photo by MC Quinn via flickr (CC BY 2.0)
Paraphrased from a homeschool math discussion forum:
“Help! My daughter struggles with arithmetic. I guess she is like me: just not a math person. She is an outstanding reader. When we do word problems, she usually has no trouble. She’s a whiz at strategy games and beats her dad at chess every time. But numbers — yikes! When we play Yahtzee, she gets lost trying to add up her score. The simple basics of adding and subtracting confuse her.
“Since I find math difficult myself, it’s hard for me to know what she needs. What’s missing to make it click for her? She used to think math was fun and tested well above grade level, but I listened to some well-meaning advice and totally changed the way we were schooling. I switched from using workbooks and games to using Saxon math, and she got extremely frustrated. Now she hates math.”
Not a Math Person?
Please don’t tell your daughter she has to be either a math person or a language person. It is quite possible to be both. It sounds to me as though she has a very mathematical mind, if she is so good at strategy games and chess. Numbers are only a tiny part of math, even if they are the part that fills elementary textbooks. And if she can analyze a word problem, she is way ahead of many kids her age!
Since her problem shows up in adding and subtracting, it could be a couple of things. Perhaps she does not understand the concepts of putting things together or taking them away — but surely that is NOT true, because she does well with word problems and was doing well with the workbooks you used before. Maybe she loses track of the numbers, especially when she tries to count in her head. If she isn’t sure of her math facts, she probably gets flustered when she has to deal with larger numbers.
Here’s my best guess: I think your daughter’s problem is that she has not quite internalized the place value system. She knows it on a surface level, but she needs to know it down in her bones. This is a key to understanding more math than you would think at first glance.
First Steps to Recovery
- Drop the Saxon textbook, if you have not already done so. That book carries too much emotional baggage at this point.
- Go to the library and check out Family Math if they have it, or The I Hate Mathematics! Book or Math For Smarty Pants, for a more interesting approach to mathematical thinking. Order them through library loan if you have to. Play around with math for awhile before you attempt to do textbooky work again.
- Meanwhile, pick up a cheap workbook for practicing with numbers, or try a few online worksheets from my math resource page.
- Whenever you are ready to try another textbook — next school year, perhaps? — look for one that will focus on conceptual understanding and word problems. I like the Primary Math series, but as you found out before, what works for someone else will not necessarily work for your daughter. If you get a chance to attend a curriculum fair, you may want to take her with you to look around at all the possibilities. Once you decide which math program to try, be sure to use their placement test, so you start working at just the right level.
Learn Math by Playing Games
- Because the number 10 is the foundation of our place value system, your daughter needs to work on the sums that make 10 until she knows them instantly. If you say “6” she needs to be able to say “4” right back at you. At her age, this won’t take long, but it is super-important.
- Practice with a math card game like Tens Concentration.
- Practice the math facts until she is confident, and then practice them some more. Try the game that is worth 1,000 worksheets.
- Play some of the advanced games at the end of my number bonds article.
Practice Mental Math Skills
- Talk about how the pairs that make 10 can help her with mental addition and subtraction. If she needs to add 5+8, she knows that:
and
So
- Or here is another way to look at the same problem. (There are many ways to approach any math problem!) To figure out 5+8, your daughter could ask herself, “How many more does 8 need to make 10?”
- If she needs to figure out 13-7, she can do it backwards:
So
Be sure to notice that you are taking away the 3 and the 4, not taking away the 3 and then adding the 4! - It may help to use M&Ms or toothpicks to model the numbers, so she can move them around and find the 10. Practice this until she starts thinking in 10s and can immediately recognize them:
or
or
And so forth. - “Finding the 10” may sound too simple for a student your daughter’s age, but this is the most important step, because our number system is set up in tens. In our base 10 place value system:
and
Etc.
Moving On to Bigger Numbers
- Now use these same tricks to add or subtract some larger numbers, like her Yahtzee scores. Work in place value columns, but do it differently from what the textbook had her doing. No “carrying” allowed!
- If she is going to add, say, 273+596, have her work from the bigger parts of the numbers to the smaller:
That should give her 7 hundreds, 16 tens, and 9 ones. She can even write it that way, with the 16 in the tens place, as an interim step — have her write the numbers with a wide space between place value columns to allow for this. And then she can easily see that those 16 tens are the same as one more hundred plus 6 tens. - For subtraction, try the same sort of trick. The next time she needs to subtract something like 462-175, work from the big part to the small part. Start with the hundreds:
Does she understand that 3 hundreds and 6 tens is the same as 36 tens? Now she is ready to take away the 7 tens.
