Anki

In education research, three things have my attention: one-on-one tutoring, Zig Engelmannism, and spaced repetition.

Anki is a flash card program. Each card has a front and a back. When you are reviewing the cards, the program shows you the front (maybe it is a word in a foreign language or the symptoms for a disease) and you try to remember what is on the back (maybe it is the word in your language or the name of the disease). When you remember it, or give up trying to remember, you press a button and the program reveals the answer. You press another button to record whether you got it right or wrong, and the program shows you the next card to review.

Which cards does it show you each day? The program schedules them for you. When you tell that you got one wrong it plans for you to see it again soon. When you tell that you got one right, it plans for you to see it again later, maybe 10 minutes later if you last saw the card 5 minutes ago. After using it once the spacing is measured in days instead of minutes. After using it for a while the spacing is measured in months and years.

Where do these cards come from? There are many available for download, curated by language learners and med students and hobbyists. When Maria was a baby, while I had free time but my work and sleep were diminished, I tried out some trivia for fun: countries of the world, famous paintings. While it lasted I enjoyed it and it worked well, but I didn't keep it up.

A few years later I read an article about Anki by Michael Nielsen, and I realized I had been using it poorly. Before I was conceiving of someone compiling a table of information about a topic (columns in the table for the definition, the spelling, the pronunciation...), and then formatting the table into a "deck of flash cards," which Anki could help you memorize. Maybe this deck of cards could be shared online with others who wanted to learn the topic.

I learned from Nielsen's article a more effective procedure: create bespoke and personal cards, few or one at a time, that remind you of the latest things you've learned and want to keep hold of. Don't organize them and don't share them with anyone — it's hard to sum up the reasons why not.

Last year I was writing a hundred words a day about ancient history for Maria's reading lesson. I was sensitive about the pointlessness of it. My five-year-old didn't know how to read a map, dimly understood how many days in a year and how many lifetimes in a thousand years, and lived far away from the Tigris and Euphrates. It seemed likely she would retain no details and maybe no substance from the readings. 

It didn't seem like wasted time as long as I liked preparing them and she cooperated in reading them. Still, could she get more out of it?

We started an Anki practice. For a year I've picked a tidbit from each reading, and made a card out of it.

Tyre had a reputation for being smelly, because of how the Phoenicians made {{c1::purple cloth}}

The first pharaoh to be buried in a pyramid was {{c1::Djoser}}

 In the mountains close to the Indus River, ancient people were {{c1::mining lapis lazuli}}

The notation {{c1:...}} is Anki's way of making a cloze. I searched the web for an image to include on every card: a murex snail for Tyre, the step pyramid for Djoser, a pile of blue rocks for the mining operation. 

As of November 2021 I've made about 400 cards. Eleven months ago, I had only made two dozen, and the program showed Maria three or four of them every day. Lately it shows her as many as thirty every day. She gets through them in not much more than 15 minutes. The 15 minutes are fun.

The inventor of spaced repetition software, Piotr Wozniak, wrote a long essay expressing a lot of pessimism about using the software with children. He marshals some evidence and also includes this anecdote:

I recall my mom's heroic efforts to make sure I could speak German as a kid. She would speak to me in German. She would send me to a German class in kindergarten, and in the primary school. She tried all tricks in the book. She gave up after some 8-9 years of trying in vain. I was roughly 12 years old! I never learned any German. All her efforts were to naught. Kids need to want to learn or they will learn little. Coercive learning is wasteful, may result in toxic memories, and may ultimately lead to a hate of learning

Those are reasons to be cautious. I was briefly incautious and tried to use Anki to help Maria learn not German but her "math facts." It was unpleasant for both of us and a waste of time. I dropped it pretty fast. 

But showing her history this way is not unpleasant at all, just the opposite. At the start of each reading lesson, I read aloud what she read yesterday. Then she reads the new installment to me. The flash cards remind her of something we've read to each other. The experience of showing them to her on the couch is very much like reading a picture book on the couch. I'm optimistic that they're also educational.

What I suspect so far about Anki with children:

1. One new card a day is plenty.

2. Never tell that she got one wrong, just quietly press "again"

3. Put an image on every card.

4. Put the cloze at the end of the card: "Besides Tyre, another Phoenician city was {{c1::Carthage}}", not "Tyre and {{c1:Carthage}} were two Phoenician cities."

5. Be quick to drop any tiring or stressful card.

If-Then Propositions

Mathematics is a deductive science. That means that you can go from a simpler form to a more complicated form if you play according to the rules of the game. To make the correct move, however, you have to make the right kind of statement. Almost always, the statement can be expressed as an if-then proposition: "If 8 plus 1 is 9, 80 plus 10 must be 90." "If 8 plus 8 is 16, 8 plus 7 must be 15." "If 8 times 5 is 40, 8 times 6 must be 48." "If 8 equal the fractions 8 x 1/1, 8 x 2/2, 8 x 6/6, it also equals the fraction 8 x Dg/Dg."

Lewis Carroll wrote a funny dialog in 1895, between the runners in a Zeno paradox. The tortoise teases Achilles that he understands an assertion A, and he understands that A implies B, but he still does not believe B. The Engelmanns' instructions for training a 4-year old to follow these inch-long deductions reminded me of it. I felt a little demented discussing it with 5-year-old Maria: "If this is a book, then this can't be a book." I sometimes had to keep from laughing. But ignorance of logic is a normal state of mind, and I don't mind having relied on something pre-logical (not demented at all) to cure that ignorance.

