Subtraction: More Than Just Taking Away - Page 2 of 14 - Well-Trained Mind


Subtraction: More Than Just Taking Away

Can you tell what these three word problems have in common?   

  • I had 8 cookies, and then I ate 5. How many cookies did I have left?
  • I have 8 cookies. 5 are chocolate, and the rest are oatmeal raisin. How many are oatmeal raisin? 
  • There are 8 chocolate cookies and 5 sugar cookies in the cookie jar. How many more chocolate cookies are there than sugar cookies? 

Yes, they’re all about cookies. But these word problems have something else in common, too. These situations feel very different from each other, but you can solve them all by subtracting 5 from 8.

Many textbooks only present subtraction as taking away. But there are actually three different interpretations of subtraction:

  • Taking away
  • Part-whole
  • Comparison

When your child learns all three meanings, he’ll understand subtraction on a deeper level. He’ll be able to solve real-life subtraction problems with confidence, and the three meanings will even help him master the subtraction facts more easily. In this article, you’ll learn all three interpretations so that you’re equipped to teach your child this vital skill.

3 Essential Meanings of Subtraction  
Taking Away 
In take-away subtraction problems, items are removed in some way. Pencils are lost, cupcakes are eaten, pennies are spent, and so forth.

I had 8 cookies and then I ate 5. How many cookies are left?
Thinking of subtraction as “taking away” makes subtraction easy to understand and concrete. That makes it the perfect way to introduce subtraction to young children. As they begin to explore subtraction, they can solve simple problems by physically “taking away” objects to find the answer.  

Starting with take-away subtraction also helps children understand that subtraction is the opposite of addition. Since removing objects is the opposite of adding objects, they can concretely see and feel the difference between these two operations.

Part-whole subtraction problems don’t involve removing anything. Instead, you know the total number and the size of one part, and you subtract to find the size of the other part.
I have 8 cookies. 5 are chocolate, and the rest are oatmeal raisin. How many are oatmeal raisin?
To solve part-whole subtraction problems, children don’t physically remove objects. Instead, they separate the whole group into two parts and find the size of each part. In real life, part-whole subtraction often comes up when you have a group with two smaller subgroups and you want to know how big one of the subgroups is. (For example, a flock of birds with robins and sparrows, or a class of children with boys and girls.) 

Comparison subtraction is the most challenging interpretation for most children. In comparison subtraction problems, you subtract to find out how much bigger (or how much smaller) one set is compared to another. For example, we often use comparison subtraction to tell how much more one person has compared to another: How many more toy cars does Daniel have than Jacob? Comparison subtraction also arises frequently when measuring: How much shorter is this rope compared to that rope? How much older is this person than that person?

There are 8 sugar cookies and 5 gingersnaps. How many more sugar cookies are there than gingersnaps?

We call the answer to a subtraction problem the difference because of this comparison meaning of subtraction. When we subtract, we are literally finding out how “different” two numbers are from each other–how much greater one number is than the other. 

How to Teach Subtraction Like a Pro

Here’s how you can help your children develop a deep understanding of all three meanings of subtraction.

  1. Include all three subtraction meanings in your teaching. Begin with simple take-away subtraction problems with your kindergartner or first-grader. Introduce part-whole problems in the later part of first grade, and finally tackle comparison problems in second grade. This clear, gradual progression will help your child make sense of all three interpretations without feeling overwhelmed.
  2. When reading subtraction problems aloud, say “minus” for the minus sign and not “take away.” For example, when you read the problem 6 – 4, say “six minus four” and not “six take away four.” This helps children think more flexibly about the role of the minus sign and mentally prepares them to understand interpretations other than take-away.
  3. Use manipulatives to act out subtraction problems. You can use commercial math manipulatives, but simple household items like buttons, blocks, or small toys work well, too. Nothing helps children understand subtraction better than physically acting out problems with real objects.
  4. Word problems about unfamiliar situations can feel very abstract and difficult to children. When you’re first introducing a new subtraction meaning, keep word problem situations as relevant to your child as possible. Kids especially enjoy word problems that include their interests. For example, if your child loves soccer, you might ask questions like:
    • I brought 12 juice boxes to the soccer game. The team members drank 9 of them. How many juice boxes were left? (Take-away)
    • There are 10 kids on the team. Six of them are girls. How many are boys? (Part-whole) 
    • My team scored 7 goals, and the other team scored 5 goals. How many more goals did my team score? (Comparison)
  5. When your child solves numbers-only problems (as opposed to word problems), encourage her to use whichever subtraction meaning makes the problem easiest to solve. For example, take-away subtraction is best suited to problems like 12-3 which involve subtracting a small number. On the other hand, comparison subtraction is very helpful for problems like 9 -7 where the two numbers are close together. These strategies will help make mastering the math facts much more manageable for your children.


