What's the percentage of "adders-across" in Numeracy?

In the past, I've given diagnostics before a unit so as to be able to compare pre- and post-instruction scores. Now, in the spirit of differentiation, I'm going to go one step further.

The next unit is about adding and subtracting fractions and mixed numbers. On my diagnostic, I wanted to see what percent of the students are still "adders-across" (#25 down: snakes that are bad at math). That would be 68/80, or 85%. The remaining 12 students could all do the basic algorithms, but most stumbled on the more complicated mixed number subtraction problem.

So here's the plan. In each class, I will assign one of the non-adders-across (NAA) to an adder-across (AA), tasking the NAA to help the AA learn over the coming lessons. If I see that they remain on task during practice time, the NAA will not have to take the quizzes, earning an automatic 100% on them. This seems reasonable, since they have already shown me they know the skill. Additionally, if the AA passes the quizzes (i.e. becomes an NAA!) then the NAA helper will earn some oh-so-coveted extra credit points. This way, the NAA has strong incentive to help, but there is no penalty if the AA doesn't make enough improvement.

Since almost no students showed mastery of the mixed number subtraction problems, every one will need to take that quiz when we get to it.

Now, the only thing that remains is to pair up the NAAs with the AAs effectively. I need to factor in personality, motivation, and so forth. Also, this experiment really highlights the imbalance between classes, even though we try to avoid any tracking (a constant difficulty in a small school). Here are the numbers of NAAs by period... Period 1: 5, Period 2: 4, Period 4: 2, Period 6: 1.

My mini-whiteboard love-hate relationship... Can you help?

I've been using mini-whiteboards daily in my numeracy classes all year. Students use them most of the time, except when I have a worksheet for them to do (and even then, they tend to use them for scratch work).

Positives:

• I can see, from anywhere in the room, what students are doing, and if they are on task.
  
• Students enjoy writing on their whiteboards more than on paper.
  
• Students don't have to waste paper for scratch work (this is especially helpful for those students who have still not mastered the art of bringing school supplies to class).
  
• And I don't have to make worksheets for every single task either.
  
• It makes collaboration easier during pair/group work tasks.
  
• It's great for quick checks of understanding - put a problem up, students do it on their boards, and then immediately lift them up for inspection.
  

Negatives:

• We burn through markers like nobody's business, and the ones that are low-odor cost about a buck a piece. I've tried the cheaper ones, but they run out really fast, or have fumes that cause much complaining of headaches.
  
• Tables and hands tend to get really messy (for some students more than others...) Our beautiful white laptops are getting covered in whiteboard marker smudges.
  
• "Mr. Greene, can I please go wash my hands???"
  
• Some students not able to respect materials, destroying markers by pounding in their tips, or writing with them on paper till they run out.
  
• Some students unable to stop drawing beautiful works of art when I am presenting material. Or maybe this is a positive because I can see that they are off-task, whereas if they were doing plain old paper-and-pencil doodling, I might not notice?
  

I was wondering if anyone had any ideas to help with the logistical issues of mess and expense? Remember the Magna Doodle?

4.58 x 10,000 = 4.580000

Most of my numeracy students remember that helpful rule from middle school: "Multiplying by 10 means adding a zero", and so we get results like the title of this post. This is one of those fundamental place-value problems, the type of thing that betrays just how little some students really get about the number system. It's taken about two weeks of practice to get them comfortable with the idea of shifting the decimal place left and right (and remembering which way to shift it, depending on the operation).

We are also currently struggling with the issue of the missing decimal point... when there is no point shown in a number, where is it really? Some of my students still think that you put the point at the front of the number. Why do they think this? I'm not sure. Before break, we spent a whole lesson on what the decimal point means, and it seemed to go well. Since we've been back in the second semester, the question of where the missing decimal point goes has been asked and answered many times each class period. They are getting better at comparisons: if I ask them to compare 473 and .473, or .4 and .39, or .4 and .04, they are usually getting it right. And yet, when faced with the problem 473 ÷ 100,000, some students seem to forget it all and start with the decimal at the front of the 473 (or sometimes between the 4 and the 7), forgetting that this changes the value of the number.