Finally, take away the 5 ones.
- She can work in her head if she wants, but she will probably want to write down the numbers as she goes through the steps, at least until she gets used to working this way. The main thing is to give her a different approach from what the textbook did — no “borrowing”! — and set her free from those negative feelings about math.
Please let me know if these ideas help, or if you have any more questions. Best wishes to you and your daughter on the great adventure of learning math!
Denise Gaskins' Let's Play Math 
One difference between Asian education and American is the belief, instilled early in a child, that they can learn math if they work at it. You overhear adults at dinner parties confess; ‘I just can’t do math” but you would never hear them say; “I just can’t read.”
Students in other countries aren’t naturally better at math, and they don’t have a math gene. They are taught that they can do math if they work hard.
What this child needs to know is that if she works through the math, it will get easier for her, as it gets easier, it becomes more rewarding. (If this weren’t true, we’d never get a 3rd grade boy to read 😉
“If this weren’t true, we’d never get a 3rd grade boy to read.”
LOL!!
It’s interesting how much difference the culture can make on this sort of thing. We live and breathe such different assumptions all our lives — and we so rarely even notice them. I think it is important for our students to realize they CAN learn, but it is even more important to let them know that, as you say, “…as it gets easier, it becomes more rewarding.” I don’t know how it is elsewhere, but in the USA, most people find it impossible to believe doing math could be enjoyable.
“When we do word problems, she usually has no trouble. She’s a whiz at strategy games and beats her dad at chess every time…
She used to think math was fun and tested well above grade level”
Excuse me? This is a description of a person that is good at mathematics. She is currently having a problem with arithmetic. That’s all. This is certainly something you need to address, but for goodness sake, please don’t saddle a child who obviously has a talent for much of mathematics with the “no good at mathematics” label simply because of a (quite possibly temporary) issue with arithmetic.
I happen to be very well acquainted with a very accomplished associate professor in mathematics (he taught me, and he is, indeed, a very accomplished and talented mathematician, widely published).
He needs a calculator to add 14 and 9 (I know this, because I watched him pull one out to do it when we were discussing a particular issue, and he literally didn’t know what they added up to).
Help her, but for goodness sake, someone who is good at chess and at word problems is a demonstrably able mathematician.
Preach it, Efrique! I agree 100% — but it brings up the interesting questions, “What do we want from mathematics education? And how can we get that result?” Especially when most American parents and (at least in elementary school) teachers don’t understand that there is a difference between arithmetic and mathematics.
You might want to take a look at Minus One Sheep http://www.lulu.com/content/2150336. (Disclaimer – I’m the author.) It’s a fairly short book (around 120 pages) aimed at getting kids to _understand_ maths rather than simply practicing rote learning of number bonds and recipes. It’s full of experiments instead of sums and readers really get involved in the unfolding story.
Thank you for this suggestion, I’m looking for ways to help my girls learn math and looks helpful!
Best wishes to you and your girls, Heather! You may also want to consider using the Buddy Math system (it works with any math curriculum). It made a huge difference for my kids.
Oh, how I wish I had known you when I was still homeschooling. My son, in particular, survived my “well, let’s try something ELSE” approach, but not without many tears and much gnashing of teeth (and that was just ME!).
Your approach is so refreshing!
We live by the “let’s try something else” method where my 11 year old is concerned. Math is tough for her, but just knowing that I am willing to work around it and throw away (put away) any books seems to help her relax.
Hi, Kim and Sheri!
The “Well, let’s try something ELSE!” approach is a great way for the teacher to learn, even if it can be a bit trying for the student. And I find that most kids are resilient — don’t you? As long as I eventually find a way to explain the concept that makes sense to them, they can ignore a few teaching missteps along the way.
Who in their right mind would actually recommend Saxon? Talk about “undoing” the great strategies this young lady has already learned through games at home, etc. When someone says Saxon…my advice is to run as far and as fast as you can in the other direction.
KP, your comment made me laugh. “Saxon? Eeek! Run away!” 😉
I don’t care for Saxon, either, but it is a dominant player in the homeschool market.