Present if-thens this way:

1. Make a green mark on a piece of paper with a crayon. Next to it, make another green mark. Explain, "We're going to figure things out a new way. We're going to if-then." Point to the first mark. "If this is green . . . [point to the second mark] this is green. We can say that because they are the same. Let's do it once more. If this is green, this must be green." Repeat with two blue marks and two red marks.

2. Introduce something a little more complicated. Take two books. Don't tell the child what they are called. Just show that the first book has pages with printing on them. Show that the book has a cover. Now show that the second book also has page, printing, and a cover. Conclude. "They're the same, just as the colors were the same. So I can say, 'If this is a book, this other one must be a book.' Your turn . . ."

3. Put a green mark on your paper. Next to it make a red mark. "Here's a new kind of if-then. If this one is green, this one can't be green. It can't be green because it's not the same. We can't figure out what it is, but we know that it can't be green." Repeat with different color combinations. Then use the book again. Place it next to a block, and make the line of if-then reasoning more specific. "If this is a book, this can't be a book."

"When we if-then, we always start out with what we know. We know that a book has pages, has printing, has a cover. Does a block have pages, printing, and a cover? No. It can't be the same."

4. Repeat with various examples, comparing spoons with dishes, lamps with rugs, and so on.

5. Demonstrate some basic if-thens with number-type concepts. Put a row of 4 circles on your chalkboard. Beneath it put another row of 3 circles.

◦  ◦  ◦  ◦

◦  ◦  ◦

Have the child count the circles in the top row. Then say, "If there are 4 up here, there can't be 4 down here  because these rows aren't the same." Show why they're not the same by drawing a line from every circle in the top row.

◦  ◦  ◦  ◦

|  |   |  |

◦  ◦  ◦

"If these lines were the same, there would be a circle below for every one above. But there's no circle for the last circle."

Give similar examples in which the line below is either the same or different from the line above.

6. Introduce if-then with addition. Start with the familiar problem, 3 + 1 = 4. Represent it with circles.

◦  ◦  ◦ + ◦ = 

◦  ◦  ◦    ◦

Next to the original problem present 3 + 2 = . Modify the 3 + 1 diagram.

◦  ◦  ◦ + ◦┃◦ = 

◦  ◦  ◦    ◦┃◦

The difference between 3 + 1 and 3 + 2 is now evident. A circle has been added to the right of the heavy vertical line, changing the 1 into a 2. The top row is no longer equal to the bottom row. To make it equal we have to add 1 circle.

◦  ◦  ◦ + ◦┃◦ = 

◦  ◦  ◦    ◦┃

We can translate the diagram into an if-then proposition by covering up the portion of the problem to the right of the vertical line and saying, "If 3 plus 1 is 4 . . . "

◦  ◦  ◦ + ◦

◦  ◦  ◦    ◦

Now uncover the top row to the right of the vertical line, ". . . 3 plus 2 can't be 4. It has to be 1 more than 4." Uncover the bottom row to the right of the vertical line. "It has to be 5." Repeat, "If 3 plus 1 is 4, 3 plus 2 is 5."

◦  ◦  ◦ + ◦┃◦ = 

◦  ◦  ◦    ◦┃◦

By using the diagram properly, you can show the child just exactly why 3 plus 2 can't be 4, and why it has to be 1 more than 4 (because the circles in the top row must correspond to the circles in the bottom row).

Present six to eight if-then problems during each of the next fifteen sessions. Base these on familiar +1 and number-pair facts: 6 + 1 =, 3 + 3 =, 1 + 4 =, 5 + 5 =, 5 + 1 =, 2 + 2 =. Always change the second number, never the first. If 6 + 1 = 7, 6 + 2 = ? Take the child through the steps in reasoning. Don't expect him to pick up the patter overnight. He won't.

7. Use diagrams during the first four or five sessions. Then drop them and work from the problems, using a modified approach.

2 + 2 = 4

2 + 3 =

"Look. We start out with 2 plus 2. In the second problem we don't have 2 plus 2; we have 2 plus 3. We know that 3 is bigger than 2—1 bigger." Draw a line from the 2 down to the 3, and put a +1 in the middle of it. "We added 1 . . .

2 + 2    =  4

   + 1 ) 

2 + 3    = 

"Here's a rule. If we add 1 on this side of the equal sign, we have to add 1 on the other side of the equal sign.

2 + 2    =    4

   + 1 )     ( + 1

2 + 3    =    5

What's 4 plus 1? Sure, 5." Write it, completing the bottom equation. "Let's go through the whole thing. We start with 2 plus 2. If 2 plus 2 is 4, 2 plus 3 can't be 4. It must be five." It helps to phrase the if-then the way we suggest here. The child finds the line of reasoning easier to follow if he knows that 2 plus 3 can't be 4. He can then see that it has to be 1 more than 4. It has to be 5.

Keep the child responding. As he becomes handier with if-then reasoning, let him do more and more of the talking. "What are we starting out with? How are we changing it? What's the rule? If we add 1 here, we must . . . Good. So give me the whole if-then. Fine."

8. After the first two-weeks, introduce if-thens in which the first term changes.

3 + 3 = 6

4 + 3 =

Show that they diagram the same way as the original problems and that they follow the same rules.

◦┃◦  ◦  ◦ + ◦  ◦  ◦ = 

  ┃◦  ◦  ◦    ◦  ◦  ◦

They should present no real stumbling blocks.

By the time the child has been working if-thens for two weeks, he should be able to phrase an if-then conclusion and he should have reached the point where the words are starting to mean something. But it will be a while before the full meaning comes through.

