two step problem.

Throw an integer in, like 2 1/3, and you've got the other transformation to get to 7/3.  We never go in the other direction, from 7/3 to 2 1/3.  When I see this in a book, I comment that this is lame.  In higher order math, we only work with 7/3, or 142/25, and never mixed fractions.  Also, as I mentioned before 6 ÷ 3 = 6/3 (this is impossible to write in a vertical line, but basically I'm writing division problems as fractions and never using ÷ again).

Step 2 - The Other Arithmatic Operators
Once we 'get' fractions and practice transformations, we have to tackle addition and subtraction, then multiplication and division.

Addition and subtraction involves transformation.  We can't add apples and oranges.  We have to transform one or both.  This is why transforming and comparing fractions is a prerequisite.

Pictures might help if you didn't spend any time doing step 1.  We usually just skip to the hard parts, but you need to read Step 4 below to see why.

Note that this is a 2 or 3 step problem.   These types of problems reward a child who works slowly and a parent who doesn't expect correct answers.  If the child is expected to do a lot of problems, expected to get them correct, and expect to do them quickly, the child will fail at multi-step problems.  Because of this, I have settled on one or 'a couple' of problems as our daily routine until the child builds speed.

If this child was 10 years old, I would expect the child to devise and explain a formula for adding fractions.  Before this age I never even hint that there is such a thing as a formula.  I want the child to go through the 5 or 6 substeps every time, using working memory, because amazing and surprising subskills will develop in that child's brain that will pay off in a big way later on.

For an 8 or 9 year old, I would want to see a picture and an explanation of what is happening.  I would also try out 1/2 + 1/3, 1/3 + 1/4 etc from the list I explained above.  But I would do this every time he was stuck on something like 5/11 + 2/3, because this age desperately needs intuition number sense and now's the time to develop it.  This is really going to slow down the topic, but if you do it right, you'll save many years later on not having to explain math topics.

Multiplication and division require starting all over again with this article, both parent and child section, with each operation.   What does it mean to divide 3 by 1/2 or 7 by 1/2?  What does it mean to divide 1 by 1/3?  How about dividing 4/5 by 2/3?  The same basic questions are asked about multiplying fractions.  What is 1/3 times 3/4?  Before algebra in about 6th or 7th grade, I would want this child to think through the meaning of these problems every time instead of just turning 4/5 x 2/3 into (4 x 2)/(5 x 3), because if the child skips thinking through these each time, they will get to algebra ready to calculate but unable to understand.  This approach precludes some problems and precludes lots of practice.  This approach involves a few problems over a much longer period.

Diagrams work really well in understanding multiplication and division.  These will be articles on their own so I'm not going to cover it here.  Have you ever read a history book that starts with the beginning of time, evolution, 40,000 years ago etc until it gets to the main topic, which might be 1972?  That's how I handle these topics.

How Bad Can It Be?
The biggest challenge with teaching your child math is coming to terms with how stupid your child is.  You're doing something that you just did the day before, and your child not only forgot what he learned the day before, he can't even add.  He does a single problem in 30 minutes and it's totally wrong.  There are 29 problems on the page that are not completed.  It's a disaster.

This is the make or break moment in your child's academic career.  You have the choice between a future surgeon with join doctorate degrees in Sumerian literature and Bioengineering, or a kid who drops out of community college to form a rock band.  The choice is yours.

I'm usually pretty pleased and announce that will pick up problem #2 the next day.  I can do this because in the futile mess I see cognitive skills developing.  Within a few months, my child is making adequate progress and I'm looking for books on Sumerian literature on Amazon.

Sometimes  I am discouraged and ask how he could possibly screw up such a simple problem.  After I say something like this, he will spend the next few weeks perfecting a base guitar riff.

How Good It Can Be
Once you've taken on a few topics like this once, each successive topic is easier and more fun.  The key is that 6 to 8 months of hard work pays off, and you can see that doing a single problem for 2 weeks and getting nowhere is normal and leads to ripping through pages down the road and eating math for lunch.  For a parent, it requires nothing short of faith to get through the first few weeks.

For those of you who took my advice to do EDM Grade 2 in Kindergarten, you already know this.  For those of you who do TPM, which is not all that mathy but is really thinky, you're ready to start.  Unfortunately, in both cases, nothing ever gets easier and you still have to go through the whole painful learning curve with new maths.  But doing Algebra II with a 9 year old and going through a painful learning curve is much more gratifying than doing decimals and going through no learning curve.

Last week, my child was struggling on a problem from his Algebra I final exam.  We stopped using math books altogether and just take tests, figuring things out on the spot.  Sometimes, we'll take a break and do some worksheets on a new topic.  Anyway, there were 4 maths involved in this topic, and he didn't know 3.  He didn't even know the formula for the area of a circle.  It took us over an hour to do a single problem, what with all the backtracking.

Then I realized I accidentally grabbed the Algebra II final.  When we went back to the Algebra I final, he had 6 questions of the form "What is 42% of 66?" and didn't know how to do them.  Arrrgggghhhh!

In each case, we took apart, figured out, and mastered new topics on the spot.  This is the skill set that I want.  This is the skill set behind the MAP test, for very important reasons.  If you can get this skill set down early on, say fractions, then it's just a matter of plowing through pre-algebra, functions, algebra, geometry, trigonometry, calculus (AB and BC), linear algebra, real analysis and series, and then statistics. 

Post Holiday Math

We are in day 2 of a 2 week holiday break.  Day 1 was a holiday and I have a hard time convincing anyone to do any math.  My kids sat around all day having fun, eating, chatting and helping with chores.

Math starts today.

Daily math is a prerequisite of the kids doing anything fun.  The kids say, "I don't want to do anything fun and I'm not doing any math!"  Then they read, do crafts, engage in an imagination-building-problem solving activity like Legos in order to not do any math.  It's quite amusing to me when I walk by their room, and they are sitting there reading for hours, and they look at me like 'Ha, ha, I don't have to do your stupid math, I'm just going to read.  I win.'

Reading is way more important than math.  The jokes on them and I'm not telling.

Daily math started with the simple thought, "If a child becomes a strong reader and thinker because he reads daily, how is he going to become at STEM?"  The answer was daily math.  Around third grade, I thought "There is way to much homework to do each night.  We'll just do daily math on the weekends" and that's where we've been ever since except summer and breaks.

Math contains more than math, of courses.  It contains anything I think they need to succeed at the time.  This usually contains math.  On Saturdays in the summer, this can be math, vacuum the basement, practice your instrument and do a reading comp question, fix the toilet, replace light bulbs. 

This year, the April MAP test is on our radar and I'm becoming slightly more organized with daily math.  We overdid vocabulary between SSCC and 2nd grade and haven't done much in this area other than define and discuss any unknown word found in reading or reading comp.  I am reintroducing vocabulary as part of math.  With a vengeance.

I like the MAP.  It has a lot in common with the COGAT.   The cognitive skill set is slightly different, but in both cases there is an advantage that can be gained from working on these skills simply because school doesn't really teach cognitive skills.  Doing lots of practice, ala Kumon doesn't help at all, and learning algorithms ala Singapore sets up a train wreck (like ending up in the 90th percentile or less - I never really defined what a train wreck is but that's it).  The problem with any program at all is that the child can get ahead and doing well, and the parent thinks that success has been obtained.  The clock is ticking.  Any time a child is practicing or applying or using things taught, learning may or may not happen, but skills building is not part of the deal.

