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# Summer Math Class Day 1

I had my first class yesterday. I had 8 students, 4 girls and 4 boys. All were eager and coopertive except one boy. This can certainly color what the class can do. I started with some activities to gauge where the kids were. I was actually very pleased:

• They could count backwards and forwards by 2, 5 and 10.
• They knew their place value and could add 1s, 10s, and 100s in their heads.
• And they really liked the games that required them to compete against a partner.

### place value game

We played a place value game where they had to roll a dice and add that number to a game board. The board was divided in 100, 10, and 1. So when they go more than 10 in the 1 box, they had to trade them in for a 10. The first one to 100 wins. They played this for 20 min and no one quit before someone won. I didn't even get to most of my activities.

I'm going to concentrate on one digit addition and subtraction for the next class.

### the trouble starts before fractions: algorithms & math facts

Many readers are commenting on fractions as the place where kids really begin to slip behind. It is true that the ability to do fractions is where the problem becomes glaring. However, I believe that it starts much earlier. Algorithims in any subject are really just teaching children to generalize a solution to a problem. If you teach an algorithim, you are telling them that any time you see this combination of numbers and symbols, this is the quickest and most accurate way to get the right answer.

But they can't learn the algorithim if they haven't memorized the math facts necessary for the algorithim. For example, I am tutoring a student in long division. She never memorized her subtraction facts. So even though she knows her multiplication table, she can't do long division because she can't subtract. Think of all the facts that need to be memorized and all the algorithims that have to be mastered before you get to fractions. A gap in any one of these things can keep kids from mastering fractions.

-- AnneDwyer - 16 Jun 2005

# June 18, 2005

### At Home Summer School

I also put together a summer school plan for my kids.

I have two children.

My oldest is 12 and just finished the fifth grade. He is developmentally delayed and has just been reclassified autistic spectrum disorder by the school district. He does

• two pages of Singapore math (currently 3B)
• cursive handwriting practice

My daughter is 7 and had just finished the third grade. She does

• at least 4 pages a day in Singapore math. (She is on 1B and it is really too easy)
• reading at least one chapter per day in a grade level appropriate book
• two pages of a spelling and vocabulary workbook.

## Goals of a Summer School Program

My goals for each child are different.

For my son, his math goal is to be able to manipulate fractions. He will be doing all of Singapore math book 3B this summer. Right now, he is working with lengths and converting cm to m, m to km, ft to yd, and in to ft. This is an excellent precurser to fractions. This book spends 40% of its time on this subject and converts all units of measure (length, weight, and volume).

His writing goal is to consistently use cursive. His fine motor skills are really his weakest skill. His printing is terrible. Think kindergarten. But cursive, because it requires joining letters, forces him to be neater. It can actually look, at times, like middle school handwriting.

My daughters goals are much simpler. I want to reinforce the basic skills she learned in 2nd grade and pull her ahead. In reading, I want to make sure she doesn't lose ground over the summer. In spelling, I want to make help her be an independent speller.

## Tutoring

I have several students that I am tutoring this summer. The one having trouble with long division is actually going into 9th grade. She is very delayed in math and, like Daniel, has been in special education classrooms.

I could write a long discussion on the pros and cons of separate special education classrooms. I'll just comment that one thing that I have noticed is that a parent must monitor the academic program to make sure that the student is not doing work that is too easy and is progressing.

For example, Daniel just brought home his end of the year papers and workbook. In with all the books on reading comprehension (which was at his correct level) was an Explode the code book.
For those of you not familiar with this, it is a series of workbooks that helps children who are delayed in reading understand phonics and how to sound out words. Daniel has done some of this before. But this was at first grade level and using 3 letter words. A total waste of his time.

Anyway, this 9th grade student and my son are at almost the exact same level. She was a little behind, but is now catching up. My goal is to get her through fractions this summer also.

# Summer Math Class Day 2

I had my second math class on Wednesday. It takes a couple of days for me to replay what we did and think about what worked and didn't work. It also takes a couple of days to get comments from my daughter.