I can testify something similar as what Efrique said. I’m math professor and I’m having more and more troubles to do basic arithmetic operations mentally, but yet I believe that my math skills are just growing in time….
Wow! I praise God for the internet when it brings me to pages like yours! My daughter & I are using Singapore Math & they follow many methods similar to yours. I have simply been at a block as how to get the math facts comprehension & memorization. I am looking forward to putting up the book for a couple weeks or months & playing the games that you have suggested. I know my son who is a year behind my daughter will also appreciate the games. This should be a blast. Thanks for sharing your knowledge.
Hi, Colleen!
Thank you for the encouraging words. Best wishes to you and your children in the adventure of learning math!
This is another great post. My third grader takes a little while to get a new math concept but then she does ok. I’ve been trying to figure out what is the problem and I think the place value is the key for her getting these math concepts. I’m going to give that a try and I’m going to add yet another one of your blogs to my 3rd grade math word problems binder. Thanks again! http://www.livebinders.com/play/play?id=1619
My oldest daughter had trouble with fourth grade math – her problem was fractions. She had her math facts down, but just couldn’t get fractions. We were using Abeka and she cried when we had to get the book out. My lovely mother came over for her weekly visit one day in the middle of one of her crying fits. She told her “That’s okay, I guess you’re just not a math person – like me!”
This was a horrible lie. Mom made A’s in algebra back in high school. She also worked in retail for over twenty-five years and could add up a whole cart full of groceries in her head, plus figure the tax, and have her money ready – to the penny – before the cashier could ring her up! But I digress…
About three months before the end of the school year we dumped the Abeka book and I got her Life of Fred Fractions. She loved it and understood fractions. We ended up changing curriculum this past year (to Saxon, actually) and she’s doing fine. We’re still working on Life of Fred Decimals.
Hi, Maria,
I’m glad to hear you and your daughter found a way around your problem. It’s amazing how much difference changing the book can make sometimes, isn’t it? Fractions are fractions — the math doesn’t change — but a new style or a new explanation can make the light come on for a student.
I agree that understanding place value is so important. You have some great ideas!
Here are a few websites that have fun place value instruction and games:
http://www.funbrain.com/tens/index.html
http://www.dositey.com/2008/addsub/Mystery10.htm
I know your frustration when you said my child has a problem with math. I have students that just hate math. I have began to bring more games in the classroom which has began to simulate my students minds and they have began to become more active and challenging when it comes to working math problems. I teach the concept or skill first, then I have a math challenge either through individual activities or through group activities. The students are very competitive and before they realize it they have actually reinforced the skill taught. This strategy has also helped with classroom discipline.
How disappointing it is to still here people say that I’m not a math person. We know how to balance checkbooks, determine if we have been short-changed at the grocery store, know a good deal when we see one. Math is not a choice, it is a part of life. Humans were built to know math. It is natural.
The difference is that some of us are willing to put forth the effort in understanding the theory behind math. Theory in any field can be very challenging and take a lot of time and determination to sit in front of a single page for days to understand some proof.
Let your daughter know that there will be days when you try to study hard but you feel you learned so little compared to the time you put in. If you decide to study something challenging, there will be days when you want to quit and even cry because you feel so frustrated. Everyone that chooses to study something difficult goes through that. (It divides the weak and strong, if you will. That’s why those fields are highly paid.) But stick to it, because in the end you will be a stronger person and have gained such great skills that you can only gain from studying a difficult subject.
That’s my two cents. I tell this to all my students that really want to learn math, young and old.
By the way, don’t fret about teaching your daughter the name of the place values before she understands how to add and subtract using the real number line. Draw it for her, it marvels my students every time they see it. After they are comfortable adding and subtracting, then I teach them place value names, and continue to show them in great detail what is happening when you “borrow”. I don’t use that term however, because you don’t really borrow anything! I hate that we use that term in America.
This is so great! Thank you, we have been working with our 8 year old nightly and it has been a struggle. I am sure that this will help. Thank you!