No wonder scientific notation is such a bear to teach in Algebra... To reinforce both concepts, I've been teaching scientific notation (with positive exponents only) in this unit, and it's finally starting to work. From the start, my students could tell me that 10^6 was the number 1 followed by 6 zeros, but they couldn't see the relationship between the problems 9.02 x 10^6 (which was totally confusing) and 9.02 x 1,000,000 (which is finally becoming easy). Converting a number into scientific notation is starting to make more sense to them now, since I've finally figured out another flaw in some of the students' understanding: they don't really get the significance of the equals sign. I would show over and over why 302,000,000 = 3.02 x 10^8, and some kids just weren't catching on. But then, when I asked them what they would get if they multiplied 3.02 x 10^8, they were surprised to see that it was 302,000,000. I would get lots of "ohhhs" as they realized that the two parts of the equation had to be the same, and that you could multiply to check your answer. The main problem I still have is getting them to remember that the first part must be between 1 and 10. But at least we're making progress! Though we have been learning dividing by powers of 10 at the same time, I don't want to introduce scientific notation with negative exponents now (since they have never seen negative exponents before). I want to give this time to sink in, and maybe come back to it later in the year.

We have the rest of this week off for winter break; when we start next week, I think it's time to move on from this percent and decimal concepts unit and start in on fraction operations. 1/2 + 2/3 = 3/5, here we come! (One of my favorite things to show numeracy students is why this equation doesn't make sense.)

Welcome

Most of our students are English language learners, but most have Spanish as their native language. As of a few weeks ago, we have a new student who is a refugee from Myanmar - she showed up in my SSR period and in my Numeracy class. Not only is language a huge barrier, there is also her difficult past. Working in her favor, however, is a massively strong desire to learn.

An article came out in today's paper which gave us all more insight.

Here is the text of the article (if the link is bad).

Orphans survive wars, find safety in Bay Area
REFUGEE CHILDREN ADOPTED THROUGH UNITED NATIONS
By Rebecca Rosen Lum
Bay Area News Group
Article Launched: 02/18/2008 01:33:04 AM PST

Kate's smooth brow buckles when she thinks about the soldiers who muscled their way into the house where she lived with her grandmother - plundering belongings, forcing their attentions on her and ordering them to prepare meals.

"The soldiers make me too sad," said Kate, discriminated against as an ethnic minority in Myanmar. "I don't like."

One day Kate, now 16, fled to the home of sympathetic friends in a neighboring town. She learned soon afterward that the soldiers killed her grandmother in retaliation.

After a desperate flight through underground channels of Southeast Asia, Kate has found a lasting safety: She now lives with a family in San Jose. "Baba" and "Mama" are the Rev. Ben and Anne Daniel; she has three siblings.

As rain pounds on the roof of Ben Daniel's church, Kate sits comfortably between her new parents, a delicate girl with shiny black hair and a wide open smile. She has been here little more than a month, but she says this is home.

"Everything OK," she said. "Not tired. Not scared. I happy."

Kate is one of a trickle of refugee orphans finding homes with Bay Area families through a special program of Catholic Charities, one of two agencies that contracts with the U.N. High Commissioner on Refugees to place the children.

In such countries as Liberia, Uganda, Sri Lanka, Myanmar and Nepal, children have been driven out by armed conflict or pressed into service by government militias and rebel groups - as combatants, sex slaves and virtual pack mules.

If an adoption always includes risk and reward, these adoptions offer a double dose of both.

Preparing food is now a source of surprise and delight for Kate. She likes oatmeal with hot sauce. At first, she dissolved in giggles at the sight of Baba popping up a skillet of popcorn on family movie night. (Men don't cook in Myanmar). Now they fix dinner together.

Kate dropped out of school after her fourth year to help her grandmother farm corn and beans. She asked to start school the morning after she arrived: "I want right now," she said, laughing. She studies music with Anne and says she hopes to become a minister, like Ben.

