We've been plodding along with math at a slow and steady pace for about 7 years now, and I'm starting to get the hang of it.
My original thought was if reading each day helps reading, wouldn't a child need to exercise math each day in order to be adept at math? This was before I knew that reading alone teaches all thinking skills, although too gradually to pass a GAT test before 3rd grade. This was before I realized that core thinking skills and working memory are much more important to math. This was before I enumerated cognitive skills.
I generally spend half the year doing thinking related at home work, and the other half of the year applying that thinking to math. The the other half of the year doing way too hard math at a slow and steady pace. That makes 18 months in a year for us.
As I mentioned a few months ago, the soon-to-be-3rd-grader and I are covering fractions. I decided to do decimals as well. I give him a single problem every other day, and make him do the ridiculously mind taxing working memory problems from Test Prep math the other days. For his fraction problem, we talk through it. We might talk through a problem and solve it a few different ways over a 20 or 30 minute period. I've found that 30 problems on a worksheet are useless, but a single good problem teaches a lot. 6 problems for younger children is also good, which is what Every Day Math does.
I can now fully appreciate the Every Day Math curriculum. When we break down pre-algebra problems, there isn't a single step or skill that wasn't taught in the EDM books. Curriculum is moving toward a problem based approach, and away from the type of practice in EDM. I know this is a good idea, but this is going to frustrate educators and parents and the pendulum will probably swing back in the next paragraph.
Last year, my son was supposed to memorize his multiplication math facts, but at my suggestion, he didn't, or what he learned he has already forgotten because I think math facts are a way to teach bad habits and I was openly negative about the task. We can't really do fractions without math facts, and pre-algebra is impossible. If your child is getting killed in pre-algebra, you should start with math facts, even if it means taking a Kumon course or doing worksheets or flash cards.
He needs his multiplication facts for next year's curriculum, so the last month in summer will include this exercise: For each multiplication problem 2 to 12, devise a rule and apply it. No memorization. I may hold up flash cards and have him shout out his transformation.
To prove my point, I asked him to add 6 + 7. He said 13 immediately. Then I asked him to multiply 6 x 7 and after about a minute, he said 48. I know he was adding 6 over and over again.
Then I asked him how he added 6 + 7. I expected something like 6 + 6 +1. He replied, "I put 7 tick marks on my brain board, and then 6 more tick marks, and then I count. That's why I stare or close my eyes." Holy cow. What kind of mutant wierdo method is that?
When we first started to learn math, back at age 3, I used 2 groups of blocks (like 2 blocks and 4 blocks, for example) and told him to tell me what changed. I call this Pre-COGAT. I noticed that he would assign personalities to the blocks. I recommend doing this once a child learns some numbers, like 3 or 4. It builds an internal visualization of numbers, and this will be very useful in graduate school in a variety of fields.
When I was learning 3 + 5, my mother showed me how to draw three little points with the pencil, and then 5 little points, and count them. Both I and my sister did this during the SAT in high school. I don't recommend this, but it's no different than the brain board.
With the older one going into 6th grade next year, you might be wondering how we could consistently work on math all year long without giving up at some point. One answer might be that I vary the material constantly and have a minimum of 2 books for them to choose from. One answer might be that we spend a lot of time looking at each problem and talking about it, and it's not really boring. The real answer is video games. I don't like them playing video games each day, so I made a rule - no video games unless you do your math for that day and practice your instrument. I think if I had daughters instead, I would get time either iPhones or Facetime on an iPad or something similar.
Saturday, July 23, 2016
Monday, July 18, 2016
Raising the COGAT Score After 4th grade
A reader recently asked what to do about the COGAT with a 5th grade child. I've been thinking a lot about this lately because our next big testing event is 7th grade. This is the year children are tested for high school enrollment in Chicago.
There is really very little good material after the 1st grade, and many children did it all before 1st grade anyway because grade school enrollment is so competitive. Fortunately, the COGAT tends to become more verbal after the early years and starts to look more like the SAT. In addition, the SAT has changed dramatically in the last few decades and now evaluates a similar skill set as the COGAT, because it's a better predictor of school success than 350 vocabulary questions.