How to Teach Addition and Subtraction with Play Money

Hands-on math materials help children understand math concepts better and become more fluent at solving problems. Plus, they make math time a lot more engaging and more fun.

Making your math lessons more hands-on doesn’t mean you have go out and buy a lot of specialized math manipulatives, though. In fact, you probably have one of the best math manipulatives spilling out of a crumpled cardboard box in the back of your board games closet.
In this video, you’ll learn how to use play money to teach your children how to add and subtract multi-digit numbers with confidence. With the help of play money, your children will learn not only how to add and subtract larger numbers, but they’ll also understand why the process works.

After you’ve watched the video, make sure to fill out the form to grab your free cheat sheet. It’ll jog your memory on those mornings when you haven’t had enough coffee and you’re trying to teach math with your toddler hanging on your leg.

Video Transcript

Hi, I’m Kate Snow from the Well-Trained Mind Press and Kate’s Homeschool Math Help. And today I’m going to show you how you can teach your kids addition and subtraction with play money. Using play money from your kids’ cash register or from board games you have around the house makes learning addition and subtraction a lot more fun and help kids understand what they’re doing a lot better. Instead of just carrying the one or borrowing a ten, they’ll really understand what they’re doing when they attack these problems with two or three-digit numbers or even more.

So, before you go ahead in teaching your kids multi-digit addition and subtraction, there’s a couple things that your child should know first. We don’t wanna jump ahead of ourselves here. First of all, your child should know the concepts of addition and subtraction, of course, so they understand the idea that, with addition, they’re joining things together, with subtraction, they’re taking things away. Your child should also know the math facts well before starting these kinds of problems. They should know that 9 plus 8 is 17 or that 15 minus 6 is 9. Having those basic facts mastered before they try to do these more complicated multi-step problems will make them go a lot more easily. And then, finally, your child should already be familiar with place value and regrouping. This is the idea that if we have 10 ones we can regroup them into a 10, or we can take a 10 and break it apart into 10 ones that are very important for this multi-digit addition and subtraction.

You won’t need very much for this. All you’ll need is paper, pencil, and then the ones, tens, and hundreds in play money. So to start teaching your child… We’ll start with addition first, and then we’ll go on to subtraction. To start, you’ll need a piece of paper to use as a place value map, and to make a place value map, I know it sounds kind of fancy, but all you actually need to do is just make a really simple chart on here that we’re gonna use to keep our play money organized. We’ll label one column tens and the other column ones. And so we’re going to start with a problem that doesn’t require regrouping. We’re gonna start with 24 plus 31. And so to do this, we’re gonna start…first, we’re going to act it out with the play money, and then we’ll record what we do. So we’ll start here by representing the 24 with the play money. We have two tens and four ones. So I’m just gonna put those out with the play money. I’ve got two tens here, and then I’ve got four ones. So that will be my… Oops, I’m moving things around here from my bank. So there’s my two tens and my four ones. I wanna add on to it 31. Three tens and then one one. Okay, so now I’ve got these all out.

And so now to teach your child what to do, we’re gonna start and go through step by step. So we’re gonna start here with the ones. We have four ones plus one one. That’s these four ones and one one more. And so those four plus one add up to five ones, which I’ll write one here right here in the ones column. Now, over in the tens, we have two tens plus three tens. We’ve got two tens plus three tens, which adds up to five. So we have five tens. So the total answer is 55. That’s an example with no regrouping. Now, once your child is comfortable with those sorts of problems, then you want to move on to ones that are a little bit more complicated that do require regrouping or what you might have learned as carrying the one when you were in school.

