Introducing Grammar for the Well-Trained Mind

Introducing Grammar for the Well-Trained Mind!

After you’ve watched this video, check out our other videos for Grammar for the Well-Trained Mind: Teaching Tips for Grammar for the Well-Trained Mind and A Sample Lesson from Grammar for the Well-Trained Mind.

Want to dive in? Here is the first six weeks for free!

Transcript (with a few annotations):

I’m Susan Wise Bauer, and I want to introduce you to our brand new grammar program. It’s called Grammar For The Well-Trained Mind.  This has been in process for I think the last eight years and we’re so excited that it’s finally finished and ready to bring to you.

Grammar For The Well-Trained Mind is a four-year grammar program. You can start at any time after [THE START OF] fifth grade. Once the student has gone through all four years of it, they will have all of the grammar knowledge that they need to go on to the study of rhetoric, which is really just the application of grammar to written and spoken word. So each year of this program, you use four books.

The first book is the Core Instructor Text. This has in it all of the lessons that you will be teaching the student. These lessons are scripted. We give you dialogue so that you don’t have to figure out how to best teach the grammar concepts. We’ve already done that for you. You’ll use this same Core Instructor Text for each of the four years of study. So one purchase, you’ve got your core text for all four years.

The second thing you need then is a Student Workbook. This Student Workbook has in it all of the rules and examples that the student will be learning. And full practice exercises to drill every single concept that the student is learning.

Each Student Workbook comes with a Key. These Keys don’t just have the answers to the Student Workbook. They also have explanations for you, the parent, so that if something is a little bit ambiguous or difficult, you’ll have additional resources to explain to the student why it works the way it does.

So, Core Instructor Text for all four years. Each year a separate Student Workbook and Key. And then finally we have a Comprehensive Handbook of Rules.  So this is, again, a one time purchase that the student will use for all four years. It has all of the rules and examples in it for the student’s reference.

Now this grammar program is different than others because it is based on the three essential things that have to happen in order for our minds to really comprehend grammar.

First, there has to be repetition and memorization. One we reason why we use the same core handbook [Core Instructor Text] every year is that every year you can use the same words to explain the grammar concepts to the student. The same rules, the same examples, every single year.  That repetition will help build the information, really solidify it in the student’s mind. So the first thing is just memorizing the rules.

Second, whenever you memorize a grammar rule, because grammar is something that governs the way real language works, you need an actual example. So every year the student repeats the same examples that illustrate the rules. That way they always have a concrete example of how the rule works to link to the actual memorized rule in their brain.

The third thing the student needs after this repetition of rules, repetition of examples is just lots and lots of practice. So although each Student Workbook has the same rules and the same examples for every year, each one also has a full new set of exercises of practice for the student to go through.

These exercises are mostly drawn from real literature, from novels, from histories, from biographies, from science books. We really encourage the student to take the rules they’re learning and then look at how they operate in real literature. So, Grammar For The Well-Trained Mind, four years, all the grammar you need through high school. Each year you’re going to have the Core Instructor Text, one book for all four years. Each year you’re going to have a new Student Workbook and Key. And then the student is going use the same Comprehensive Handbook of Rules as a reference work all the way through.


How to Teach Your Wiggly Preschooler Math

Preschoolers are a ton of fun. They’re energetic, curious, and immensely proud of their new skills and abilities. But their short attention spans and high energy levels can make it difficult to get these sweet wiggly little people to sit and focus.

That doesn’t mean that you can’t give your preschooler a great math foundation for kindergarten, though. It just means that you need to work with your preschooler’s natural tendencies rather than against them. Instead of trying to force your preschooler to sit down to worksheets and drills, here’s how you can teach your preschooler math in a way that’s both fun and developmentally appropriate.

1. Find the math in your child’s interests.
One of the most adorable aspects of preschoolers is the way they get obsessed with specific topics. When my son was four, his ruling passion was Thomas the Tank Engine. He would zoom his trains along the tracks for hours, and he had an encyclopedic knowledge of the name of every character in the series. So, we did lots of train math. We counted lines of trains, added rows of trains together, and figured out how many empty spots there were in the roundhouse.

My daughter’s preschool interest was her beloved stuffed animals. We’d sort them by type and count how many were in each category. (“Three dogs, four cats, one sheep…”) She also loved to have tea parties for her stuffed animals, so we would count out play cookies for each guest and make sure each had an equal number.

Whatever your child’s interest, try to include counting, comparing, and simple addition and subtraction when you play together. Incorporating numbers into your child’s favorite activity will not only build his math skills but also help him appreciate the importance of numbers in daily life.

2. “Mom, look at me!”
Preschoolers love to show off all of their new physical skills: turning somersaults, walking on tiptoe, hopping on one foot, and so on. You can use their need to move to get them interested in numbers, too, especially when they can use numbers to quantify how amazing they are: “I can run around the whole playground in just 14 seconds!”

For a child who’s still learning to say the numbers in order, offer to count how many times she can do something. “Want me to count how many times you can hop on one foot?” Listening to you say the number sequence will help your child internalize the correct order.

For a child who already knows how to count, use a mix of questions to help him practice this vital skill. Sometimes, ask, “How many times can you…?” so that your child does open-ended counting. Other times, ask questions with a fixed amount: “Can you do three somersaults in a row?” Learning to stop after a certain number is a key skill in helping children learn to count out a fixed quantity.