Before you begin working on the COGAT score, ask yourself what the test this test is all about. Schools use it to find children who will do really well in school. In the case of the COGAT, the test makers spend a lot of energy find questions that predict future success, as opposed to current success, in the hopes of identifying children who may not be in a good school, who may not come from a privileged background, but have the skill set that would make them top students in a rigorous program. In the US, this would benefit non-immigrant minorities, and this makes the COGAT authors feel good about themselves. In my opinion, since the COGAT authors spent the last 30 years making millions of dollars testing for skills that they don't tell anyone about, instead of figuring out how to teach these skills, they are evil.
The way top students go about figuring out challenging and novel problems differs from how most kids (even top students) work through grade level academic material. Unfortunately, by the end of 4th grade, a child has spent 1,000's of hours doing things the wrong way, probably getting straight A's while doing it, and has accumulated a lot of bad habits that need to be undone before middle school, regardless of the COGAT.
The first step is to find material. I recommend Continental Press Level E (which is 5th grade) reading comprehension. Maybe level F for advanced readers. I use reading comprehension to teach the skill set. A good vocabulary book certainly helps. I recommend Vocabulary Workshop. Vocab only takes about 30 minutes a week. If your child did this for 4 years already it's optional. By that point, in my experience, kids learn vocab words on sight because they are little vocab machines. Memory is like a muscle, and it needs exercising.
The summer after 4th grade, we used the SAT publisher's prep book. It only cost $10, and we only used it for math. If this sounds insane, and it is, then there are many good pre-algebra books by Kumon. If you use a pre-algebra book, do not read the introduction about how to solve the problems. If your child figures this out on his own, that's OK.
The first set of skills we want our child to develop are taking a long time reading and thinking through the question, considering the answer choices (if there are any), possibly eliminating a few answers, and then reading the question a few more times. If the child is adept at this approach, then she can learn the higher order skills on her own.
With math, I handed my son the SAT book and asked for him to find 4 problems that he thought he could do. That forced him to spend most of his time looking through the questions.
Then, as he began to run out of easy problems, I started asking him to explain each problem to me. It's really hard for a parent not to focus on the solution or teach him math, which changes the goal, but instead just walk through questions with him until he understands them. Usually this involves reading and rereading the question. Lately, we've been doing a little geometry and some functions (he's older, obviously), and we spend almost all of our time just trying to figure out questions. Once the question is well understood, the rest of the problem is just arithmetic and this goes quickly.
If you take a peak inside of Test Prep Math Level 2, and read the introduction, you'll see the core skills laid out. (Google Test Prep Math COGAT on Amazon). This presentation was motivated by problem solving skills set for High School geometry and an effective approach to middle school math that is now changing curriculum. So don't be put off if it shows up in a book for 2nd and 3rd graders. The intro in Test Prep Math Level 3 is very similar, but I wrote Level 2 later, so the introduction is better.
Most of the work in the COGAT is evaluating the question and evaluating solutions. This is not to say the actual figuring out isn't hard, but many points are lost by kids who jump right to the figuring out and miss something that they need to know first. With the COGAT, some times part of the question shows up in the solution. Speed hurts the score.
With Reading Comprehension, the goal is to figure out what the question is asking and a good way to answer it. This is a different approach than answering the question. On top of that, there is a lot of material that taxes working memory, and reading the question involves, well, lots of patient reading. For this reason, reading comprehension is excellent test prep. The best way to do reading comprehension is to spend lots of time figuring out why a question was answered wrong to get insight into what the question is trying to do, and this reinforces an extremely important test skill called trying again. It's an important skill on the COGAT because problems are designed so that the child gets the wrong answer (in my opinion), so a child who is frustrated because they spent the last 4 years begin taught that correct answers determine how smart you are is going to do poorly until they get over it.
Working memory is a big factor in tests, as is vocabulary. For many years, I packed my kids brains full of vocabulary. They didn't realize how bad it was because Vocabulary Workshop is a lot of fun for kids. But vocab and working memory are really 2 different things. At the end of each Test Prep Math book, I put in a section of working memory arithmetic, and made it 2 or 3 times as hard as I thought the COGAT would be. I wouldn't pay for this book for a 5th grader, but if there was a little brother in the house, I'd make the 5th grader do the problems as well. Expect tears for older kids. It's offensive to some kids that a simple arithmetic problem would take 15 minutes to figure out. That's 15 minutes for an adult with years of graduate math.
