Along with the rise in expectation of greater student achievement has come a search for effective interventions for those students whom we have failed to prepare for success. After spending 15 years as the Resource Specialist at a small, alternative K-12 public school, I was asked to teach two classes of Algebra at our district’s lowest performing high school. These classes would mainly consist of students from the school’s Resource Specialist (RS) and Special Day Class (SDC) programs, plus a few regular education students who had previously failed Algebra. The hope was that by continuing to act on the assumption that all students are brilliant, I would somehow be able to instill in them the same kind of confidence—belief in themselves--that I had helped to create among my former special education students. I would like to describe what I have tried thus far and also share my concerns and questions.
I started off the year with fewer than five students in each class, and it was hard for me to think about what to do to keep them coming back each day. The truancy problem at this school is huge, yet I was hesitant about beginning a course of study until we had at least 10-15 students. As the weeks went by, more students dribbled in.I noticed that they seemed to be lacking several crucial Habits of Mind:being able to persevere, that is, to struggle with a problem until they figured it out; being responsible for their own learning; and working cooperatively and collaboratively. I knew these attitudes would be essential when I started to teach them Algebra, because I planned to use College Preparatory Math (CPM). CPM is designed to foster mathematical competence and understanding by developing individual problem solving skills via a group process, while encouraging students to take increasing responsibility for their achievement in mastering the subject matter.
What did I do with the few students I had during the first six weeks of school?
1.) The first thing I did when I entered my new classroom was to ask the custodian to replace the individual desks with long tables and chairs, explaining that I was only going to be doing group work. I also found an overhead projector and moved it into my classroom.
2.) When students arrived, I asked them to write a letter of introduction, telling me what they liked about school, what was hard, how they learned best, and something about their math history. I explained that we’d begin using our text once more students enrolled.
3.) I made available units on Fractions, Decimals and Percents by Key Curriculum Press and asked them to browse and find what they needed to work on. (This decision was based on my observation that these three areas of math exploration are often poorly understood, yet very necessary for daily living). I specifically told them not to waste their time doing what they already knew, and emphasized that this class was a place where I would support them in their struggle to learn. As long as they were working, I didn’t pressure them to get a particular amount done, but rather asked them to keep on struggling. I also had simple, four-function calculators available.
4.) I scrounged up some classic math problems which had little accompanying text, such as Magic Squares, in which the students had to place the digits 1-9 in a 3x3 square so that each direction—vertical, horizontal and diagonal--adds up to 15. I gave each pair of students small pieces of paper with the numbers 1-9 written on them so they could move them around both in class and at home, and I insisted they work together on these before showing me their solutions. As I walked around the room, I praised their attempts to figure things out and kept asking students to turn or move their chairs so they were not working alone.
5.) I taught students how to combine like terms by making up problems on the board and having them come up and solve them, and then I had them make up some for us to solve. Everyone was happy to come up to the board. Sometimes I used ‘x’ and ‘y’ and sometimes squares and triangles. The students came up with very creative ideas of their own.
6.) I wrote each number from 1-25 vertically on the board. Then, after we figured out the average age and year in which the students were born, we used those four digits to find ways to get 1-25 by just using the four basic operations, such as 1+9+8+5=23 and 1(9+8+5)=22.It was so great to see them come in and run to the board to write their solutions, and it was also my sneaky way of teaching them Order of Operations; in addition, it was also a way to show different approaches to the same problem.
7.) I made copies of a calendar for September that I had gotten at a CMC conference where each day’s date is the answer to a math problem. I told them that they had until the end of the month to complete the project, which allowed them plenty of time to work on it collaboratively to explain how they got each answer. I never told them not to copy, but rather encouraged cooperation.
8.) I gave students some number patterns and sequences to complete—and this was the only time I asked them to struggle alone and not to work with a partner. The progressions start off easy and then get quite challenging. They really loved this challenge and were eager to show me how they got their answers when I stopped by to ask how they were doing.
9.) I photocopied some pages from pre-Algebra textbooks so that they could practice turning words into algebraic expressions. First, I read the problems and wrote the answers on the board, then the students wrote on the board as I read them, and finally, they worked on the problems in pairs at their seats.
