# 3-Three-3, Differentiating an Old Fashioned Math Worksheet

Math Girls

A great way to differentiate math when you are stuck with an old-fashioned math computation worksheet is to not require its completion.  Instead ask the students to simply choose the three easiest problems and the three most difficult problems from the worksheet to solve.  Be sure to ask them to explain why they found the first three to be the easiest and why the second three are the hardest for them.  It is also important to ask students to make up three additional problems that would fit with what is on the worksheet, but that are not included – and of course ask them to explain why they think they would fit.  This approach gives you a lot of information about each student’s mathematical thinking, their ability to solve the type of computations being focused on, and their understanding of the concepts surrounding the problems.

I found that having these questions and available workspace on a separate piece of paper was beneficial in keeping the students focused on what I was asking of them (some students still wanted to complete all of the computation problems and that was okay with me, as long as they completed the assignment first).  To do this I created the 3-Three-3 worksheet and found that I could use it in a great number of situations across the math strands that I was teaching.  If you would like to give it a try, I have included a link to a full-size copy of the worksheet in a pdf format.  Use the comment section to let me know how it works for you.

# Independent Learning Projects

A collaborative independent learning project in the form of a puppet play being presented to the class.

In my intermediate multiage classroom when students finished their work in a couple of self-paced areas, they were eligible to do an independent learning project. Students loved doing these projects perhaps because they had so much choice in their creation. They decided on the topic they wished to learn about, the product they would produce, and the means of sharing what they’d learned. I did not grade any of these projects but I did support and encourage the students.

Below is a copy of my Independent Learning Contract. When students were ready and had an idea of something they would like to learn, they had a brief conference with me to discuss it. During that conference I helped them refine their idea and begin filling out the contract. They then took it home to discuss the project with their parents, bringing it back to me signed and with any changes. I would add my signature and give any assistance they needed in gathering materials and in getting started. After that they worked on their project independently at appropriate times during the day, both at home and at school. I checked on progress from time to time and when the project was complete we would set up a time for it to be shared with the rest of the class.

Independent Learning Contract

name: __________________
date: _______________

______________________________________________________________

______________________________________________________________

______________________________________________________________

2. Possible resources you will need (may include places to visit, people to interview,
books, computer databases, etc.).
______________________________________________________________

______________________________________________________________

______________________________________________________________

3. How will you show what you’ve learned?
______________________________________________________________

______________________________________________________________

______________________________________________________________

When will your work begin? __________

When do you think it will end?__________

When do you wish to present your finished project to the class? _________________

______________________________________________________________

_________________
student signature

_________________ _________________
parent signature(s)

_________________
teacher signature

# Students’ Daily Schedule

When I team taught a 2nd and 3rd grade multiage class, my teaching partner and I realized how important it was for students to know where and when to be at certain places and for them to have a general idea of what they needed to have with them. To help with this every morning we had our entire class sit down and go through the day’s schedule. All of our students used the “Daily Schedule Form” to write down/fill in/check off which activities they would be participating in for that day. We found that this little piece of organization helped tremendously. Not only did the students know what to do and when, but they didn’t come to us at various times asking, “What do I do now?” Going through the day’s schedule and filling out the form took our class about 10 minutes each morning (of course at first, it took us between 20 minutes and half an hour).

At the end of the day students brought their schedule with them to a whole class meeting and they wrote a short reflection. Those reflections frequently began with, “One thing I liked doing today was…” or “Something I learned today was…” After about 5 to 10 minutes of quiet writing time students were randomly asked to share what they had written. It was a nice way to end the day.

When we had run out of time, students put the schedule with their belongings to take home that afternoon to share with their parents. In this way parents were able to get a lot of details about their child’s daily school life.

In my self-contained intermediate multiage classrooms I still found it to be important to have students know when and where they need to be at various times during the day. So for all the same reasons I had them fill out a very similar schedule.

# Helping Students Stay Organized

My multiage students have always moved from one working group to another throughout the day. Frequently they also worked on individual tasks. Due to the flexibility this kind of movement requires, I chose to use tables throughout the classroom instead of student desks. Because students had no desk in which to keep their supplies, I converted a number of lower bookshelves into student cubbies. Before I was able to purchase plastic containers, the local hospital donated numerous dish pans that served as the main container for supplies such as crayons, pencils, scissors, and other small items.