Dorothy Sayers

Dorothy Sayers gave a speech in 1947:

"I want to inquire whether, amid all the multitudinous subjects which figure in the syllabuses, we are really teaching the right things in the right way; and whether, by teaching fewer things, differently, we might not succeed in “shedding the load” (as the fashionable phrase goes) and, at the same time, producing a better result."

She thought her coevals were not educated as well as literate people of the middle ages. The early part of the speech gives examples of sloppy speaking and writing and thinking that disappointed her. They don't disappoint me in 2021, actually I don't find them easy to react to at all. But farther into the speech, her description of what school could be like is interesting:

Let us now look at the mediaeval scheme of education—the syllabus of the Schools. It does not matter, for the moment, whether it was devised for small children or for older students, or how long people were supposed to take over it. What matters is the light it throws upon what the men of the Middle Ages supposed to be the object and the right order of the educative process. The syllabus was divided into two parts: the Trivium and Quadrivium. The second part—the Quadrivium—consisted of "subjects," and need not for the moment concern us. The interesting thing for us is the composition of the Trivium, which preceded the Quadrivium and was the preliminary discipline for it. It consisted of three parts: Grammar, Dialectic, and Rhetoric, in that order.

Was it really like that? Anyway if you are responsible for educating a young kid it is food for thought as a prescription. Decades later it was influential for America's homeschoolers. When you search google for "homeschool" you find some of those influences. Two of them were the authors of The Well-Trained Mind, quoted at the end of this post.

Maria, now six years old, has 12 grades ahead of her. Sayers paints a picture of them: 4 grades of Grammar, 4 grades of Dialectic, and 4 grades of Rhetoric. I can't anticipate even 12 months ahead, but I can recognize some of my six-year-old in this passage:

My views about child-psychology are, I admit, neither orthodox nor enlightened. Looking back upon myself (since I am the child I know best and the only child I can pretend to know from inside) I recognise in myself three states of development. These, in a rough-and-ready fashion, I will call the Poll-parrot, the Pert, and the Poetic—the latter coinciding, approximately, with the onset of puberty. The Poll-parrot stage is the one in which learning by heart is easy and, on the whole, pleasurable; whereas reasoning is difficult and, on the whole, little relished. At this age one readily memorises the shapes and appearances of things; one likes to recite the number-plates of cars; one rejoices in the chanting of rhymes and the rumble and thunder of unintelligible polysyllables; one enjoys the mere accumulation of things.

If a young kid has some enthusiasm for repetition, how can you make the most of it as a tutor? More on that later. The speech describes kids drilling the times tables, Latin vocabulary, and the names of constellations and of backyard plants and insects. I haven't tried to do that. This part, just as tendentious, had more appeal:

The grammar of History should consist, I think, of dates, events, anecdotes, and personalities. A set of dates to which one can peg all later historical knowledge is of enormous help later on in establishing the perspective of history. It does not greatly matter which dates: those of the Kings of England will do very nicely, provided they are accompanied by pictures of costume, architecture, and all “every-day things,” so that the mere mention of a date calls up a strong visual presentment of the whole period.

Geography will similarly be presented in its factual aspect, with maps, natural features and visual presentment of customs, costumes, flora, fauna, and so on; and I believe myself that the discredited and old-fashioned memorising of a few capital cities, rivers, mountain ranges, etc., does no harm. Stamp-collecting may be encouraged.

I found this idea expanded and made practical in The Well-Trained Mind, by Jessie Wise and Susan Wise-Bauer (mother and daughter):

Over the four years of the grammar stage, you'll progress from 5000 BC to the present, accumulating facts the whole way. These four years will be an exploration of the stories of history: great men and women of all kinds; battles and wars; important inventions; world religions; details of daily life and culture; and great books. ...

World history is divided into four segments, one segment per year of study. In the first through fourth grades, the child will study history from 5000 BC though the present day. In fifth through eight grades (the logic stage), he'll study it again, concentrating on cause-and-effect and chronological relationships. In grades 9 through 12, he'll repeat it again, this time studying original sources and writing thoughtful essays about them.

Planning for the shorter term, they recommend covering "ancient history" in one year, "medieval history" the next year, and "the renaissance" and "the modern period" in years three and four. We've learned a lot taking an even shorter-term view. In history (we still call it "reading lesson") each day it is easy to solve the problem of what to teach: review yesterday's reading, and then inch forward in time.

It has not been as easy to decide what to do each day in math lesson.

Zig and Therese Engelmann on Algebra for 4-year olds, 1966

I wrote a half dozen entries here in October, November and December last year. I'd like to pick it up again.

In the second half of 2020 I felt that I had caught on to something exciting. When Maria was younger, and even before she was born, I had a dream of homeschool that seemed unrealistic. Over the course of teaching her to read a realistic picture of it emerged for me, not all at once. I did some reading and some experiments, and in the fall I could feel the urgency to write down the good ideas I had got and the good advice I had found. I was going to forget them.

I have forgotten them. Writing is slow and growing up is fast. But the curriculum from July 2020 still seems fresh and relevant in July 2021. From the beginning of July until the end of November, I presented the math lessons in Give Your Child a Superior Mind pretty faithfully. Some of the headings from that book tell a story, a cryptic one, about what this material was and what order Maria learned it:

Showing the Relationship between Counting and Adding — Reading Addition Problems — Word Problems Involving Adding 1 — Number pairs — Algebra Problems — Algebra Problems Involving +1 —Algebra Problems Involving Number Pairs — If-Then Propositions — Rule for +2 — Continuity Game — Subtraction —

I've copied the section on "Algebra Problems" below. I enjoyed explaining it to a kid who could work out only a few sums (6+1 and 6+6 by rote, 6+4 on her fingers — from the earlier sections copied in another post) and who didn't yet know the minus sign.

