Thursday, December 31, 2020

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?

 

Tuesday, December 1, 2020

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."

Anki

In education research, three things have my attention: one-on-one tutoring, Zig Engelmannism, and spaced repetition. Anki is a flash card pr...