By counting physical motions, you’ll also teach your child that numbers can be used to count more than just sets of objects. This will help your child understand the very abstract idea that numbers can be used to count intangible things, too.

3. Snuggle up with a book.
You probably already have a time each day when you cuddle up on the couch with your preschooler and read. This read-aloud time is very important for developing your child’s literacy skills, but you can build your child’s math skills at the same time, too.

One of the easiest ways to do this is to simply check out a math book or two each time you haul a stack of picture books home from the library. Look for books about numbers, shapes, time, measurement, and patterns. Many libraries have a special section for these books (or a special label on the spine) so ask your librarian how your library organizes these books.

Some math books are very dry, but there are many engaging ones that teach mathematical concepts along with an entertaining story. To help you get started, I’ve put together a list of 12 wonderful math picture books for preschoolers, along with suggestions for how to gently draw your child’s attention to the math in the story. (You can download it by following the link below.)

Preschool math doesn’t have to mean formal lessons or worksheets. By incorporating numbers into your child’s interests, physical play, and read-aloud time, you can enjoy this sweet age and give your wiggly preschooler a playful, developmentally-appropriate foundation in math, too.


Teach Your Young Reader how to Choose Books

How to Use Home Library Organization to Build Vital Literacy Skills

A woman walks into a book store. She’s in the mood for an escape through an imaginative story, so she browses the fiction section. Within this section, she navigates to the fantasy aisle. Sooner or later, she is bound to stumble across an author that she recognizes, with a story that is worth exploring. She sees her old friend J.R.R. Tolkien. “I must be in the right place,” she thinks.

Whether she realizes it or not, she is about to select a book using sophisticated strategies like author and theme navigation. A young reader has not yet developed the ability to navigate a mysterious and magical place, like a book store, school library, or household bookshelf. Children, especially beginning readers, can’t tell the difference between multiple books on a shelf. But if you organize your library the right way, you can teach your student how to select the books they actually want to read.

Here are three organizational strategies that you may wish to use in your library. These strategies will work at home, at the co-op, or at school, and will work both with books that you own, and books borrowed from the library.

Before You Get Started
You will need bins to organize your book collections. Stackable bins can be purchased at the dollar store. Remember that children’s books tend to be larger and thinner than adult books. You may need to bring a book with you when picking out tubs in a store. Lakeshore (and other online educational websites specializing in learning materials) have nice bins, or you can make bins at home. If you’re familiar with a magazine rack, you understand the shape of a book bin. DIY book bins can be fashioned out of cereal boxes using an exact-o knife.

Bins are easy to store and label. For the youngest readers, there may be a picture with the label. Older learners may have bins with written descriptions. You will have an idea of how to label the bins when you decide how you will organize your library.

Organizing by Author
You want to make your student aware that real people write and illustrate books. A child doesn’t automatically know that Eric Carle painted watercolor pages with many colors, cut them out with scissors, and made the pictures in The Very Hungry Caterpillar. But arranging books by author is the primary method of organization for many public libraries. If you organize your household collection by author, your child will feel at ease when they enter a school or public library.

Begin by grouping your book collection by author. Don’t worry if you only have two of Eric Litwin’s Pete the Cat books. When your child checks out Pete the Cat and the Cool Cat Boogie from the library, you’ll know right where to put it! You’ll soon notice that your child will know just where to put the Clifford books you found at a local yard sale. He’ll put the vintage edition of Robert McCloskey’s Make Way for Ducklings that he received for his birthday right next to Blueberries for Sal and One Morning in Maine.

Talk with your child about why she likes an author. Conversation improves comprehension, and helps young readers connect with authors. Find out if your child likes an author because he is funny, like Tedd Arnold (“Fly Guy” series), or because the author writes about important life lessons, like Anna Dewdney (“Llama Llama” series). The more you know about what your child likes to read, the better your recommendations will become. A fun challenge would be to set a goal to read all of the titles by an author your child is really into, like Mo Willems, Syd Hoff, or Jon Klassen. Organize by author until your child is confident searching and successfully finding authors on her own in a public library.

Organizing by Theme
Organizing by author has its limitations in a classroom or household library. You may not have 20 books by Rosemary Wells. And chances are the only Crockett Johnson book you own is Harold and the Purple Crayon. Another option is to organize your home library by theme.

It is easy to organize books by theme once you have a few books that are related to one another.

Some themes from my own classroom include “Humor,” “Non-Fiction Animals,” “Science,” and “Back to School.” In my “Humor” bin, I have David Shannon’s “No David” series, DogzillaDragons Love Tacos, and Dragons Love Tacos 2. In a tub labeled “Family,” I have Llama Llama Red Pajama, I Will Always Love You, No David, The Kissing Hand, and Goodnight Moon. All of these books are about family, comfort, and love.

Organizing by theme allows for a lot of flexibility. You can move books out of author bins that are related to specific themes easily. For example, you can take Clifford’s Halloween out of the Norman Bridwell tub and move it into the Holiday tub for the month of October. Themed bins can also reflect what is going in your kids’ lives. If you choose to focus on a particular character trait, like honesty, you can organize books representative of that behavior into a tub. A week or two before a family trip, you could organize books about travel (fiction or nonfiction) into a tub. This organizational method will then allow you to have a conversation with your young reader about why you chose to place a book into a specifically-themed bin, and what the author’s intent was for writing this particular book. This will also help your students in a future library visit when they move outside of the children’s section.