I remember when my goal was simply to cheat my kids into a GAT program.  What actually happened was that we just ended up spending a lot of quality time together and I learned how to be a parent.  The long term formula for academic success is Cognitive Skills + Interest + Will.  At this age, and in the succeeding years while we caught up, it was all Cognitive Skills at the expense of Interest and especially Will.  You can burn a kid out with daily math every day every year, so I tried (and failed) to take some years off.  To compensate, I completely changed the approach to my formula of Baffled + Spending Time on the Question not the Solution + Get it Wrong + Check the Work.  This created an environment of Zero Expectations and No Progress, and in that environment magic happened.

Somewhere along the way, 'Will' came back, most likely because of chores or instrument practice, and I'm doing my best to stay as far away from 'Interest' as I can so as not to ruin it.  A child can only develop interest in a vacuum that does not include the parent.  Unless the parent is super sneaky.

I'm thinking about 'Interest+Will+Skills' a lot because for the older child, my goal is that he does really well in AP Language Arts and/or History, with assumed A's in math of some kind.  All of the math education is pointing in that direction for this child.  I found that at one of the selective enrollment high schools in Chicago, a child can take Calculus as a freshman, followed by Linear Algebra/Multivariate Calculus, a course that's no longer on their website which I will demand be reinstated, and AP Statistics, and assumed A's in AP Language.  This is 4 years of college credit math.  We're going for it.

The only way I can possibly think of achieving these goals is to do something creative, unusual, and different.  Something that is more looking at things from a fresh perspective than hard work.  Hard work is not going to do it.

The Makings of a Thinker

Here's a rough non-copyright violating approximation of a figure matrix question from my favorite COGAT practice test, grade 2. 

This is the last question in the book and the hardest.

In this article, I'm going to show you how much mileage you can get from a single question.

When I coach, usually at the behest of a parent who provides a compelling reason or academic puzzle that I want to add to my research, I'll start with whatever material they have available and do a single question.  There are many other things I do with a practice test besides a single problem, but my favorite Academic Coaching Session Agenda is the Single Problem because the student picks up the most skills.

This may be the only time I'm working with the child, and my primary goal is to train the parent who is lurking nearby, and I want an impact, so I do it exactly like I would with my own children.  Like this:

Step 1:  I instruct the child to do the problem.  Take as long as you like, and before you answer the question, I want you to tell me that you're ready to answer the question but not what the answer actually is.  I will probably announce that this is a really hard answer and I'm totally confused so I hope that the student can do it because I sure can't.

Step 2:  The child either announces the answer or announces that they are ready to answer. a) If they announce the answer and it's correct, I'll tell them I think it is the 2nd one and be prepared to prove your answer* b) if they announce the answer and it is not correct, my favorite case, I announce that they are wrong - try again and c) if they just announce that they have completed the question and are ready to answer, I announce that they probably got it wrong so go back and double, triple, and quadruple check the answer, followed by a) or b) when they announce the answer.

*At some point during this training, the child will learn to check their answer.  I am going to encourage this behavior in multiple ways including saying 'Check your answer'.

This approach is the birth of skills.  If the child answered incorrectly, then we're going to get double the skills from this exercise.  It's not clear to most parents what these skills are.  These skills are the skills of kids who will go into an accelerated history or reading course, teach themselves, and do well.  

When I announce to the child that they are wrong, they are probably wrong, or their answer does not agree with mine, the child can sense that I'm happy about this situation, and I genuinely am happy because we can learn something.  I love mistakes, even the ones I make.  Mistakes drive learning and it's one of the 5 core skills.

Step 3:  Explain the question to me.  First of all, I want to know what the transformation is.  The first shape undergoes 3 transformations.  Zooming through problems is the way to miss subtleties like the height of the shape diminishing by about 10% before it is rotated 1/4 turn counter clock wise.  Some kids say rotated 'to the left' which is OK with me provided 'to the right' always means clockwise.

In this phase, we're learning how to see, the names of things (like rotate 1/4 turn counter clockwise or decrease in height slightly).  I will correct the child's grammar or terminology, expecting that they eventually use the adult level words that I do in adult level sentences with multi-clauses.  It's the opposite of Baby talk and the reason why my books have that awful looking graduate text book themed covers.

When the child thinks they are done, I'll point out that explaining the question includes explaining what is happening in each and every answer.  I would like to know what transformation took place to make each of the answer choices, or what transformation failed to take place.  That's 4 additional problems as far as I'm concerned.  

I've never found a problem in a COGAT book that can't be solved with a thorough out loud explanation.  Sometimes when I'm working with my own material, I get the problem wrong, repeatedly, and I look at my answer and wonder what the heck I was thinking.  Then I go through it the way using the steps I expect a student to use, and oh year, it makes sense again.  When you say the transformations out loud (problem and answer choices) hard problems are turned into easy problems.  I can't over stress the importance of this technique.  This is why Shape Size Color Count is so verbal

I call this skill 'Reading The Question' because most kids can't do it without a lot of training, and most parents lack the patience to wait.  I know as a parent I used to lack the patience, and sometimes I still do.  To accommodate my coaching inadequacies, I'll just turn over the material and go clean for 20 minutes before the teamwork begins, shouting out things like 'Read the question again' while I do my work.  

There is a prerequisite skill I call 'Seeing' that children have to develop.  In this case, 'seeing' is visual and includes proportions and the ability to mentally rotate images.  It takes some practice.  In an academic household, those places of non-stop learning that produce GAT standouts, this practice started at age about 2.  For the rest of us, COGAT practice is as good a time as any.

I should point out that this is not a hard problem because it's missing the magic of the COGAT.  The quadrilateral lacks symmetry.  A problem like this would be practice for K.  This is why practice tests are practice for the format of the test and not the thinking of the test.  Also, there are 3 transformations, which you'd think would be good for working memory, but the shading transformation removes answer choices right away, making the problem easier, not harder.

Step 4:  If the child can't get to the correct solution on their own, I'll mark the page and come back later.  This question is still holding learning.  If I have to announce that the shape is shrinking in height before turning, I just destroyed the learning opportunity.  If there are 10 more questions with this transformation, I'm stuck having to announce it.  It's a judgement call and depends on how much time is remaining before the big event.  If you have a lot of time, you can back track by drawing 10 or 12 shapes, and ask your child to shrink one dimension and turn it 1/4 turn in one direction.  Backtracking in this way is a version of finding an easier problem to solve before tackling the harder problem to solve.  No branch of mathematics can withstand this approach, and every single super hard complicated advanced problem can be solved in this way if needed.

For one child, we spent a solid 4 months doing cognitive skills training (including BTS and much much harder material of my own making).   When we finally came back to math, we followed this approach from that point forward through SAT and calculus.  I learned that these core skills are universally applicable.  This is probably why the COGAT is such a great predictor of academic success.  Take any topic, like fractions or exponents or roots of a 2nd degree polynomial, or multiplication or anything, and at one point we slowly went through a few problems using this approach and learned months worth of material with a small amount of effort.

