Numeracy Curriculum

I finally got around to posting my old Numeracy curriculum on my box. This is by no means a final draft of what I think our 9th grade basic math support class should look like, but it is where I left it last year. I didn't decide to start using Keynote presentations until the third unit, so that's why there aren't any Keynote files in the first couple of units. This year, I am not teaching this class, so I haven't had the opportunity to keep developing these lessons. I'm sure I will get around to reworking this stuff at some point. For now, I'll throw it out there for people to look at, borrow, critique, steal, and so forth. I hope someone finds it helpful.

A winning review activity

My students always complain that we don't play enough games in class. I know they love games, but most of the games I've seen are quite ineffective. "Showdown" is one of my favorite review activities for my older (more mature, more motivated) students in Algebra 2, but it doesn't work so well for my freshmen in numeracy.

To many of my students, tic tac toe is a riveting activity to be played surreptitiously during a dull lesson, or after a test. I thought I would capitalize on that, and so I present to you:


The students are broken into pairs that collaborate against other pairs. Each group of 4 is given a game board with several tic tac toe grids on it. One pair picks X and the other gets O. Turns alternate with each problem. When the problem is shown, both teams should work on it. If it is X's turn, if they are right, they get their square. If they are wrong, and the O's are right, the O's get to steal a square. (Students took a while to get this - at first they all thought it was unfair). When a game is won, the winning team gets a point. At the end of the activity, whichever teams won more games get a prize.


The benefits:
- All students are engaged on every problem. Even if it's not their turn, they can steal if the other team is wrong.
- Students have a partner to collaborate with, so weaker students are not put on the spot and can learn during the activity.
- Pairs monitor each other for cheating - they can only get the square if they've shown their work.
- Tic tac toe is the funnest game on the planet. Apparently.
- Generic mechanical pencils in fun colors come 30 to a pack for $5. Great prizes! Mini candy bars work too.

Enjoy! Let me know if you play it and it works (or doesn't work!) for you.

Prepositional Nightmare: Anywhere a Cat Can Go

One of the problems with block scheduling is that, when you lose a day of school, it throws your whole system off. Due to community day on Thursday, and spring break starting on Friday, periods 1 - 4 met twice this week, but 5-6 only met once. So, it was time for a slush lesson. Sorry, I mean "enrichment". I find these hard to do well, because if it is something worthwhile - such that you can justify spending 80 minutes of time with periods 1 through 4 - then you want the other periods to see it to. And if it isn't worthwhile, then why not just have a pizza party or something? But you'll never catch me throwing away a lesson like that. There's just no time to waste.

So I decided to experiment with correcting a linguistic problem that bothers me, but is not necessarily mission critical. That is the reversal of terms when saying division and subtraction problems out loud, confusing divided by with divided into, and my personal favorite, "subtract 7 to both sides". I know that part of the problem here is the somewhat arbitrary nature of prepositions, and I've been told that fluency with prepositions is one of the last things to develop when a person is learning a new language, and can take many years of practice. When students make these mistakes in class, I tend to repeat their words back to them, using the correct language, but not making a big deal out of it. My thinking here was that I could do a lesson on it, and then, when they make those mistakes in the future, I can just say "remember the correct way to say that?" and jog their memory, instead of launching cold into an explanation again and again.

I did the lesson. Nothing fancy - just some explanation, some practice, a little board wars (which I typically shun, but it's a slush lesson, so what the hey) and some delectable Easter candy prizes. Yesterday, the students were pretty non-enthused about working on prepositions (shocking, I know), and board wars was so-so, although there were quite a few kids who were very motivated to win the giant bunny lollipops. Today, I had some pig- and ducky-shaped candies to give away, and I think I struck gold, because the minute I showed them to my class, they freaked out and got super-focused. I don't really like bribery, but I think it's probably ok to break form on the day before vacation.