—Review —Multiplication — Ones — Tens — Twos — Fives — Nines — Eights — Fours — Sixes — Sevens — Threes — Multiplication Problems — Multiplication Involving Algebra —Area of Rectangles —Area Problems Involving an Unknown — Telling Time — Fractions —Subtracting Fractions —Adding Parts to Make a Whole —Rule for Fractions That Are Equal to One Whole — Rule For Fractions That Are More Than One Whole —Algebra Problems With Fractions — Division — Money—Column Addition — Column Subtraction — More About the Relationship Between the Terms of a Problem

Those are the section titles between pages 193 and 296, and there are also many untitled sections. I was pleased with how they worked. If there are things I now suspect I should have done differently, they weren't in this section:

Algebra problems

No, algebra is not necessarily a high-flown, abstract subject. In fact, the child already knows algebra. He simply doesn't know that he does.

1. Write this problem on the board:

3 + 1 = B

It is an algebra problem with one unknown—the B.

2. Explain that we read the problem as, "3+1 is B."  Then give him a rule for interpreting an unknown. "Whenever you have a letter in a problem, it asks you, 'How many?' This problem asks, '3 plus 1 is how many?' It asks, 'How many do you end up with?'"

3. Work it as you would any other +1 problem. "We're adding 1, so we say, 'What comes after 3?' Good: 3 plus 1 is 4."

4. Explain that you know what B equals when you answer the question, "How many?" "When you add 3 plus 1 you end up with how many? You end up with 4. So we can say B equals 4." Write:

3 + 1 = B

B = 4

Give the child six to eight algebra problems of this form during each session for about a week. Use familiar +1 and number-pair problems. Use a variety of uppercase letters for unknowns, but avoid I, O, Q, S, and X. Here's a typical exercise:

Problems            Solutions

3 + 3 = K            K = 6

8 + 1 = R            R = 9 

1 + 13 = D          D = 14 

6 + 6 = M           M = 12 

4 + 4 = U            U = 8 

4 + 1 = E             E = 5 

Algebra Problems Involving +1: The unknown in an algebra problem asks the question "How many?" regardless of where it is in the problem. The equation 3 + 1 = F tells you that if you start out with 3 and add 1 more, you'll end up with—how many? You'll end up with 4. So, F = 4.

The problem 3 + F = 4 tells you that you start out with 3 and add how many to get to 4. So, 3 plus how many is 4? If you answer the question "how many?" you solve the problem: 3 plus 1 is 4. F = 1.

1. Present the problem

6 + C = 7 

2. Explain, "This is a new kind of algebra problem. See? The C is here in the middle. But it still asks 'How many?' This problem says, '6 plus how many is 7?' You say it. Good." 

3. Show the child how to use a variation of the rule for adding 1 to solve the problem. "Look. We're adding something to 6. How many? We're adding something to 6 and we're ending up with 7; 7 is the next number after 6. We're adding something and we're ending up with the next number. We must be adding 1. Because what's our rule? When you add 1 to a number, you end up with the next number,"

The pattern of logic involved in this kind of deduction is basic, so basic in fact that we, who are used to it, have trouble reducing it to words. The child is not used to it. He has to learn it. And he'll learn it more rapidly and more thoroughly if you show him the steps you're taking. Study the sample explanation we just gave. Practice it. 

4. Summarize the problem so that the child can see the relationship between the familiar +1 fact and the present problem: "6 plus how many is 7? 6 plus 1 is 7. Say it. Good." 

5. Write the answer, 

6 + C = 7 

C = 1 

Give the child six to eight problems of this kind during each session for the next week. Put the unknown first in some problems and second in others. Here would be a typical exercise.

A + 2 = 3

9 + C = 10

4 + R = 5

V + 6 = 7

F + 6 = 7

1 + G = 2

The explanation is the same whether the unknown comes first or second. For instance, to explain A + 2 = 3, you point out that you're adding something to 2 and ending up with 3. How much are you adding? Well, you're ending up with the number that comes after 2, so you must be adding 1. How many plus 2 is 3? 1 plus 2 is 3. A = 1.

The child will soon spot a shortcut for solving these problems. He'll learn that all he has to say is "C equals 1," or "A equals 1," without really knowing why. The best way to handle this is to say, "That's right. And how do we know that A equals 1. We know it because we're adding something to 3 and ending up with the number that comes after 3. Right? Let's hear you say it. How do we know that we're adding 1?" Help him through the explanation.

Algebra Problems Involving Number Pairs: Introduce these problems only after the child is used to relating algebra problems to addition facts. "3 plus how many equals 4? 3 plus 1 equals 4."

1. Present the problem, 4 + H = 8.

2. Show that the rule for adding 1 does not apply. "Look. We're adding something to 4 and we're ending up with 8. Are we ending up with the next number after 4? No. So we're not adding 1." 

3. See if the child can answer the question "How many?" "Think hard: 4 plus how many is 8? ... 4 plus ..."

4. If he doesn't know, show him how you can figure it out. Represent the problem this way:

        ◦    ◦    ◦    ◦   +   _________ =

        ◦    ◦    ◦    ◦        ◦    ◦    ◦    ◦   

Explain, "We start out with 4 and we add. We don't know how many we add, but we must end up with 8. The problem tells us that we started out with 4 of those 8. Let's draws lines to show which ones we started out with."