Conversation with daughter the day of class:

"Did you have fun?"

"You know I don't like math"

EOC

By the second or third day, she'll tell me what she liked and didn't like.
BTW, I got most of my games and ideas from two sources. The first was a book I got out of the library. Games for Math by Peggy Kaye. The second source is Primary Mathematics 2A Home Educator Support Guide by Jennifer Hoerst It turns out that the Singapore Math website sells these separate books by Sonlight Curriculum.

I started out with some board games where the students had to solve addition and subtraction problems to find out how many spaces to move. They played it, but were not very interested. And it doesn't really work with a group of five or so because they don't really get enough problems to solve. And I am beginning to agree with Catherine on the whole flash card thing. They really need to do pencil and paper calculations. And it turns out, they like that stuff the best anyway!!!

They wanted to have a 'race' so I took out my mad minute sheets. These are 30 simple addition problems on one page and they have one minute to do it. I gave all the kids except my daughter the simplest sheet. (Problems like 0+4 and 1+2) My daughter is on problems like 9+9 and 7+8. Anyway, only one student could finish 30 problems in one minute. But they really liked it.

One of the things they liked best was strategy games. And at the end of class, I introduced them to bubble puzzles. These are well known puzzles. Picture a plus sign with a circle at the end of each line and a circle in the middle. (That's 5 circles.) Then they have to take the numbers 1,2,3,4,5 and make the three vertical circle add up to eight and the three horizontal circles add up to eight. I was surprised that they had so many problems with these. None of them could solve it until I gave them a hint about the middle number. We'll do more of these next week and I'll go over some strategies for solving them.

So, what have I learned so far?

• they like games where they compete with one another
• they prefer pencil and paper exercises
• they like to figure out puzzles

Until next time, if anyone has any ideas/suggestions, just post me a comment.

# Summer Math Class Day 3

I thought today went better than any of previous classes. I put my three main conclusions (see previous day) to work.

Everything we did was pencil and paper. They either competed against their own score or another student or student group. The upshot: not one child said they were bored and I had to tell them when class was over!!!

Erin, of course, said she did not enjoy herself.

Let me add one more thing that I have learned: You cannot work on mastering addition and subtraction facts in a once a week class. These have to be practiced with pencil and paper everyday until they are mastered. But you can introduce games and exercises that interest them and require them to use the addition and subtraction facts and give them a feel for the language of mathematics. That is what I am going to concentrate on for the rest of the class.

# Summer Program Update July 1, 2005

I used the color purple for my heading because purple is my daughter's favorite color and today's update starts with her.

Today, she had a playdate with a girl in her summer math class. We went to the pool. Afterwards, they were eating a snack. Do you know what they asked to do???

That's right. Mad Minute on a playdate for no reason other than the sheer joy of a math race. Now that tells you that kids don't find drills boring at all.

As for my son, he has been amazing this summer. He is more than half way through 3B in Singapore math. Almost half of 3B is devoted to the cool problems that Catherine talked about in her post. He is learning the metric and English measuring systems. He has to measure length and capacity (we are cheating on this) and read weight scales (hard for him). Then he has to add and subtract in those measuring systems. Capacity is next. But right now, we are finishing off a unit with word problems. They are multistep and he is getting them!!

Next, we are going to do some hands on research on the topic of learning to draw. My daughter bugs me all the time to take her to the local mall to see the puppies in the pet shop. This time, there was a new place (not strore) call Art Experience Inc.

I go in to look around and find that they are a non profit group that specializes in teaching art to developmentally challenged children and adults. They actually have an artist in residence who is autistic. All of the staff is trained in art therapy.

I told them about Daniel and they are eager to meet him. They want to talk to him and get a feel for what he can do before they put him in a group. They want to make sure he will be successful!

I talked to Daniel about it and he is wary. He doesn't want to give up an afternoon at the pool for art class. But he is willing to go in a meet the teachers. And Erin is happy because she gets to visit with the puppies.