Below the “Pink Tub,” as the students called them, were kept their 3-ring binders, spiral notebooks, and clipboards. This system worked very well. However, student cubbies did get a bit too messy from time to time just as desks tend to do. In response to this I created a hand drawn poster and had the students color it at the first of the year. Additionally I reminded them routinely that being organized would help them with their school work and that a clean cubby is part of being organized.

Another part of my system that has helped with students having the right stuff at the right time of the day has been the use of 3-ring binders. They carried these with them most of the day as it included nearly everything they needed, organized in various sections. The pocket or “Pee-Chee” type folders I had them bring at the first of school were turned inside-out and punched with a 3-hole punch. They were then labeled (mostly by subject) and used as pocket dividers in their binders. I also asked that students bring a pencil pouch to keep snapped into their binders. This helped them keep track of their pencils.

# My Teaching Philosophy

I have been a teacher of children ages 7 to 12 years old in both public and independent elementary learning environments.  I have taught both single-grade and multiage classes.  Two principles have always been at the heart of all that I have done as a teacher. First, is my believe that education should be student-centered and second, that fostering an enjoyment of learning is probably the most important thing educators can do with their students.

Learning and not teaching has been the focus in my classrooms. This simple statement is filled with implications. If teaching is the focus of a classroom, then control over student academic, social, and behavioral actions becomes the job of the teacher. Management of the details of individual students, of the classroom, of academic progress and pace, and even of recess behavior becomes the primary concern. An information delivery model in which knowledge is passed from the textbook company or state department of education through the teacher to the student would be the major route for knowledge acquisition in a teacher-centered classroom. This type of classroom supports a “culture of silence” as Paulo Freire has observed, a programming of conformity if looked at on a societal scale. It is also, as I see it, a Behaviorist way of managing instruction in which management of learning is reliant on forces external to the child, including reinforcement and an emphasis on consequences administered by the teacher.

In contrast, a learning focused classroom, as I see it, puts the primary emphasis on the process of learning and the use of knowledge. The teacher’s job is then one of helping students acquire learning skills and the practicing of those skills in ways that are useful and meaningful to the student. The teacher does not act as some sort of all-important filter of information determining all of what each student needs to know, but rather becomes more of a guide, setting up learning and practice situations and showing or modeling new ways of acquiring and using information. For instance, in my classrooms I have used many ideas from Gestalt Psychology when helping my students in math. My instruction has been based on a problem-solving model in which I guided my students to find a variety of solutions to various problems, then set up experiences for them to transfer their new skills to similar but different situations. I also used Vygotsky’s concepts of the zone of proximal development, scaffolding, and social interaction in my classrooms. By using peer tutoring and modeling thoughtfully along with my own support of the individual student, children frequently showed tremendous cognitive growth. Thus I believe that the learning environment is not limited to the classroom and to what the teacher can always control but rather is made up of the tools of learning, including social interaction.

In looking at how I came to believe such a classroom is most conducive to student learning, it is important to first visit some of my past classrooms.

In the first 3 years of my teaching career I was learning how to survive each day. My task was to get through the curriculum while still keeping school as enjoyable for the students as possible. Thus I concentrated on orchestrating a classroom full of students, keeping them “on task,” rewarding good academic and social behavior with material and nonmaterial rewards, and including negative consequences for “poor student choices.” Overall students and their parents responded well to this. Part of this orchestration was to present instruction in a variety of teacher-determined ways, including the occasional learning project. I created many “systems” in the classroom for everything from independent reading to small group management. Although I did this with input from students (I’ve been known as a very democratic teacher), I still put an emphasis on my being in control of each and every situation. I believe that nearly every conflict or negative situation could be traced to my overemphasis on teacher control, on being the giver of consequences.

After having survived that year, I reflected quite a bit on my beliefs about teaching and learning. I came to realize that children are complex and that they need to have some control over what and when they learn, that they are, and need to be, an active participant in the learning process. It finally occurred to me that I had been keeping the focus of my classroom on myself, it was a teacher-centered environment and that in order for students to really learn and to enjoy school, I had to refocus on the children, to make my classroom a child-centered one.