                    ◦    ◦    ◦    ◦   +   _________ =

                    |    |     |     |   

                    ◦    ◦    ◦    ◦        ◦    ◦    ◦    ◦

"These are the four we know about. The ones that are left are the ones we must have added. How many are left? Let's count and see . . . 4." Fill in the diagram so that it looks like this:

                    ◦    ◦    ◦    ◦   +   ◦    ◦    ◦    ◦ =

                    |    |     |     |        |    |    |    |  

                    ◦    ◦    ◦    ◦        ◦    ◦    ◦    ◦                        

"The circles that are underlined are the circles we must have added. 4 plus how many equals 8? 4 plus 4 equals 8."

5. Refer to the original problem.

4 + H = 8

      H = 8

The sample problem above shows the relationship between the answer term and the unknown in a way the child can appreciate. Use it liberally during the first three or four number-pair sessions. Then discontinue it, and encourage the child to solve the problem by remembering the addition fact. "Come on. You can remember that if you think hard: 3 plus how many is 6? 3 plus . . . " If he doesn't remember, tell him.

Include a few +1 problems in with the number pairs. Also, introduce the number-pair problems in which the unknown comes first, C + 6 = 12. The child should catch on to these algebra problems rapidly. He should be able to work all +1 and all number pairs involving an unknown a month after you introduce the first algebra problem. 

"The child should catch on rapidly," I think Maria did.

Science, July and August 2020

Over the summer she practiced reading nonfiction for 5-year-olds. I practiced writing it. I browsed wikipedia and fed her little bits of it like she was a baby bird. A reading might tire her out but she could read it and understand it. If two weeks later I showed it to her again she could read it faster and understand it faster.

She was building a skill and a habit, but I don't know if the facts stuck with her. Today she can read well but I doubt she remembers anything about Aragon or Castile. I have kept on including details like that anyway.

Whether or not it was sticking, in the second week of June I found a way to double her enthusiasm for a reading lesson: I could put her name in the last paragraph. She would spot it when she was sizing up the page before we started. "Why does it say 'Maria' here?" and I would tell her to read it and find out. Once it was about dinosaurs and the last paragraph told a story about our outing at a museum. It was a true account that she remembered, but written in the third person.

I only pandered like that some of the time. Here is a less personal passage that I wrote for her after two sessions about dinosaurs-and-Maria:

Paleozoic Mesozoic six sixty million change changed living growing ocean forest swamp reptile frog

Life began in the ocean

All of the dinosaurs died out sixty million years ago. That was a time before people.

There was also a time before dinosaurs! The time of the dinosaurs is called "Mesozoic time." The time before dinosaurs is called "Paleozoic time."

During Paleozoic time, there were no dinosaurs. At the start of Paleozoic time, there were no living things on land at all! All of the plants and animals lived in the ocean.

Plants and animals changed a lot during the Paleozoic. At the start of it, everything lived in the ocean and nothing lived on the land. At the end of it, the land had lots of forests and swamps with tall ferns growing in them, and lots of bugs and frogs and reptiles living in them.

But dinosaurs came after.

I changed the subject to light, with the idea that I was going tell her about Newton working with prisms during the plague. I dropped it before getting to Newton, we spent September-October-November-December doing something else.

Earth planet universe sky why light right night eight whole bounce bounces minute minutes

Light from the sun

Some light comes out of light bulbs. But most of the light that we see comes out of the sun. The light starts out at the sun and then it comes to our planet. Do you remember what our planet is called? Yes, Earth.

Light moves very very very fast. It is the fastest thing in the whole universe! But the sun is very very very far away. After the light comes out of the sun, it does not get to Earth right away. It takes eight minutes to get here.

We need light to see. At night the sun is not in the sky and we don't see most of its light. That is why it is harder to see at night. But a little bit of the sunlight bounces off of the moon onto Earth. We call that light moonlight.

The next day

of off bounces bouncing different color strawberry together warm white blue brown green red goes eight

Light and sight

Last time we read that it takes eight minutes for light to get from the sun to Earth. At night, some of the sunlight bounces off of the moon and onto Earth. That way we get a little bit of light at night.

Light is always bouncing around. That is how we see things! We can see the moon when light bounces off of the moon and into our eyes. We can see a strawberry when light bounces off of the strawberry and into our eyes.

Light comes in different colors. The strawberry looks red to us because only red light bounces off of it. All of the rest of the light goes into the strawberry and warms it up a little bit.

When you mix green and blue and red paint together, you get brown paint. But when you mix green and blue and red light together, you do not get brown light. You get white light!

Two days later

Earth bounce bounces sometimes magnify magnifying off glass eye eyes through strange focus focusing water

Light and glass

Light comes from the sun to Earth. We see an apple when the light bounces off of the apple and into our eyes. Only green light bounces off of green apples. Only red light bounces off of red apples.

If you use a magnifying glass, some of the light goes through the magnifying glass before it goes into your eyes. The magnifying glass bends the light that goes through it. It bends it in a strange way, that we sometimes call focusing. That strange bending of the light can make small things look bigger!

Light also bends when it goes through water. Sometimes when you look at something through water it will look different. It might look like it is in a different place than it really is.

I will show you a magnifying glass after you finish reading this.

A couple of weeks later

-scope telescope through magnify magnifying probably spread eye

Telescopes at Night

When light goes through a magnifying glass, the light spreads out. When that spread-out light goes into your eye, the thing that it bounced off of looks bigger. Four hundred and twelve years ago, someone invented the telescope using two magnifying glasses.

We don't know the name of the person who invented the telescope, but they were probably Dutch. That Dutch inventor put one magnifying glass at the end of a long tube, and the other magnifying glass at the other end of the tube. When you looked through that tube, things that were far away seemed closer.