I'll keep you posted.

# Summer Math Class Day 4

Today was not a good day in the summer math class. It was very cold this morning, and the kids did not go to swim practice. I forgot my folder and had to wing it.

The first half hour was fine. We couldn't start with Mad Minute. So we went straight to a strategy game with large numbers.

The kids pick the number of digits (we usually start with 5). They put 5 dashes on their paper. I turn over 5 cards in a deck one by one. They have to decide where to put the numbers. Then each kid reads their number to me while I put it on the white board. The kids with the highest number wins.

For some reason, they love this game. On the next round, we go up one digit. Today, we went all the way up to 100 million.

It's a great game.
• They gain familiarity with large numbers. They get a lot of practice with reading large numbers out loud and hearing large numbers read out loud while it is being written on the board.
• They have to use strategy. In some games, we have a lot of high numbers at first which every kid puts in the same place. Then, they winner is the determined by the numbers in the ones and tens place. Conversely, sometimes we have a lot of low numbers in the beginning. Then the winner is determined by the highest digits. Much more interesting is when we have medium and low cards. Then, they have to do a lot more thinking about where the cards go.
• There are very concrete results from this game that allow us to explore numbers even further. In one game, 5 out of 8 kids had the same highest number. So we talk about why and when does this happen? In one game, we had one winner that was a lot higher than anyone else. When does this happen?

The next game we played was 5 to make 10. They get 5 cards and they have to make as many equations that equal 10 as they can. They write them down on a piece of paper. This game teaches:

• Properties of mathematics. For example, 6+4=10, 4+6=10. And that is 2 equations in our game.
• Strategy. What are the best cards to get? What are the worst. When you make an equation with 3 cards, the number of equations you count goes from 2 to 6. So a mix of low and high cards is the best. A hand of all high cards is disaster. I should experiment with allowing them to swap cards.

But, after that, they just couldn't concentrate any longer. They didn't agree with my suggestions of what to do next and they couldn't agree on what they wanted to do next. I had to let them got out to play after 45 min.

The kids sit around a big table. This allows them to partner and share the material. But when things start to go bad, having them all around a table multiplies the problem. This is what I dislike so much about the grouped desks in school. It distracts the kids even in the best of circumstances.

##### Summer Program Update

Daniel has been doing very well in Singapore Math. He is halfway through 3B. But today he had some review word problems. What I find is if Daniel can't do a specific word problem, I help him as much as needed because the same type of word problem will come up again. And when he is ready to process that type of word problem, he'll get it. But if it is still difficult for him, he won't be able to do it 100% if the time.

In Daniel's IEP, his goals are usually written such that he has mastered something when he can do it 80% of the time. If he hits a certain type of word problem in Singapore Math that he is very comfortable with, he can do it 100% of the time. If he comes up on one that he understands but is not comfortable with (problems with 3 numbers freak him out because he knows it is going to be multiple steps), then he'll be able to do it 8 out of 10 times...but those last 2 times are total trauma.

It could be he is having a bad day or he is tired or something is bothering him. And it doesn't pay to dwell on the problems on days like that. Just do your best and go on.

I had him listen to a story in Read Naturally and then read it to me. And he did terrible. So I told him to listen to it again. When I tried to rewind it, I found out that the batteries were dead, so he didn't listen to the story. He didn't want to tell me because he was afraid he had broken it.

After he listened to the story, he did much better (10 more words per minute and no mistakes). So, listening to the story which models good fluency does make a huge difference in how he reads.

Erin is finally finishing up 1B. She is working on money. She has always had problems counting money. She can skip count by 5 and she can skip count by 10. But she couldn't skip count by 10 if she was on a number that ended in 5. I tried several methods to help her but nothing worked until we started working on the white board. I wrote out skip counting by 5 and underneath I wrote out skip counting by 10. And finally, in frustration, I suggested that when she had to increase the number by 10, she should skip count by 5 twice. Well, that just totally clicked with her. She got it and has never had another problem counting money. I never would have 'taught' it that way, but it totally worked for her.