Here is a list of themes that you might try in your library:
Being Tidy
Counting (Early Math Skills)
The Earth/Environment/Nature
Overcoming Fear
Unconditional Love
The Word “No”

Pay attention to what your student is interested in; this will help you to create specific tubs that will thrill your child, like “Butterflies,” “Construction Equipment,” or “Making Pizza.”

Organizing by Academic Subject
The last organizational method that could work for you is to organize by academic subject. These titles would be tied to current subjects that the child is studying. If a student is beginning to learn about numbers, then counting and computation books like 5 Little Ducks, The Very Hungry Caterpillar, or 10 Apples on Top would tie reading to early math skills.

Organizing by academic subject also leaves room for branching off into further study. If a student is learning about animals in science, you may discover that the student loves whales and dolphins. Then you, as the curator of the classroom or household library, could create a tub for ocean animals. Say your student is learning about how a town works, and you find that he is interested in public servants. You could create a bin centered on men and women who serve as police and emergency responders. If a student is reading about Thanksgiving and gets really into why Americans celebrate by eating turkey, you could create a bin for books about various food traditions.

Literature often gives students the experience of connecting with a subject and a character at the same time. For example, a student may be learning about medieval times and also be into “Fly Guy.” When a student reads a book like Fly Guy presents Castles, her interests come together in a way that makes reading, and learning about history, even more exciting.

However you choose to organize your library, remember that the end goal is to help your student associate books with themes and ideas. Children who learn this skill at an early age will be able to choose books easily as adults.



Five Events That Began the Renaissance (Or Ended the Middle Ages)

The Middle Ages (the favorite historical period of 9 out of 10 young history students!) is generally thought to occupy the years between the collapse of Rome and the beginning of the Renaissance—between, more or less, 400 and 1600 AD.

Most of us were taught, in history class, that the Middle Ages ended when the Renaissance began. But that’s not quite right.

In fact, it’s a little bit like saying, “My childhood ended when I started liking spinach,” or “He stopped being President when he got married.” It’s putting two completely different things side by side, and pretending that they are the same.

For one thing, the Renaissance isn’t a historical period, like the Early Modern period; it’s an intellectual movement.

And for another, the idea of the “Middle Ages” didn’t even exist until historians began talking about the “Renaissance.” In the eighteenth century, scholars increasingly focused on the reawakening of interest in ancient art, ancient philosophy, and ancient literature that began in the 1300s (or later, depending on which eighteenth-century historian you’re reading).  Once they began to identify this re-awakening, they had to find a name for the years that came between ancient culture itself and the return of ancient culture. These “in between” years didn’t have any quality of their own—they simply lay in the middle.

Hence, the Middle Ages.

So instead of trying to figure out what “ended” the Middle Ages, let’s try to identify what changed, between 1200-1500 or so—what actual events began to create something new in history.


I. The Failure of Holy War

To properly set the stage for the Renaissance, we need to start with the failure of the First Crusade (1095-1099).

You might think the First Crusade succeeded–it did, after all, recapture Jerusalem.

But it failed in another drastic way. The First Crusade fatally cracked the accepted Western consensus that the cause of Christ was the highest and most paramount loyalty on earth.

Those cracks had to widen and fracture apart before Renaissance humanism could establish new loyalties.

Here’s a quick review: Just before the First Crusade, the Greek-speaking Christian empire of Byzantium was under the rule of the emperor Alexius Komnenos. And Alexius Komnenos was alarmed by the gathering power of the Ottoman Turks over to his east.

The Turks were wandering tribes, originally from central Asia, who had been filtering westward for centuries. About fifty years before, they had united together into a coalition–and their unified power had grown. They conquered a good deal of Byzantine land in Asia Minor; in 1077 they took the city of Jerusalem away from the Fatimids, the Islamic dynasty that ruled Egypt;  and by 1095 Alexius was seriously worried.

So he sent a message to Pope Urban II, in Rome, asking for mercenaries to help him push the Turkish threat back.

Urban II had soldiers to spare–he was not only the head of the Christian church, but the political ruler of the Papal States, a chunk of land which had been given to the church bit by bit over the past 300 years by various rulers of Italy–and the Papal States had its own army.

Instead of sending mercenaries, Urban II decided to issue a call to every Christian man in the west. Here is what he said:

The Turks and Arabs . . . have occupied more and more of the lands of those Christians, and have overcome them in seven battles. They have killed and captured many, and have destroyed the churches and devastated the empire….On this account I, or rather the Lord, beseech…all people of whatever rank, foot-soldiers and knights, poor and rich, to carry aid promptly to those Christians and to destroy that vile race from the lands of our friends….Christ commands it….[And] All who die by the way, whether by land or by sea, or in battle against the pagans, shall have immediate remission of sins.

 The goal of this Crusade, Urban II said, was to liberate Jerusalem, which Alexius Komnenos hadn’t actually mentioned.

When the western noblemen and knights summoned by Urban II began to show up in Byzantium, on Crusade, most of them paid no attention to what Alexius Komnenos actually wanted—help defending Constantinople. Instead, they started to take chunks of Alexius’s empire for themselves.

A Sicilian adventurer named Bohemund captured the Turkish city of Antioch, at the upper right hand corner of the Mediterranean Sea, and kept it for himself. He called it the Principality of Antioch.