I was labeled as a child. No one tried to help me learn math, reading or what have you. Now, I am 49 years old and in school. I never tried before because I was convinced that I could not learn and would fail. I have no idea where my courage came from to try to learn, but I am in my 3 semester and heading on to a bigger university. I currently have a 4.0 gpa. I so wish I had not been labeled as a child. I am currently struggling with some math that in my opinion is elementary. I never had it and am trying to understand it. I no longer believe that I am a mildly retarded person. I am not. I hate it when I hear someone say that their child is just not a “math” person or a “frilly lace” girl, or how about this one, my child is not one for college, we are thinking about good trades for her now…my goodness the girl was all of 12! What in the world are parents thinking? Is it that easy to sweep kids to the side? Well, to all you web site producers, not all the folks stopping by the “how to do the easy math” are kids…some of us are grown up and trying our best to get this, and you know what? WE WILL! Thanks for making a site for people like me.
Congratulations on your courage to learn! I hope you will keep working to understand the “why” of the math you study, because each topic you understand becomes a foundation on which you can build future learning.
I’m in the process of completing a three year study on the matter of struggling math students. Based on my literature findings, not a whole lot of emphasis has been placed on looking at support systems placed in classrooms where 1 teacher is responsible for 20 plus students who are simoultaneously trying to figure math stuff out.
Trying harder can only get you so far. Vygotsky focused on a theory he developed called the Zone of Proximal Development – essentially, students working at their low level of ability are doing so because they are working independently, whereas students working at their highest level of development do so based on some sort of guided practice. This practice does need to lead to independent work, but I do feel (and my research does back this up) that too early a release to independent practice leads to a lot of the math problems we see in primary classrooms.
I’m currently working on different strategies that help decrease the 20 to 1 or so teacher to student ratios in class in order to make the guided, sustained support possible for struggling students.
Absolutely agree that Saxon = stigma…. Math–especially for a student that struggles–needs to be made fun and less laborious whenever possible. As you suggest, connections are key; whether it be chess or music–anything to ignite that spark! Thanks for a great blog entry….
what are the ways of teaching a 12 hr clock and a 24 hr clock to students of 4th grade.
I assume you mean to teach them how to read a clock with hands? That gets more difficult yearly, as more students have digital watches or phones with which to tell time.
The best way I know to teach any kind of clock is to put one on the wall and use it throughout the day. The logic of the clock is not hard to explain, but don’t go directly to reading minutes. First teach hours, then round to the nearest quarter-hour. That’s pretty good for estimating how much time you have left to do something, and it’s much easier to do on a 12-hour clock with hands than digital, so the students can see an advantage to learning it.
I don’t know any young person who doesn’t enjoy playing pretend. (There probably are some, but I’ve never met them.) So the romance of using “spy time” should make using the 24-hour clock attractive — and of course, they have to learn the 12-hour version in order to do that!
Parents should start teaching children math from a young age. Playing games and learning math facts are fun for children when they are young. However, the older the child gets the more difficult it gets. I know children that have mastered times table to 12 by the age of 7. Great blog – lots of great ideas here for me to try with my children. I also recommend times tables tricks such as teaching that, 8 times 5 is half of 8 ,4, plus a zero on the end (40) 7 times five is half of seven , 3.5, without the decimal point (35). If it’s even, then add a zero. If it’s odd, take away the decimal point. Thanks Nina
Hi, Nina!
Your times table trick is my daughter’s favorite way to multiply. But I wouldn’t teach it as an abstract “trick” — rather, as a logical result of the fact that 2 fives = 10.
This articles does make sense. There are some families that feel that they weren’t good at math so their children won’t be either. I like the use of adding and subtracting in groups of 10. That makes so much sense and would be very useful to students. The game of Math War is fun for kids.
Wow, parents like that make my blood boil. We should never tell a child they a “math person” or a “non-math person”. What a stigma to saddle a child with. It seems this student is good at math (chess! strategy games! my goodness!!!) I think the parent is transferring their own fear of math to the child….what a shame.
Knowing that some kids are not that good at something means they should exert effort on those things. And that requires more self discipline and I know it would be very hard for them. Parents should also exert extra effort in supporting their kids so that they can be inspired in some way.
That’s true, Soph. Improving one’s weak areas does take effort. But also, especially in a weak area, we need to work efficiently.
Most people who think of themselves as “not a math person” see math as a series of rules and procedures they have to memorize — so if they try to exert effort, they work harder at stuffing more and more rules into their brains. That’s a recipe for confusion, and it’s a big reason why so many people hate math.
Much better to work at learning to see relationships, patterns, and WHY the math works. That’s the most efficient way to improve weak skills.