# Summer Math Class Day 5

OK, I think I am starting to figure this out. In a one hour class, I have their full attention and their brains fully engaged for about 35 minutes. Then, for the last half of class, I have to have short, quick things to do to keep them from getting restless.

# Working with Whole Numbers

I like to give them number puzzles. I think puzzles make them think about numbers and how they fit together. If you tell them, let's do a puzzle, most of them will groan.

Plus, I have been thinking about Carolyn's question about good sources for pre-algebra. I talked about using equations with boxes. But I also think it is important to give kids practice with equations. So here was their first assignment:

Using the white board, I asked them, how many equations can you make that add up to 3 using only addition and the numbers 1,2, and 3. You can make 3 equations: 3 by itself (they didn't like this so I had to let them use 3+0), 1+2, and 1+1+1.

Then, I erased that and put the number 6 on the board. I said that there are 10 equations that you can make that add up to 6 using 1,2,3,4,5,and 6. Let's see how many you can find.

There was total silence and concentration as they all put together their equations. When they were done, they each contributed an equation or two on the white board. I believe everyone got 10 or close to it.

Now, they wanted to do more. They asked about the number 10, but there are 42 equations. So we agreed on 7 (15 equations). Again, there was total silence and concentration while they wrestled with this problem. After about 10 minutes, they asked to work with partners so they could find all 15 equations. By the end, everyone had all 15 equations.

# Suggestions for Pre Algebra Class

If these kids had been older and it had been a regular prealgebra class, I could certainly see this as a nice exercise for introducing unknowns.

Before I did that however, I would teach the identities with unknowns. This is how I do it in a tutoring session. I say to the student, "Any number plus zero is the number. So what is 1+0?" And they, of course, answer zero. Then I go up, What is 2+0, what is 10+0, what is 100+0. By now, they are looking at me like I'm crazy. Then I say, "What is a+0?" Most students will automatically answer "a". Then they hesitate. "That's right", I say, "a+0=a and a can stand for any number." I do the same thing with a*1.

Now, put all 10 equations for the number 6 on the board. And substitute "a" for one of the numbers. For example, use 3+2+1 and substitute a for 1. Then the equation becomes 3+2+a=6. And 1+1+1+1+1+1=6 becomes a+a+a+a+a+a=6.

From here, you can show them how to manipulate the unknown in an equation. Since they already are familiar with the equation and they know what a stands for, you can show them, for example, that combining a+a+a+a+a+a=6 into 6a=6 is the same equations and has the same solution. This way, you are not introducing so many new things at once.

That is actually my philosophy when I tutor. I try to connect a new topic to an old topic in a very concrete way.

# Discoveries in Art Class

Daniel is taking an art class during these next two weeks. I have two reasons.

• I think it will help me see how he sees the world.
• I hope it will help his visual perception skills and his fine motor skills.

It's a great class. The first thing they did was make a portfolio to hold their work!!

Today was the first time I saw anything of his. They did a still life drawing with charcoal pencils. There was a violin, a book, a feather, and some other things. I was very proud of him. The art therapist told me that he sees everything in great detail so he had a very hard time just picking out certain things to draw for the picture.

I can see where it would be a problem if you saw all the details and had a hard time picking out certain details. That's probably why he likes the detective show Monk so much!!! He sees all the details that no one else picks up.

# Summer Math Class Summary

Well, summer math class ended today. I did eight sessions and covered the following topics:
• one digit addition and subtraction for speed
• up to three digit addition and subtractions with and without carrying and borrowing
• place value up to 1 trillion
• measuring length in inches (to 1/16") and cm (to .1 cm)
• fractions (most common 1/4, 1/2, 3/4)
• coin money including adding money and making change
• word problems (both one and two step)
• strategy in games and puzzles

This is what I learned:

• kids love to do pencil and paper work in math
• kids like to compete against each other
• one hour is too long for kids at this age. Better is 45 min.
• do all the hard work or introduce new topics in the first 30 min.
• Ten kids sitting around a table who all know each other is not the ideal setting for doing math. Especially when the temp hits the 90s and there is a trampoline in the back yard.

