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 insituted 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.


Spelling Power

Subject: Spelling

Grade level: 3-12

Publisher: Castlemoyle Books


Description:  This one-volume spelling resource, recommended in earlier editions of The Well-Trained Mind, is designed for use in any grade (third and after). Rather than taking a workbook approach to spelling, Spelling Power is list-focused; it divides a list of 5,000 words (“most frequently used and misspelled by children and adults”) into eleven levels, based on frequency of use, and then further separates those levels into groups by phonetic principles. Students are taught to visualize and trace words in the air before writing them; a selection of games and activities expand the method to include the other senses. The parent needs to spend preparation time on the lesson before teaching it.

Pros: Affordable, multi-sensory.

Cons: Requires more preparation time than Spelling Workout but provides less detailed instruction than All About Spelling or Sequential Spelling.

Why it’s not in the book: Spelling Power is still a good resource, but All About Spelling and Sequential Spelling are more complete programs for dyslexic and spelling-challenged students, while Spelling Workout is simpler to use for students who don’t struggle with spelling. Spelling Power “falls in the middle.”

What parents say: Visit these forum threads to find out more.


Memoria Press Online Academy

Memoria Press, a classical curricula publisher with a strong focus on Latin and classical literature, offers live online classes for grades 3-12 across the curriculum.

Enroll for Latin, Greek, logic, and other subjects, taught by qualified instructors.

Memoria Press Online Classical Academy is explicitly Christian but does not require students to hold their beliefs in order to enroll.

Home page:


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