Others followed his lead. Soon there were two other Crusader kingdoms in the east, the Kingdom of Jerusalem and the County of Edessa–all of them enemies of Alexius Komnenos.

In fact, it wasn’t long before Bohemund declared war on Constantinople.

He planned to sail back to Sicily to recruit more soldiers from his homeland. Alexius got wind of this and sent Byzantine ships to hover in the Mediterranean, just off Antioch, to intercept him on his way. So Bohemund told his officers to announce that he was dead, and after a lot of very public hair-tearing and weeping they stuffed him into a coffin with air-holes in it, ostentatiously loaded the coffin onto a ship, and announced that they were taking him home. Alexius’s biographer, his daughter Anna, says, “[I]n order that the corpse might appear to be in a state of rare putrefaction, they strangled or cut the throat of a cock and put that in the coffin with him. By the fourth or fifth day at the most, the horrible stench was obvious to anyone who could smell. . . . Bohemund himself derived more pleasure than anyone from his imaginary misfortune.”

And that was the heroic ending of the First Crusade.

These Crusader kingdoms were the most lasting legacy of the Crusade. They were a constant reminder that whatever Christian unity had existed at the beginning of the First Crusade was no longer in operation. There was no longer a “land of the Christians,” as Pope Urban II had put it. The “consensus of Christendom” had been revealed as an illusion.

 II. The Rediscovery of Aristotle

Over on the Spanish peninsula, the Christian kingdoms of the north had been fighting against the Muslim dynasties of the south for centuries. By 1144, the Christian kingdoms of Spain had managed to push as far south as Seville and Cordoba. This gave them control of a score of major Muslim cities–and what they found in those cities were a lot of unfamiliar books.

In those books were Greek works of philosophy, medicine, geography, and science that the West had almost forgotten.

By 1200, Europe was largely ruled by monarchs of German, Frankish, and Roman descent. Their courts, and the scholars and noblemen who helped run their countries, spoke (and read, when they read, Latin), and also their own local dialectcs–the early forms of French, German, and Italian.

They did not speak Greek. And they also had a very different approach towards learning.

The Greek intellectual tradition, which had been kept up in Byzantium, was heavily tilted towards philosophy and natural science–understanding the universe for the sake of understanding it.

But the Franks and Germans were heirs to the Roman, Latin intellectual tradition–which was a very practical-minded one.  The Romans did not treasure learning as an end in itself. Natural science and philosophy, for the Romans, were leisure-time activities for the wealthy and the independent–hence the label “liberal arts,” arts for free men to indulge in as they were able.

Grammar, logic, and rhetoric, on the other hand, were highly valued by the Romans, because they were useful. You could win a case in the law courts, or get elected, if you were good with words.

So the Latin tradition was literary and practical, not philosophical or scientific. And the Franks and the Germans had no particular reason to keep up their Greek–or to translate Greek texts.

However, thanks to the proximity of Islamic lands to the Greek-speaking East, Muslim scholars did read the Greek texts.  Aristotle, Plato, and many other Greek authors had been translated into Arabic, and brought across North Africa into the Muslim lands of Spain.

When the Christian kingdoms took control of the Arabic cities in Spain, they opened up the libraries of Cordoba, Seville, and especially Toledo to western scholars–most of whom didn’t speak Greek, but many of whom could read Arabic.

Beginning in the middle of the twelfth century, Italian, German, and Frankish scholars started to travel to Spain, which they could now do, semi-safely, to read and translate these texts. There were dozens of them, but one of most prolific was Gerard of Cremona, who single-handedly translated over seventy Greek works of philosophy, natural science, and mathematics from Arabic into Latin.

And among the works translated in the twelfth century were many, previously very little read, of the works of Aristotle.

Why was the re-introduction of Aristotle back into Western thought so significant?

Because Aristotle offered a complete world-view, a way of understanding the universe, man within it, and man’s place within it, that was internally consistent, convincing—and had absolutely nothing to do with Scripture or with Church authority.

Whether or not you believed in Aristotle’s world view, his complete works demonstrated that it was possible to reach an understanding of our existence, without reference to revealed Christian truth or Church authority.

To many who had just seen the unedifying spectacle of the First Crusade, this was a welcome idea.

And Aristotle’s rules of logic pointed out something even more revolutionary. They promised that any right-thinking man could find and demonstrate truth. Peasant, knight, king, shepherd, priest, layman–it didn’t matter. The rules of logic yielded truth, no matter who made use of them.

III. The Black Death, 1338-1353

Bubonic plague began just west of China, which we only know from headstones found in a little village graveyard south of Lake Issyk-Kul. Hundreds of them date from the years 1338 and 1339, and they read, “These died of plague.”

After this, sickness seems to have moved silently west, along the trade routes between the Chinese cities and the markets of India. In 1344, an army from Delhi was wiped out by a “pestilence.”  Death moved steadily westward in 1345 and 1346. In 1347, an army laying siege to Caffa, a trading port in the north of the Black Sea, suddenly died; a chronicler says, “[They] died as soon as the signs of disease appeared on their bodies: swellings in the armpit or groin caused by coagulating humours, followed by a putrid fever.”

And then plague entered the city of Caffa itself. From there seems to have spread by ship, by way of the rats that lived in the holds and ran in and out of cities unchecked. It burned across Europe.  Millions died, some from the bubonic form of the disease, many more when it settled in the lungs and caused pneumonia–often killing within 48 hours.

