Fellowship Paper

Experimenting with Open-Ended Questions:
One Teaching Pair’s Experience

By Gena Merliss and Daniel Noel

Francis W. Parker Charter Essential School

Massachusetts Charter School Association
Fellowship Program

2003

Experimenting with Open-Ended Questions:
One Teaching Pair’s Experience

In the field, mathematicians and scientists need to think critically, observe, plan experiments, find patterns, ask challenging questions, and communicate solutions. To prepare them for such work, math and science teachers need to teach students critical skills, and they must provide students with the opportunity to learn and practice thinking like mathematicians and scientists. This type of thinking and skill development is not only for students who make connections between what they learn in school and outside the classroom. All children can learn to think like mathematicians and scientists, if they are coached and given multiple opportunities to practice.

Background: Parker, Goals, and Research

The National Council of Teachers of Mathematics (NCTM) holds that for years many students have been learning information without truly understanding what that information means. They write, “Learning without understanding has been a persistent problem since at least the 1930’s” (NCTM 2000, 20). Certainly, relevant vocabulary and symbols are important, but they are not enough. Students used to memorizing vocabulary words or rules, and students asked test questions that simply require them to write recalled information, often struggle when asked to apply what they have learned.

According to NCTM, the goal of instruction in mathematics is to enable students to gain mathematical power. “This term denotes an individual’s abilities to explore, conjecture, and reason logically, as well as the ability to use a variety of mathematical methods effectively to solve nonroutine problems” (NCTM 1989, 5). In order for students to be able to gain mathematical power, NCTM has laid out specific goals for all students including:

• They learn to value mathematics;
• They become confident in their abilities to do mathematics
• They learn to communicate mathematically
• They learn to reason mathematically (NCTM 1989, 5)

Similarly, the Massachusetts State Frameworks in Science has the following benchmarks for students in the middle grades:

• Design and conduct an experiment specifying variables to be changed, controlled, and measured.
• Present and explain data and findings using multiple representations, including tables, graphs, mathematical and physical models, and demonstrations.
• Draw conclusions based on data or evidence presented in tables or graphs, and make inferences based on patterns or trends in the data.
• Communicate procedures and results using appropriate science and technology terminology.
• Offer explanations of procedures, and critique and revise them. (Science and Technology/Engineering Curriculum Framework 2001, 5)

We believe that our students can achieve these goals when doing both mathematical problems and scientific investigations.

We’ve developed our perspective while teaching in Coalition (for Essential Schools) schools, attending related conferences, and working in an environment where the use of open-ended questions is supported and encouraged. We teach at the Francis W. Parker Charter Essential School (Parker), a charter school that has been open for nine years. It’s a part of the Coalition of Essential Schools and is located on the old Fort Devens. The school serves grades 7-12 and enrolls about 350 students from 40 surrounding communities. Students are randomly selected from a heterogeneous pool of applicants. We, the authors of this paper, teach Division One (primarily 7th and 8^th graders with some 9^th graders) Math, Science and Technology (MST). We integrate subjects as much as possible; Dan Noel is the math specialist, and Gena Merliss is the science specialist, but we are both in the classroom at the same time, team teaching. We teach block classes and meet our students for a total of 9 hours per week.

When we began teaching together at Parker, we sought to enable all our students to achieve meaningful success as mathematicians and scientists. We wanted to find methods of teaching that would not only help students understand the concepts they studied, they’d also enable students to apply these concepts, by solving varied and complex problems. Our research led us to discover a process called “Open Ended Problem Solving.”

The Open Ended Approach

We found an “open” approach to teaching that held promise for us. It has been practiced in Japan and researched since 1971 (Becker and Shimada 1997). In this open approach to teaching, problems are selected which either use a diversity of approaches to solving a problem (the process is open) or encourage the development of multiple correct answers (the end products are open). There is also an emphasis on having students develop new problems or investigations that relate and extend their understanding (ways to create and solve problems are open). Student ideas and experiences are then used as a basis for exploring new concepts and skills that will lead them into deeper understanding (Becker 2002).

What does this open approach to look like? In solving open-ended problems students are asked to solve problems in three distinct ways (as summarized by Dan Noel, from Becker 2002):

1. Students are asked to find several or many correct approaches or ways to solve a problem where there is one problem, one solution, and many ways to reach that solution (the process is open).

2. Students find several or many correct answers to a problem where there are numerous correct solutions to the problem (end products are open).

3. Students formulate or pose problems or investigations of their own, related to a specific concept or idea, where there are numerous problems and where there may or may not be numerous solutions.

Our research led us to believe that this type of lesson structure, an open approach to teaching and learning, would allow our students to be at the center of the problem solving process and would allow them to genuinely think and conduct investigations as mathematicians and scientists. We felt that an open-ended approach had the potential to help students hone their skills of problem solving, communication, and reasoning, while pushing them to understand required concepts and skills. We also felt it served as a way to help students gain mathematical power.