Kate's odyssey hardly seems likely for a child, but it is mirrored throughout countries where war and strife have made homelands unlivable. Many have been persecuted for religion, ethnicity, or political affiliation. They have been separated from their parents or seen them killed. The children escape brutality by guts, wit and luck, walking for miles, hiding in jungles, riding on the backs of sympathetic elders to safety - mainly, in refugee camps.

Five million refugees have fled their homelands, according to Refugees International, a non-profit organization. If one includes those who are trapped in their home country, such as in Darfur, that number balloons to 14 million. They can't go home in many cases because home is no more; their villages have been destroyed.

Tracy Weiss read all she could get her hands on about the conflicts that racked the Eastern coast of Africa after she agreed to adopt three siblings from Monrovia, Liberia.

When she picked them up from Mineta San Jose International Airport, Sadiki, the eldest and tallest, stood in front, "scanning everyone, looking for danger in every direction." His sister Maryama tucked in behind him, holding a bag, the U.N. signal for a refugee arrival. Antimana, called "Ansu," crouched behind his two siblings. They wore donated clothes - Ansu, a 1930s-era man's suit.

"I said, 'Hi. I'm your new mom,' " Weiss remembered. "Ansu was the first to break into a grin."

The trio has been living with Weiss in Los Altos for three years and - Maryama counts on her fingers - six months.

Rebels executed the children's Mandingo father, as well as Sadiki and Ansu's mother. The children and Maryama's mother ran from rebels, living in the bush, moving constantly, sometimes getting separated. They settled for a time in Bo, a village in Sierra Leone. Sadiki - he thinks he was 3 or 4 - made many friends there.

"Then things got bad if you are not a citizen," said Sadiki, now 18. "We had to find a way to stay alive."

Sadiki's earliest memory is of a village in chaos, with people running everywhere to escape the approaching rebels. Alone, he held up his arms in hopes someone would carry him to safety. Someone did.

He thinks the family spent five to seven years on the run.

Chatting one afternoon, Sadiki's new mother asked him if he had any photos from his earliest years.

"Mom," he said evenly. "You are running with a whole stack of things on your head. You step and you fall in the river, everything gets ruined."

They eventually made their way to the Bandajuma refugee camp, where his stepmother died from complications of diabetes.

It took them some time to get used to the idea that they could make the four-block walk through their wooded suburban neighborhood to school without getting mugged, that loud pops were not likely to be gunshots. Weiss had to quickly abort a July Fourth trip to see fireworks in San Francisco when the multiple blasts badly shook the children.

While life here brings a sense of safety, negotiating the social minefield of a new culture can prove dicey.

Language is a separator at the outset. Then come the mutual misconceptions of American kids and the newcomers.

The refugee orphans are surprised to see all Americans aren't wealthy and white. Alternatively, few Americans have had to run for their lives.

"One kid said to me, 'Did you ever fight a lion?' " Sadiki recalled, howling with laughter. "I said, 'Yes, two.' "

Many don't even know where Africa is, Maryama said, and they know much less about the violence that devastated her homeland and scarred her family.

"I can't be angry at them," she said. "They don't know. When they know, they care."

Divide by Zero?

This is pretty cool. I've never heard of the animated lego genre before, but I guess it's pretty popular. Most of these films are shorts, but some are feature length! Wow..

Caught Being Good

If you're a Harry Potter fan, you've probably noticed that classroom management at Hogwarts isn't much of an issue. Sure, they get to fly around and do magic all day. And parent involvement seems quite strong. But what else do they have that keeps the young'uns in line and focused on getting to a four-year wizard college? An entirely hassle-free incentive system. I'm talking, of course, of the Hogwarts' House Cup, and the constant cries of "10 points for Gryffindor" and the like.