In Constantinople, at least half the city’s population perished. In Marseille, fifty-six thousand people died in a single month. Eight hundred souls died every day in Paris.

It crossed over to England and Ireland in 1348. Nearly half of England died. “Many villages and hamlets were deserted,” the Leicester priest Henry Knighton noted, “because everyone who had lived there was dead, and indeed many of these villages were never inhabited again.”

In 1349, a hundred thousand Egyptians died in Cairo alone.   In Sienna, Agnolo di Tura wrote,  “Giant pits are being excavated for the multitudes of the dead and the hundreds that die every night. And I . . . have buried five of my sons with my own hands. . . . Everyone believes it is the end of the world.”

The Black Death is often given too much credit for the Renaissance. I can’t tell you how many survey texts I’ve read that explain that the plague destroyed faith in God and that opened the way for humanistic thought to flourish.

I don’t see that. For one thing, a good part of Europe thanked God on its knees when, in the early 1350s, the plague finally slowed and stopped. What the Black Death really did was shake up the social structures of Europe. Given that so many noble families were dead, it became much easier to move up through the social ranks, from craftsman to minor nobility.

And, even more vitally, the working class of Europe started to gain a new power. There were a lot less of them. It was a lot harder, if you were a surviving nobleman with an estate, to find enough hands to tend your fields, milk your cows, and shear your sheep. Workers were suddenly in demand–and this gave more power to the people.

This leads directly to the next event…

IV. Power to the People

In England, something odd happened in the spring of 1381–tax collectors noticed that about five hundred thousand or so laborers, shepherds, and farmers seemed to have simply disappeared from England since 1377.

Proof of their existence was gone. No birth records, no baptism records. On paper they no longer existed.

There was, in fact, a massive tax revolt underway and it was widespread and organized.

In the wake of the plague, taxes had been hiked–fewer taxpayers meant less revenue–and the people of England were fed up.

At the same time that they were revolting, a Oxford professor named John Wycliffe had begun to argue that the Christian sacraments, protected by the church and administered by the church, were not necessary for salvation. A contemporary chronicler writes, “He publicly asserted…that the church of Rome is not the head of all the churches . . . that the pope at Rome does not have greater power than any other ordained priest . . . [and] that the Gospel is sufficient guide in this life for any Christian.”

These were messages that took power from the hands of the privileged and handed it over to the men and women in the pews.

Even more startling, Wycliffe began the massive task of translating the Bible from Latin into English– a shockingly anticlerical project that grew out of his conviction that “no simple man of wit should be afraid to study in the text of holy Writ.”

This was a conviction guided by Aristotle: that any man could come to correct conclusions on his own, as long as he followed the rules of logic.

John Wycliffe gained a large following, and his theological teachings and the tax revolt came together in the preaching of the “mad English priest” John Ball. Ball began travelling through the countryside, calling for a radical reorganization of English society:

My friends, the state of England cannot be right until everything is held communally, and until there is no distinction between nobleman and serf, and we are all as one. Why are those whom we call lords masters over us? How have they deserved it? By what right do they keep us enslaved? We are all descended from our first parents, Adam and Eve; how then can they say that they are better lords than us, except in making us toil and earn for them to spend? They are dressed in velvet and furs, while we wear only cloth. They have wine, and spices and good bread, while we have rye, and straw that has been thrown away, and water to drink. They have fine houses and manors, and we have to brave the wind and rain as we toil in the fields. It is by the sweat of our brows that they maintain their high state.

This was a startling new message of egalitarianism and equality.

Ball gathered huge crowds. And meanwhile, the royal council which was ruling England for its underaged King–that was Richard II, who was 14 at the time–decided to go out and find and arrest all those missing peasants.

When the first arrest was made, in June of 1381, the English peasants armed themselves and went to war. Many of them had fought in France, as the Hundred Years War was still dragging on, and they knew what they were doing.

The revolt failed, of course; they did manage to capture London, but then were wiped out by the royal armies, and the teenaged Richard had some final words for them– “Peasants you were, and peasants you are. You will remain in bondage, not as before, but in an incomparably worse state. For as long as we are alive to achieve this and by the grace of God rule this kingdom, we shall work our minds, powers and possessions to keep you in such subjection.”

He didn’t know it, but he was fighting a losing battle.

Wycliffe’s teachings had spread; they were picked up by Jan Hus, who was the rector of the University of Prague; Prague was in the little independent kingdom of Bohemia, which was within the German borders, and sometimes was under control of king of Germany, and sometimes not. When Hus made his support for Wycliffe’s ideas public, he was fired from his job.

He went on teaching and spreading “Wycliffism,” and when he too began to accumulate a large public following, he was invited by Church authorities to come and explain himself to the Council of Constance, a massive church council which was meeting in the German city of Constance to resolve a number of issues, among them the embarrassing existence of three separate popes, which we don’t have time to talk about just now.

Hus was guaranteed safe conduct. Instead, he was arrested, tried, imprisoned, and then burned to death.

At this, his followers in Bohemia gathered up their weapons and went to war. They called themselves Hussites, and for nearly fifty years, the carried on a bloody, extended war against the authority of the Church, the emperor, the king and the pope.

It is out of this Renaissance happening that Martin Luther’s 95 Theses, the Protestant Reformation (usually considered post-Renaissance), and ultimately the Age of Enlightenment grew.