We don't know the name of the person who invented the telescope, but the first person to use a telescope to look at the night sky was named Galileo. When Galileo looked through a telescope, he saw things that no one had ever seen before. He saw the rings around Saturn!

She had seen those rings before. I didn't think she would know about Jupiter's moons.

The next day:

build building straight might eight heavy light Galileo Saturn Earth Italy Italian Pisa

Galileo in Pisa

Galileo spoke Italian. He lived in a city in Italy called Pisa. In Pisa there is a very famous building. It is a tall building that does not go straight up. It leans to one side. It looks like it might fall over! But it has not fallen over for eight hundred years.

That building is called the leaning tower of Pisa. There is a famous story about Galileo and the leaning tower of Pisa. Galileo was a real person but this might not be a true story. But it is a nice story.

Galileo wondered whether heavy things fall faster than light things. To find out, he went to the top of the leaning tower of Pisa, and he dropped a small stone and a big stone out of the window. People thought that the big stone would fall faster. But Galileo saw that the two stones fell at the same speed! He dropped them at the same time and they hit the ground at the same time.

A week later

-scope telescope microscope Galileo interesting -tion nutrition respiration digestion use used

Telescopes and microscopes

A piece of glass that can bend or spread out light, like a magnifying glass can, is sometimes called a "lens." When light went through Galileo's telescope it went through two lenses. Then it went into his eye. That way things that were really far away looked like they were closer. When he used the telescope to look at the sky he saw the rings of Saturn, and the moons of Jupiter.

Saturn is very, very big. It is a planet and it is bigger than the Earth! But you can also use lenses to see things that are very, very small. That kind of "scope" is called a "microscope." With a microscope, you can see cells!

Even though cells are so, so small, they are not simple. They have an inside and an outside, and there are lots of interesting things inside of them. They are alive! They need nutrition to live. They do respiration and digestion.

We are made of cells. What are cells made of?

 

Adding one

Even last year, Maria knew how to count on her fingers to find the sum of small numbers. When we were staying home together in March and I asked her if she'd like to practice math, I learned that she was using this method not just to find out what is 3+5 but even to find out what is 2+1. "Do you think you could find out 2+1, and 3+1, and 4+1, without using your fingers?" The question made her tense, and I backed out of it right away.

The reading lessons had started to go really well. People can be fragile about math and I thought we had time to wait for a more perfect moment to start. But for a few months I was nonplussed about how I was going to teach someone to compute 4+1. Was I supposed to give her a seminar about +1?

"Give your child a superior mind" came in July, and I saw that the answer was yes. Here is Zig and Therese Engelmann's detailed outline of a lecture series for 4-year-olds (but Maria was 5) on how to add one:

Showing the Relation between Counting and Adding:

1. Draw 2 small circles on the board 

                    ◦    ◦

2. Count them, pointing as you count: "1...2..."

3. Let the child count them. "Your turn."

4. Define adding so that it ties in with what the child knows about counting. "If I add 1 more, I'll have...:" Draw another circle, and count them.

                    ◦    ◦    ◦ 

"Three circles." Repeat. "I have 3 circles. If I add one more, I'll have..."

                    ◦    ◦    ◦    ◦

"...1...2...3...4. I'll have 4 circles. If I add 1 more, I'll have..."

                    ◦    ◦    ◦    ◦    ◦

And so on. Add circles until you reach 20. Repeat the procedure several times.

As soon as the child catches on to the idea that when you say, "Add 1,:" you're going to the next number in the counting series, start speeding up the operation. "I have 7 circles and if I have one more, I'll have...yes, 8. I have 8 circles. If I add 1 more I'll have...Good."

5. Present the rule for adding 1: Draw 5 circles on the board. "Here's the rule for adding 1. You first see how many you start out with. How many are here? Sure, 5. And you say, "I have 5. If I add 1 more I'll have...if you don't know ask yourself, what comes after 5? What's the next number? Sure, 6. So if I have 5 and add 1 more, I'll have 6."

Reading addition problems: 

1. Present 3 + 2 = 5.

2. Identify the + as plus, the = as the equal sign.

3. Then read, "We read just the way we do in the book. We start over here [left] and say, 3 ... plus ... 2 ... equals ... 5—3 plus 2 equals 5. Your turn. ..."

4. Point out that you can add any two numbers together. Demonstrate with various examples. Let the child give the numbers and you write them down. "Pick any numbers you want."

5. After he can read addition problems and properly identify the terms (which may take at least an entire session), show him how to interpret addition problems. Begin by pointing out that in addition you start out with on number and end up with another. Put the problem 3 + 2 = 5 on the board. "The problem tells us that we start out with 3 and end up with 5." Give the child practice on several other problems. Then present the problem

         

                    4 + 5 =

"What about this one? We start out with 4 and we end up with—you don't know. That's what we have to figure out. I know that when you start out with 4 and add 5 you end up with 9, so I can write . . ."

                    4 + 5 = 9

"I solved the problem. Read it . . ."

Repeat with examples until the child catches on to the idea of starting out and ending up. This is an extremely important notion. Make sure he learns it.

Translate symbols into the operation of adding.

                    3 + 1 =

"Look at what the problem tells you. It tells you to start out with 3. Then the plus sign tells you to add. It says, Get more numbers. How many more? [Point to the 1.] Look. It says get 1 more. Start out with 3 and get 1 more. And when you do that, you'll end up with 4.

                    3 + 1 = 4

"Read it."

Keep the presentation straight and simple: start out with the number on the left (at least in the present problems); the plus sign tells you to get more, the number after the plus sign tells you how much more; and the number after the equal sign tells you what you end up with.