At DCP, we've only formally developed negative consequence systems (detentions, referrals, contracts, etc.). These work to an extent, but not for all students, and not in all cases. For a while now, I've wanted to get a positive system put into place as well. I thought that this would help increase student buy-in, especially for the freshmen making the transition to DCP and becoming a college-prep student. So, combine this need with our students' love of getting points, use Hogwarts as a model, and presto-hey you've got the "Chalice of Pride"!

I got some other teachers together, and we made a plan for this at the beginning of the year, but we haven't been able to get it off the ground until now (time, time, time...). We originally had a more complicated setup, but the lack of magic wands put a damper on our plans - the system had to be totally easy for the teachers, or it wouldn't fly. So here's how it works: each freshman tutorial class (there are 5) is competing for the Chalice of Pride. Students can earn a Ganas Point (i.e. a "caught being good" ticket) for any behaviors that really demonstrate one of the school values. This is simply handed to the student by the staff member. The student then must put the slip into the clear plastic locking drop-box that is assigned to his or her tutorial (these are attached to the wall in a central location). At the end of each month, we tally up the slips in each box, and the winner gets to display the "Crest of Community" and claim bragging rights. At the end of the year, the tutorial with the most total points earns the "Chalice of Pride" and a field trip (like a day at the beach, or whatever floats their collective boat). We're in day 3 of the system, and I'm starting to see slips in the boxes already. I'm looking at this as a kind of experiment in positive incentive systems - clearly, it can be done more effectively - but we have to start somewhere.

Continuing on with the positivity trend, we've also updated our homework checker system for the new semester. For the past few years, students have had to carry a homework checker with them to each class; it got marked each time they didn't have their homework, and then the checker would be looked at by the tutorial teacher and the parents. Of course, this would cause students to "forget" their checker on days when they did not have their homework. Therefore, we gave detentions to students for not bringing their checkers (to force them to produce them), and this really never led anywhere good. So, we made a simple switch to stamping their checkers when they do have their homework, with some simple rewards attached to getting a certain number of stamps over the 6 weeks. The rewards and reward-levels were created by the student council: 85% = prizes like stickers, candy, etc.; 90% = a free homework pass; 95% = free dress day and double lunch. We also started this on Monday, and so far, it seems to be working well. Students really want the rewards, and they are making sure to get their stamps. Even if they miss a homework, they will still be more likely to produce the checker the following class period so they can get the next stamp (in the past, they knew we would just mark all missing homeworks when they finally brought out the checker, so some students never would).

Who knew how effective stamping and stickering could be in high school?

A Jewish-Italian Hannakristmas

I'm in Cleveland at my dad's house for the annual event... the homemade gnocchi and sauce were great, the kids got lots of noisy plastic crap to forget about by tomorrow, and the vegan chocolate-peanut butter cake from Mustard Seed cafe was tasty: I had two, ok three, slices.

After desert, board games, and being shot at with nerf guns (which have gotten scarily high-caliber), I was talking with my 10-year-old half-brother about his school. Specifically, I wanted to see how he is doing in math. He is (not surprisingly) unable to articulate exactly what he is doing in math, so I was asking him specific questions to see what his math is like. Last year, I was surprised to find out he knew square roots already - I explained about cube and higher roots, and he picked it up instantly.

I wanted to see what he knew about fractions as a 4th grader at a typical Cleveland-area public school. I asked him if 1/2 or 3/4 was bigger.. way too easy. I asked him if 2/3 or 3/4 was bigger. He got it right, and quickly, but couldn't really explain why. I then asked him if 3/7 or 3/8 was bigger, and he said 3/7 immediately. I asked him to explain how he knew, and he looked at me like I was stupid, saying, "a seventh is bigger than an eighth, so..". I asked if he had worked with mixed numbers, and he hadn't, so I asked him to figure out what 3 1/4 - 1 1/2 is. He couldn't do it in his head, so I told him to get paper and draw a picture. That's all I said - he drew fractions circles correctly, crossed off a whole, the fourth, and then another fourth from a whole, and came up with 1 3/4.