V. The Fall of Constantinople, 1415-1453

In the 350 years since the First Crusade, the power of the Turks had continued to expand.

The particular Turkish confederation that was running the Turkish empire in the 1450s was known as the Ottoman Turks, named after the Turkish chief who had seized control of the Turkish cause back in the thirteenth century.

Turkish attacks had eaten away away at the Byzantine empire until it had lost everything  except Constantinople, a little bit of land around Constantinople, and the southern tip of the Greek peninsula.

The emperor of Constantinople had become so desperate that he had even asked the pope to send soldiers to drive the Turks away. This had never worked before but he had no other options.

It didn’t work; there were no more soldiers to send.

In 1453, the emperor of Constantinople was Constantine XI; the sultan of the Ottoman Turks was Mehmet II, a young, very aggressive sultan who was determined to finally end the Byzantine existence. He assembled a huge army to besiege the city–60,000 archers and 40,000 horsemen, cannon that shot out stones as heavy as 1800 pounds and as large as 7 feet across.

He began his bombardment on April 4, 1453, and for 55 days, the cannon battered the walls with up to 120 stones every day.

He couldn’t sail his warships right up next to the city, because the inlet there, the Golden Horn, was blocked off with a chain. So he outfitted 70 ships with wheels and took them across land to slide into the water, on the other side of the chain.

Early on May 29, just after midnight, he threw every single cannon and man he had against the walls, and broke into the city.

Constantine XI was killed–his body never found. An Italian soldier who was there said that there were so many bodies, both Turkish and Byzantine, that they floated in the Sea of Marmara “as melons float through the canals of Venice.”

By the morning of the twenty-ninth, Constantinople belonged to the Turks. And that was the end of the Byzantine empire.

With their city in Turkish hands. scores of Greek-speaking scholars fled away from the Turks and into the west.

The result was a massive acceleration in the knowledge of the Greek traditions.

They brought some of their treasured texts with them, but this wasn’t mostly a matter of new texts. It was primarily a matter of knowledge–and it was knowledge that particularly accelerated progress of science.

Since the twelfth century, the west had had access to many of the Greek science texts, such as Ptolemy’s Almagest, which explained how the universe was structured, earth at the center, heavens rotating around it.

But they hadn’t been able to do much with it. The calculations were too complicated. Centuries of language-centered education had resulted in a Europe full of scholars who weren’t practiced in the complicated geometrical skills needed to truly understand Ptolemy. The foundation of scientific learning had well and truly decayed, and rebuilding took time.

What the fleeing Greeks brought with them was mostly knowledge of the language, facility with the figures, mathematical and astronomical expertise, and also the conviction that the Greek intellectual legacy, little developed in times of security, was now endangered and in need of preservation. And their presence in the west brought a whole new renaissance in Greek culture to France, to Italy, and to Germany.


Bias, Compromise, and Teaching History to Children

In the fall of 2005, history professor Larry Schweikart was on the book-promotion circuit, talking up his “unbiased” history of America, A Patriot’s History of the United States.           

Schweikart wrote his book as a corrective: “I found bias, ranging from merely tolerable to exceptionally bad, in almost all of the major textbooks,” he told the Carolina Journal.  To find out whether a history text is “biased,” Schweikart suggested in 2010, apply the “Reagan test”: consult the index, and find out whether Ronald Reagan is given full credit for ending the Cold War. If he isn’t, the book is “politically slanted” and untrustworthy.

Whether or not you agree with him, Professor Schweikart was simply exchanging one set of biases for another. Yet he continued to insist that The Patriot’s History was entirely spin-free. “Ultimately,”  he wrote, in an essay about the book published by the Mackinac Center, “learning ‘just the facts’ of the American past leads a student to inevitably conclude that the United States is the best place on earth, and that it has acted…far better than any other nation at any other time.”

The popularity of The Patriot’s History (which temporarily topped best-seller lists) suggests that quite a few parents and teachers were searching for a “just the facts” approach to history—and took Schweikart’s assertions at face value.

 But history is never a simple chronological narration of “facts.”  Even when it claims to be “unbiased,” history always has an argument at its center: a basic assertion about how and why human events unfold as they do.

“The United States…has acted…far better than any other nation at any other time” is a fairly obvious argument. But a look at any history for young readers will turn up others—sometimes much more subtle. Consider, for example, this excerpt from Hendrik Van Loon’s classic The Story of Mankind. About the Phoenicians, Van Loon writes,

They bought and sold whatever promised to bring them a good profit. They were not troubled by a conscience. If we are to believe all their neighbours they did not know what the words honesty or integrity meant. They regarded a well-filled treasure chest the highest ideal of all good citizens. Indeed they were very unpleasant people and did not have a single friend….They were practical business men and could not spend hours engraving two or three letters. They set to work and invented a new system of writing which was greatly superior to the old one. They borrowed a few pictures from the Egyptians and they simplified a number of the wedge-shaped figures of the Sumerians. They sacrificed the pretty looks of the older system for the advantages of speed and they reduced the thousands of different images to a short and handy alphabet of twenty-two letters.

 Certainly this passage contains “just facts”: the Phoenicians streamlined the Sumerian writing system and created our alphabet. But it also contains an argument. When Van Loon writes that Phoenicians (all of them) “were not troubled by a conscience,” he is telling young readers that there is such a thing as a racial character. (The French are snooty, the British are drunks, Americans are rude.)