10.) We did a lot of work with Venn diagrams. Sometimes I’d draw circles on the board and have the students fill them in and sometimes I’d give them a phrase and ask them to draw the appropriate circles; we worked with “all...” “some...” and “no...”. This was really hard for them, but became easier when we used numbers. I was particularly pleased when they were able to make up their own categories, like “school” and “fun” being two separate universes.
11.) I made up a rubric which gave 4 points for trying everything, 3 for trying most, 2 for trying some, 1 for trying a few and 0 for trying none. At the bottom of the page is a list of “things one can try,” such as attendance, perseverance, cooperation, respect, class work, homework, preparedness, etc., and a column for each week of the marking period. In very big bold letters at the top are the words: College Preparatory Algebra. (They love it; I told them this did not mean that they had to go to college, but that it would prepare them if they chose to.)
By the end of September, things were going fairly well. The students were learning to struggle and to work in groups, some were doing homework, enrollment was slowly increasing and attendance was becoming more regular. Suddenly, nearly all of my students were removed because another teacher had too few students. Now I was back to three students per class, who were getting tired of the wait for a real Algebra class (with a real textbook) to begin, as was I.
At the end of six weeks I had to give out grades, and decided to assign a Portfolio Assessment because of my conviction that good organizational skills were a necessity for success in this class. I gave each student a folder and asked them to put their pages in chronological order, make up a table of contents and write a summary of what they’d learned so far, including what they’d struggled with. Then, on at least four pages of their work, I asked that they write: “This was easy because...” or “This was hard because....” Though they moaned and groaned, they did a beautiful job. They kept asking, “Do we have to?” and I kept saying, “Yes.”
By the seventh week of the semester--the beginning of the second marking period—the students and I were pretty frustrated about not having real Algebra class, so I camped out at the counselors’ office till they agreed to program more students. They were given a list to draw from which included all RSP and SDC students who scored at least 4th grade on a math test—though some had reading scores well below that.I also went to each and every math teacher and asked if there were any students who they thought might benefit from an Algebra class taught by an experienced special education teacher. (I will NEVER do that again with a staff I don’t know because they “dumped” some students on me that they just wanted to get rid of.)
The new students and textbooks arrived at about the same time that I found out I needed to have surgery that would keep me out of school for several weeks. That meant that a substitute would be taking over for a while, and I desperately wanted to set the tone for the CPM before leaving the classes. I knew that establishing rituals and getting them into a rhythm would be crucial to their success. Luckily, the teacher who shared my job (already working in my stead one day a week) agreed to work during my recovery, so we set about planning how to support our non-believers and make them into converts, into believing that they could succeed in Algebra.
College Preparatory Algebra for Special Education Students
The very first thing I did was to go to a recycling place and get a whole lot of sturdy dividers and make three holes in them. I also dug up old binders at our district warehouse. Every student was given a binder with dividers and told what to write on each: textbook, class notes and tool kit; class work and homework; tests and quizzes; graph and lined paper. They were then called up to get their Algebra books. I made them sign a paper that said, “I understand that I am getting a brand new CPM Algebra book. I promise to take good care of it and return it at the end of the year (or pay for it) in order to get a grade.” Everyone signed it and to this day, only one student lost his book.
We started off very slowly. I insisted that they put their name, date and problem number at the top of each page and then put it in the correct section of their binder when they were through. I also insisted that they bring their binder (which had the Algebra book inside) and pencils daily. I worked very slowly with them, reinforcing any and every positive thing they did, and praising their struggles to the sky. Frederick Douglass looks down on us from a poster above the blackboard, saying, “Without struggle, there is no progress,” and I point to him often. As they work, I walk around the room helping them to get started and encouraging them to keep going. When they get stuck, I listen to their doubts and remind them they are brilliant and capable by pointing to some work they have already done.
1.One of the first things we learned about was the diamond problem.This is a prelude to factoring, where the two open spaces at the sides of a large ‘X’ have numbers in them—say a 2 and a 3—which are then multiplied to produce the product (6) which is written in the top space, and the sum (5) written at the bottom. The students did well on these. From there we covered the following topics:
2. Reading and creating graphs. Most of them had no trouble at all, though some of the scales were backwards, as were some of the answers in the diamond problem, but many learning disabled students have spatial orientation challenges.