At some point during the actual test, the child will come to the questions that differentiate the 97th percentile from the 99th percentile.  These are the questions that differentiate those kids who probably would do well at Stanford with a little effort from those kids who will be sitting in a GAT program next year because of the ridiculously high cutoffs in almost all states.  The kid who gets these question correct will either be the child who is already 99% because his parents both have PhD's from those who have learned the skill set and go super slow on these problems:

• They are not the slightest bit discouraged by not knowing the answer right away or being confused.
• They take a long time to thoroughly investigate the problem
• They have a few techniques to fall back on when it gets really, really hard.  
• They are not discouraged when they don't see their expected answer in the pick list.  They try again as a matter of course.
• They check their answer, and all the answers, at least twice if not more.

I think the best way to teach these skills is to approach the training in the way I described above.  You should see how the approach is consistent with these skills.  It should also be clear that the other approach, I call this the school approach - explanations and lots of routine practice in the hopes of memorizing or mastering a set of question techniques - is not consistent with the skills needed at the top.

For parents a week or two from the COGAT who reach out to me the first time for help, and have done zero of anything before that, this approach is the way to go.  Of course, if you plan ahead, you'll be able to go much, much farther, but the approach is roughly the same.

Problem 123

Testing season is in full swing in Chicago right now with the majority of test takers in K grade, followed by 1st grade.

While sitting in the testing center, you may notice a members of a tiny but super intelligent articulate species talking to their adoptive parents about the composition of the earth's core.  Then on the drive home, your child may sit in the back seat telling you in explicit detail about each problem he missed.  These are both good reasons to buy a math book that your child won't see for 2 years and make him do it.  It made me feel better.

In this article, I'm going to demonstrate how to help your child work through material two years in advance.   Problem 123 is short for the last problem in EDM Grade 2 book on page 123, and the context is going to be a 5/6 year old in Kindergarten who made it to page 123 despite not completing K math and having skipped 1st grade math.  You can apply this context to other grades and other material (like a 2nd grader doing fractions), but if your child has been going to an after school math program for the last 2 years this is not going to produce experience for the child nor the same set of cognitive skills and you'll have to find a different challenge to achieve the same results.

I owe a reader a discussion of fractions, and I'll use this article to warm up.

Let's begin with my favorite email from parents and my common response.  Here is a brief summary of the email:  "This isn't working and I don't know what the heck I'm doing.   I don't know how to teach math.  What should I do?"

Here is my response:

• You are not teaching math.  Focus on teaching the core learning skills and the child will teach herself math in the case you are blessed beyond belief with daughters, or himself math if you're like me and stuck with a bunch of boys.  
• The 1st few pages in the book took us about 3 weeks.  Any page could take a week.  Acceleration happens later in the process.
• Our error rate was about 50% on a good day.
• After about 30 minutes on this exact problem, I just gave up and made a note to come back to this topic at some point in the future (which was next week).  I'm going to do it fully below because it shows you how to teach math to yourself which will make you a better math coach in the future.  

At this age, we're going to focus on the most important skill of Being Baffled, which is comprised of numerous subskills.  Then I'll talk about the 'Reading the Question' subset which you will focus on through 4th grade.  The other core skills like Getting the Problem Wrong (aka Making Mistakes) and checking your work are not discussed.

Page 123, Lesson 5-6, #3:
Connect the points in order from 1 to 3.

Find and name 3 triangles
__________________________
Try to name a fourth triangle
________________________
Color a four sided figure.

Step 1:  Be Baffled
Say 'This is a hard problem' then leave your child alone for a minimum of 15 minutes to do the problem.  I started this approach on page 1.  Somewhere between page 1 and page 123, 15 minutes of doodling, yelling, and complaining became 10 minutes of thinking and trying and 5 minutes of doodling, yelling and complaining.

Step 2:  Backtrack
The first challenge is that section 5-3 discusses the naming of line segments, like AB, problems 1 and 2 in this lesson connect shapes with lettered dots, but it's left to the child to make the leap to naming triangles. A Kindergarten kid is not only not going to make the leap, but by this point they never mastered (or even got) the whole line segment naming business.

Over the years, I've come to appreciate that 'Being Baffled' is a mandatory problem solving step, because it sets up the rest of the process, especially in BC Calculus.  Being baffled relaxes everyone (especially the parent) and opens the brain to thinking.  The opposite of 'Being Baffled' is frustration, impatience, and a subpar performance.

Fortunately, the example at the top of this page (not shown) has the same triangle without the numbered points, so we need to backtrack a bit.  Ask the child to name the line segments in the example triangle.  We should get AB, AC and BC.  Then ask the child to come up with a way to name the triangle.

I'm rarely severe on vocabulary.   At some point, I might just say that a triangle is named just like a line segment.  A line segment is AB, but a triangle is ABC.  What is the difference between BCA and ABC?  Does this triangle have any other names?  If the child is 8 years old and a boy, I would be disappointed if the child didn't say 'Bob'.

If this were a problem like 72 - 49 = ?, backtracking might be a 1st grade workbook for a day or two.

Step 3:  Dig into the question.
What is a triangle?   Ask you kid to define it.  It's a shape with 3 sides.  How do you make a triangle?  You put three sides together.  Show your child 3 lines that don't touch and announce you created a triangle.  Each side has to touch 2 other sides at its end point.  I'm meandering through the question starting with the Stone Age and working my way back to 2017.

There is a whole set of skills that formulates the skill of 'Seeing'. Some kids can do it, other kids have a lot of work to do.  In this particular problem, there are 4 triangles.  Two are obvious, one is not obvious, and one is hidden.  This problem will show up on most competitive math tests in one form or another.  Seeing is a big part of math and reading and science and innovation and internet startups.  It's also one of the main skills of the COGAT.

Ask the child to find all of the line segments in this picture.  I see A1, 13, 3B for example.  Then how many ways can you take 3 line segments that each touch 2 others at the end?  We gave up after 3 named triangles.

Step 4:  Give Up
You will give up on something.  You are not working with a 2nd grade child, but a 5 or 6 year old.  At some point, it's time to move on, and you have not achieved mastery over some math topic.  Fortunately, EDM has some repetition so you'll see some topics again, just not this one.  Fortunately, your child is going to get this material again in school, and they'll look like the smartest person on the planet when they see it again and figure it out quickly.

After doing this for 8 or 9 months, children should be completing the work with reasonable accuracy in a reasonable amount of time, but I need to stress this child will never complete the work like an 8 or 9 year old would.  My my goal of 'reasonableness' was met, and we stopped at about the 1/2 way point of book 2.  That was good for 99% on the MAP for a while.

Think carefully about what I did.  I got a child to sit and work alone for 10 to 15 minutes on material he wasn't taught and didn't know before I would jump in and start helping.  As the months go by, he gets less and less help, just more questions.  I taught him (because math is a team sport and I was the missing team member as needed) to be baffled, to spend a lot of time on the question and to backtrack as needed, to make mistakes and be totally OK with that, to try over and over again and to check his work because he got most things wrong on the first try (not demonstrated above).