I've only ever taught at DCP, so I don't have much of a frame of reference for knowing if he is above average or not, but this is the kind of thinking that all students must have to be successful in high school math. This is the kind of numeracy ability I want my students to develop; this way, when they get to a new problem, instead of giving up, they can reason it through and at least make progress. I struggle daily to get them to pay attention, to care, to think, to not give up when a problem is hard, and their mathematical progress is painfully slow. In a couple weeks, when we start reviewing for finals, and half the kids don't even remember what an integer is, it will be painful. But I know that, by the end of the year, most of my students will have improved their math abilities in many ways. Never as much as I want, but it will have to do! I just gave our grade-level equivalency test before break, and the median score has improved by 1.1 grade levels (from 5.9 to 7.0) and the average by 1.65 grade levels (from 5.76 to 7.42) since the summer. If I can squeeze that kind of growth or better out of them during the second semester, most will be in pretty good shape for next year.

A case study: freshmen's ability to listen...

(Background: cell phones are not permitted. If seen/heard, they are confiscated until parents pick them up.)

{Phone rings; I answer}

Me: C, they need you at the front desk with your cell phone, because your sister got hurt and they don't have your mom's cell phone number.

C: (Looking angry) But I don't even have a cell phone!

Me: They don't want to take your phone, they just need your mom's number.

C: But I don't have a phone!

Me: Just go...

(5 minutes later)

C: Mr. Greene, do you know why they wanted me? My sister hurt her leg, and they needed my mom's number!

Me: That's exactly what I told you.

C: You did?

More decimal and percent work

For the next lesson in Numeracy, I wanted to keep building students' ideas about what percents and decimals are, and how they relate to fractions. I've mentioned before that I am teaching the students to use bar modeling to solve word problems, but I haven't been posting the problems, and some examples would probably be nice. This week, I've started incorporating percents into the problems, which is, of course, throwing the students for a loop. But, they will get it eventually (and some already have), and I think that continually reinforcing the visual connection between percents and fractions is important. So here are the three problems I'm using this week:

Lesson 1

Diego and Dora both took a test in Algebra 1. Diego got 70% of the questions correct, which was 42 points. Dora did very well, and even got the bonus problem right, so she got a 105% on the test. How many points did Dora score on the test?

Lesson 2

By the end of tutorial, Mariana completed 45% of her homework. She spent 54 minutes working (the rest of the time, she was giggling with Gricelda). If she works at the same speed at home, how much longer will it take her to finish all of her homework?

Lesson 3

At the school dance, 70% of the students were girls, and the rest were boys. Ms. Vasquez wondered why there were so few boys there – she counted only 36 boys. How many students were at the dance in total?

We only do one problem like this per lesson (3 lessons per week - block schedule), because it really takes 15 - 20 minutes for the whole process (more when the students are unfocused) to play out. What's nice about these problems and this method is that it naturally connects percents to the work students have been doing for months drawing whole number bars (first) and then fraction bars. Right now, lots of students are still struggling, but I think it's more due to the proximity of vacation affecting their ability to care about math than a conceptual problem. We'll pick up with this after break as we review for finals, and I think it'll go better.

For this lesson, after finishing the problem solving portion, we did another class activity. I gave each group a set of 10 post-it notes with various decimals and percents, all between a pair of consecutive whole numbers. They had to stick their post-it notes to the board (where I had blue-taped up a long number line) drawing arrows with marker to indicate more precisely where the number should go. This only took about 5 minutes or so, and then I had everyone sit back down so we could evaluate how we did. I told them that they would earn 2 team points (whoopie!) for each number in the right place, and 1 bonus point (what can I say? Freshmen love their points!) if they could find a mistake in another team's positioning. We went through team by team, and I asked the class to point out any mistakes. The mistakes that were pointed out lead to additional discussions and modeling, until students seemed satisfied that everything was in the right place.

