PSST!! Pass It On . . . Lessons from the Field: Mary Jo Rasmussen

Part of the NCA Commission on Accreditation and School Improvement Journal of School Improvement, Volume 3, Issue 2, Fall 2002

PSST!! Pass It On . . . Lessons from the Field

Mary Jo Rasmussen

About the Author: Mary Jo Rasmussen is an Associate State Director for NCA CASI in Michigan. She has ten years of elementary teaching experience, as well as seven years as a school board member. She can be reached at mjrasmus@umich.edu.

Perhaps the reason you are reading this article is because your school is compelled by the evidence in your school profile to tackle a math problem-solving goal. It might even read something like this: "All students will increase their mathematical problem solving skills across the curriculum." Maybe you are on the math problem-solving committee and wonder how on earth do we assess this? How does the art teacher infuse lessons dealing with math problem solving? The orchestra director? Physical education and health teachers?

I have worked with many schools, teachers, and goal committees to tease out practical solutions to these and other questions that will make a difference in student learning. The ideas contained herein are not mine-indeed, you will see they come from a variety of sources across several states and one neighboring province. That is because we have talked with educators, visited schools, attended workshops and conferences, read books, and found on-line sources seeking answers. I have merely consolidated many of these wonderful ideas into a package that seems to work for schools. It is hoped that you, too, find success with a few of the following ideas, and pass them on!

First, define what problem solving means at your school. What skills will you focus on? Look at your data-what problem solving skills are the most challenging to your students? Observing? Organizing information? Analyzing relevant and irrelevant data, questioning, inferring, reasoning (is this solution reasonable?), evaluating, to name a few. Choose a reasonable number of skills to work on, perhaps three. Remember, these are the skills you want everyone in the school to use very frequently in their teaching. Spending the time to tease out this information now will make a huge difference in the success of your goal.

Second, present this to the faculty. Discuss how every teacher, no matter what subject area he/she teaches, can use these skills in their day-to-day teaching. Principles and Standards for School Mathematics (2000) tells us that students should have frequent opportunities to formulate, grapple with, and solve complex problems that require a significant amount of effort and should then be encouraged to reflect on their thinking. By learning problem solving in mathematics, students should acquire ways of thinking, habits of persistence and curiosity, and confidence in unfamiliar situations that will serve them well outside the mathematics classroom. In everyday life and in the workplace, being a good problem solver can lead to great advantages.

Providing enough professional development so teachers are comfortable is key. Assure everyone that you will provide multiple learning opportunities so that using problem solving in the curriculum will become second nature.

Third, adopt a schoolwide problem solving framework for students to use while grappling with problems. Students need a problem solving roadmap regardless of their age, and it must be consistent throughout the school. The framework may be applied a bit differently in each subject area, or from grade level to grade level, but the steps are identical.

Developing and using a schoolwide framework becomes a key intervention in your school improvement plan. The following sample student framework could be used as a starting point. It contains four steps with prompts for students to think about as they work through a problem and can be used across the curriculum.

Sample Student Problem Solving Framework

1. Clarify the problem or task-I can explain the problem.

Clarify concepts and language

Understand the vocabulary

Restate the problem in your own words

Identify the question to be answered
Gather data

Identify pertinent information or components

Recognize when sufficient information has been obtained

Understand conditions and restrictions

Attempt to classify the problem

2. Brainstorm possible means of solution-I can list some ways to solve the problem.

List the possibilities

Guess and check

Write a number sequence

Find a pattern

Make a simpler problem

Work backwards
Make connections to prior problem solving experiences
Evaluate pros and cons

3. Select a solution strategy and "give it a go!"-I am trying this plan and seeing if it works.

Make an initial choice
Put the plan(s) to work
Monitor progress toward a solution
Collect evidence of progress and/or success
Modify and adjust plans/strategies as necessary

4. Debrief/Reflect-Let me explain what I did, what happened, and what I might do next.

Provide opportunities for discourse, primarily public

Solutions/conclusions

Identify, name, and model specific strategies
Reflection, primarily independent

Written reflections: journals, logs, trackers

Solutions/conclusions

Process

Attitude and Effort (Sperling & Elsholz, 1997)

It is often challenging to devise ways to have students to reflect on their learning. The following rubric from Problem Solving in the Limelight (Sperling & Elsholz, 1997) allows a student to self-assess in a way that is not too risky. Maintaining the rubrics in a log format will allow students to follow their own progress. Collected as a whole class or the entire school, the student rubrics could give your committee valuable insights about how confident your problem solvers are.

Student Problem Solving Rubric

Name___________________________________________Date________________________

Problem:

How difficult was the problem?

Very hard

Okay

Very Easy

(please circle)

Understanding the problem or situation:

1	2	3
I have no idea what the problem is about	Some of 1 and some of 3	I can explain the problem well
I do not know what the facts are		I know the facts
I do not know what the question is		I know what information is needed
I do not know what kind of answer to look for		I know what kind of answer I am looking for

Applying strategies:

1	2	3
I can't think of ways to solve the problem	Some of 1 and some of 3	I can think of ways to solve the problem
I stick with one strategy even if it doesn't work		I change strategies when it is not working
I don't know how to use different strategies for different problems		I use different strategies for different problems
I don't know how to explain the strategy I used		I can explain what strategy I used and how

Checking the problem:

1	2	3
I didn't check the problem for accuracy	Some of 1 and some of 3	I checked the problem for accuracy
I don't know how to estimate if the problem is reasonable		I make good estimates to check if the answer is reasonable
I can't explain why the answer makes sense		I can explain why the answer makes sense

Fourth, select additional interventions. Interventions mirror the skills you chose when you analyzed your data in the first paragraph. For example, if students are having difficulty organizing information in a problem, teaching students how to use graphic organizers may be an intervention your goal committee wants to explore. When chosen as an intervention, every teacher in the school is then expected to infuse the use of graphic organizers in their lessons on a regular basis. Teachers could use graphic organizers to:" Illustrate and explain relationships found in textual material.