So here we have a slightly harder problem, 24 plus 38. I have already laid out all of the dollar bills, so we have 2 tens plus 4 ones for 24, 3 tens plus 8 ones for the 38. Again, we’re going to start with the ones. We’ll move around the play money first and then record what we do. So we have four ones plus eight ones. Four plus the 8, and 4 plus 8 equals 12. And so here’s a case where we need to regroup. We need to take 10 of our ones and trade them for a 10 dollar bill. So that’s just what I am going to do here. I’m going to take this 10 down here that I need to select up and to get rid of. I am posting this on the screen, but with play money, you do wanna take these and put them down here in your bank. And then take another 10 and move it out here, because we traded the 10 ones for that 1 10. Now, we’ll record what we did. We took 4 plus 8 is 12. We took 10 of those and made that into a 10. Oops, lost something here on my screen. Sorry, there we go. So that’s the 10 that we traded, and then we have 2 ones left here in the ones column.

Now, we’re gonna look at the tens column. We have these 2 tens, the 3 tens, and then the 1 extra 10 we added. One plus two plus three that’s six tens. So our final answer is 62. And then once your child is dealing good about that, then you could extend it, and I do problems where you need to regroup the tens. Again, I’ve laid out the dollar bills already to match the numbers. So 85 plus 32. Eight tens and five ones plus three tens and two ones. We’re going to start with the ones again. We have five plus two equals seven ones. No regrouping necessary there. But now we get to the tens. Eight tens plus 3 tens is 11 tens. So we need to regroup again. This time we are going to add a column to our place value chart. We’re gonna add the hundreds to the place value chart. And then we’re going to take 10 of these tens and trade them for 100. So, again, count up 10 of these to put back in the bank, and you trade them for 1 100. And to write down what we did, 8 plus 3 gave us 11 tens. We kept 1 of those tens and traded 10 of them for 100. So we put a one in the hundreds place. And so we have a total of 117, and see how that matches what’s left on the place value map, 100, and a 10, and a 7, 117. And so that’s how you can use play money to teach addition, and now we’re going to reverse that and use the same ideas for multi-digits subtraction.

Again, we’ll start with problems that don’t require regrouping and then move on to ones that do require regrouping. So here we have 46 minus 33. This and what I’ve done is I have put up 4 tens and 6 ones to match our top number, the 4 tens and the 6 ones in the 46. And now we’re going to take away 33 from the 46. There are several different ways to interpret subtraction, but this one is easiest for this procedure. So we want to literally take away three of the dollars, and then we write down what we did. We did six, we subtracted three, and that left us with three ones here. Now, moving over to the tens column, we start with four tens and we wanna subtract three. So, again, we will just simply take away 3, 1, 2, 3, leaving us with 1 10 left. So 46 minus 33 equals 13. One 10 and 3 ones. So it’s pretty easy when you’re not regrouping. But the play money is a great way to show regrouping for subtraction as well. You might think of this as borrowing a 10 that you learned that way.

So now this is the more difficult one, with 83 minus 45, so with 8 tens and 3 ones. And we need to subtract four tens and five ones. So let’s start with the ones place again. We have three ones, but we need to subtract five. So what I need to do now is I need to take one of my tens here. I’m going to put it into the bank and trade it for ten ones over here. These are gonna get a little crowded, but I think I can just fit them in 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10. Just gonna end it in there. Okay, so now what I’ve done is…now we’re gonna write down what I did. I had eight tens, but now I only have seven of those tens left. I took that other 10 and traded it for 10 ones, so now I have 13 ones over here in the ones column of my place value chart. And so I write the 13 here, and I like to write it out entirely so that kids see that it’s really 13 ones. And now I need to subtract five of those ones. I’m gonna take five of them away, one, two, three, four, five. And that leaves me with eight ones left in the ones column. Now, we move on to the tens column. Now, there’s seven tens in this tens column, and we need to subtract four of them. So, again, we’ll subtract here. One, two, three, four, and that leaves three of the tens left. So 83 minus 45 equals 38.

So that’s how you can use play money to teach multi-digit addition and subtraction. Make sure your kids know the concepts of addition and subtraction, the addition subtraction facts, and place value before you start. And then teach this incrementally step by step, and your kids will be fluent in no time. Thanks and happy math.