My daughter is in first grade. She’s supposed to be learning her addition math facts (up to 12) with “automaticity.” She still doesn’t know 6+4 and 3+2 without counting on her fingers. I was ready to sit her down this very evening and have her write 6+4=10 until her little arm fell off. Good thing I consulted Google – your math concentration game sounds more fun, though I think we’ll try with the cards face up until she’s better with the arithmetic.
Having the cards face-up sounds like a good modification for a beginner, Joey. Here are a few more ideas to teach her.
(1) Rather than counting both numbers, count up from the larger number. Whether it is written 6+4 or 4+6, you can start at the 6 and count 4 more numbers: 7, 8, 9, 10. This is an application of the Commutative Property (or “Any Order” Property), and it will help limit counting errors.
(2) Nobody learns all the math facts at the same speed. When she sees an addition problem she doesn’t know, if it’s close to one that she can remember, she should use the one she knows and adjust it. For instance, once she learns 6+4=10, then 6+5 is the same as 6+4 plus one more.
(3) Whenever possible, try to make a 10. Because our number system is based on 10, tens are easier to work with than other numbers. So for something like 8+6, we can think. “8+2=10, so 8+6 is 8+2+4, which is the same as 10+4.”
Fantastic!! I have finally found someone who understands the way I do my math and now I can explain it properly to my son who struggles in math. I just spent an hour going over this page with him and he gets it!! Thank you so much and I cant wait to go through the rest of your material.
I’m so glad to hear my blog was helpful. Best wishes to you and your son!
This post offered a ton of great ideas and easily accessible resources. There are a lot of ideas that I would use in my own classroom. The parent could also seek out a fellow homeschooling parent. This would be an option to send the student to that parent for her math instruction.
I have always told the parents of the kids in my classroom ” fake it”….you can pretend you like math.
Well done! Awesome post – have to admit, I take the same attitude when I’m tutoring students, particularly when they are feeling down about maths.
It takes a bit of effort, but when you inspire the kids, it makes a big difference.
My daughter will be 12 the end of February, she reads on an 11th grade level and does quite well in all her subjects but math. She is doing fractions and whole numbers now. She has struggled with math since she was in 1st grade, we need help.
Hi Chris,
If you’d like, I work on a blog in my spare time – I develop maths resources and that sort of thing. That’s at http://www.mathematicalmischief.com, and I attempt to help with all sorts of maths (by going over answers to questions, working on my own resources, reaching out to kids that need help). Feel free to drop by anytime – I’d be happy to try and help you out, 🙂
I am feeling frustrated when trying to teach my 10year old how to do Maths. She refused for me to teach her step by step & claimed that she knows. However, when i leave her to do them herself, she is struggling & feeling frustrated. Still, she will not let me help or teach her.
How can I help her when she doesn’t even want me to?
Hi, Leesh,
I’m sorry it took me so long to respond to your question. The email got lost in my overflowing inbox. I hope you and your daughter have found a system that works for you, but just in case you’re still looking for ideas…
By far the most effective way I know to help a student who is struggling with math is to work with her buddy-style (which you can do with any math curriculum). It has made a world of difference with my daughter. I explained how it works in this blog post: Buddy Math.
Best wishes on your learning & parenting adventure!
I believe that these ideas are really helpful. I have tried a few of these on my students too, and they produced awesome results. Thanks Denise 🙂
I also have a Mathematics website of my own. I would like you to have a look at it. Please provide me with your suggestions and feedback. Thanks.
I had a friend ask me why I had multiple calculus texts. I’d bought them at yard sales. I told him that, when I was taking calculus in college, I found the official textbook to be difficult to understand. So, I simply bought some other textbooks. The math didn’t change, just the problems and the explanations. If I didn’t get it from the official textbook, I just read it in another one. As long as I got the homework done and passed the tests, the instructor really didn’t care what book I used.
Finding alternate explanations for a tough topic can really make a difference — and not just for struggling students! I think anyone benefits from seeing a new topic from several points of view.
Hi my daughter will be 11 in November and really struggling with her maths and will use her fingers the teacher told her to find another method can any one help
Without knowing more about your daughter’s history, I can’t offer much advice. But finger counting doesn’t keep a person from doing well at math. I still count on my fingers, too! Please take a look at my Times Table Series of blog posts, and try a low-pressure conversation game of playing with numbers together with your daughter. Or check out the many math games to build number skills.