In any case, we were well into the first round of board wars when the phone rang. When I picked up the receiver, I heard some students say "Mr. Greene, we're in Algebra class right now and we have a question." I was pretty confused, until their teacher came on the line. He had them on speaker phone, and said, "My students are telling me that I'm not speaking like a mathematician." (Speak Like a Mathematician is the phrase I use with them for all matters linguistic.) They were all laughing in the background. I finally got what was going on, and said, "Hold on, let me put you on speaker phone here." When I did that, his class erupted in a cheer, which my class could hear, and they were shouting hellos back and forth (although nobody knew who was in each class). They quieted down, and I had them ask me the question - it seems that their teacher said "subtract by 7", and not only did they notice the mistake, they had enough confidence in themselves to call him on it. So I settled it for them, all the kids shouted goodbye to each other, and we went on to an excellent board wars competition.

Later, when talking about it with the other teacher, he told me that he had actually read a problem that said "reduced by 7", but the students swore he said "subtracted by 7" and he decided to play it up for them and call me since he knew I'd been working on it with them. Moments like that are really cool (and potentially powerful), and they can't really be planned out. I love when the last class before a break is a really good one.

p.s.
Does anyone else remember the phrase "anywhere a cat can go"? I still remember it from 7th grade French.

p.p.s
Funny cat videos. My classes loved these for the physical humor. But if you've ever had a cat, you'll see that the cartoonist captures their behavior really well. Enjoy!

Don't tell, but I learned something on YouTube

I've been using Keynote this semester as an experiment, to see how it could work in my Numeracy class. So far, it's gone pretty well - especially after I bought a remote mouse so I could control it from anywhere in the room. Combined with the mini-whiteboards, it's been a really efficient way of getting students to do work. After presenting a concept, I can have them practice a few problems right away by showing the next slide, and having them work on their boards. There is no time wasted passing out worksheets. Also, I can make sure all students are focusing on a specific set of problems (versus on a worksheet, where they tend to start jumping around right away, based on what seems easiest). Then, I can show work/answers on the slide without having to pull out a transparency.

Since I've got the projector reserved and set up now, I can easily insert fun and interesting images, sounds, and video clips. I've recorded myself and other teachers singing little ditties (like the infamous "Don't add across"). I've started scouring YouTube for interesting stuff... though the ratio of total crap to interesting stuff is quite high, I've found a couple of gems. I even unearthed my old calculus professor from college, who recorded a "top ten algebra mistakes hit parade" as well as "all of calculus in 20 minutes".

So I'm in my fraction adding unit now, and we've been working with fraction circles to understand adding. Now, we're taking a break from that to do some work on prime factorization, reducing fractions by canceling common prime factors, and finding LCM. Once they get all this mastered, we can go back to adding fractions using common denominators. I hope they don't forget it all over spring break... I've always found it difficult to teach factors and multiples, and GCF and LCM because students confuse these concepts very easily. Part of the problem is their difficulty with the language of division. Just about every student I have says "divide 6 by 40" when they mean 40÷6. If I ask "does 3 go into 12?", they'll say yes. But they'll also say yes if I ask "does 12 go into 3?". (Aside: I think I'm going to devote an entire lesson to this issue - along with the whole "subtracted from"/"subtracted to" issue.)

In any case, I YouTubed LCM and GCF to see if there was anything interesting out there. I was surprised to find a method for finding both LCM and GCF at the same time using Venn Diagrams that I'd never seen before. It's mathematically equivalent to looking at the prime factorizations and picking the right factors, but it provides a nice structure for students to remember which is which. So I designed a lesson to practice finding factors and multiples, and then using this model to find LCM and GCF. It went quite well. I don't know how much will be retained over the weekend, but we'll practice more on Monday/Tuesday because I want them to have LCM down solid. Here are two of my slides, and then the original video I got the idea from.

Math & Art: Big Numbers

Check out this site. Really cool images.

"Depicts one million plastic cups, the number used on airline flights in the US every six hours."


"Detail at actual print size:"

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.)

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.

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.