6. Diagram the problem.

                    ◦    ◦    ◦    +    ◦ =

                    ◦    ◦    ◦          ◦ 

The top row shows you what you start out with and what you add. The bottom row shows what you end up with. The bottom row is equal to the top row because you can draw a line from every circle in the top row to a corresponding circle in the bottom row.

                    ◦    ◦    ◦    +    ◦   =

                    |    |     |          |   

                    ◦    ◦    ◦          ◦

To find out how many you end up with, simply count the symbols in the bottom row. You can see where each one came from. You can see the relations between the three terms.

Stress the idea that the term you start out with and the term you add are reflected in the answer. Point to the bottom row, "See, here's the 3. And here's the 1. 3 plus 1 is 4." 

Rule: For every circle in the top row there must be a circle in the bottom row.

7. After the child understands how to read an addition problem and how to interpret it (which will probably take three sessions and 40-50 examples), introduce the rule for adding 1 to other numbers.

                    3 + 1 = 

 "What do we start out with? . . . Yes, 3. And what do we do (pointing to the + sign)? . . . Yes, get more. And how many do we add? . . . Yes, 1. . . . And what do we end up with? . . . We don't know. But I know a way to figure it out. Is there a 1 in this problem? . . . See if you can find it. . . . Yes. So I draw a circle around the 1.

                    3 + 1⃝ =  

 "We're adding 1 to which number? . . . To 3. [Pointing to the 3.] When you add 1 to a number, you end up with the next number. We're adding 1 to 3 so we're going to end up with the number that comes after 3. What comes after 3? . . . Sure, 4. So we're going to end up with 4. Let's write it down."

                    3 + 1⃝ =  4

Summarize the problem as an addition fact. "3 plus 1 is 4. Say it."

8. Follow with a series of problems. Include some in which the 1 is first (1 + 3 =) and last (7 + 1 =). Have the child read the problem. "7 plus 1 is . . . we don't know." Then ask, "Are we adding 1?" Have him locate it. Circle it. "What's the rule when you add 1?" Help him state the rule. "When you add 1 to a number, you always end up with the next number." Have him find the number to which 1 is being added. He should have no trouble, because it's the only uncircled number in the problem."What comes after 7? Good. So we'll end up with 8." Write the answer.

                    7 + 1⃝ =  8

Diagram the problem with circles.

9. Don't introduce 1 + 1 or 1 + 0 until after the child has worked on +1 problems for a week or so. Solve them the same way you would any other +1 problem. In 1+1=, circle only a single 1 and ask what comes after the other 1. In 1+0 =, explain how to handle the zero )which the child already knows vaguely from counting backward). "We call this zero. And zero means that you have nothing. When I say I have zero shoes, it means I don't have any shoes." Give other examples. Then tell the child that 1 comes after 0. Tie this in with what he knows about counting backwards.

10. Demonstrate that the rule about the next number applies only when you're adding 1. The child will probably get the idea after the first few sessions that addition is simpler than it actually is. Find the 1; look at the other number; then say the number that comes next. Pretty easy. To keep him honest, present a series like this.

                    1 + 4 =
                    5 + 4 =

                    5 + 1 =

                    6 + 11 = 

                    1 + 9 =   

"Now remember, we can't use the rule for adding 1 unless we find a 1 in the problem. Do we find a 1 ion the first problem? Good. Then we can use the rule. Let's do it. Do we find a rule in the second problem? No. So we can't use the rule. The problem 5 + 4 is 9 is one you just sort of have to remember. Say it . . ." 

11. Present continuity exercises. For the next month, spend about five minutes at the beginning of each session on the +1 problems. Here are several series of problems you can present. They show the relation between +1 and counting. 

                    0 + 1 =                     9 + 1 =

                    1 + 1 =                     8 + 1 =

                    2 + 1 =                     7 + 1 =

                    3 + 1 =                     6 + 1 =

                    4 + 1 =                     5 + 1 = 

                    5 + 1 =                     4 + 1 =

                    6 + 1 =                     3 + 1 =

                    7 + 1 =                     2 + 1 =

                    8 + 1 =                     1 + 1 =

                    9 + 1 =                     0 + 1 =

Stress the addition facts "2 plus 1 is . . . well, what comes after 2? . . . 3. So, 2 plus 1 is 3. Say it . . . 3 plus 1 is . . . What comes after 3? . . . 4. So 3 plus 1 is 4. Say it . . ." These are quite similar to the What comes next? games the child played during the previous year.

Word Problems Involving Adding 1: Introduce word problems after the child has been exposed to +1 problems for at least two weeks. The idea to get across is that the rule for adding works with dogs, sheep, cows,  or anything else the child can name. The idea is not that these prove the rule. You wouldn't try to convince the child that a red ball proves the concept red. It is merely an example of red, just as adding pies is an example of adding.

Present problems like these: "I have 8 dollars. Then I get 1 more. How many do I have?" "A farmer buys 4 sheep. Then he buys 1 more. How many does he have?"

1. Tell the child specifically how to translate the words into mathematical symbols. The parallel between the every-day language we use and the language of mathematics is not obvious—not even logical. Conventions: The first number you mention is the one you start out with. If the first number mentioned is 7, you start out with 7. The words more, adds, gets, buys tells you that you're adding. You indicate these on the chalkboard with a plus sign. Then you add as many as the farmer gets or adds or buys. If he steals 1, you write 7 + 1. Next, you put an equal sign, and ask the question the problem poses.