After doing this with my first period, I wasn't sure if the activity had been all that useful. I asked students if they found it useful (many did) and to share something that they had learned. This lead to some good questions and observations - the key one being that 2.45 is less than 2.5 (I still had the papers on the board from the previous lesson to refer back to) because of the value of each number; similarly, several realized that 2.45 is not more than 2.5 even though it has more numbers in it. Other students said that it helped them see better how a percent and a decimal could both be plotted together on the same line (yes!). So this made me feel better about the whole thing, and I didn't modify the lesson for the other periods.

Now that we've used manipulatives and done a couple of class activities, in tomorrow's lesson, students will be doing a worksheet I put together to get some solid independent practice time. They will start by drawing paper pieces (as in the last lesson) to represent a variety of decimals, fractions, and percents. Then, given a number in one form, they will have to write it in the other two forms. Finally, they have a bunch of problems where they must compare a pair of numbers - written in any form - to see which is larger.

The decimal point's job... where's the one?

This has been a long and stressful week, but it ended well yesterday. The biology classes were on a field trip, so my Algebra 2 class only had about 5 kids in it. I didn't bother trying to teach anything new - I just helped them with the homework, and then we spent the rest of the time looking at different problems they needed help with, and talking about stuff that was on their minds... some "remember when", since I taught most of them as Freshmen - they remember amazingly well things that I said over two years ago, as long as they are not math related, apparently... we talked through some of their fears about going to college... we reviewed fractions... I wish there were more times available to just sit and chat with students.

In my Numeracy classes, the goal of the day's lesson was to learn how the base-10 positional system works, with a focus on the decimal side of things. Specifically, I wanted them to be able to represent fractions and percents as decimals, and to understand how each decimal place relates to a specific base-10 fraction. This lesson went very poorly on Thursday with my first two classes, due to management issues (i.e. they were behaving terribly). Is there ever a point in time when you can get through a week without feeling like a first-year teacher again? Anyway, I didn't rewrite the lesson for my Friday classes, because I was confident that it was actually a good lesson, and worthwhile.. I just tightened it up, and made sure to stay on top of things behaviorally, and it really paid off. The discussions in both periods (and these are my two weaker classes) were quite rich and productive. On my board, I had taped up papers as in the graphic above, and told students that each smaller piece had been made by cutting the previous one up into 10 pieces. I asked how big each was, and they easily saw that each paper was 1/10 of the one to the left. I then asked which paper represented one whole. Some students thought it was the 10 big pieces, some thought it was the single full-size piece, some thought it was one of the smaller ones. They discussed it, and as I kept asking more students what they thought, in each class, several students started saying that any one of the pieces could be one whole. They convinced each other without much of my prompting - it was like magic! I then pulled out my magic Decimal Point cutout, and told them that the decimal point has one and only one job - to determine which shape is worth one whole. I taped the point up on the board (making the full sheet of paper the one whole for this lesson), and we then proceeded to label all the place values relative to one whole. All of a sudden, why the tenths are called "tenths" started to make sense for some students.

I showed what the fraction 71/100 would look like by taping up 7 strips and 1 tiny square in the right columns, and then translated this to the decimal 0.71. We did a couple more examples (I called students up to try some), and then they began asking about adding zeros at the end. Some thought it was ok, and others thought it would change the value. So I had them debate for a while if 0.6 and 0.60 were the same or not. Eventually, they were convinced that they were the same. So, to push them, I asked why 6 and 60 are not the same, which allowed me to show them that the 0's only change the value when they push a digit into a different place value. For each example, I also went back to the definition of percent (a fraction out of 100) to show that the tiny squares must therefore be considered 1% each. This made it easy to see why 0.36 would be the same as 36%, as they could see the 36 little squares on the board (comprised of 3 strips and 6 squares). It also made it easy to explain why 0.237 is the same as 23.7% - you have 23 little squares, and 7/10 of a little square.

I saw quite a few light bulb moments over the course of this lesson. I need to find a way to get my first two periods back on board now... We'll review this on Monday; I hope that, by the end of next week, students will all be able to compare and order decimals, and easily move back and forth between decimal, fraction, and percent representations. That would be quite an accomplishment, and a good way to go into break.