How to Teach Subtraction Facts That Stick

In this instructional video, the Well-Trained Mind’s math expert Kate Snow (a homeschool mom herself, and author of three books) gives you practical, simple tips and techniques for helping children master the skill of subtraction.

In this instructional video, the Well-Trained Mind’s math expert Kate Snow (a homeschool mom herself, and author of three books) gives you practical, simple tips and techniques for helping children master the skill of subtraction.

All the slides from Kate’s presentation can be found here.

Kate is the author of Preschool Math at Home, Addition Facts That Stick, and Subtraction Facts That Stick…easy-to-use books for parents who might feel intimidated by math but want to give their children a strong foundation in the subject.

For more great math tips, visit Kate’s website or check out her courses for parents at the Well-Trained Mind Academy.


Maturity & The Real Child, Part II: Strategies for the Age-Grade Mismatch

If your child’s maturity level doesn’t happen to coincide with the (artificial) grade level that matches his age, what strategies can you use?

First, do your best to separate out the different “subjects” that make up the child’s curriculum and think of each of them separately.

On the most basic level, most students find either language based (reading- and writing-based subjects) or symbolic (math and related subjects) learning to be more natural, and will progress more rapidly in their preferred subject type.

Don’t use either the “slower” or “faster” subject as a way to locate the child within an entire grade.

I’ve often spoken to parents who are frustrated (for example) because their fourth-grade aged child is reading at a much higher level, but is struggling with second- or third-grade math skills. The tendency is to focus on the child’s slower areas, to spend more time on those in order to move the child into a higher grade. But the result can be that the child ends up evaluating himself by his weaknesses, not his strengths. The most damaging thing about our grading system is the way in which it can obscure natural gifts, requiring children to spend untold hours laboring away at subjects they dislike at the expense of learning in which they excel.

A century ago, Montessori educator Dorothy Canfield Fisher wrote a popular children’s novel called Understood Betsy in which nine-year-old  Elizabeth Ann leaves the big city and her urban school, where age grading has been thoroughly instituted: “In the big brick schoolhouse,” Canfield writes, “nobody ever went into another grade except at the beginning of a new year, after you’d passed a lot of examinations. She had not known that anybody could do anything else.” Instead, she goes to live with country cousins and attends their tiny rural school, still one room and multi-age, led by one teacher who praises her reading skills, but realizes that she needs work in math:

After the lesson the teacher said, smiling, “Well, Betsy, you were right about arithmetic. I guess you’d better recite with Eliza for a while. She’s doing second-grade work. I shouldn’t be surprised if, after a good review with her, you’d be able to go on with the third-grade work.”

Elizabeth Ann fell back on the bench with her mouth open. She felt really dizzy. What crazy things the teacher said! She felt as though she was being pulled limb from limb.

“What’s the matter?” asked the teacher, seeing her bewildered face.

“Why—why,” said Elizabeth Ann, “I don’t know what I am at all. If I’m second-grade arithmetic and seventh-grade reading and third-grade spelling, what grade AM I?”

The teacher laughed at the turn of her phrase. “YOU aren’t any grade at all, no matter where you are in school. You’re just yourself, aren’t you? What difference does it make what grade you’re in! And what’s the use of your reading little baby things too easy for you just because you don’t know your multiplication table?”

“Well, for goodness’ sakes!” ejaculated Elizabeth Ann, feeling very much as though somebody had stood her suddenly on her head.

“Why, what’s the matter?” asked the teacher again.

This time Elizabeth Ann didn’t answer, because she herself didn’t know what the matter was. But I do, and I’ll tell you. The matter was that never before had she known what she was doing in school. She had always thought she was there to pass from one grade to another, and she was ever so startled to get a glimpse of the fact that she was there to learn how to read and write and cipher and generally use her mind….[I]t made her feel the way you do when you’re learning to skate and somebody pulls away the chair you’ve been leaning on and says, “Now, go it alone!”

Grasping this thoroughly yourself, and then articulating this reality to the child—giving her a sense of normalcy over the variety in her abilities—can begin to defuse frustration. 

Second, take a good hard look at the child’s physical development.

Age-grading is based on a mean—and if you’re not a math person, “mean” is simply one way to express “average.” In math, you find the mean by adding a list of numbers together and dividing them by the number of numbers. Here’s what’s important about that: Often, the mean is a number that didn’t even appear on the original list.