This is an argument about human nature—and, as it happens, a particularly loaded and dangerous one. It reappears prominently in American white supremacist writings, such as Wade Hampton’s infamous 1890 essay “The Race Problem”:

[T]he white and black races are essentially different, not only in physical organization, but in mental characteristics. This assertion is not made by way of reproach or opprobrium, nor does it apply to those of the race in this country—and there are many of them—who have proved their capacity to be numbered among our reputable, estimable, and valuable citizens…they are the exceptions which prove the rule, which, from time immemorial, has shown that they are not fitted to govern that great race before which all others have gone down—the masterful, the conquering and the unconquerable Caucasians.

Reading Van Loon certainly won’t turn young students into white supremacists. But Van Loon’s casual assumption that “racial character” exists has the potential to make such arguments more convincing.

Consider another example: H. E. Marshall’s explanation for the founding of the League of Nations at the end of World War I, found in Our Island Story (a classic, and still widely used, history of England).

 It does seem as if the love of fighting was born in us. It is nothing unusual for a boy at school to fight. No one is surprised if he comes home with a cut lip or a black eye. Indeed it is taken as a matter of course. It is all part of the game of life, and a boy who can use his fists often gets on very well at school.

But when a boy becomes a man he changes. If he wants to get on well in life he no longer uses his fists but his brains. If in his profession or business he wants to get the better of another man he does not throw off his coat and offer to fight him. He sits down and thinks.

 And even as children grow so nations grow….

 Men are only now beginning to see that just as in the old days no baron had the right to break the peace of his country, so now no state has the right to break the peace of the world. And this has led them to the League of Nations.

Underlying this explanation (and most of Marshall’s history) is the assumption that humanity as a whole is maturing over time: that earlier societies were less advanced, more barbaric than our own, that the human race is moving, inexorably, towards greater and greater perfection.

This is an argument. Children who read Marshall are absorbing it, with or without their recognition.

 So how do we teach history to our young students? What should they be taking in, along with the “facts”? Do we continue on our search for bias-free history? Do we hunt for the history that coincides most closely with our own beliefs about human nature?

 No. Instead, we should compromise.

To understand my suggested compromise, take a brief side-trip with me into historiography.

The earliest forms of historical writing are the accounts kept by ancient kings—Assyrian, Sumerian, Egyptian—of their victories (not, generally, of their defeats). These military chronicles have historical value, but they resemble press releases more than historical accounts; they have a single, limited purpose, the immortalization of one particular king.

“History,” in the sense of a coherent story written to illuminate the past, first appeared among the Greeks, when the great ancient triumvirate of historians—Herodotus, Thucydides, and Xenophon—began to write the story of men. These histories were hero-centered, much concerned with leaders and generals. “History,” for the Greeks, happened when great men exercised their ambition, planned, schemed, fought, and either triumphed or showed themselves wanting. There was an argument in this method: History happened when extraordinary men (and, very occasionally, women) acted.

This scheme persisted until Augustine, who led medieval historians into a new way of looking at the past. Augustine’s City of God approached history not merely as the story of men and women, but as the story of God’s eternal purposes being worked out on earth. Greek history had seen the past as a tangled linkage of related stories, each telling of a great man’s life, with no single overarching purpose that related these tales to each other.

But Christianity shook those tangled links into a straight chain, a line pointing forward, unbending as an arrow, to the End: the time when God’s will for His creation was finally completed.

This was enormously clarifying to historians, who could now make sense out of what had previously seemed shapeless. Faith in the eternal God led historians to fashion a story with creation at its beginning and the world’s remaking at its end. All of God’s acts in history pointed to the birth of Christ, and then past it to the time of Christ’s return. History was a progression forward.

By the Renaissance, this sense of history marching inexorably forward towards its predetermined end had led historians to develop the concept of primitive: undeveloped, because it took place earlier in time. As John Lukacs points out, the idea of “primitive” (inferior because it is far away in time) replaced the Greek idea of “barbarian” (inferior because it is distant in space, far away from Greek land).

In the eighteenth and nineteenth centuries, the assumption that history is moving steadily forward, from primitive to modern times, from less developed to more complex, evolved into three different but related historical methods: providential, Whiggish, and Marxist.

Providential historians saw God’s hand at work in history; the ultimate explanations for historical events lay in the divine realm, not the human. History was moving from fall to redemption, from a less to a more perfect end.

Whiggish historians were secular providentialists, seeing history as an inevitable, evolutionary progression towards perfection, with no reference to God.

Marxists saw the past as a constant struggle between “the people” (honest workers) and “aristocrats” (corrupt and dishonest tycoons); after a time of fierce conflict, this clash would lead to a more perfect society.

All four of these historical methods (hero-centered, providential, Whiggish, Marxist) are still in use, and are particularly prominent in histories for children.

Greenleaf Press publishes Famous Men of Greece, Famous Men of Rome, and three similar titles; the Childhood of Famous Americans series, the Landmark Biographies, and Grosset & Dunlap’s Who Was…? series. All fall into the Greek great-man tradition. There is, in these histories, an implicit argument about what drives human events: the will of the extraordinary person.