With that skill set, and continued refinements over the next few years, it is reasonable of me to expect that he gets 99% on both sections of the MAP from this point forward, can handle accelerated work in all subjects with little or no help, can teach himself instruments and other things of interest to him, and go to Stanford for graduate school.

On the other hand, what if I trained and drilled him on math topics during this period?  What would I expect from a child who spent 4 years zipping through math because he was expertly taught and trained on math concepts?  This is what school does really poorly and what after school math programs do really well.  But it's not the skill set I want. You wouldn't notice a difference between either approach if you just looked at math and you just looked at a 2nd or 3rd grade performance on a math test of some kind.  The difference will show up elsewhere and it will show up later.

Advanced Math and Little Kids

I have about a dozen questions from readers that have been swirling in my brain, all on the topic of casual work-ahead At Home Schooling in math.  I've been trolling parent forums and reading amazon reviews while a new round of 1st through 3rd curriculum shows up from my latest buying spree.

Let's take the first question first.  How do I teach my child fractions?

Here is my step-by-step*:

1. You do a complete inventory of all of your child's skills and your skills as a parent that are required for your child to teach herself fractions.
2. You fix the ones you can fix immediately and work on the rest at the appropriate pace and the appropriate material.  You can work on fractions if you want while you do this.
3. Your child teaches herself fractions.  You help by reinforcing the 5 core skills which you can see while your child struggles with the material on her own, with no help learning the actual math.

#1 is the problem, of course.  It's also the problem with parent forums and helpful parent advice.  It is also a problem with teachers, even good ones, but not the really great ones who have taught for 20 years.  #2 is easy once you get it, and looks impossible before you actually see it work, then it's total magic.  #3 is our goal.

*I will present a more detailed step-by-step but we've got a lot of ground to cover first.

Back to parent forums and book reviews.  Parents are blind to the cognitive skill set of their child and where this fits relative to other children, not to mention their own skills as an At Home academic coach.  They find something that works and then state with no further thought that it should work for other parents.  Maybe, maybe not.  If the parent mentions either a) my child reads 6 hours a day or b) my child got 99% on both the COGAT and the MAP or c) my child got 99% on the MAP but didn't do so well on the COGAT then I have a pretty good idea where this child is on the skill spectrum.  a, b, and c are three totally different places, but I've spent enough time investigating so many children in these three cases that I can just prescribe the medicine.  The rest of the world needs more analysis.  

Wouldn't it be great if you could follow really 100's of successful parents around for 10 years and take notes and build a program based on what they did to put their kids at the top of the heap?  That's exactly what I did, and not just in math.

Recently I've been getting questions related to a certain famous math curriculum.  I haven't seen this material in 5 years since I reviewed it and then gave the books to a tiny little test case and followed up every week.  It wasn't right for my children, but I found a little girl who I thought would benefit for her specific case and she did.

The books are arriving and I'm really disappointed.  It's not about the core skills at all.  It's about explicitly showing the child how to do mathematical operations.  It skips learning.  Even worse, the questions tend to be the one-shot deal, as in one sentence that is pretty clear that the 2 numbers have to be added.  The inevitable result is a child who is told how to do math, never develops the skill set for #3, does pretty well on tests, and then has to be taught fractions.

In the last few months, I've gotten to personally know the Amazon drivers in Chicago because they show up at my house so much delivering material.  The last time I did this I was so disgusted that I wrote Test Prep Math.  Not much has changed. I've also pulled down at least a dozen curriculums (sic) from the web and gotten to know their creators from doing a little research.  I've come to the conclusion that the Test Prep Math series is the best material math material anywhere.

This is hard to say.  Authors have warned me that once you publish, you face a life of insecurity from that point on.  They were right.  I've freaked out when one mother told me that her child who's at the 99% found the books easy.  OK, I can deal with that.  The book is designed to get the child to 99%.  Just skip ahead until it gets hard. There is a review on Test Prep Math 2 where the reviewer slams me because the book is confusing and the answers are wrong.  As explicitly stated in the introduction, it is supposed to be confusing, and even I get the answer wrong when I speed through it and forget that it was designed for multiple readings on purpose, for you to see you skipped something or blindly assumed the wrong thing.  Those are core skills #1 (dealing with confusion) and core skills #2 (spend more time with the question - a lot more time - like 3 weeks if that's what it takes for the skills to emerge for the first time).  The book was returned and I feel personally responsible that the reviewer's child is going to eventually fall short in school.  

I've gotten a lot of emails and a few comments from readers who state 1) my child finished TPM Level 2 and is finishing TPM Math Level 3 and 2) what do I do next?  When I get this type of email, the questioner probably has no idea that they have a friend for life.  I'm planning to put TPM Level 4 on a free website, mostly because it's going to take me a lot of time to piecemeal the material out there and my new friend for life won't have time to wait, and I'm still weeks away from TMP Level 1 and it's taking up time.

By the way, in my ongoing effort to make kids so ridiculously smart that they blow away the COGAT, which was my original goal before I decided a math chair at MIT was also a good idea, I've finally perfected my ability to deliver figure problems to 6 year olds that are 3 times harder than anything they'll ever see again.  It's much easier with older children to take away the net.  Never underestimate the importance of the COGAT.  It measures skills that kids need to teach themselves fractions.  It doesn't care if they can actually do fractions or any other type of math.  The COGAT wants kids who already know how to learn and can go from Kindergarten to fractions in one year, which is what happens when you enter certain gifted and talented programs.

Test Prep Math 4 launches the math career.  It's all about math.  The skills continue to refine and develop, and the fifth core skill (problem solving skills) becomes wider and deeper on it's march to passing the AP exam in BC Calculus.  When you child chooses a joint major in English and Music instead of a STEM career, those problem solving skills explode yet again and you discover why so many CEO's and law firm partners have English or music backgrounds, but you wanted a doctor so we blew it.

Here are the Test Prep Math Level 4 milestones.  By 6th grade, your child will have finished all of the practice math tests in at least one SAT book.  You will have administered at least a rigorous Algebra 1 final where they will encounter some pre-algebra and many algebra topics for the first time.  They will have been introduced to important concepts in high school geometry, Algebra 2, trigonometry and calculus and you're holding off on the ones that require maturity to grasp.  If you've ever seen TPM, you won't be surprised to find out that TPM 4 includes the reading comp portion of the SAT as well, but you have to go a bit slower because of all that unfamiliar vocabulary. If you were fortunate enough to do Pre-K Phonics Conceptual Vocabulary and Thinking, and followed the directions with regard to the Word Board, the SAT vocabulary goes pretty quickly.  Some day, when my youngest completes his 7th and 8th grade high school enrollment nightmare, I'm going to spell out in detail why we're doing this.  Until then, just go with the flow.

We're not even going to look at the SAT until the summer after 4th grade and really get into it a year later.  Before then, we've got a lot of ground to cover, and it includes fractions.

I'm going to need 2 articles to do it, and they'll probably be long.  The first article is going to lay down the ground rules that apply to math starting in Kindergarten and that you will use thereafter if you want your child to learn.  

Pick the Right At-Home Math Curriculum

I spent the last few days thinking about the comment I received from Anonymous asking about current+2 curriculum for a 2nd grade child.  The last two articles on this topic were experimental and not helpful, and I'll delete them some day.