Nevertheless, I think this model is something I want to explore more. So this fall, I will be offering math enrichment classes through the local recreation department. It will be 45 min classes. I will offer one for grades 1&2, one for grades 3&4 and one for grades 5&6.

### One and Two Step Word Problems

Today, we worked on one and two step word problems. I took them from Singapore Math levels 2A and 3A. I worked specifically on getting them to write the equation for the problem and then get the answer. I saw some interesting things.

With the easy one step problems, the more intutive kids worked the problem in their heads and then wrote the opposite equations.

The problem was: After giving away 10 comic books, Rami has 36 left. How many did he have to start with?

I expected the equation to be 36+10 = 46 and most kids wrote it that way. However, two kids solved the problem in their head and then wrote 46-10 = 36. They did this repeatedly for easy problems.

Also, I think this is an interesting statistic: I had 6 kids in the class today. All could solve the single step problems even when they involved 4 digit numbers. Three kids could solve the multistep word problems with equations. Three of the kids could not solve the multistep problems and really didn't understand the solution once it was presented to them. This makes me think that there is a developmental age for being able to do this kind of problem. The ones who could not get it were the youngest kids in the class by at least 6 months.

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Anne, it's weird little things like click -- that skip-counting trick -- that mean that the best teachers are those who can attack a problem from a number of directions, and those who can be flexible and let things go.

I need to take a page from your book, when it comes to that last thing.

-- CarolynJohnston - 07 Jul 2005

Anne, thanks for keeping us posted on your summer class! Will you also write about your enrichment classes this fall?

3 classes -- that's a lot of classe for one person. But I guess you are building up a knowledge base.

One last question -- what were the ages of the kids in your summer class?

-- CarolynJohnston - 28 Jul 2005

Carolyn,

I am looking forward to my classes so I can gather data about this model of delivering instruction to students.

My summer class had all kids going into third grade and one going into second (her brother was in the class). But my daughter's birthday is in August and many of the kids had birthdays in the Jan - March range. The going-into-second-grade student couldn't do the word problems either. So it seems at this age that a 6 month or more difference in age makes a difference. The two step word problems were from second semester third grade Singapore math book.

-- AnneDwyer - 28 Jul 2005

I have a quick off-topic question.... (and I realize I'm behind reading your journal PLUS I need to get this pulled 'up front.')

I ordered a READ NATURALLY book, and it's fantastic. Just amazing.

I'm now having Andrew listen to one story each day, the one about the peacock & the peahen. I hold his finger and point it to each word as we go along.

But now I'm stuck.

He basically doesn't talk at all--though he's now learning--and I can't quite think how to test him on the content.....

Do you have any ideas?

I'm a little 'stuck' with Andrew overall, and have been for a long time. Somehow the fact that he doesn't speak gets me gummed up; I can't quite envision the alternatives.

Fortunately this year the school taught him to read using Edmark, so that jogged my brain loose a bit.

Anyway.....if you have thoughts, I'd like to hear.

(I guess first of all I'm wondering whether I ought to try to ask questions using pictures, as they do in Edmark...)

-- CatherineJohnson - 28 Jul 2005

Oh gosh, you've jogged my memory. There's a new study out in SCIENTIFIC AMERICAN showing paper and pencil to be far superior to manipulatives for young kids--incredible.

I'll get it posted today.

-- CatherineJohnson - 28 Jul 2005

Anne--I think your experience probably jibes exactly with the work described in the new SCIENTIFIC AMERICAN article, so I'm adding the two posts I wrote about it here:

-- CatherineJohnson - 28 Jul 2005

Wow--I didn't realize how behind I am!

Fantastic.

I'm wondering if I ought to have some Mad Minutes on hand for that very purpose. (The Saxon pages are a little more intense, since they're 5 minutes per page....)