At about this time I discovered the ideas surrounding multiage educational practices. For me the two mixed quite well. My needs to change the underlying beliefs and structure of my classroom were a near perfect match with the beliefs and structure of multiage educational environments.

In addition to “student-centeredness,” I also believe that school should be enjoyable. I realize that not all learning tasks are going to be viewed as fun for all students, but that if school in general is considered fun, then learning is fun and the motivation to learn more is enhanced. I believe that engendering a love of learning is probably one of the most important things I can help elementary age children achieve. With that, every other learning task comes relatively easily to a student. As John Dewey stated in Democracy and Education, “Study of mental life has made evident the fundamental worth of native tendencies to explore, to manipulate tools and materials, to construct, to give expression to joyous emotion, etc. When exercises which are promoted by these instincts are a part of the regular school program, the whole pupil is engaged, the artificial gap between life in school and out is reduced, motives are afforded for attention to a large variety of materials and processes distinctly educative in effect play and work correspond, point for point, with the traits of the initial stage of knowing” (p. 195)

As a result of my classroom being both student-centered and a place where learning is enjoyable, the educational opportunities I gave my students were varied. Learners and the learning process are very complex. I believe that there is no one formula for teaching that fits all situations or all students. For this reason I believe that a variety of approaches thoughtfully geared to both the content to be learned and individual students is the best way to structure a learning program. This matches Howard Gardner’s Multiple Intelligences Theory, there are many different ways knowledge can be gained and learning can be assessed. Hands on learning activities will work well with many students and with a number of content areas. Research projects jig-sawed among the class members facilitate learning best for some students and some subject areas. A discovery approach in which the tools for a learning task are given to the students, a simple question is asked, and the procedure is left up to the students to discover works well in other situations and with some students. Again, variety of learning opportunities allows for the most number of children to both enjoy learning and, if choice is incorporated into the activity, to give children some control over their education.

I believe that structuring classroom management to allow for a wide diversity of children and for shared student control of the classroom environment is at least equally important as academic goals. The degree to which a classroom is student-centered or teacher-centered shows up in its management structure. During parts of each day I had students take over various management tasks. Students managed the opening portion of each day (attendance, reading the schedule, discussing the weather forecast). Students also ran a discussion/transition time in which we discussed the various books we were reading. These are things in which I could monitor but that didn’t need to be done by the teacher. Students in my classrooms also had the privilege of choosing their seat. It was not uncommon for a child to be seated at four or more different places in the classroom during the day. They were given the responsibility to choose a place where they could best learn and where they were most comfortable. What a difference from when I assigned seats and decided whom would sit where based on my needs and my perception of each student’s needs. By giving students some control over their school lives, I supported a belief within the students that they, and not some external force, have control of their behavior and their learning. Also that through their own effort they can achieve success. This matches the idea from Attribution Theory that learning is acquired through constructive effort.

As you can see, the two principals of student-centeredness and enjoyment of the learning process underlied all aspects of my classrooms and are two cornerstones of my philosophy of teaching.

Appendix – Multiage Education

The term “developmentally appropriate” is used a lot by multiage educators. The idea, from the work by Piaget, that certain educational practices are appropriate for children of certain age spans but not for those of different age spans, allows the educator to design lessons that match the individual child. This becomes important in a classroom of children of diverse ages. The interesting thing about Piaget’s developmental levels and multiage education is how the idea is used to determine the age spans appropriate to be placed in a single classroom, and how multiage teachers use the diversity to help children transfer from one cognitive level to another. Multiage programs are most often considered “primary” or “intermediate,” roughly matching Piaget’s Concrete Operational Level, divided in two. This recognizes that some children transition from one level to another at different times and that there is a substantial qualitative difference between cognition at the Preoperational Level and the Formal Operational Level, the levels bordering the Concrete Operational Level. The transition from one developmental level to another is supported through peer interaction and modeling and takes into account the four factors that allow movement from stage to stage, maturation, experience, social interaction, and equilibration. The heavy use of peer modeling by multiage educators is supported by Albert Bandura’s Social-Cognitive Learning Theory. By using these “live models,” the students in a multiage classroom learn vicariously through the success of the model. In this way instruction is enhanced and learning is engendered through more than simple reliance on the teacher as model.