Happy Tofurkey Day

If you've read my blog in the past, you may have noticed I haven't been writing much lately. I've had some bad health problems over the past few months, which have made it difficult for me to get anything but the essentials done. It's finally under control, and I am feeling relatively human again. I am thankful for strong drugs and medical advances!

So, my lack of energy along with the typical November doldrums have made my freshman Numeracy classes less effective and positive than I'd like. I hope that this mini-break will give us all a chance to recover a bit, and come back to end the semester strong. Here are some of the major things I need to work on:

1) Multiplication Tables.
I decided not to focus on teaching the tables this year to the whole class, as it is a waste of time to the 1/2 to 2/3 of the kids that know them. And, I'm not sure how to do it really effectively for those that are in high school and still don't know them. ALEKS doesn't deal with multiplication tables.

I've given students who need them 12 x 12 tables to keep in their binders and look at as they do things like reducing fractions. I need to find a way that they can actually work on improving, and I think this will be different for each student. Some combination of flash cards, games, incentives, and quizzes will be needed. But how to work this seamlessly into the class? Hmm... Needs more thought..

2) Classroom Culture / Readiness
I've posted about this in the past, and my readiness checker was working very well. But, recently, class hasn't been getting started very efficiently. The students recognize this, and freely admit that, now that they are well into school and have made friends, there is a lot more temptation to hang out between classes and avoid getting into their classroom or seat in the room until the last possible second.

I don't sweat it if we miss out on the first minute or two of class, but more seriously and annoyingly, school supplies seem to be growing ever more sparse (kids don't have pencils or binder paper, and act shocked when I ask them to take these things out!). It's like they buy some new stuff in September, and when they're out, that's it for the year. Also, binders have gotten to be a mess (i.e. holy terror) again - and that's for the kids who still have binders.

I need to come up with an incentive system to get this all back on track - and a way that forces me to stay on top of it.

3) Problem Sets - Singapore Bar Models
I overshot greatly with my initial stabs at assigning students problem sets. My goal was to get them started on longer-term planning, while assessing their ability to use the bar model method to solve word problems. My initial idea was to assign ten problems on Monday, due the following Monday. I would grade and return them, and they would have till the following Monday to do revisions to increase their scores. Sounds reasonable? Well, it was still way too much for them to handle. The second time around, I made mandatory progress checks during the week, to help keep them on track. I got more turned in the second time around, but they were still not very good. And, there is a big problem with cheating. Unfortunately, students don't really understand all the time what cheating is, and it's hard to get them to see it. I want them to work together to help each other; many of them think that copying someone's answers who is "helping you" isn't cheating. This is going to require a lot more coaching. Here are my initial ideas for how to run problem set #3:
- Reduce the number of problems from 10 to 6.
- Continue with the progress checks, for regular homework credit.
- The day the assignment is due, give a one- or two-question quiz with selected problems from the assignment, but with numbers changed. This will help me see which kids actually understand the work, and which kids copied.

Don't worry, I'm not dead...

My Numeracy students are now about a month into their ALEKS experience. I started all students out on the third grade standards level (the lowest ALEKS goes), and on average, my students scored around 50% mastery on their initial diagnostics. At this point, some of the students have completed level 3 and are onto level 4, and many others are close to completing the level. There are the stragglers too, of course. I'll do more detailed stats later on. The goal I've set with students is that they should try to complete 3 entire levels by the end of the year (i.e. 3 years of growth in math ability). I was skeptical at first, but after seeing how the students interact with the program, I have much more hope. ALEKS is not a creative, fun, snazzy program. Essentially, students get a sample problem to try. If they don't know how to do it, they read an explanation and try again. When they get a certain type of question right 3 or 4 times in a row, without asking for help, the concept is added to their pie chart. Periodically, they are re-assessed by the program, and concepts they no longer know are pulled back out of their pie chart.