Taking a step back, here is a better version of the question: "What is the best way for a 2nd grader to work through 4th grade math so that she (or he) obtains all of the grit related benefits from doing so, learns more math, subject to the following constraints":

1. You've only got so much time to help and you're not a teacher
2. You need a high MAP score and teacher recommendation for the GAT program
3. You may or may not have to pass the COGAT this year
4. This child is only 7.  And not necessarily good in math. 
5. You can't afford a tutor or an after school math program.  Plus you hate driving.
6. If your child does get into a GAT program, you want them to be the best.
7. If you run into problems, you're going to send 19 emails a day to getyourchildintogat@gmail.com so this advice better be good.
8. Math curriculum from US publishers stinks

I updated the article How To Create A Math Genius to be more clear about this situation. You might want to refer to the content starting at first grade.  In this article, I'm going explain why my curriculum choices are counter intuitive and logically valid. 

My top 2 choices for curriculum are Go Math from Houghton Mifflin and Eureka Math.  A few years ago, a teacher suggested I review Eureka Math for 4th grade and I had a pdf of the whole book but I can't find it.  It's totally spoon feeding math, not only in the book but in the problems.  Go Math has a more intuitive approach, which means more concepts and less actual math.  For a kid who's already been through the advanced math exercise, he can do the Go Math homework for current+1 on the bus while playing Minecraft and discussing Star Wars memes.  And get them all correct.

If I was more worried about the MAP, I'd go with Eureka.  If I was more worried about the COGAT I'd go with Go Math.  I would probably pick Go Math anyway.

The target of Eureka and Go Math, and the rest of US curriculum, are the 50% of below average kids in the US with parents who know nothing about math and don't care.  This is perfect for a 2nd grader attempting 4th grade work, because the 2nd grade is starting way, way below average and her parent has zero experience teaching 4th grade math to a 2nd grader.  Really great 4th math curriculum is designed for bright, talented, engaged 4th graders with a parent who knows something about 4th graders, or at least has had 9 months of experience with a 3rd grader. 

If your 2nd grade child works through 4th grade math, and you follow the rules, #1, #2, #4 and #5 are taken care of.  #8 makes this possible.  For #3, you need more material beyond advanced math.  The COGAT is looking for kids with generalized problem solving skills who will be strong academically in the future, not kids who are ahead now.  But if you want your advanced math to impact the COGAT score, start with 100 (average) and add 1 point for every leading question you ask, add 5 points every time your child makes a mistake and you just shrug your shoulders because you don't care, and subtract 1,000 points every time you tell your child how to do something.  This will be an indication of their final score on the COGAT.

#6 will happen on it's own.  Most GAT programs only go 1 year ahead on math so your child would see the exact same math for a second time.

I'll take care of #7 right now.  "My son/daughter has been working on one of these books for 3 weeks and gets them all wrong and has only done 2 pages."  This is exactly what I expect.  This is the path to gifted.  The secret is just to keep going even though it doesn't make sense.  This is so counter intuitive that only about 10% of parents are willing to try it, and only 1% of parents are willing to follow the guidelines of an encouraging learning environment at home under these conditions.  That's why only 1% of children make it into the top 1%.

Teaching Algebra To A Fourth Grader

I'm really excited about yesterday's algebra post and the worked the followed last night proving I was totally wrong.  I'd delete the post entirely if it wasn't chock full of good advice that sets the stage for today's topic, which is teaching Algebra to a 4th grader.  Which is going super well, thank you, and after I complain that this is insane and ruining our children, I'll show you how.

There is a branch of math curriculum that is not very mathy.  It's fuzzy and intuitive and wholelistic and verbal and it doesn't really care if the kid currently or will ever know anything about math.  I love this type of math.  It's used in our school, chosen by a group of teachers who have about 9,282 years of teaching experience between the 4 of them.  One is named Yoda, just so you know.

What I've found is math that works for average or below average children in the US will work for your slightly above average child much earlier.  I think the best math curriculum of all time, even surpassing Sylvan's Kindergarten math book is the middle school math Jo Boaler created.  It involves no math at all, and then she just walks kids into 'WHAM' real math.  Her problems show up in CMP math, which is what our school uses.

On the other end of the spectrum is Singapore math.  I've decided that Singapore math is now Public Enemy #1, replacing Kumon as the worst thing you can do to your child.  One of my early famous articles describes Anti-Kumon, a program that I felt so strongly about that it ended up being the Test Prep Math series.  I also hate Mathasium and Level One.  I'm going to hunt down Singapore math grades 5 and 6 ASAP and start using them.  I've already recommended Kumon pre-algebra grade 6, a good book once you rip out the part at the beginning that spoon feeds how to do each problem and the part at the end called Solutions.

What Singapore found out is that you can train your child on advanced math and they'll look pretty capable as a result.  The top high schools in the country are full of overstressed, overanxious kids spending long hours doing homework and beginning the teacher to just please please tell them exactly what to do to get an A because they've never been given the opportunity to learn  and don't know how.  The Singapore material itself isn't such a bad idea (which is why they're about to get an order for books from me), it's what parents do with the material that is detrimental to their child's future.

Recently I conducted a search for problem solving books.  I've developed my methodology based on Poyla's 1945 book and was wondering if anyone else came up anything else that was helpful.  (The short answer was The Art of Problem solving that came out 10 years later and is helpful for wealthy people who live in suburban Connecticut.)  For the 8 weeks prior, all of my Google searches were 'Algebra II problems', for the older brother, and Google's search engine is nothing if not intuitive, so it gave me an extensive list of books and websites devoted to detailing the step-by-step solution to every algebra problem ever devised.  I was horrified.  This is where math training leads.  A drug addiction to solution guides.

Anyway, I'm going to apply hard core Poyla to 2 problems.  The first problem is below, and the second problem is that you have to get your child to learn it on their own, no cheating.   In yesterday's post, there were two problems, and children naturally gravitate to the the second one.  I should have recognized this one as a necessary step for the first one but I blew it. 

Here we go with the harder question.

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? 

We followed exactly the problem steps from yesterday's article, including a diagram that was totally unhelpful, but there was no way to solve this equation without extensive backtracking and #7 the missing element.   I hate the missing element technique.  It is absolutely fundamental to geometry proofs, but it takes a lot of work for a child to derive the missing element in 1 hour that took brilliant mathematician's 250 years to derive.  The secret is #8 use everything you ever learned that points to the missing element, and if you haven't actually learned it yet, you need backtracking.

Step 1:  Backtracking
For backtracking, I used IXL and Khan Academy algebra problems with parenthesis.  I did this a few months ago. These are real powder puff exercises, like 4(x - 10) = 23.  I look for parenthesis because kids who grow up with wholelistic language-based thinking math take many months to remember how parenthesis work no matter how many ways you spoon-feed it to them.  I finally created Kumon style parenthesis worksheets and told them just to memorize it.

I'll tolerate estimate-iterate for a while (is x 3?  How about 30?  What about 12.2?) because it's good arithmetic practice and builds the type of number sense needed for statistics, but eventually I'll resort to something like 1/x(23 - x) = x^1/2 so they quit guessing and ask for help.

The missing element is the equation x = 3(13 - 2).