Recently, I discovered that adding mixed fractions have a similar load on working memory, provided that your child does a small number of fractions and does then mentally with no writing. Bad way to do math, great way to build working memory for the COGAT. As a bonus, doing 4 fractions problems (instead of 30) with no help (well, maybe a little discussion), is a great way to learn fractions as well. Encourage your child to come up with creative ways to think through the fractions to compensate for the lack of a pencil and paper. This helps problem solving skills. I'm doing this right now with my other child, and he's developing whatever is the next level of number sense with fractions. I'll write about this when I'm finished because it doesn't have a name yet.
These problem solving skills are the next level of COGAT skills (after patience and reading the question thoroughly, doing it over, etc). As far as I'm concerned, it doesn't matter what the application because problem solving skills are problem solving skills, and the COGAT problems are all new and unexpected anyway, even if you have a test prep book that purports to be just like the COGAT. Pre algebra is good and fractions are good if you do only a page or a small number of problems and take your time. Somewhere in my blog I discuss George Poyla's work and the problem solving skills are there.
I've made it very clear to my children that I'm not impressed if they solved a complicated problem, but I'm very impressed if they broke it down into 2 or more really simple problems to solve. Math is all about cheating. Solving fractions is not a useful skill, but cheating will help a child go far in life. This works really well only if I remember to rip out and dispose of the solutions in the back of the book and hide the calculators. If they figure out how to really cheat, then my fallback plan is for them to explain how to solve the problem, and it better be really creative or they'll end up with "Oh yeah, then do these 3 pages Mr. Smarty pants".
Since I've eliminated the solutions, this raises the probability that I'll get a problem wrong when asked to help, or have to stare at the problem for 20 minutes trying to figure it out. There are a few really good studies that demonstrate that this is the single biggest factor in raising smart children, and therefore, I'm usually totally baffled when we work together.
Finally, after all of this, a practice test is useful for a few points because familiarity with the test format saves time and headaches during the test.
I hope this helps the reader who asked. It's certainly doable to spend a day on some math and a day on some reading comprehension during the summer. If the parent approaches this work with the right attitude, and enforces the correct habits with their children, the results in school will be dramatic.
There is really very little good material after the 1st grade, and many children did it all before 1st grade anyway because grade school enrollment is so competitive. Fortunately, the COGAT tends to become more verbal after the early years and starts to look more like the SAT. In addition, the SAT has changed dramatically in the last few decades and now evaluates a similar skill set as the COGAT, because it's a better predictor of school success than 350 vocabulary questions.
Before you begin working on the COGAT score, ask yourself what the test this test is all about. Schools use it to find children who will do really well in school. In the case of the COGAT, the test makers spend a lot of energy find questions that predict future success, as opposed to current success, in the hopes of identifying children who may not be in a good school, who may not come from a privileged background, but have the skill set that would make them top students in a rigorous program. In the US, this would benefit non-immigrant minorities, and this makes the COGAT authors feel good about themselves. In my opinion, since the COGAT authors spent the last 30 years making millions of dollars testing for skills that they don't tell anyone about, instead of figuring out how to teach these skills, they are evil.
The way top students go about figuring out challenging and novel problems differs from how most kids (even top students) work through grade level academic material. Unfortunately, by the end of 4th grade, a child has spent 1,000's of hours doing things the wrong way, probably getting straight A's while doing it, and has accumulated a lot of bad habits that need to be undone before middle school, regardless of the COGAT.
The first step is to find material. I recommend Continental Press Level E (which is 5th grade) reading comprehension. Maybe level F for advanced readers. I use reading comprehension to teach the skill set. A good vocabulary book certainly helps. I recommend Vocabulary Workshop. Vocab only takes about 30 minutes a week. If your child did this for 4 years already it's optional. By that point, in my experience, kids learn vocab words on sight because they are little vocab machines. Memory is like a muscle, and it needs exercising.