13, 17, 28, 52, 71

Added together: 181

Divided by 5: 36.2 (the mean)

The “average” fourth grader is 4’3 3/4” tall and weighs 70.5 pounds. But in any given fourth grade class, there may be no students who are actually this height and weight–plus ten-year-olds who weigh anything from 40 to 90 pounds and range between four and five feet in height.

Physical development affects learning. Children who are on either end of this completely normal range often struggle with “grade level” work. Very small children need time to catch up; children who are on the larger side often need the same amount of time to figure out how to manage their bodies,; they can be like large uncoordinated puppies, growing towards an imposing presence but with no idea how to manage their limbs. When you’re trying not to trip, struggling to keep your pants up and zipped, and having a hard time fitting into your desk, your attention isn’t going to be on your essay assignment.

If your child is on the low or high side of the “average” for his or her “age grade,” consider that you may have a serious maturity mismatch.

Third, if you’re dealing with younger students, be very careful about accelerating them.

It’s very tempting to jump a bored kid ahead by one or two grade levels as a quick fix, but consider this: The biggest maturity leaps happen between 6-10, and again between 13-16. If you leap your second grader ahead into third grade because she’s more mature than the other second-graders, there’s a very real possibility she’ll find herself, at thirteen, in a group of more-mature students and struggling.

It’s the nature of our school system that it is much easier (and less emotionally fraught for the kid) to move ahead than to drop back. Dropping back is traumatic, even when it’s necessary. So think very hard about the wisdom of starting a child early or accelerating them before they reach puberty.

And think about the results of accelerating, on the other end: a student who reaches high school early will not be old enough to drive (when all of his friends are) or take part in other age-graded activities.

You may also end up with a sixteen- or seventeen-year-old college freshman. Some students are mature enough to benefit from college at those ages, but fifteen years of university teaching has convinced me that most are not. You need to be not just intellectually, but emotionally mature to benefit from college—and emotional maturity can’t be rushed; it happens when the earth has gone around the sun the appropriate number of times.


Maturity & The Real Child, Part I: The Problem With Ages and Grades

On my Virginia farm, I raise livestock; lambs and kids born on the same date rarely clock in at the same size, wean themselves, or eat the same amount of hay and grain on any given day. Daffodils bloom, baby birds fly, and puppies stop chewing on chair rungs when they’re ready—not when the calendar dictates.

But we generally don’t extend this same consideration to our children.We’ve been so conditioned to accept the pattern of infancy, toddler, preschool, elementary, middle, high school, college that it’s almost impossible for us to break out of it and think: What makes me think that  this tiny human being should mature on the exact same schedule as the rest of the tiny human beings born at the same time?

The common-sense answer is: Nothing convincing.

It is far too easy for us to consider the speed with which our children march through the grades as some sort of natural measure of their intelligence. In fact, we consider fast movement through the grades to be a positive good: Fast means smart.

Thank carefully about this assumption. It makes speed to be a positive good–when, in fact, it should be morally neutral. I’ve written about this elsewhere—most recently, while debunking the value of speed-reading in The Well-Educated Mind:

The idea that fast reading is good reading is a twentieth-century weed, springing out of the stony farmland cultivated by the computer manufacturers. As Kirkpatrick Sale has eloquently pointed out, every technology has its own internal ethical system. Steam technology made size a virtue. In the computerized world, faster is better, and speed is the highest virtue of all. When there is a flood of knowledge to be assimilated, the conduits had better flow fast.

Our general approach to life is too often shaped by the combined factory-computer ethic: More and faster is better.

Think about how you refer to the computers in your house. The fast computer is the “good” one; the old slow one is the “bad” one that no one wants to use. Or the checkout line at the grocer store: the bad line is the slow one. I’m not suggesting that speed is completely unimportant, particularly if you need to get your groceries bought before dinner, but the ease with which we assigning the morally loaded words “good” and “bad” to a span of time should give you pause.

Now circle back to the child who is maturing at her own perfectly normal rate, but has been slotted into our Prussian age-grading system. As parents, we too often take pride in our children working “above” grade level—assuming that the faster you move through the grades, the more accomplished the child is. (In fact, in many home schooling circles, graduating a child at fifteen or sixteen and sending them off to college early has become a validation of how well the parents have done their job.)