What makes x = 3(13 - 2) a better problem than 1/x(23 - x) = x^1/2 or  4(x - 10) = 23?  The answer might take multiple 30 minutes daily discussions.  The answer is that in the easy equation, x is on one side and all of the numbers are on the right.  In fact, the easiest equation of all is x = 33.  x on the left, a number on the right.  The goal of algebra is to get x on one side and the numbers on the other side, in the cheatiest least effort way possible.  The goal of prealgebra is to handles all types of numerical operations including exponents.  Algebra adds 'x' and mixes things up.

From that point forward, we had an 8 week battle to see whether or not the stubborn kid could solve the problem without resorting to algebra, no matter how long it took, or whether he had to learn the fundamental principle of algebra:  you can add/subtract/multiple/divide/power up/power down each side of the equation by the same factor (whether it's 5 or (x - 2)) and the equation will take a step in the direction of 'easier' if you didn't screw up the parenthesis again.

We spent so much time analyzing what was wrong with equations (the x is not on the opposite side as numbers) that it qualified as a principle on which to build.

Step 2: Derive the Equations
In the problem above, this wasn't an issue because our math program is founded on convoluted complex word problems with double reverse logic.   We lost a few minutes because one of the unfortunate side effects of this approach is a kid smart enough to point out how stupid the problem is.  "Who buy's 2,685 jelly beans?  Like they're going to sit there and count them.  This is a dumb problem."

For some kids, backtracking might include writing equations from word problems.

So we got 3B = A and A - 2685 = 1/2B. 

The second equation was rewritten as B = 2A -5310.  The reason is that at one point in our backtracking, I told him if he see's x in an equation (aka a variable), then there is a 100% probability he'll have to work the equation with transformations to derive the answer, so stop wasting time trying to solve it in your head. 

The three important principles for this step that we haven't come to terms with fully are:
a) The best way to determine the correct equation is to write down the crap you know is wrong and fix it
b) don't write the two equations buried in a bunch of pictures
c) if the older brother wants to interrupt math with the new Avengers trailer, you're going to lose 20 minutes

No matter how many times I encourage mistakes and do overs, each new step up the math ladder is greeted with this expectation of getting things right the first time.  Mistakes are the fastest way to the goal.  Perfection is a hard stop on the road to learning.  We would save a lot of time if he would just write 2,685 - B = 2A, realize it's wrong, and fix it.

Step 3: Wait for the Leap
At some point during this problem I started cursing Anonymous for putting me in this position.  This would be a great problem for a long weekend.  To solve it, my kid has to figure out how to solve simultaneous equations, on his own, and all we've got are our problem solving techniques.

I am 100% sure that 100% of Singapore kids are told what simultaneous equations are and shown how to solve them, then they can practice this technique, get high test scores and great grades, without ever have experienced true learning.  It's like taking a Grade A steak and grinding it into dog food.  For my buddies from Southern India, I don't have a good analogy.  I once made Indian food and proudly brought it to work.  My coworkers told me it was 'bachelor food'.  All those great spices mixed into a tasteless mess.  That's what happens to Singapore math when it's trained and not learned.

My son pointed out that he can't solve the equation, and then complained and glared at me.

Why?  "Because it's got a B and an A.  It could be anything."

I asked him to specifically point to what is wrong with the equation.  After about 5 minutes, he pointed to "2A" in the equation B = 2A -5310.  So I asked him to fix it.

We had already established algebra is about fixing equations.  He knew the way to do this was transformations.  In the first 7 or 8 minutes, he just stared trying to determine how to transform the equation.  No luck.  Then he got really intense because somewhere in the pictures of his bear and a girl named 'Amy', he could sense 3B = A plays a role.

In Poyla, one of the foundations of understanding the question is 'use ALL available elements of the problem'.  This becomes really important in geometry.  We haven't spent much time on it.  I asked him if anything else could help.  Since 3B = A was buried in doodles, I asked him to show me all of the pieces of this problem.  I'm not sure this was necessary, but it was getting late and he had science homework and my spouse was yelling at me.  (Solution strategy #9, when your spouse is yelling about how late it is, start asking questions that direct your child.)

We had 7 or 8 minutes of silence and I could see he was becoming really excited in an intense concentrating way.  He said "2A is 6B" and wrote 5B = 5310.  When you're excited about learning, you can do 3 transformation in one step and I'm not going to complain.  This is how brainiacs get to the point where they solve things mentally to the consternation of their teachers.

What did I do?  I did three things.
1.  I didn't look up the solution and explain it to him.
2.  I didn't help other than ask questions and suggest one of the 8 problem solving techniques.  In this case, I suggested all 8 and we used all 8.  I will continue to do so until I'm banned from helping by my son, which is scheduled for middle school, at which point I will solidify my role as the dumbest, lamest parent on the planet and my child will reach self sufficiency.
3.  I waited, and waited, and was prepared to wait for the next 3 weeks if that's what it takes.

I was rewarded in 3 big ways.
1.  I concluded the whole session by mentioning that 2 equations with 2 variables is called 'simultaneous' equations.  I pointed how that he taught himself how to solve simultaneous equations and this is a big deal.  He already knew at this point that he taught himself and it was a big deal to him.
2.  3 months ago, it was horribly painful for him to transform x - 3 = 6 by adding 3 to both sides.  Now he was doing 4 steps in once (multiplying 3B = A times 2, substituting 6B for 2 A, subtracting 6B from each side of the equation and multiplying each side by -1).  I have repeatedly told parents to look for this effect, starting with phonics and first math when you get 3 weeks of zero and want to quit.  It's nice to see anyway.
3.  As a parent, I took a big leap myself in problem solving skills under the problem of how NOT to teach my child how to solve simultaneous equations even though Anonymous put me into this awful spot.

We are not going to have to practice simultaneous questions to perfect it. It's been earned, not trained. I don't like perfection, it removes the problem solving aspect that will be gained the next time the topic comes up, which will probably be this weekend with 8th grade simultaneous linear functions because I'm totally psyched.

In the last 6 weeks, I've come to the realization that the approach behind Test Prep Math is not at all compatible with Singapore math before grade 4.  Test Prep Math tries to avoid math at all costs while building up the skills underlying math, logic, and reading convoluted problems to earn the first 3 foundation problem solving skills that I covered in yesterday's article.  These are 2 wholly different world views.  I'm betting the farm that by middle school and then again in high school I will inevitably be proven correct.  I'm standing on a mountain of research, logic, and common sense from qualified teachers that I stole (problems solving technique #10).  By why wait until then?  4th grade is a great time to crush a few years of Singapore math.

Teaching 4th Grade Math to a 2nd Grade Child

I received this great question from Anonymous that deserves at least one post, if not a book.  Your child and skill level may vary, but from my stand point, it's the same question. 

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.

I'm going to provide a step-by-step guide, and this is going to be a long article.  Brace yourself.

There are many great reasons to teach a 2nd grade child 4th grade math.  Here they are ordered from most important to least important.