                    7 + 1 = 

How many does he end up with? We don't know. That's what we're going to figure out. 

2. Diagram the problem on the board, using circles for each number in the problem. If the farmer has 8 bulls and buys 1 more, diagram the problem this way: 

                    ◦    ◦    ◦    ◦    ◦    ◦    ◦    ◦    +     ◦   =
                    |    |     |    |     |    |    |     |           |

                    ◦    ◦    ◦    ◦    ◦    ◦    ◦    ◦           ◦   

Explain that each circle stands for a bull. "For every bull we have in the top row we must have 1 in the bottom, so we draw them in the bottom row and count them—that's how many bulls the farmer has."

Introduce one or two word problems during each session for at least two weeks. Don't be afraid of repeating problems framed in the same situation (such as bull-buying). The only way the child will learn the key words is to hear them over and over. 

That is about 6 pages, ending at p. 202, of a 300 page paperback. The instructions are much more dense than "Teach your child to read," but I was able to follow them without confusing myself or confusing her. It took two or three weeks to get through it, half an hour a day at the dining table with a tiny whiteboard. 

The rest of page 202 is another outline, even more compressed, to teach the number pairs: 1 + 1 through 5 + 5 at first, and 6 + 6 through 10 + 10 after another two weeks. On page 203 a new section starts with the heading "Algebra Problems."

ReadWorks

She was reading. Could I just get her a library card, and let her educate herself? 

In June she was still unsure about words that end in y, and about double effs and double esses and double ells. In November, so far, I'm still not raising a bookworm. I once in a while catch her reading the side of a cereal box, or one of her old board books. But I'm glad we kept on doing reading lessons.

If it was complicated, she couldn't read it. Not in June. But if most of the words were familiar, and the sentences were short, and there was lots of repetition, she would bring all her concentration to it. At least for one page of big print. She might read it from start to finish, or she might stop after every sentence and react. Talking to her is not the same: she zones out or interrupts or tries to change the subject. Our conversations are fun but they are not didactic. Reading lesson was a new channel of communication between us.

What should I send down this channel? Most of the world's written knowledge is news to a five-year-old. When I was counting the days until I ran out of the "100 easy lessons", I found this idea on a website called ReadWorks:

Article-A-Day is a 10-15 minute routine designed to be done every day to build background knowledge, vocabulary, and reading stamina. Article-A-Day complements a broad range of curricula and is recommended for kindergarteners to eighth-graders. ... The Article-A-Day sets are grouped topically or to systematically build vocabulary. Find sets that gradually become more challenging as the school year progresses and are coordinated by topics across grade levels.

They don't cost money but you have to make an account to see their articles. I was disappointed when I did. What they have that is pitched at kindergarteners is meant to be read aloud by a teacher. The kids fill out a worksheet after. "If the students cannot write yet, they can draw their responses." For a new reader it wasn't great for practice, and I wasn't excited about the content.

But I liked the idea of a little bit of nonfiction every day. After one false start it wasn't hard to write 100 words about a subject that was new to her, in words she could handle. I made it easier to write that much the next day, and every day, by having a big part of each passage recapitulate yesterday's passage.

I wanted to write about history but she knew very little about calendar dates, or even about numbers. The day after "Cats and dogs," I didn't make her read anything but this:

five, zero, ten, one, two, hundred, thousand

Each of those in large handwriting, an arrow drawn underneath, and a box to the side for her to print numerals. It was a worksheet. She already knew in speech the words "hundred" and "thousand," but only now in November is she maybe starting to know their magnitude. "Five" and "zero" were established favorites for her. The irregular spellings of one and two drew out a nice conversation. "A thousand years ago I think they actually said it t'whoah but we don't say it like that anymore."

Here's the lesson from the day after that: 

Spain, Genoa, India, Bahamas, America, ght, caught, light, bought, hundred, tried, year, years, month, people, explore, explorer, exploring, think, thought, heard 

A new way to get to India

Five hundred years ago, the king and queen of Spain paid an explorer to find a new way to get to India. That explorer was from Genoa and his name was Columbus. The king and queen thought that he was good at exploring.

The old way to get to India was by land, but Columbus looked for a new way to get there. He tried to get there in a ship. Two other ships went with him.

After one month, their ship found land. There were many people living on that land. He thought that he had found India and that those people were Indians. But that land was not India. It was the Bahamas.

The Bahamas are close to America. Until then, the king and queen of Spain had never heard of America. No one in Spain or Genoa had heard of America.

and the day after that:

Columbus, Ferdinand, Isabella, Spain, India, Aragon, Castile, first, paid, country, marry, married, merge, merged, treat, treated, people, prince, princess, remember

The first king and queen of Spain

Remember the king and queen who paid Columbus to find a new way to get to India? Their names were Ferdinand and Isabella. They were the first king and queen of Spain. Before they got married, there was no country called Spain.

Ferdinand was the prince of a country called Aragon. Isabella was the princess of a country called Castile. They were born almost six hundred years ago. They got married and then later they merged Aragon and Castile into one country. That country was called Spain.

Ferdinand and Isabella were a prince and a princess once, but they were not heroes in a story. They treated some of the people in Spain badly. Sometimes you get that from kings and queens!

If I had read them aloud to her she wouldn't have listened. She engaged with them when I made her read them aloud to me. If part of the reading went over her head, I sometimes would expand on that part the next day—it didn't have to be important to me, it just kept away writer's block.

For one session a couple of weeks later, I had too little time or maybe I did have writer's block. I didn't have something new ready for her so I had her read "The first king and queen of Spain" again. It took her almost 40 minutes the first time and only 10 minutes the second time.