The summer after 4th grade, we used the SAT publisher's prep book. It only cost $10, and we only used it for math. If this sounds insane, and it is, then there are many good pre-algebra books by Kumon. If you use a pre-algebra book, do not read the introduction about how to solve the problems. If your child figures this out on his own, that's OK.
The first set of skills we want our child to develop are taking a long time reading and thinking through the question, considering the answer choices (if there are any), possibly eliminating a few answers, and then reading the question a few more times. If the child is adept at this approach, then she can learn the higher order skills on her own.
With math, I handed my son the SAT book and asked for him to find 4 problems that he thought he could do. That forced him to spend most of his time looking through the questions.
Then, as he began to run out of easy problems, I started asking him to explain each problem to me. It's really hard for a parent not to focus on the solution or teach him math, which changes the goal, but instead just walk through questions with him until he understands them. Usually this involves reading and rereading the question. Lately, we've been doing a little geometry and some functions (he's older, obviously), and we spend almost all of our time just trying to figure out questions. Once the question is well understood, the rest of the problem is just arithmetic and this goes quickly.
If you take a peak inside of Test Prep Math Level 2, and read the introduction, you'll see the core skills laid out. (Google Test Prep Math COGAT on Amazon). This presentation was motivated by problem solving skills set for High School geometry and an effective approach to middle school math that is now changing curriculum. So don't be put off if it shows up in a book for 2nd and 3rd graders. The intro in Test Prep Math Level 3 is very similar, but I wrote Level 2 later, so the introduction is better.
Most of the work in the COGAT is evaluating the question and evaluating solutions. This is not to say the actual figuring out isn't hard, but many points are lost by kids who jump right to the figuring out and miss something that they need to know first. With the COGAT, some times part of the question shows up in the solution. Speed hurts the score.
With Reading Comprehension, the goal is to figure out what the question is asking and a good way to answer it. This is a different approach than answering the question. On top of that, there is a lot of material that taxes working memory, and reading the question involves, well, lots of patient reading. For this reason, reading comprehension is excellent test prep. The best way to do reading comprehension is to spend lots of time figuring out why a question was answered wrong to get insight into what the question is trying to do, and this reinforces an extremely important test skill called trying again. It's an important skill on the COGAT because problems are designed so that the child gets the wrong answer (in my opinion), so a child who is frustrated because they spent the last 4 years begin taught that correct answers determine how smart you are is going to do poorly until they get over it.
Working memory is a big factor in tests, as is vocabulary. For many years, I packed my kids brains full of vocabulary. They didn't realize how bad it was because Vocabulary Workshop is a lot of fun for kids. But vocab and working memory are really 2 different things. At the end of each Test Prep Math book, I put in a section of working memory arithmetic, and made it 2 or 3 times as hard as I thought the COGAT would be. I wouldn't pay for this book for a 5th grader, but if there was a little brother in the house, I'd make the 5th grader do the problems as well. Expect tears for older kids. It's offensive to some kids that a simple arithmetic problem would take 15 minutes to figure out. That's 15 minutes for an adult with years of graduate math.
Recently, I discovered that adding mixed fractions have a similar load on working memory, provided that your child does a small number of fractions and does then mentally with no writing. Bad way to do math, great way to build working memory for the COGAT. As a bonus, doing 4 fractions problems (instead of 30) with no help (well, maybe a little discussion), is a great way to learn fractions as well. Encourage your child to come up with creative ways to think through the fractions to compensate for the lack of a pencil and paper. This helps problem solving skills. I'm doing this right now with my other child, and he's developing whatever is the next level of number sense with fractions. I'll write about this when I'm finished because it doesn't have a name yet.
These problem solving skills are the next level of COGAT skills (after patience and reading the question thoroughly, doing it over, etc). As far as I'm concerned, it doesn't matter what the application because problem solving skills are problem solving skills, and the COGAT problems are all new and unexpected anyway, even if you have a test prep book that purports to be just like the COGAT. Pre algebra is good and fractions are good if you do only a page or a small number of problems and take your time. Somewhere in my blog I discuss George Poyla's work and the problem solving skills are there.