Worse than that, we manage to convey a very clear message to our children that if they do not advance through the grades at the correct ages, they are “slow,” behind, failures. Even when it is perfectly clear that a child needs some extra time to mature and to master fundamentals, we feel that providing them with that time risks separating them from friends, giving them a sense of failure, putting them “behind.” Slow, like fast, becomes a moral judgement–an evaluation of the child’s worth–rather than a simple measure of maturity.

What are the signs of a maturity mismatch between a child and a grade level?

The prime symptom of immaturity is nonverbal frustration. A child who weeps, or resists but won’t say why, or slouches and refuses to make eye contact, is signaling that something is wrong—but cannot articulate what it is. Children confronted with work that is too advanced for them are usually incapable of saying, “I’m sorry, but this is developmentally inappropriate and my mind isn’t yet able to grasp it.” In fact, a child who says, “This is too hard!” is probably actually working at close to grade level, because she’s able to understand the task even if it’s difficult.  The child who just bursts into tears isn’t ready to do the work in front of her. She can’t yet comprehend how to do it, or find a way into the task.

A child who is working right at the top level of his maturity can also be derailed by physical factors—allergies or a bad case of flu, suddenly expending a lot of physical energy in a new sport, puberty. What was once difficult suddenly becomes impossible. If a child stalls or begins to go backwards with work that had previously been doable, consider that he might be bumping up against a maturity ceiling.

And remember that abilities doesn’t develop evenly in children, any more than their bodies grow at an even rate.  In our highly structured school system, students are expected to be at grade level in math, science, reading, and writing. But these require very different thinking skills, and it is far more common for students to be working at two or more grade levels across the curriculum. It is normal for a fifth-grade aged student to be writing at a third grade level, reading at a fifth grade level, and doing math at a seventh grade level. A child who prospers at two subjects and cries over the third may still be showing immaturity—and the answer may be to drop back to a lower level in only the third subject.

When learning stalls, particularly if it’s across the board, always consider evaluation by a learning specialist. But in many cases, a child who’s struggling simply needs the earth to circle the sun one more time.

If there’s a mismatch, what strategies can you use?


How to Teach Addition Facts that Stick

In this instructional video, the Well-Trained Mind’s math expert Kate Snow (a homeschool mom herself) gives you practical, simple tips and techniques for helping children master the skill of addition.

In this instructional video, the Well-Trained Mind’s math expert Kate Snow (a homeschool mom herself) gives you practical, simple tips and techniques for helping children master the skill of addition.

If you missed any of the slides in Kate’s presentation, you can find them here.

And to get started now with your own children, try Kate’s easy-to-use books Addition Facts that Stick and Subtraction Facts that Stick. Samples of those products are included in the product descriptions, but who has time for two clicks these days, right? Your kids are flooding the bathtub while you click that second click. So here is a sample right NOW.

To learn more from Kate, check out her courses at the Well-Trained Mind Academy, or visit her website.


Fourth Edition: Resources Update

Below, you’ll find a continually updated list of resources recommended in the fourth edition of The Well-Trained Mind that have changed in their format or availability. If you’ve discovered others, please email us at [email protected]!


DATE: December 20, 2016

RESOURCE: Latina Christiana II (page 233)

CHANGE: This product has been discontinued by the publisher, Memoria Press. Memoria now recommends that you progress straight from Latina Christiana I into First Form Latin (as recommended on p. 489 as an alternative path; it’s now the only path).


Why Do Six-Year-Olds Go to First Grade?

In contemporary education, “What grade are you in?” has become synonymous with “How old are you?”  But the age grading system that shoves six-year-olds into first grade, seven-year-olds into second, and so on up isn’t remotely natural.

Nor is it based on sound educational principles. 

It dates, in the U.S., only back to 1847. Before then, teachers in one-room schools taught mixed-age groups together, with no standard curriculum, and students moved to more difficult material when they were ready, at widely varying times. (The medieval predecessor of the American one-room schoolhouse, the European cathedral school, typically had students from age 8 up to 21 or 22, all chanting the same lessons until learned.) 