I have been impressed by how self-reliant the students are being. They are managing to read the explanations and figure out the problems on their own. Some students are really getting into it, and are bragging to each other about how much of their pie they have completed. They have also figured out that getting a problem wrong, or clicking on the "explain" button causes the program to require more correct problems to add the concept to the pie. For that reason, they are actually trying harder to get the problem right the first time. The immediate feedback has been very helpful for the students. My favorite moments are now at the end of class; sometimes, when I tell students they need to log off, a few will be like "oh wait, let me just get this one last problem so I can add it to my pie".

Right now, I am just assessing them on time spent on ALEKS - not on the actual amount of progress being made; it seems to be effective enough, and the whole point is to allow students to work at their own pace. We'll see if I need to modify that policy in the future.

On a different note, we have been working on bar modeling to solve word problems every class for 15-20 minutes. I assigned the first problem set as homework last week, and I graded them this weekend. They were quite bad. It's always a bad feeling when you realize your students are a lot farther behind than you thought. I've pushed ahead into more complicated problems, but I just realized that many students are still having trouble with the basics. That's ok.. we'll just cycle back to the beginning and have another go at it.

In Algebra 2, we've started in with the basic idea of logarithms, using the Big L notation I wrote about in an earlier post. I think it is working well. We have been focusing on the similarities between roots and logs: in a root, the index tells you the exponent, and you are looking for the base. In a log, the subscript tells you the base, and you are looking for the exponent. Last year, many students had trouble in power expressions determining when to use a log or a root; I think they will have a much better understanding of it this year.

Are you ready for this?

I quickly mentioned the Readiness Checker idea a few posts ago, as a new idea I got from another teacher. I want to revisit it here, now that school has been under way for a few weeks. I have to say, it is simply excellent. In the past, I've done a variety of readiness checks, with various degrees of success. This is working better than any other method I've tried.

So here's how it works. To be ready, students must, *by the time the bell rings* do the following:

• Take out binder


• Take out homework (and Readiness Checker)


• Have a pencil sharp and ready


• Put backpack in back of the room


• Begin working on the Do Now

If all of those things are completed, the student earns a sticker or stamp on their Readiness Checker. When the checker is filled in (I currently have 9 spaces on mine, but will probably extend it to 12 for the next round), it turns into a "get out of homework" pass. This has the benefit of putting a nice positive incentive on being ready for class, no negative consequence, and it is not directly tied to points in the grade.

It has been working like a charm in my 9th grade Numeracy classes. Most of the time, I have at least 3/4 of the students earn a sticker (often more), which lets class start quietly, focused, and on track. Of course, there are off days (like Friday afternoons), but overall this has been a fantastic new innovation. If you teach students that have difficulty getting started, I highly recommend a system like this. A couple of our 12th grade English teachers are doing this too (I was surprised, but they say the students love it.) Best of all, there is no added management on your part - if the student loses their checker, give them a nice new blank one. (So far, though homework sometimes "gets lost", I haven't had a single student lose their Readiness Checker. What a surprise! :)

Another discrete learning moment

In Numeracy, we have so far been working on two concepts: solving word problems with the bar model method and adding integers with and without manipulatives.

The bar model work has been quite interesting, and I'll post more on how it's going later. Adding integers has gone pretty well, as it is not that difficult of a topic for most students. The hard part, as always, is breaking students of their deeply ingrained habits of wanting the "rule" or the "shortcut" that will let them solve the problem faster. They can't seem to figure out that they have learned these rules again and again over the years, and that they haven't stuck yet. And, even though they may think they know the rule, they might not. Several times today I heard "a negative minus a negative is a positive". But I digress...