What do you think of Kumon or Enopi? (After school Math and Reading Program)
I don’t know enough about Kumon or Enopi to have a valid opinion about them.
Hats off to a parent who can understand and accept that his/ her daughter is weak in mathematics. Half the war is won ! Great work !
My Blog on Vedic Mathematics
I am struggling with this now and have actually come to a solution lol. I was taught math very differently than how schools are attempting to teach it now. I learned in a different country where my parents were stationed and when we returned to the usa I was graded harshly and often failed, even when getting the right answers, because I didn’t do it the right way. This was so frustrating! So I gave up and failed math..well d’s so I technically passed haha. In college I dreaded math. Found out the professor didn’t care how I got the answer as long as I showed how and got the correct answer. I got an A in every math class I took (which was a lot since I started in basic math). Never did I dream I’d get an A in College Algebra!! Just freeing myself of having to learn it one way opened up the joy of learning math, and yes, I did enjoy it! So with my daughter now we are exploring different ways different countries teach 🙂 I learned to subtract by adding to the bottom rather than borrowing from the top…I still to this day, do not understand borrowing fro the top..sounds crazy but when I try I always get the wrong answer lol. It’s ok! I know how to subtract and manage my checkbook. I don’t understand all the pressure to learn it one way and one way only!
Thanks for sharing, Jenna!
I’m glad you were able to find the freedom to learn math in a way that fit you. I think the “Do it my way only” pressure comes mostly from adults who don’t understand math themselves — and this unfortunately includes many teachers. If they did understand math, they would know that you can do any calculation all sorts of different ways. As long as what you do makes sense, it’s valid!
One of my friend daughter is struggling with the basic math numbers, she remembers even numbes easily, but when it comes to 5, 7, 9, 10 she identity it correctly but sometimes she confuse herslef in the same. Is there any possible way to rectify this pblm.
Hi Isha! It depends on the age of the girl. If she’s young, then confusion is normal and will probably settle itself out as she grows up. Your friend might enjoy this new book: Preschool Math at Home, which offers great ideas for playing with numbers with young kids. If the child is older, then she may have a learning disability relating to numbers (dyscalculia) and should probably be tested.
You are right – math is a game and if played right it can be easy and fun. An example:
to square any number that ends in 5 – the last 2 digits of the answer will be 25 because 5 x 5 is 25 OK.. easy right? Now lets move to the rest of a problem …
35 squared would be (can’t get superscript on here) 35(2) = 1225 – OK so what you do to get this – 3 times the next number- so if counting up the 3, 4, 5 … – so 3 x 4 = 12 and so 35(2) then is 1225.
Now try any number …. 55 would be 5 times 6 which is 30 and the 25 at the end …so 3025. Now you try it and have fun. You can have fun with every number if you know the tricks.
You can also have fun figuring out why the trick works. In this case, what is it about the number five that makes multiplying work this way? How is it related to our system of writing numbers in base ten (ones-tens-hundreds-etc.) place value?
hello, my little sister hated math since she was in kindergarten, and all her siblings including me were excellent students from the start. i tried to engaged her into math by doing all sorts of games and exercises but her grades keep on falling. she’s 11 now, and i tried every approach but she still isn’t interested in math. how can i help her understand that math is not that hard and is essential in the future ?
Actually, math IS hard — at least, the school math she’s facing in her classes. And the fact that all her siblings have done well adds to the pressure. But everything in the world that’s worth doing is hard.
She might find encouragement in some of the materials at YouCubed.org, especially the student interviews. And if she likes those, perhaps she’d be willing to work through Jo’s free course on How to Learn Math.
Love the Leave the Saxon math comment..it is the most horrible book in the world! I never understood why the learning support kids got that math book, while all the regular kids got the colorful and more real, everyday math book! Something was not right there!
Hi, Lauren! I have known some students who appreciated the “just the facts, Ma’am” approach of Saxon. I don’t know what the elementary levels are like, but the upper-grade books I’ve seen are very heavy on memorization.
The idea of giving Saxon books to learning support kids reminds me of a W.W. Sawyer quote: “A widespread fallacy about teaching is the idea that remembering is easy and understanding difficult. John is a bright boy, we will teach him what the subject really means; Henry is dull, he will just have to learn things by heart. Now exactly the opposite is true: to remember things which you do not understand is extremely difficult.”