Wednesday, November 29, 2017

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.

1 comment:

  1. I asked my 4th grade tester to do these problems. He did the 2nd problem in 14 minutes using estimate and iterate. 10 of these minutes were complaining that he had other homework to do. The second problem took 26 minutes, again 10 of which were complaining.
    He got to 3B = A and A - 2685 = 1/2 B after I badgered him to use algebra and then he complained he didn't know what to do. Refactored to B = 2A - 5370 and complained. I asked him to explicitly point to what the problem in this equation is and he pointed to 2A. So I told him to fix it and agonized for 15 minutes while he looked around for some way to fix it. Finally, he saw 2A is really 6B and then finished the rest of the problem.
    This is a great book for 4th graders who are on the cusp of using algebra to solve simultaneous equations and parent willing to sit silently while the kid figures it out himself.
    There's no way this is suitable for 2nd grade.