Today, I began integer subtraction in two of my classes; subtraction is, of course, much more difficult for students to master. In my first class, there was a lot of buy-in. First, I showed them how to do problems where the second number is smaller in magnitude than the first number (8 - 5, -6 - -4), which is easy to show with unit cubes and an integer mat. I like showing how the second example is no more difficult than the first when you understand what you are actually doing. Then, things got really interesting when we moved to problems like 6 - -4, 4 - 7, and -4 - -9. I showed them why and how we add zero pairs to be able to subtract. After I went through it once, a couple eyes lit up. After the next problem, a couple more. And after the third, a few more. I could actually witness students engaged in the act of finally learning a concept. This is one of the joys of teaching basic math to older students. One of my repeating students raised his hand and said, "I don't get it. Why is this so easy? Last year this made no sense, and now it's easy." I think I was able to convince him that the fact that he was paying close attention throughout the lesson was the answer to his question (I didn't teach him last year, but I know he almost never engaged in his class). I'm not sure if this meta-knowledge will stick, but if it does, I think he may now be set to finally learn some math and pass algebra. For sure, when he does lose focus in the future, I'll remind him about what he discovered today.

But with all successes come setbacks (I didn't say failure! I must be getting less cynical). In the next class, the lesson did not go over so well. A couple kids showed me the bright-eyed look of victory, but most were just playing with their cubes. I think I need to invest in unit cubes that do not lock like legos... Some of the students know the "rule", and though they don't know why it works, they wanted to keep using it and not try the blocks. I wouldn't mind it so much (for the few who really do know how to use the rule), except that it prevents students who don't know the rule yet from seeing the value in using the manipulatives. It's like creating a short-circuit. I have two more classes to go on this lesson, so we'll see how the others react. I am still getting a feel for the different character of my different periods, but certain patterns are already surfacing.

Writing in math rocks my socks

I handed out and used the reflection journals in my Numeracy class for the first time today. Of course my first class was too fast for me and half the covers were tagged before I could even react; but I learn fast and outlawed tagging script for the rest of the classes. "Aw man, no tagging??" But they listened for the most part..

I know countless other teachers do the reflection journal, but it isn't seen as much in math. And I've never tried it before. The kids were a bit unsure what to write, and I was a bit unsure what to tell them. My prompt today was something like "write about what you learned today in class and what you feel that you still need help in". I also told them they could choose to follow my prompts, or write something else. General ideas: what are you understanding? What are you confused about? How did the class go? Is there anything you want or need to let me know? I did confirm for one girl that the writing should, indeed, be at least tangentially related to what is going on in class. She seemed to find this reasonable.

So far, I am totally into this. The last 5 minutes of class are silent, as kids process what they just learned, and think about what they still don't get. At the end of the day, I read through 3 classes worth (~60 students), in about 20 or 25 minutes, and responded to what they wrote. The immediate feedback was awesome. Most found the Bar Model method long and seemingly difficult, but they almost all conceded that it helped them to understand the problem better and make it easier. The kids who were totally confused let me know. One girl said she was proud of herself for having learned the new skill. Another told me that I talk too fast sometimes but that she thinks I'm going to be a good teacher anyway and is looking forward to the year. One boy told me his stomach hurt from lunch and that he needed to use the bathroom (he's in Numeracy for the second time - but I dig his sense of humor).

I'm going to try to commit to reading their journals every Friday at least.. I think that the more I write, the more they are likely to write to me.

I collected their math autobiographies today (only 3 or 4 kids didn't do them!) and I am looking forward to reading them later on. I'll probably post a few choice excerpts.

Wha?

The first day came and went in a blur. We started the day with a special assembly, in which all the departments came up with a little skit to present themselves to the students. There was a lot of energy, and the kids had a good time. The math department came out as if we were doing an encore to a show - we brought costumes and real instruments, and rocked out to "Cult of Personality" (i.e. Cult of Math Ability). Then we introduced ourselves, including our new stage names. This year, I am "Bass 10". Hah!

I saw 2 Numeracy classes today (80 minute blocks), plus SSR and homeroom classes. Lots and lots of freshmen everywhere. They are always so different in the first week or so of classes: silent and afraid to stand out as their brains are processing all of the new social cues and are trying to make sense out of their new world. Or something. I promised I'd know all of their names by the end of the week, but that may be pushing it.

