Written By: Leslie Koske, Curriculum Specialist, Ginnings Elementary School, TX
Response to Intervention (RTI) begins with both high-quality instruction and universal screening tests for all students to determine levels of learning competency. Intensive interventions in small group settings are then provided to support students in need of assistance with mathematics learning. Student responses to intervention are regularly measured to determine whether students are making adequate progress within the three-tier model.
Beyond the “bare facts” approach, the use of a well-designed mathematical performance task like those developed by “Exemplars” may reveal how well a student has grasped and applied the math concept in an intervention or lesson(s). The performance task rubric is critical in providing the intervention team with information as to how to help the student continue to increase problem-solving thought patterns. It also provides the interventionist and other school personnel with data that can be used to place students in groups within the three tiers of RTI instruction.
While common skill assessments can identify and direct remediation of math weaknesses, it is a leap of faith to move the student into the arena of open-ended problem solving. Unlike a student armed with the tools of math facts and basic computation skills plus adequate reading skills, the RTI student may be undertaking a complex task with minimal skills in all areas.
So, with heart in hand, we begin to delve into the world of creative problem solving with tons of scaffolding to keep the students engaged and afloat.
First, we approached Exemplars not as a “math problem” (immediate defeat), but as a “math story” full of fun. Students begin by analyzing the meat of the text with verbs and action, armed with their best reading strategies (seeking main idea, keeping summary and inference with character and plot in mind) and using the famous five W’s: who, what, when, where, and why. “Who is this story about?” “What do we know?” “What are we looking for?” “Why and when did this happen?” “Can we predict what will happen next?”
We chart out information from the Exemplars math problem on a four-quadrant chart loosely referred to as “UPS Check” model borrowed from Polya’s work: Understand, Plan, Solve, Check.1This framework supports the organization of a complex math problem by directing the student to “chunking” the parts: understand and paraphrase the question, set up a solution plan (t-chart, number line, picture, labels, etc.), actually solve the question, then evaluate and justify the answer. This method is often a group project with four students, each one taking a fourth of the quadrant. A weaker student may need to copy the problem and ask for help reading it, while students with other strengths will tackle the “plan, solve and justify” quadrants.
Believe me, just understanding where to begin is a major and very risky undertaking for the struggling student. We usually work in pairs or small groups in order to spur ideas. We also incorporate another problem-solving strategy called “RUBIES” in the “understand” quadrant that is a problem-solving acronym we borrowed from the science people: Read and Reread, Underline to understand the question, Bracket information, Identify key Elements. This is yet another support to clarify deeply connected math embedded within fictional text.
RTI students need many structures to support and verify their thinking as they investigate possible solutions. I provide “wipe boards” to sketch out solutions, because mistakes can be wiped away without fuss and muss. Students select from a variety of manipulatives to give physical evidence to their thinking. I also feel it is comforting to begin the process with a whole-group experience as the teacher and students plunge into analytical thinking together using “wait time” (be quiet and wait for students to ponder) and “think aloud” (model thought processes out loud so students don’t think teachers were born with answer keys in their heads) and other “active listening” strategies to demonstrate the process of true problem solving as being a walk-in-the-dark to new ideas and not a quick answer. Additionally, I give great attention to modeling different approaches to problem solving and relish using the student work that you [Exemplars] provide to show students the many ways that a solution can be discovered. During this time, we discuss the process: Working backwards, we make a table or chart, find a pattern, and use simpler numbers and so on until students no longer need this structure.
What follows is an example of scaffolding the integration of a well-known perimeter investigation with a similar Exemplars math problem.
INTEGRATED “SPAGHETTI AND MEATBALLS FOR ALL” WITH “SEATS AND TABLES”
ENGAGE: Read Spaghetti and Meatballs for All by Marilyn Burns 2 to the students. Use color tiles to model the various table arrays to find different seating arrangements as the teacher reads.
EXPLORE: Students will color the models of their tiles on centimeter paper and draw conclusions as to the effect of the dimensions of the arrays on the number of people at the table.
(This is an introductory activity with all the same shapes.)
EXPLAIN: Teacher asks students to reflect on the table arrangements and the number of people per table. Does the length or width of the array effect the seating? Are there hidden sides? Develop definitions for perimeter, square, rectangle, array, sides, and edge. (For ESL students, a pre-teaching of vocabulary for this lesson is recommended.)
ELABORATE: Exemplars Task: Seats and Tables (click to download task)
“You are in charge of setting up a classroom with 20 places for people to sit. You can use any number of tables and any combination of 3 kinds of tables. A hexagon-shaped table has 6 places. A square table has 4 places and so does a rhombus shaped table. How would you set up your tables so that 20 people have a place to sit?” Show how many people can sit at each of the tables and how do you know there are places for 20 people.”
- You may use pattern blocks.
- Pretend the paper is a miniature room.
- You need exactly 20 places.
- Provide: graph paper, colored pencils
REAL-WORLD CONTEXT: We have four different kinds of tables in our room (rectangle, hexagon, circle and small rectangle private office). During lunch and work time, there are specific numbers of people allowed at each table. This creates social strains and naturally gets kids talking about the classroom set up on a daily basis. They initiate their own discussions of how to maximize their contact with people or minimize it with others. I decided to introduce this problem because it is a familiar topic for them and they seem interested in solving their own classroom seating issues.
WHAT THE STUDENTS DID: The students took the shapes and tried various arrangements to get to 20. They had a hard time remembering to match sides — not vertices — when making their arrangements. Students traced the shapes and really experimented with all kinds of structures.
Some students lost the questions and went to 20 pieces — not 20 sides or “seats.” They did not really relate to the shapes of the tables in the classroom and needed redirection to relate this activity to the real-life situation around them.
To solve the problem, students used the shapes of tables in the classroom. They traced and counted sides, and then added a different shape (triangle, for instance) to reach 20 seats.
Some students placed numbers at each angle instead of at the sides. They added two squares are eight and then added up the squares and hexagons (8 + 12 = 20). Some students multiplied the tables, which represent the same amount of people (5 x 4 = 20) and used equations and counting to add them up.
Students tried to use all the shapes and changed their minds when the numbers did not count up to 20. Some students traced shapes that were a correct solution, but were not able to write an equation and/or the numbers.
EVALUATE: Using Exemplars rubric categories and Task-Specific Assessment Notes, the student’s work is evaluated.
At this time our efforts are modest as we venture into the waters of true explorers of math thought and away from canned textbook algorithms. I believe our partnership with Exemplars is rock solid and can only lead to mind-expanding experiences through the wonder of thoughtful questioning.
1. Polya, George. 1945. How to Solve It. Princeton: Princeton University Press.
2. Burns, Marilyn. 1997. Spaghetti and Meatballs for All. City: Scholastic Press, Inc.