I've made it very clear to my children that I'm not impressed if they solved a complicated problem, but I'm very impressed if they broke it down into 2 or more really simple problems to solve. Math is all about cheating. Solving fractions is not a useful skill, but cheating will help a child go far in life. This works really well only if I remember to rip out and dispose of the solutions in the back of the book and hide the calculators. If they figure out how to really cheat, then my fallback plan is for them to explain how to solve the problem, and it better be really creative or they'll end up with "Oh yeah, then do these 3 pages Mr. Smarty pants".
Since I've eliminated the solutions, this raises the probability that I'll get a problem wrong when asked to help, or have to stare at the problem for 20 minutes trying to figure it out. There are a few really good studies that demonstrate that this is the single biggest factor in raising smart children, and therefore, I'm usually totally baffled when we work together.
Finally, after all of this, a practice test is useful for a few points because familiarity with the test format saves time and headaches during the test.
I hope this helps the reader who asked. It's certainly doable to spend a day on some math and a day on some reading comprehension during the summer. If the parent approaches this work with the right attitude, and enforces the correct habits with their children, the results in school will be dramatic.
Wednesday, July 6, 2016
How I Teach Math To Kids
Lately, for lack of math topics, I've been teaching fractions to a 7 year old. It's going well. I'm basically doing what I did when I taught counting, addition, and multiplication.
We have 3 modes with math, and, like all of my articles, mode #3, which appears at the end, is the most important of all..
Mode #1 is heads down workbook catch up. By catch up, I mean I want the child to catch up to his natural level which is somewhere between 1 and 3 years ahead of school curriculum (in the US). I do this once or twice in the early years, depending on the kid, for 6 months at a time.
Any more than that will backfire, since the child isn't learning anything by becoming an expert at arithmetic or decimals. I could care less if my child has memorized the times table and gets 100's on all tests. I discourage this. This is not learning. Of course, if this is a GAT evaluation year and grades or standardized tests count toward a GAT program, I could care less about learning, because the priority is to pass the GAT requirements and then learning can happen next year.
The math brain is robust to setbacks. A child could take 2 years off of math and, with a bit of work, catch up and surpass his peers. This can't be said about reading. The rest of this article doesn't apply to a 7 year old who has carefully stepped through all of the right steps and is already in the 99th percentile. It applies to every child, especially the ones at the bottom of the ladder just starting out.
Mode #2 is thinking, figuring out, solving puzzles. I used to call this "test prep season", and we spend half a year on it whether there was a test or not. I now call this "cognitive skills season" and my goal is a bigger brain that can teach itself math without my help. I consider 2nd through 4th grade "cognitive skills season" because 90% of the math is useless math facts at this time. If I were in charge of curriculum, math would end in 4th grade (with 2nd grade math) and then start again in the 5th grade. In lieu of actually taking 3 years off of math, I did create my own curriculum and you can find it on Amazon if you google Test Prep Math Cogat. I suppose that the word COGAT ended up in the search criteria because I used the COGAT as inspiration for my methods. You'll see 2 books with graphics of an army ranger on an obstacle course on the cover.
Mode #3 is where actual math learning takes place. Lately we've been doing fractions. Like I did with their math predecessors, adding and subtracting, I provide my child a single problem about every week or so. That's right, one single problem. There's no hurry. Children magically learn things on their own, and when it's one single hard problem, it's a magical learning experience. This is what Mode 3 is all about.
A worksheet is something different. A worksheet requires a set of skills to achieve a different objective - finishing the darn worksheet. It's not so much about the wonder of math concepts, it's about finishing the darn worksheet. Math is an ancillary objective, and not the most important tool when worksheets are involved.
The first problem for younger kids is something like 3 +3 or 2 x 3. Then I just leave them wanting more. Later, we do 5 x 5 or 6 x 2, or whatever I can think of that is easy. Hopefully, I'll get wrong answers, and if I don't, I may do 10 x 3. Something harder, but something with a pattern to it. I'm waiting for light bulbs. The key is a brand new concept that is a stretch for the child. If the child is prepared for multiplication, this step will be incremental and not a leap, and you can just hand them a worksheet. Mode 3 needs a leap, like addition and subtraction for a child who just learned to count.