1. You didn't think to teach your child 2nd grade math when they were in Kindergarten.  Frankly, it won't matter by middle school when you begin, but the earlier you start, the more time you have to block out all of the memories of frustration until you just remember what a great idea it was.  An earlier start imparts more technical skills and a later start imparts more grit, but grades are high in science and language arts, which is what you really want.
2. You want to imbue your child with unmatched grit and generalized problem solving skills so that the rest of their academic career will be easy no matter what the challenge.
3. You want your teacher to notice that your child is bored in math and recommends your child for an advanced or accelerated program.
4. You are blatantly cheating your way to a high score on the MAP test.

This problem is from Singapore math.  Be very careful with Singapore math because like Kumon, it shows the kids how to do the math and undermines a host of more important skills like how to think.  There are problem solving guides that come with these types of math courses and they short circuit learning.  You can destroy your child's thinking ability in one shot and it's hard to undo the damage.

One more thing to keep in mind with Singapore math.  4th grade math compares with 5th grade or 6th grade math the way we normally refer to math curriculum in the US.  I've seen 2nd graders do 2nd and 3rd grade level Singapore and come out ahead.  You might want to think about switching to 3rd grade Singapore math or 4th grade lame standard US math.

Rule #1:  Don't, under any circumstances, teach math.  

You don't want your child to learn math.   If you focus on the more important skills, they will learn really advanced math on their own.  But if you try to teach them math concepts to solve these two problems, they are not going to learn math or anything else.  It's not about math.  The child is in charge of math, and you are in charge in an environment and experience where learning will explode.

Rule #2:  It's going to go painfully slow at the beginning.

It's really hard to watch a child tackle a problem that requires basic problem solving skills while they pick up basic problem solving skills.  It's painful.  If you want your child to learn how to learn, you can help by being confused, by being patient, by asking questions, but you can't just tell them how to do it.

It does not surprise me when a child takes 2 or 3 days to get past the first problem.  It does not surprise me that they forget something we did or said 10 minutes ago.  But I'm always totally shocked that in a few months they're zooming through 4th grade material like a slightly below average 4th grader, and I'm pleasantly surprised that test scores are now 100% across the board.

I'm always happy to receive an email from a parent that starts out with "I was doubtful at first because we got no where in the first 3 weeks..." because I know exactly where it's going.   If your child does ballet every day, they will probably become adept at ballet.  In the same way, success is inevitable on 4th grade math.  Give it time.

I like to say "of course your child can't do 4th grade math, because she is only a 2nd grader".  But she will.  These problems, however, are challenging for a 6th grader.  At Math House, we've worked through much more inappropriate problems, so I say go for it.

Rule #3:  Let's teach something besides math.

Language is probably the most important.  In the 2 problems above, there are at least a dozen words that your child could read and not understand, at least not in the context of the problem.  I'm going to provide some solution strategies that will help you in the first few weeks, but you need to get to a discussion of the problem as the primary way to work through it, not just because you want a high reading comp score as a bonus, but because understanding of math and language are linked.  I'm not sure math itself is linked to language, probably, but understanding math definitely is.

In the first few passes of each problem, invite your child to explain it to you, word-by-word and sentence-by-sentence.  For many parent-child teams, this will be total culture shock.  It takes changing gears and practice.  If your child can't articulate the question on the 7th try, word by word, you may ask for a picture or try again the next day.

Being confused, having to read a question 5 times, and getting it wrong are 3 important skills that have to be practiced and developed.   If your child doesn't become an expert at these 3 skills, and you as the primary academic coach aren't totally on board, more advanced work is going to be a real struggle. 

Rule #4:  You need solution strategies to survive.

You, the parent need the solution strategies.  My kids know all of them and are ready to tackle graduate study of Lie Groups, but if they use them, they use them behind my back.   I've never met a problem anywhere that can't be solved by these, so when they are stuck, I just shout out random solution strategies and we're back in business.

Now about that solution.

The challenge with the 2 problems above for a 2nd grade child is "2685" and "five times".  I don't care if my 2nd grader picks up an understanding of 4 digit numbers and multiplication/division.  That's his problem.  I want him to understand the essence of the logic and problem definition.

If the child understands the problem, in second grade, we're way ahead of the game.  Moving forward with strategy and solution will follow in time.   I prefer the child to get there when they get there, on their own.

By the way, you can just google these problems, tell your child the solution framework, and set your child up for failure down the road.   It's your choice.  

Here is the parent tool set:

1. Draw a picture.  This doesn't work really well with 2,685.  Plus, this strategy is appropriate to geometry and should only be used as a fallback when your child is really frustrated.  Drawing is relaxing.   In this case, I would ask them to draw a diagram to show me the before and after (with colored bars instead of cards) just so I could see that they understand the problem.  Given the difficulty level of these problems, a drawing is inevitable, or acting them out with a stack of pennies.
2. I tried algebra.  Total failure in 2nd grade.  The 4th grader is now starting to get it because I told him it's total cheating.  Yahoo answers recommend algebra for 4th graders, but if you are successful, by the time your child gets to 4th grade they will just look at the question, stare at it silently, and announce the right answer.  They will be using elements from the rest of the list.

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?

This problem needs #3: Make a simple problem.  In competitive math and math after calculus (like infinite series), a simple problem is followed by incrementally harder problems until we've developed a generalized algorithm.  In this case, we just want to understand the problem.

There were three times as many jelly beans in Jar A as in Jar B. After 25 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?

Now we've got a problem that a 2nd grader can work through, although it's going to take a few days at 30 minutes of concentration time per day.  I would recommend getting a bag of jelly beans after the first day.  Hopefully, you have lots of pennies, but now we've got a problem that deserves a picture.  Regardless, going back to 2,685 is going to add nothing to the problem for 2nd grade.

How did I pick 25?  I estimated and iterated (solution technique #4 which kids get really good at for problems like this after a few months of work).

Lay out the 25 sold jelly beans, and ask your child what we don't know.  (Many readings of the question later and some discussion) and we don't know how many beans are in Jar B and how many beans in Jar A were not sold.  You can do this on a 3 part diagram and place the sold beans in part 2 of Jar A.

Then invite your child to start putting down beans in the 2 missing places (#4 estimate) until we've got the beans left in Jar A to equal those in Jar B.  Finally, have your child read the question out loud and explain the answer to you.  Here's a tip.  Start with 1 bean in A for the part left after the sale (solution strategy #5 - start with 1) and ask how many need to go in B to establish twice.  Ask whether or not 1 in A and 2 in B satisfy the initial condition.  Your child is going to go "What does initial condition mean" so you have to read the problem again and write down the 2 conditions the beans have to satisfy.  As your child adds beans so that the part in A that is not sold is 1/2 of the part in B, see whether or not you got the solution.

In this way, a 2nd grader will build number sense, learn multiplication/division from the ground up, and have to concentrate really hard to get through it.  All great skills.  If you throw in discussion skills, your child is going to make a lot of progress.  It is unlikely that your child will get any where near competent on 4th grade Singapore math.  This has never been part of teaching current+2, but eventually it will happen.  The first year is mainly about grit.

On to the next question.  Solve these in order:

Aileen and Barry had an equal number of postcards. After Barry had given Aileen 1 postcard, Aileen had two times as many postcards as Barry. Find their total number of postcards.

Aileen and Barry had an equal number of postcards. After Barry had given Aileen 2 postcards, Aileen had three times as many postcards as Barry. Find their total number of postcards.