Prepare effective lectures and demonstrations.
Help visual learners to perceive abstract ideas.
Assist students who have a limited vocabulary in organizing ideas before writing.
Provide visual linkage of thinking skills programs to content learning.
Design bulletin boards, murals, or multimedia presentations. (Parks & Black, 1997)

Students need multiple opportunities to use graphic organizers in every class. Once information and relationships have been recorded on graphic organizers, students then use the pictorial outline to form more abstract comparisons, evaluations, and conclusions. Students could use graphic organizers to:

Record relationships in textual material for more abstract examination and evaluation.
Depict information as a prewriting tool.
Organize ideas in preparing essays, reports, or oral presentations.
Understand and manage their own thinking and learning.
Prepare displays and demonstrations.
Improve memory of factual information. (Parks & Black, 1997)

This example also illustrates how your goal committee could present ideas to the entire staff to show the flexibility of interventions. Many teachers may already use graphic organizers to some extent and can share their experiences with the faculty.

Fifth, devise a quality locally developed assessment. Rick Stiggins (2001) maintains that a balanced assessment program works only if both classroom assessments and standardized tests are of the highest quality. Quality assessments are those that:" Arise from, and accurately reflect, clear and appropriate achievement expectations for students.

Are specifically designed to serve particular purposes, both users and uses.
Provide a representative sample of student performance that is sufficient in its scope to permit confident conclusions about achievement.
Are designed, developed, and used in such a manner as to eliminate sources of bias or distortion that interfere with the accuracy of the information they provide.

Most schools give some type of standardized test, either state or national, that assesses students' math problem solving abilities either through short written responses, multiple-choice items, or both. What to do about quality locally developed assessments? The following example comes from Edmonton, Alberta, and contains a scoring rubric. The sample problem solving framework above would work well with this problem. The problem could be worked out individually, or groups of students could work together on the problem, and then answer the two questions individually.

SAMPLE MATHEMATICAL REASONING PROBLEM AND SCORING GUIDE

Problem:

Estimate the number of beans in the bucket.

You should have:

Large bucket of beans

Small cup

Magic markers

Scales

Tray

Instructions:

Using any estimating strategy or population sampling techniques, estimate the number of beans in the bucket.

Explain the strategy you used to estimate the number of beans.
About how many beans are there in the bucket?

Holistic Scoring Criteria

3		Problem Solving	Communication
3	B E Y O N D	Analyzed and readily understood the task Developed an efficient and workable strategy Showed explicit evidence of carrying out the strategy Synthesized and generalized the conclusion	Rich, precise and clear all the time (mathematically correct, correct symbolism) Representation is very perceptive (chart, diagram, graph) Explanations are logical and appropriate
2	A T L E V E L	Understood the task Developed a workable strategy Inferred (some evidence) but not always clear Connected and applied the answer	Appropriate most of the time, accurate, mostly clear Representation is accurate and quite appropriate Explanations are mostly clear and logical
1	N O T Y E T	Partially understood the task Appropriate strategy some of the time Possible evidence of a plan-not clear Partial connection of answer	Appropriate some of the time, but may not be clear Uses representation but not too precisely Explanations have some clear parts
0	B E G I N N E R	Totally misunderstood Inappropriate, unworkable strategy No evidence of carrying out a plan No connections of answer Blank	Unclear or inappropriate use of symbolism Incorrect use of representation Explanation is not clear Blank

Alberta Learning, Edmonton, Alberta. Mathematics Performance-based Assessment

Many educators I have worked with emphasize the importance of making the scoring rubric public. Rewrite it in age appropriate language, and visibly post it everywhere. Youngsters are then able to check their own work and respond to fellow students' work. They know exactly what the target is every time. Additionally, adult scorers need to be able to consistently apply the rubric to student work-which takes staff development!

Sixth, organize staff development. Determine what staff development is needed, and what forms it should take. Do you have in-house experts? District, regional or state expertise? Invite an expert to your school to model lessons for teachers and students in classrooms. Visit neighboring schools and classrooms for hands-on demonstrations. Examine lots of student work to determine whether your interventions and rubrics are working. Attend conferences, contact higher educational institutions in your area, and comb the Internet. The list is endless, of course.

Final thoughts. Above all, coordinate with your other school improvement goal committees. Schools have found that at least one link exists between problem solving and the other goals-by all means cross-pollinate. Often you will be able to use common interventions and/or assessments for multiple goal areas. And it's much more fun that way, too.

What a great group of problem solvers NCA folks are. Challenging ourselves to solve the problem of teaching students to become better problem solvers is the ultimate problem to be solved! Rick Stiggins once told me that a good problem solver is someone who knows what to do when they don't know what to do. Well, now we've got some ideas about what to do. And I hope you'll pass them on!

References

Mathematics Performance-based Assessment. (1997). Edmonton: AB. Alberta Learning. Author.

Parks, S. & Black, H. (1997). Organizing thinking. Pacific Grove, CA: Critical Thinking Press & Software.

Sperling, D. & Elsholz, R. (1997). Problem solving in the limelight. Ann Arbor, MI: Unpublished.

Stiggins, R. (2001). Leadership for excellence in assessment: A powerful new school district planning guide. Portland, OR: Assessment Training Institute.

Principles and standards for school mathematics. (2000). Reston, VA: National Council of Teachers of Mathematics. Author.