After we've done about 10 of these problems over the course of a few weeks, we'll try to demonstrate the problem with a picture. I should mention that for smaller children, with addition and subtraction, I start with a picture or 2 stacks of blocks. For older children, I don't provide a picture; instead, I provide a few days to solve the problem so that they have ample opportunity to visualize it first. All math should be drawn at some point. With multiplication, the picture is a the area of a square cut up into units. I have my heart set on graduate level math, and all math needs a picture.
This summer, we did 2/3 + 3/4 and 3/4 x 5/6. I might have given 2 other problems. Last weekend, we did 3/4 ÷ 5/6. That's not a lot of problems to do in a summer, but without going into technical details, it's enough to grasp fractions. By the time my child sees fractions in school, he'll have a mature understanding of what they mean. We dabbled in decimals, but decimals are boring concept I'll leave for school.
Note that working memory is essential to doing advanced math. Multiplication, for example, is not memorizing 6 x 5 = 30, but splitting 6 x 5 into much easier problems and then aggregating the result. In other words, properly done, 6 x 5 = 5 x 5 + 5 or 3 x 5 x 2. That requires working memory, which is why Test Prep Math has such goofy convoluted problems, and a whole ridiculously hard (at first) section that makes the COGAT look like a walk in the park. When you get to 2/3 + 3/4, you have a lot more steps going on than with 6 x 5, and it's all working memory driving the solution.
Next weekend, we're going to draw these problems and work them in a variety of ways. Then we're going to use language to describe them as many ways as we can and related the fraction operator to the division operator. You can do the same thing with all the elements of arithmetic for younger children.
The reason why we do a few problems over a long period of time is that I want triple learning out of my child. I want them to learn the concepts on their own, and learn how to think through things and learn to learn. These are the real goals of math. Arithmetic and fractions are just an excuse to use the brain. At some point, I jump in with lots of questions and different ways of thinking about it, but only after my child is conversant with problem and will readily get what I'm suggesting.
To summarize mode 3: a few problems to spark the imagination and leave a template for the "how to", pictures to explore the actual math concepts, and the language involved to further elaborate the operations. (The "language involved" is also the basis for common core, which is the one thing our education system got right in the last 30 years.)
Most parents see school curriculum and workbooks, and surmise that the goal of math is a lot of work to achieve a big objective, like mastery of operations. Math education researchers used to think the same way, since they developed this curriculum, until they realized that it is a complete failure in the sense that no one is learning anything and no one is choosing math majors in college. Now we know better.
What I know now is that a child who does a few problems thoroughly is going to end up with higher scores in math, and is much more likely to get 100 on every worksheet than a child who just jumps in and does a lot of problems. The road to mastery has a big detour, and I call it Mode 3.
We have 3 modes with math, and, like all of my articles, mode #3, which appears at the end, is the most important of all..
Mode #1 is heads down workbook catch up. By catch up, I mean I want the child to catch up to his natural level which is somewhere between 1 and 3 years ahead of school curriculum (in the US). I do this once or twice in the early years, depending on the kid, for 6 months at a time.
Any more than that will backfire, since the child isn't learning anything by becoming an expert at arithmetic or decimals. I could care less if my child has memorized the times table and gets 100's on all tests. I discourage this. This is not learning. Of course, if this is a GAT evaluation year and grades or standardized tests count toward a GAT program, I could care less about learning, because the priority is to pass the GAT requirements and then learning can happen next year.
The math brain is robust to setbacks. A child could take 2 years off of math and, with a bit of work, catch up and surpass his peers. This can't be said about reading. The rest of this article doesn't apply to a 7 year old who has carefully stepped through all of the right steps and is already in the 99th percentile. It applies to every child, especially the ones at the bottom of the ladder just starting out.
Mode #2 is thinking, figuring out, solving puzzles. I used to call this "test prep season", and we spend half a year on it whether there was a test or not. I now call this "cognitive skills season" and my goal is a bigger brain that can teach itself math without my help. I consider 2nd through 4th grade "cognitive skills season" because 90% of the math is useless math facts at this time. If I were in charge of curriculum, math would end in 4th grade (with 2nd grade math) and then start again in the 5th grade. In lieu of actually taking 3 years off of math, I did create my own curriculum and you can find it on Amazon if you google Test Prep Math Cogat. I suppose that the word COGAT ended up in the search criteria because I used the COGAT as inspiration for my methods. You'll see 2 books with graphics of an army ranger on an obstacle course on the cover.