-- CatherineJohnson - 28 Jul 2005

I have their full attention and their brains fully engaged for about 35 minutes

IIRC, Skinner figured out the maximum amount of time a person could concentrate purely on one thing.

It was somewhere around 40 to 50 minutes, and that was a highly-trained adult.

The story went that he trained himself to do nothing but work in his office. The instant an extraneous thought came into his head, he left. After awhile his office 'triggered' concentration, which of course never happens for kids (or for any adults I know).

But even with all that training, the longest he could go without extraneous thoughts was around 45 minutes.

-- CatherineJohnson - 28 Jul 2005

"Any number plus zero is the number. So what is 1+0?" And they, of course, answer zero. Then I go up, What is 2+0, what is 10+0, what is 100+0. By now, they are looking at me like I'm crazy. Then I say, "What is a+0?" Most students will automatically answer "a". Then they hesitate. "That's right", I say, "a+0=a and a can stand for any number." I do the same thing with a*1.

Brilliant.

-- CatherineJohnson - 28 Jul 2005

That technique is jogging my memory.....and I absolutely can't figure out what you've reminded me of. DARN.

It was something like that, where the teacher works up the rhythm & the structure so that the student basically can't NOT come up with the correct answer.

-- CatherineJohnson - 28 Jul 2005

Saxon does all kinds of fun pre-algebra things that I think are good.

He'll have a huge, long addition column, all single digits, maybe 12 of them, and there'll be one variable somewhere in the stack.

-- CatherineJohnson - 28 Jul 2005

I try to connect a new topic to an old topic in a very concrete way.

This is smart constructivism.

I haven't started posting about this yet, but it turns out there is a smart form of constructivism that went missing when people created the NSF-funded curricula.

ALL knowledge has to 'hook into' knowledge that's already there, inside the brain.

It's much better to know and use this fact than just to throw stuff at people and let them try to hook it to something they already know on their on.

-- CatherineJohnson - 28 Jul 2005

Last question: how are you 'billing' your rec department classes?

You're calling them 'enrichment'--what else did you say?

-- CatherineJohnson - 28 Jul 2005

OK, lots of questions and I am on vacation at the beach. It is heaven.

Anyway, Catherine, there was an article somewhere about an autistic girl who could not speak but could type on the computer. She still cannot talk, but she is in college and can communicate at her college level on her computer. See if you can find it from my vague info because I believe it had some good info on how she was finally able to show people that she could communicate at the appropriate age level. Before, she was totally closed off. Maybe your son could learn something like this.

As for my class, I billed it as securing basic skills and increasing self confidence in math. I call it Math Booster. I'll keep you posted.

Anne

-- AnneDwyer - 01 Aug 2005

Have fun--we'll see you when you get back!

-- CatherineJohnson - 01 Aug 2005

Anyway, Catherine, there was an article somewhere about an autistic girl who could not speak but could type on the computer. She still cannot talk, but she is in college and can communicate at her college level on her computer.

That's Sue Rubin!

She's great! (I know her & her mom a little). a good friend of mine from NAAR days was involved in the production (the NLM Foundation funded it).

I love 'Math Boosters'!

-- CatherineJohnson - 01 Aug 2005

I think the rec department approach is an interesting one.

Cool.

-- CatherineJohnson - 01 Aug 2005

I loved Mad Minutes too! Even when I was in hs calculus I'd do a few "just for fun" after school to see if I could still pass them. My mom teaches Saxon Pre-Algebra and Algebra 1 and has her students do Mad Minutes every day. They enjoy the competition, but she also gives them the reward of lunch out with her and a friend once they complete a sheet for each operation (about 8?)

-- AndyJoy - 29 Aug 2005

Andy,

What problems does she have on her mad minutes for pre-algebra and algebra 1 students?

Does she go back to basic addition, subtraction, multiplication and division or does she do something more complicated?

-- AnneDwyer - 30 Aug 2005

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