3. A big challenge for all of them was setting up Guess and Check tables, CPM’s way of introducing a format for translating word problems into tables which eventually leads students to write and solve equations. The beauty of this approach is that the students are using a process that will not let them focus on the answer until it emerges from their guesses. Even the most reluctant/terrified student was able to perform the final “guess” on his own after getting help with setting up the table and making the first few “guesses.” It was so easy for me to reinforce their struggles because they really wanted to be able to solve problems that allowed them the luxury of using their minds to figure things out.
4. One thing that we did that was new for me was probability. I have no recollection of ever learning this, so I was a little intimidated, but we worked out a system that was quite successful. Every day the number of students in attendance is different, so each day I can ask the same question:suppose I put each of your names in a hat and picked one. What is the chance of a female student getting picked to be the $1,000 winner? A male? What would my chances be of getting the money? I had them raise their hands without calling out and just kept on repeating the question till nearly everyone raised their hand. They were actually able to grin with pride about knowing the answer. There was one mishap, however, on the day we used dice--which I attribute more to teacher naiveté than to anything else—it was hard collecting them at the end of the lesson.
5. I also ask students to write a lot about what they are learning about themselves, about working in a group, etc.; and while their writing needs much improvement, they are becoming better at expressing themselves.
6. We started using Algebra Tiles--manipulatives which help students learn about variables--to deal with combining like terms—including both adding and subtracting; they didn‘t seem to have much difficulty, and all the pieces were returned.
7. Just before the winter break, I decided we needed to solidify what we’d learned before moving on, and I also wanted them to be confident of themselves, so I put ten pages of problems from the book’s assessment bank on the board and told the students to look at them and decide where they were still having difficulty. They were to get that sheet from my desk (I had copies of each) and work on it with someone. By the end of the week, most of the students were ready for their “test.”
8. I gave two tests: they could either choose to do 10 out of 15 problems or 8 out of 12. I gave them the choice. Of course, some did really well and a few did poorly, but the test was hard, and I am so proud of them I could cry.
It is now the end of December. For the most part, the majority of my students are coming to class regularly and are engaged for the entire period; even though I was unable to walk around and help them following my surgery, they managed to learn. Unfortunately, due to a high truancy rate, moving forward with the whole class is hard, as is encouraging group work, since the groups differ each day.The most important thing that is happening is that these students are learning that they can learn. They are developing good Habits of Mind. Some do homework and have learned to work cooperatively. Some have let me call their homes to brag to parents, grandparents, guardians et al, about what great students they are. In all, I believe they are moving in a positive direction, and I believe that many will actually get good grades in this class.
Much of what is happening is because of the strong belief that my teaching partner and I share regarding our mission:we teach children, as well as math, with an emphasis on the learner and the learning process.We also have had the help of a wonderful Instructional Assistant who has quickly adapted to our way of doing things and is very supportive of the efforts made by our students. We believe it is our job to get them to engage in the struggle to learn that they are brilliant and capable of mastering the material, and that any behavior to the contrary is just leftover stuff from many years of failure, criticism and lack of confidence. We just persist and never give up on any of them.
I am learning as much or more than the students are, and still have lots of questions
1. What are the big ideas in Algebra? Can these ideas be taught at all levels of school or only in an ”Algebra” class?
2. What math do students really need to know for life? I distinguish this from the new High School Exit Exam, which I don’t consider to be part of life’s necessities.
3. How can we best ensure that our students learn the most important lesson—that they can learn?
4. Does it make sense to require Algebra for all students? If so, when should it be taught? Who should teach it? What is a good student/teacher ratio? Is there an optimum teaching/learning situation?Developmental level? Knowledge base? Attitude? Organizational competence?What kind of support is needed to insure success? How do we provide it?
5. What should be taught in an Algebra class? How long should that take?Should this decision be made by publishers, politicians or teachers--who have a sense of their students’ readiness to move ahead?
6. How can we ensure that our students are learning a unified field of study, a coherent system, rather than the piecemeal approach, which seems to be so widespread? How can we improve the articulation of mathematical instruction so that it makes sense to students?
7. What is our overall goal? What is our priority?Do we want students to pass tests? To learn the beauty and excitement of struggle and engagement? To succeed in life? Should we ever make curricular needs a higher priority than the needs of our students?
I am a veteran teacher, having spent more than 30 years working with “labeled” children—those in special education, who are rarely taught much of anything, least of all, that they are brilliant and capable of passing a regular math course.As a first time Algebra teacher, I am in the process of learning, and would welcome suggestions from anyone who would like to share strategies or concerns.
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