Aligning routine tasks & the Rule of 3
Tim
Boerst 

Math Menus as a Context for Exploring the Use of Representation in Problem Solving
It in order to work meaningfully on the Rule of 3, there needed to be routine opportunities to develop understanding of representations, as well as proficiency with them, in meaningful mathematical tasks. Like many teachers of mathematics across the country, I have been provided with curriculum materials, as well as with district and state level benchmarks and objectives. Unfortunately there are many times when curriculum materials, like textbooks, do not match or cannot keep pace with the ever shifting and rising student expectations (and the instructional action they imply). As a result of this incompatibility teachers are faced with the challenge of cobbling together resources and practices to meet student needs.
My colleague and I were faced with a situation where we expected to help students become sophisticated and flexible problem solvers with little support from our text or other curriculum materials. From various sources (conferences, NCTM journals) we learned about "math menus" as a means to provide routine problem solving experiences and as a way to teach mathematics through problem solving. What follows is a brief description of math menus. They are the medium through which my students and I worked on representation in problem solving. The majority of student work contained in this website is the product of their engagement with math menus, so the information below may provide better understanding of the work and also of the idea of the Rule of 3.

Jump
to:
Defining
and Creating Math Menus
Using
Math Menus with Students
Math
Menu Work Products and Assessing Menus
Leaving
the Menu Behind, while Moving Forward with Work on Representation

Defining
and Creating Math Menus
A math menu is a set of five mathematical problems for students to solve. The problems were often drawn from a variety of sources including our texts, "problem of the day" books, NCTM journals, old preservice teacher education texts, websites, and the like, but there were occasions when we created the problems ourselves or at least altered the ones found elsewhere for our purposes. These problems were selected so that they would draw out key issues of topics that would be the focus of instruction for each week. The set of problems on any menu would always include entry points for students of different knowledge and skill. They were also designed to motivate and engage students, either through the use of familiar contexts, connections to real life applications of mathematics, or sometimes even through outlandish scenarios (like the one where a teacher speeds after a "happy vacationing student" to pass along homework or one where participants in a contest could win $100 or a stuffed Elvis doll). From year to year problems were added, altered, scrapped, or redistributed as our experience with students, pacing, and other factors came to bear. By no means should the menus shared here be viewed as finished or as exemplary, but more as a foundation from which we strive to create something better.

...problems
were added, altered, scrapped, or redistributed as our experience with students,
pacing, and other factors came to bear..... 
Using
Math Menus with Students
Each and every week of the school year students worked with new math menus.
They had the opportunity to work in a variety of groupings and with multiple
tools to solve math problems. As the name menu would suggest, students
had the option to select three of the five problems to complete based
upon their own interests (they could of course choose to do more than
that). On Mondays the class would read through the problems and discuss
questions or concerns. In some cases the content within certain problems
would precede specific instruction related to the problem. However, students
were often able to find different or less sophisticated routes to start
work on these problems, draw upon prior knowledge, or defer work on "tricky"
problems until later in the week, when more foundational work had been
done in class. Students had time each day to work on their menus, to ask
questions of the teacher, and to seek clarification in whole class situations.
On Fridays we held whole class discussions of selected menu problems,
one of the students choosing and one of my own choosing. My choice was
usually grounded in which problem had the best opportunity to explore
a variety of representations while learning core content, problems where
students had generated interesting representations, or where under utilized
representations could play a productive role in solutions and justification
of solutions.

...students
were often able to find different or less sophisticated routes to start
work on these problems, draw upon prior knowledge, or defer work on "tricky"
problems.... 
Math
Menu Work Products and Assessing Menus
Students utilize multiple representations when completing math menu problems.
While numerical and linguistic representations dominate, some math topics
and/or some problems within any topic area align themselves well with
many representations. Some problems would specifically ask for certain
types of representation, but most were openended in terms of the representations
that the students could choose to use. Over the course of the year I worked
to encourage the use of multiple representations in numerous ways, including
constructing a new assessment tool. The menu
work samples included in this website were specifically chosen for their
ability to illustrate key features of representational work. They should
not be viewed as random or in some cases even as representative samples
of math menu work. They are strategic samplesr, not in the sense that they
are exemplary, but that they are "telling" samples that support
communication of what was done and what I have learned thus far.
To
see examples of math menu work, look at the student work on these pages:
For
description of analysis of these work samples please refer to "Reflections of the Rule of 3."
Read
a menu assessment tool
(pdf) that I used to encourage attention to representation.


Leaving
the Menu Behind, while Moving Forward with Work on Representation
As I noted in the opening of this section, the creation and use of problem
solving menus in math class was primarily necessitated by a mismatch between
available curriculum materials and district/state level student learning
expectations. This year my school district adopted a new text that extensively
couches the learning of mathematics in problem solving and in meaningful/
motivating contexts. Another strength of the new text,
which I have written about and illustrated in another section of this project,
is its integrated attention to a variety of representations in mathematics
learning and problem solving (Read further
description and analysis of that text). My exploration of representation
will move forward with the new text as the medium for that work.