But over in Prussia, a new system had been instituted in the early 1800s: smaller classrooms where students were grouped by age and led by a single teacher. This strategy wasn’t driven by educational research. It was an attempt to try to restore Prussian greatness after a humiliating defeat by Napoleon in 1806.

Struggling to rebuild, Prussian statesmen decided to organize schools like military units, in order to instill the will to fight and build pride in Prussia’s historically militaristic national culture. Students were organized into platoons by age and assigned to a single “squadron leader,” (a system which made the transition into military service quite straightforward).

When Horace Mann, American politician and reformer, visited Prussia, he was serving as the Secretary of Education for the state of Massachusetts. He had long hoped to see a “common school” introduced into America, a school that all students would attend together, a school that would give American a common language and purpose, a school equally accessible to all. “Education,” he wrote, in one of his annual reports, “beyond all other devices of human origin, is a great equalizer of the conditions of men.”

But drawing the masses of the uneducated into those multi-age classrooms was a daunting task, one that would require huge resources and scores of talented and flexible teachers. The Prussian system (complete with compulsory attendance, not at that time an American practice) struck Mann as the perfect answer: the very best way to channel a large number of diverse students into a single institution with maximum efficiency.

With Mann’s support, the Prussian system finally came to Massachusetts in 1847, when the Quincy Grammar School was built with twelve separate classrooms, containing a single age graded class led by a single teacher. The new plan did indeed turn out to be highly efficient (factories generally are), and age-graded schools were soon spreading—into the rest of New England’s urban centers, westward to other cities, and then out into rural areas as well. By the turn of the century, age grading was the norm in almost all of the nation’s “common schools. Compulsory attendance laws, also modeled after the Prussian system, followed shortly after; Massachusetts again led the way, passing the first regulations in 1852.

So our strong identification of age with grade is the result of (in the words of Rick Hess) “our peculiar devotion to a model that defeated Prussian leaders developed in order to salvage the last vestiges of their shattered national pride.”


High School Transcript Forms

Which format should my child’s transcript take? Here are a few suggestions.

There is no single universally-accepted form for high school transcripts. Forms acquired from any of the following sources are perfectly acceptable. (See Chapter 41 in the fourth edition of The Well-Trained Mind for step-by-step guidance to creating a high school transcript for your home-educated high school student.)

Build your own transcript online at Transcript Maker.

If you (still) have a PC, you can use Edu-Track Home School software or Inge Cannon’s Homeschool Transcripts. Neither are currently Mac-friendly.

Janice Campbell’s Transcripts Made Easy shows you how to create a transcript with your word processor.



The Newest Edition of a Homeschool Classic

Susan Wise Bauer walks us through the differences between the 4th edition of The Well-Trained Mind and its previous versions. Text! Video! Bullet Points! Everything you need to navigate the extensively updated edition is right here.

The Well-Trained Mind: A Guide to Classical Education at Home has gone into its fourth edition! Here’s a list of the major differences in this most recent revision.

  • Each chapter has been separated into two sections: first, how to teach a subject (methods, goals, expectations, etc.); and second, what resources to use (recommended texts and curricula.) This makes the book even more flexible, since parents can use the principles of teaching even if they choose to use other specific texts or programs than the ones we suggest.
  • Completely updated book and curricula recommendations.
  • New guidance on dealing with learning challenges and difficulties. Children who struggle with learning disabilities seem to make up a much higher percentage of home educated students than in previous years, since schools often are unable to provide the support they need. As home education has become more visible and additional resources have become available, many more parents are reacting to these very individual needs by choosing to remove struggling children from the classroom entirely.
  • New online resources, including alternative curricula (not included in the book because they were too complicated, expensive, specialized or quirky—but all of which have enthusiastic support among many veteran home schoolers), additional help for struggling learners, apps and online learning games, and more.
  • Brand-new maths and sciences chapters. Classical education has often been criticized as stronger in the humanities than in the maths and sciences. Working with highly qualified experts and experienced teachers, we have overhauled our approach to provide a much more rigorous and coherent maths and sciences education.
  • Shift of quickly outdated appendices (list of suppliers and publishers, index of home education organizations, links to state laws, and other constantly changing resources) online, where they can be regularly updated.


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