Mode #3 is where actual math learning takes place. Lately we've been doing fractions. Like I did with their math predecessors, adding and subtracting, I provide my child a single problem about every week or so. That's right, one single problem. There's no hurry. Children magically learn things on their own, and when it's one single hard problem, it's a magical learning experience. This is what Mode 3 is all about.
A worksheet is something different. A worksheet requires a set of skills to achieve a different objective - finishing the darn worksheet. It's not so much about the wonder of math concepts, it's about finishing the darn worksheet. Math is an ancillary objective, and not the most important tool when worksheets are involved.
The first problem for younger kids is something like 3 +3 or 2 x 3. Then I just leave them wanting more. Later, we do 5 x 5 or 6 x 2, or whatever I can think of that is easy. Hopefully, I'll get wrong answers, and if I don't, I may do 10 x 3. Something harder, but something with a pattern to it. I'm waiting for light bulbs. The key is a brand new concept that is a stretch for the child. If the child is prepared for multiplication, this step will be incremental and not a leap, and you can just hand them a worksheet. Mode 3 needs a leap, like addition and subtraction for a child who just learned to count.
After we've done about 10 of these problems over the course of a few weeks, we'll try to demonstrate the problem with a picture. I should mention that for smaller children, with addition and subtraction, I start with a picture or 2 stacks of blocks. For older children, I don't provide a picture; instead, I provide a few days to solve the problem so that they have ample opportunity to visualize it first. All math should be drawn at some point. With multiplication, the picture is a the area of a square cut up into units. I have my heart set on graduate level math, and all math needs a picture.
This summer, we did 2/3 + 3/4 and 3/4 x 5/6. I might have given 2 other problems. Last weekend, we did 3/4 ÷ 5/6. That's not a lot of problems to do in a summer, but without going into technical details, it's enough to grasp fractions. By the time my child sees fractions in school, he'll have a mature understanding of what they mean. We dabbled in decimals, but decimals are boring concept I'll leave for school.
Note that working memory is essential to doing advanced math. Multiplication, for example, is not memorizing 6 x 5 = 30, but splitting 6 x 5 into much easier problems and then aggregating the result. In other words, properly done, 6 x 5 = 5 x 5 + 5 or 3 x 5 x 2. That requires working memory, which is why Test Prep Math has such goofy convoluted problems, and a whole ridiculously hard (at first) section that makes the COGAT look like a walk in the park. When you get to 2/3 + 3/4, you have a lot more steps going on than with 6 x 5, and it's all working memory driving the solution.
Next weekend, we're going to draw these problems and work them in a variety of ways. Then we're going to use language to describe them as many ways as we can and related the fraction operator to the division operator. You can do the same thing with all the elements of arithmetic for younger children.
The reason why we do a few problems over a long period of time is that I want triple learning out of my child. I want them to learn the concepts on their own, and learn how to think through things and learn to learn. These are the real goals of math. Arithmetic and fractions are just an excuse to use the brain. At some point, I jump in with lots of questions and different ways of thinking about it, but only after my child is conversant with problem and will readily get what I'm suggesting.
To summarize mode 3: a few problems to spark the imagination and leave a template for the "how to", pictures to explore the actual math concepts, and the language involved to further elaborate the operations. (The "language involved" is also the basis for common core, which is the one thing our education system got right in the last 30 years.)
Most parents see school curriculum and workbooks, and surmise that the goal of math is a lot of work to achieve a big objective, like mastery of operations. Math education researchers used to think the same way, since they developed this curriculum, until they realized that it is a complete failure in the sense that no one is learning anything and no one is choosing math majors in college. Now we know better.
What I know now is that a child who does a few problems thoroughly is going to end up with higher scores in math, and is much more likely to get 100 on every worksheet than a child who just jumps in and does a lot of problems. The road to mastery has a big detour, and I call it Mode 3.
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