Sunday, August 19, 2007

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.


Jackie Ballarini said...


I love the writing aspect... it forces students to think about and communicate their understanding of the topic (when they tell me they can't write about it, it tells me they don't really understand it - they are just reproducing steps). It also gives me a nice view of where they are with a concept.

In my precalc class, I'm giving them a graph of a piecewise function [made with grapher - sound familiar? :)] and I'm asking them to spend 5 minutes silently working on describing the graph. I'm hoping this will give me a good picture of where each of them stand on day one, in addition to rekindling some of what they forgot over the summer.

Have you seen the conversation going on at Math Notations the Singapore curriculum?

Thanks for the tip on the book!

Anonymous said...

It's great that you keep tinkering. One huge advantage here is that this is the "extra" class, so that any gain is very positive, and opportunities missed don't hurt as much as they would elsewhere. This helps give you the freedom to try things out.

The differentiation sounds like it might actually function... which would be in and of itself unusual. Watch, and tell us.

I don't know about the bars...

Dan Wekselgreene said...

Jackie, check out this post on GeoGebra, as it seems like it can do piecewise functions - and with holes and arrows!

Jonathan, a voice against differentiation? I can't believe it! :) Everyone seems to be pushing for it now, but to me, it raises a lot of complex issues that I haven't seen adequately addressed. Have you had specific negative experiences with it?

Same about the bars - what about them don't you like?

Anonymous said...


Saw the post, sadly I'd downloaded GeoGebra this summer...just haven't had time to play with it yet! Hopefully some time this week. Off to first day of school!

Anonymous said...

"First, I want to incorporate writing and reflecting into the daily activities. "

This strikes me as a terrible idea. As if the kids don't have enough to worry about, now they have to write in a math class.

The best rules to follow when considering a new teaching priority is simple: is it additional, and are suburban kids required to do it? If the answers are yes and no, then it's a bad idea.

I certainly hope you won't compound the error by grading the poor kids on their efforts. At least the math course should reflect the math grade.

I'm a bit surprised, though, that your students need that much work on numeracy. I've worked with students who were well behind their grade level, and in my experience they either grasp the basics quickly or they don't. The ones that don't should not under any circumstances be taking Algebra I, much less Geometry or Algebra II.

Jackie Ballarini said...


Which suburbs? What if the answers are "yes" and "yes"?

Also, you say, "... ones that don't should not under any circumstances be taking Algebra I, much less Geometry or Algebra II." What if these students do not grasp the "basics" quickly? Should they never be exposed to higher mathematics? In the state in which I work, it is a graduation requirement to have at least one year of algebra and one of geometry. How would you accomplish this?

Jackie Ballarini said...

I did not intend for my last comment to sound...confrontational. I'd really like to hear the thoughts of others on these questions.

Dan Wekselgreene said...

Jackie, don't worry - Cal loves confrontation. You can check out some of the other posts on Numeracy if you want to see more. I'm done responding to her posts because it doesn't lead to anything productive.

I'm mainly interested in the pedagogical opinions of math teachers with experience - especially working with underserved / underperforming students. Maybe writing activities are not a good use of time, and I'd love to hear from anyone who has tried it. I've read a lot about how it is a useful tool for improving mathematical thinking, so that's what I plan on trying. I will assess the value of it for my students as the year progresses, and based on that, I will make a decision for the following year.