In Numeracy today, I asked them to start by brainstorming around MATH - what images does this loaded word bring to mind? As this was the first day, discussion was hard to draw out of them. Someone would finally mention fractions, and I'd say, "Is there anyone in here who doesn't really like fractions", and every single hand would suddenly shoot up. See, you guys do have things to say! Then, I handed out a math survey to try to measure their self-perception and self-confidence, among other things. We'll repeat this at the end of the year so I can see what changes have been made.

Then, I handed out the Math Autobiography assignment. It had some questions to get them thinking about their math experiences thus far, and they need to write a full 3/4 sheet autobiography for homework! I'm sure there will be some really interesting ones, and I'll post them once they come in.

We burned through class rules and expectations as fast as possible, and then I began the whole-class unit on integers. (Differentiation will start later on, once I figure out when our laptops will be coming in.) I introduced what integers are (and we talked about applications like money, position, time zones, temperature), and then I showed them how to use unit cubes and an integer mat to model integers. We also learned about zero pairs, and how to simplify integer mats by removing zero pairs. I feel like not rushing things is a good plan. Spending a few lessons really scaffolding integer addition, I think, will pay off in the long run. The kids did a good job with the manipulatives, but reading directions is going to be (as usual) a constant challenge.

I'll try to keep posting about what is happening in Numeracy - though maybe not full lesson plans. If anyone has questions about details, always feel free to leave a comment.

Leer es poder!

In preparing for Numeracy this year, I've been reading up on the Singapore Math curriculum and philosophy. They really seem to know what they are doing. Everything has a logical, mathematical reason, and it all fits together neatly. I just read through two books which I highly recommend for anyone teaching primary level math to high schoolers. In just under 2 hours of reading, I've gotten some really good and practical ideas - both big and small picture.

Handbook for Primary Mathematics Teachers

8 Step Model Drawing

I read this next one a while ago, and have recommended it on this blog before, but it bears repeating. It is a fascinating comparison study of teachers in the US and China, and what kind of mathematical knowledge and ability is required in order to teach primary math.

Knowing and Teaching Elementary Mathematics

What books do you find useful/enlightening/interesting with regards to teaching math?

Numeracy 07-08: Project DI

If you're a new reader of this blog, it's important to know a bit about my school, which serves students 9th graders whose average grade level in math is 5th grade, and we seek to have all students ready for 4-year college in 4 (or sometimes 5) years. All students take Algebra 1 when they first arrive, even if they have "passed" it in 8th grade - which many have. The majority of the students also take the Numeracy class, which I have written a bit about before, though I mainly wrote about Algebra 2 last year.

I wrote the current Numeracy curriculum 3 years ago, but the students have not gained as much as they could from the class due to the wide range of skills (and deficits) they bring with them from middle school. My plan this year is to take some of the best elements from the old curriculum, but to differentiate instruction. The units in the old curriculum went like so: Place value and addition/subtraction facts; multiplication and factors; division; fraction concepts; fraction addition and subtraction; fraction multiplication; fraction division. The first three units were not that useful for about half of the class, while they went too quickly for the other half. No one really got what they needed. So here is the plan for the new year (revised a bit from what I sketched out in an earlier post):

All students will spend the first half of the 80-minute class working on a mandatory curriculum. This will start with a unit on integer operations, and then will move on to fractions/decimals/percents. There will be heavy use of manipulatives to ground the students in concrete understanding of the concepts (which is what they lack the most), but I will also move them to algorithmic proficiency as quickly as possible. I'm going to try this year to focus more on representational fluency between fractions, decimals, and percents, instead of teaching them sequentially.

The second part of the class will consist of shorter units that target specific skills: multiplication/division facts; multi-digit operations; place value; rounding; multiplying and dividing by powers of 10, and so on. But here, students will take a quick diagnostic before each unit. Those who need the help will work with me during this portion of the lesson. Those who don't will now work with ALEKS; this software is totally individualized, so students can choose to work on whatever skills they need most help with - and are ready to learn. This will allow me to focus on the weaker students, and to provide them with a conceptual foundation for whatever the skill is, as ALEKS is really only good at providing practice with procedural fluency. I am also looking into the possibility of having students work on ALEKS as their homework, instead of doing worksheets. This will depend on the percent of students who have ready access to the internet, and if I can make the computers accessible to them during tutorial. But if this works, and I don't need to assign and check worksheets every day, that will be a huge time saver for us all.