The following is from a brochure I produced that explained to parents what underlying beliefs I have and some of the general methods I used in the classroom.

A multiage educational program is a union of an organizational structure and unique combinations of teaching and learning strategies. The way learning occurs is made possible by the multiple age structure.

Why Multiage?

• Allows for flexibility in the grouping of children according to need, ability, or interest; not just by age.
• Problems associated with a yearly transition from one grade to another can be overcome. The teacher has a nucleus of children; trained in the details of the class organization who keep it going while newcomers absorb it.
• As the student-teacher-parent relationship develops over a longer period of time, students will receive greater support for their success in school.
• A more natural learning situation is established. Children work at their own pace. Their program is not geared to the work of a single year but can be adjusted over two or more years.
• Benefits come to the older children from the quality of leadership and responsibility they develop.
• Younger children are stimulated intellectually by older children.
• Children have a broader social experience with increased opportunities to lead and to follow, to collaborate and to make stable peer relationships.

It is my goal to use instructional strategies that:

• Expand the teacher’s role to include that of a facilitator as well as a source of knowledge.
• Produce cooperation.
• Allow students to learn from each other through peer tutoring.
• Give students responsibility and independence in both learning and behavior.
• Build understanding of action-consequence relationship.
• Provide choice to students in different areas of learning that will reflect learning-style differences.
• Allow continuous learning through the use of learning centers, group instruction, projects, and individual pacing.
• Involve parents in classroom activities.
• Encourage student responsibility and ownership of the learning environment.
• Teach goal setting from an early age.
• Build leadership skills in all students.

Note: The information on this page has developed over a number of years beginning as I was defining a shift in my own teaching practices during the spring of 1995. I have revised it many times since, its latest revision was to meet the needs of this blog.

REFERENCES:

Ballantyne, Paul F. (2006). Piaget’s Stage Theory (online) Available: http://www.igs.net/~pballan/Piaget(Stages).htm

Dewey, John (1966). Democracy and Education. New York: Free Press.

Freire, Paulo (1970). Pedagogy of the Oppressed. New York: Seabury Press.

Ginn Wanda Y. (1995). JEAN PIAGET – INTELLECTUAL DEVELOPMENT (online) Available: http://www.sk.com.br/sk-piage.html

Gredler, Margaret (1997). Learning and Instruction: Theory Into Practice. New Jersey: Prentice-Hall, Inc.

Kearsley, Greg (2013). Social Development Theory (L. Vygotsky) (online) Available: http://www.instructionaldesign.org/theories/social-development.html

# Research Supporting Multiage Educational Practices

Following are a number of links to research and online articles that support multiage educational practices.  Although some of these were published a while ago, they are still relevant in this age of STEM and high stakes testing.

# Dear Sub…,

## Introductory Letter for Substitute Teachers

I have found that my Substitute Teacher Notebook has been valuable to most subs and it was absolutely indispensable when I was team teaching and both of us were absent. Below is a copy of the first page of my Substitute Teacher Notebook that I kept on my desk at all times.

### Dear Sub…

I’m glad you’re here! You are about to work with a GREAT bunch of students! They are usually excited about learning, they want to be good, and they will do almost anything you ask of them!

Read through the information in this “Guest Teacher Notebook” and use the blank checklist for recording appropriate and inappropriate behavior if you need to. If you need help don’t be shy about asking for it. The students, neighboring teachers, and the principal will be glad to help you in whatever way they can.

As to academics, if you don’t get through everything, don’t worry about it. We can most likely make it up later. On the other hand, if you get through with things too soon, feel free to play an educational game or do something from your own “bag-o-tricks.” You’ll also find in this folder a logic worksheet that the kids haven’t done but should have fun with. In addition, if there is anything you are passionate about teaching or sharing with the students, feel free to do so, I only ask that you check with me first.

I hope everything goes well and that you will have an enjoyable time with the kids!