Among Christian educators, providential history remains popular. “Throughout this time period,” writes Linda Hobar in Volume 1 of The Mystery of History, “the Lord was unfolding a great plan to Abraham.” The unfolding of God’s plan (“God used the Roman law to protect Paul…”) is the organizing backbone of Hobar’s curriculum, as it is in Peter Marshall’s The Light and the Glory for Young Readers: Discovering God’s Plan for America and Mark Beliles’s America’s Providential History.

We have already seen Whiggish history in H. E. Marshall’s Our Island Story.

 And Howard Zinn’s A Young People’s History of the United States: Columbus to the War on Terror clearly displays the Marxist perspective. In only one of many examples, Zinn writes,

Bacon’s Rebellion came about because of a chain of oppression in Virginia. The Indians had their lands seized by white frontiersmen. The frontiersmen were taxed and controlled by the rich upper classes in Jamestown. And the whole colony, rich and poor, was being used by England. The colonists grew tobacco to sell to England, but the English set the price. Each year, the king of England made a large profit from the Virginia colony….[T]here was much conflict within the colonies. Slave and free, servant and master, tenant and landlord, poor and rich—disorder broke out along these lines of tension.[1]

For Zinn, those class tensions, and the struggles that result, drive every development in American history.

None of these histories are “unbiased.” All of them are making an argument about what moves history forward, why people act as they do: what we are.

I would suggest, first of all, that as we teach our youngest students, we should give up the search for “unbiased history.” Where a narrative exists, there are underlying assumptions, a particular point of view.

Nor, in my opinion, should we try to find materials with a bias that happens to match our own.

Children in the “grammar stage” of education—kindergarten through fourth or fifth grade—do not yet have a sense of the events of the past, the basic facts of history. Without this foundational knowledge, they will accept any convincing story; and the first story they are taught is likely to remain in their minds as the most powerful one. To teach them the providential, or Whiggish, or Marxist interpretation of history is to give them a strong, potentially distorting lens through which they will get their first look at the past—a lens which it will be almost impossible for them to glance away from, in later years.

Instead, I propose that the youngest students should be taught history by the oldest method: the Greek approach, devised during the childhood of history itself. This approach allows  the youngest history students to grasp onto concrete places, events, and people. The biggest philosophical flaw in the Greek method—the assumption that history can be explained entirely by the actions of kings, queens, and military leaders—can be partly circumvented by supplementing with some of the many elementary-level history resources that explore the daily lives of ordinary boys and girls, farmers and workers, peasants and miners. By sixth or seventh grade, a student should have a basic grasp of who, where, and when.

Even more importantly, his or her critical faculty should now be maturing. Now is the time to introduce the student to providential history, to Whiggish history, to Marxist history, and to other methods as well.

And as we teach, we must continually remind our young historians that no history is unbiased; that assumptions about what humanity is underlie every narrative.

A lengthier version of this article originally appeared in the thought journal Comment (9/4/14,

[1] Howard Zinn, pp. 42-43, 51.


Five Logic-Stage Teaching Tips for History

Teaching logic-stage students history can be a challenge. You want to encourage students to own their history studies, rather than simply following your lead. How can you do this?

Try the following five tips. These can also help you to make a simpler narrative text such as the Story of the World series more appropriate to logic-stage studies.

As you work with your middle-grade student, you should already be making use of regular history reading, outlining, producing a timeline, engaging primary sources, and writing. Think about adding two or more of the following techniques:

  1.     Make Reading Dramatic: Have your student reread a text out loud dramatically. If the particularly chapter has a story accompanying it, ask your student to act it out a bit. Tell him or her to give the characters in history a funny voice (try on a French accent for Charlemagne, or a British one for Henry VIII).

As a bonus, ask the student to deploy this technique at the family dinner table to explain a particular story to everyone.

  1.     Supercharge the Timeline: Ask your student to add pictures, whether he or she draws the picture or prints them off the internet. Some students love to draw comic strips and this can make for a great way to “highlight” important events in history, like the fall of Rome. You’ll be amazed at what your student thinks Attila the Hun looks like.
  1.     Commission a Craft: Maybe your student likes modeling with clay or wood; or drawing maps and sketching plans; or he or she sews. Whatever the hobby talent, have them model a Viking Ship, or draw the pilgrimage of Mansa Musa, or design a full-size samurai outfit.  Whenever you study a particular historical phenomenon, ask yourself: How can we make this three-dimensional?
  1.     Be the Teacher. Ask your student to pick a topic, and then require them to give you a fifteen-minute lesson. Challenge them to prepare materials, including slides and images, and walk you through history. Your student can even make up quizzes and tests–and give them to you afterwards!

For a twist, pretend your student is a famous professor and interview him or her on your news show. Record it and then share the video with relatives!

  1.     Plan a “time travel” trip. Hand your student a miniature phone booth, inform him or her that he is now a Timelord, and ask him to plan a family vacation in some or several past civilizations. Who wouldn’t want to visit the great Mayan and Inca cities when they were alive? Or see the first World’s Fair at the Crystal Palace in London? Make sure the student prepares all the arrangements including accommodations, costs (he’ll have to do some research on that!), and entertainment!

During the logic stage, students transition from solely familiarizing themselves with persons, facts, events, and dates, to making connections among these. Just as importantly, students are ready to see how history functions as the backbone of their classical education, allowing history to help organize what students encounter in literature and music, art and science.  The tips above, combined with a solid foundation, will push your student to own their study of history.


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.