Aileen and Barry had an equal number of postcards. After Barry had given Aileen 4 postcards, Aileen had four times as many postcards as Barry. Find their total number of postcards.

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.

In addition to problem decomposition (inherent in these problems), estimating+iterating, and diagramming, I recommend solving these problems in reverse.  #5 Start with the end state and see if you can work your way backwards to the initial condition.  It's good practice on an important solution strategy.

Those 4 versions of the problem are not just a variant of start easy and work your way up, but have an element of what I call 'Backtracking'.  When we do 'work ahead' these days, we'll come across something like arithmetic in the complex plane and have to take off time from the problem to practice adding etc complex numbers.   It can happen on any problem.  In your case, it could be arithmetic with multiple digits or decimals.  Be prepared.

On that note:

Rule #5:  Get a Fallback Book for Bad Days

I've used boring current+1 workbooks which just have pages of fill in the blank when we're having a bad day because at least I want daily math to be an established pattern during the current+2 year.  In your case, I highly recommend Singapore Math Grade 3, or grade 3 if some publisher stole this question, because you may find that the grade 3 book is already 2 years advanced over 2nd grade and end up switching to it.  Then get a boring 3rd grade fill in the blank book for bad days.

Plus, I can't help every day, and it's nice to have a worksheet that I don't have to grade.

Plus, we may need it to backtrack on missing math topics and a 3rd grade book would do it.

Rule #6:  You'll Never Succeed

You'll never succeed in a 2nd grader doing 4th grade math like a top notch 4th grader.   You don't want to, so don't set out with this goal in mind.  You want your 2nd grader to be an amazing kid in all subjects, prepared to take on the best of the best.  But a great 4th grade mathematician will crush him.  If you want a child to work quickly and accurately 2 years ahead at the end of 6 months (which may happen a few years in the future on its own), you'd have to spoon feed, memorize, and train, and you'd end up with a dummy who hates math.

Instead, after you get to about the 75% mark of the book (or the 3rd grade book in this super hard series once you come to your senses), when your child is only misses half of the problems and takes forever, look for amazing things in all subjects.   Take a year off of math and do other things if you can.  Then be prepared to spend the rest of grade school feeding your child advanced math so they aren't bored.

The original experiment for current+2 never got beyond adequate, although he works nicely on his own.  Sometimes he does really well with current+3 or current +5, and sometimes it's 100% wrong.  Recently, I created a new website for our Boy Scout troop.  He sat at the computer next to me because he wanted his own website. [Insert eye rolling here, because that's what I was doing.]  I sat there stunned when he typed html from scratch.  Who types html from scratch?  He certainly didn't learn this in school.  Then started adding detailed styling and animation like he has a programming gene.  The level of learning skills when he's motivated is at about current+7.  That's what I'm talking about.  I didn't give him a fish when he was hungry.  I didn't give him a fishing pole or a net.  Apparently by focusing on problem solving skills and not helping or caring about the answer to a math problem, I gave him a whole fleet of fishing trawlers.  That's what I'm talking about.

Add 20% To Your Child's Score

Here is a thorough paper summarizing early childhood studies.  It's slightly dry if you're not in to this sort of thing, but it's very inspiring how much success has followed investment in at risk children living in poverty.

The general conclusion is that taking kids who live in a home devoid of eduction and putting them in a top notch academic program is going to have a big impact.  Early studies found that when you send a kid back into the original environment, the scores and grades plummet back to where they came from.  It's nice to see later studies address this issue.

These authors ask an open question that I have already answered.  It's a really big question and has a big answer.

...many early child interventions are conducted with at-risk children living in poverty. There are many reasons to suspect that the same results may not occur if the same intervention were conducted with affluent children. 

"Affluent" in this case means a home with education and stability.  I would agree a child from an "affluent" home may not see much benefit from a program designed for an inner city child with a single parent who didn't finish high school, even though some of the at risk kids in these programs saw IQ leaps from IQ = 92 to IQ = 130.   That is one friggin' big leap.

Since I don't have any at risk kids in my home, I asked a different question "What type of radical dramatic change would I need to do around here to go from 110 to 125, or 125 to 135, or to 160 just for the day of the big test?"

Step 1:  I'm going to give myself 18 months.  It turned out that it took 14 months just for me to get my act together as a parent, followed by 2 months for my child to get past radical core skill therapy (in one case barely in time for the test) and then 2 months to ramp up to a new level.  You can get your act together on day 1, and I'd be happy to provide a list of mistakes not to make, but if you've been reading my blog I think you're past that.

Step 2:  I want my kids to experience the same shock that these at risk kids experienced walking out of poverty into an advanced academic program run by a bunch of PhD's and taught by their graduate students. 

Step 3:  We're not going back.  I am on constant watch against video games, surfing, online chatting, and fun of any kind as my kids try their best to have a normal life.  It turns out that we only need about 20 or 30 minutes a day of heads down concentration on something inappropriately hard, but I've made those 20 to 30 minutes a prerequisite of fun.

Step 2 is formalized into Test Prep Math.  I want a single shocking 25 minute problem a day at first.  (Yes, I ramp up slowly because some kids cry and more adept kids can just zoom ahead feeling confident before the 'wham'.)   I want mistakes and confusion.  This is the birth place of problem solving skills.  If you present a child with a doable problem, there is no need for problem solving skills.  How about just easing your child along with some step-by-step and scaffolding?  You're not going to get a leap of 20+ points like these studies have found taking baby steps.

But the work is not done. There are two problems I'm dealing with in my own research. 

On one end, I just got 1 started on Amazon.  The reviewer complained "the book has so many errors".  Those "errors" are alternate solutions.  I stole this directly from the COGAT and love it.  You do a problem, get it wrong, don't understand the solution, and then dig in for 20 minutes to figure out that you assumed adding but the only available solution uses multiplication.  If you don't like confusion, don't by the book, because this is the most important skill and the base of the whole GAT skill pyramid.  I'm always worried about printing issues, so I'm getting a new copy just to check the solutions for the 10th time.  My other copies keep getting 'borrowed' by neighbors.

On the other end of the spectrum are kids who have really great math training and skip right past the confusion and problem solving steps because they already know how to do the problems.  The COGAT is a big stumbling block because it demands problem solving ingenuity.   A child never learns to solve problems if they are formally taught math, and if your child goes to a great math program like Mathasium, Level One, or Singapore, they have completely different academic world view than the COGAT.  It's not a bad thing, and could be a good thing, but it's the opposite of what I want for my children.  I'm laying awake at night wondering how to fix these kids.  It's one thing to lead a horse to water and they refuse to drink.  It's another thing if the horse is drinking gallons of water and is still thirsty. 

I'm thinking of just adding more bonus question to Level 2.  I am the master of giving a child a question that he can't answer without 20 minutes of logic and solution strategies, but it would just make people like the 1 star guy more baffled.  I could spoon feed everything in the solution, but this will just help others avoid the learning process.  I hate solutions.  Too many parents think that the whole purpose of test prep and math is to have your child know something.  It's not.  Think more radically, like 20 points radically, whether this is from 79 to 99 or from 99.1 to 99.7.  Step 0 is big goals.

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.