In addition to this differentiation scheme, I plan to add in two other key components. First, I want to incorporate writing and reflecting into the daily activities. Our students even worse at explaining their work than they are at doing it! We decry their inability to explain and justify what they are doing, and to see how what they are learning connects with their other classes, the real world, and their future - and yet, we never really give space in the curriculum for them to improve at this. I read the book Writing to Learn Mathematics by Joan Countryman; it is a slim little volume, but it has a lot of good, practical suggestions. I'm going to start by having a daily 5-minute quick-write, where students respond to a prompt (or can write about something they learned or still have questions on), and then a longer journal entry every couple of weeks, where students are asked to explain mathematical concepts in more detail. I plan on reading these journals every weekend, and responding to as much as I can. I hope that this will help the students make more powerful connections, and help me understand better what they are really getting (and still needing) from the class.

Second, I plan on teaching students the bar-modeling method for solving word problems that is used in Singapore Math. If you look at some of the problems that 6th graders are expected to do in this curriculum, you'll see that many of our high-schoolers would have trouble doing them efficiently (or at all). I think the bar-modeling method is simple and powerful, and that it will be a tool my students can really use. I've purchased the series of primary math workbooks (and their series of challenge problems), and I plan on adapting these to fit my classroom. My plan is to spend a few days at the beginning of the year (before the differentiation kicks in) teaching this method with simple addition, subtraction, multiplication, and division problems. Then, as the year goes by, students will be assigned a "problem set" (in addition to their daily homework) that will be collected and graded every other week. This will give them a chance to practice the foundations of problem solving, as well as multiple chances to meet a longer-term deadline. I expect that many students will wait till the last minute for the first few assignments, and will learn how to better plan as the semester goes by.

So this is the general plan. I am interested in hearing any comments, questions, concerns, and suggestions as I embark on this new stage of my teaching practice. I don't pretend to have it all figured out - I just have a lot of ideas and a lot of hope that this will come together and help my students really, finally learn some good math.

Why should I learn math? (Take 2)

I'm back, and beginning to plan for the new year. There are going to be lots of changes this year as we revise our program. I will be working primarily on Numeracy again, which will be totally overhauled as I plumb the depths of differentiated instruction. I'll also continue to work on Algebra 2; however, we have decided to stop having a separate honors class. I will be collaborating with another teacher to create a rigorous class that our target student can pass (if they put in the work), and that also provides academic opportunities for those who want to dig deeper and prepare for pre-calculus. I'll write about both of these challenges more in the upcoming days. For now, I am starting to think about setting up my Numeracy classroom and what sorts of things I want to get up on the walls. Most of the stuff you can buy is tacky and uninspiring to students. I had the idea to create a giant butcher-paper poster with the title "Because" (in huge letters) "you can..." (in smaller letters), and then a series of hand-lettered-by-sharpie, colored-in answers to the implied question. Here is the list I've come up with so far. Any suggestions for additions or changes?

1. Design video games
2. Defy negative stereotypes
3. Become a doctor or nurse
4. Avoid getting cheated
5. Know when politicians are lying
6. Make stronger arguments
7. Graduate from high school
8. Manage your money better
9. Become a forensic scientist
10. Design and program computers
11. Help your family and community
12. Get rich in the stock market
13. Solve challenging problems
14. Show the world how smart you are
15. Get a college degree
16. Become a better thinker
17. Increase your opportunities
18. Become a teacher
19. Understand statistics
20. Become a lawyer
21. Study criminal justice
22. Design bridges, cars, and buildings
23. Run your own business
24. Discover a cure for cancer
25. Help your children with their homework
26. Fly airplanes
27. Run for Congress
28. Start a new school
29. Study psychology
30. Make your family proud
31. Change the world
32. Believe in yourself