Solving Problems, Asking the Right Questions

Whether it is called the Scientific Method, the Engineering Design Process, or something else, the method to solving a problem scientifically follows a set of general steps, so the result is reproducible. In our classroom, we wanted to solve a problem: we wanted to construct a way of working together. We had to begin by articulating our question or problem.

We noticed that during lectures, many students became restless, and during completely open explorations with manipulatives, many could not stay focused. We wanted to balance lecturing about core content and vocabulary with hands-on explorations. We wanted students on all levels to be able to begin a task and remain challenged throughout it. Our question began as: How do we engage our heterogeneously grouped students more effectively? We refined the question as: What are the right questions to ask a heterogeneous group of students, so they each remain engaged and challenged?

For us, the “right” type of question has enough structure for students to stay focused yet enough openness for students to continue thinking after finding one possible solution. The “right” type of question allows students to practice identifying a problem, taking first steps, staying focused when they are confused, connecting to their experiences, experimenting, observing, and communicating their solutions effectively. It allows students to explore concepts that are content rich and valuable, mathematically and scientifically. Finally, the “right” type of question fosters further development of understanding--leading to more investigation, questioning, or wonder.

Planning Lessons

While planning lessons with open-ended questions, we looked for topics that would lend themselves to exploration. These were often topics laden with vocabulary or formulae that were easily (or painfully) memorized but difficult to truly understand and apply. For example, the Massachusetts Science and Technology/Engineering Curriculum Framework benchmark on Heat Transfer and the Earth’s System for 6-8 grades states: “Differentiate among radiation, conduction, and convection, the three mechanisms by which heat is transferred through the earth’s system” (MA DOE 2001, 17).

We knew students needed to understand the 3 types of heat transfer to be able to differentiate between them, and we wanted students to feel the need to understand these concepts. We thought that if students had to describe something that they discovered or investigated, they would need to learn the new vocabulary that would help them do that. The Department of Education suggests that students, “investigate the movement of a drop of food coloring placed in water, with and without a heat source, and in different positions relative to a heat source” (MA DOE 2001, 17) in order to better understand convection. This is an excellent idea for an investigation, and there are many others; our students often think of investigations that are new and ingenious to us. We wanted students to have their own meaningful experience with heat transfer to which they could connect new vocabulary.

Heat Transfer Investigation

In constructing a lesson plan on heat transfer, we made deliberate decisions in order to achieve our objectives: engage students, give them responsibility, and keep them at the center of the investigation, working like scientists.

Below is the lesson we designed on heat transfer. In bold are our instructions to students. In plain text are directions to teachers. In the gray boxes we’ve included some of our reasoning behind the design of the procedure.

Lesson Plan: Heat Transfer Investigation

Objectives:

1. Students will improve investigation skills:
• Thinking of a question
• Designing an experiment
• Noticing errors or problems and working to solve them
• Making conclusions based on evidence

2. Students will have personal experience with one part of thermodynamics

Procedure:

1. Write on the board: How does heat travel? Task: design an experiment to learn about this question

   We chose a question that was simple to understand, but had many answers.

2. Ask students: How does heat travel? There are different ways that heat travels, and I could tell you all about them and you could take notes. But I would rather have you explore this question on your own first, and see what you can come up with. Then tomorrow, we will pool what everyone has learned about how heat travels and try to understand it more completely together. I'll give you some vocabulary words that will help you describe what you discover today. You will have 45 minutes to set up an experiment with the person sitting next to you, take data, record your results and draw some conclusions.

   We told students that we decided not to lecture to them, not to explain why we thought this lesson was worth their attention. We tried to use words like, “explore” and “discover” because we did not expect one specific answer, but wanted them to have a personal experience that would help them understand and remember the vocabulary.

3. Show the materials individually, telling students: Here are the materials that I have gathered that you are welcome to use: thermometers, aluminum pie plates, beakers, water, hot plates, candles, ice, and hair dryer. Please notice that these are all the supplies for the entire class so you have to share. What questions do you have? Try not to add limitations to the project, repeat that the goal is to learn how heat travels so students can be as creative and independent as possible. Tell students: You may work for 45 minutes.

   We showed each material individually to give students time to get ideas and think about an investigation.

4. Teams report on their experiments to the class. Students record a summary of the experiments on a data sheet provided:

   Students	Summary of Experiment

   After the investigation, we wanted the class to have a set of shared experiences and hoped that having students tell of their experiments would help students better understand and remember vocabulary. As a result, students recorded everyone's experiment.

5. Notice which experiments demonstrate conduction, convection, and radiation.

   While defining the terms, we wanted to refer back to the class data, so students could connect the kinesthetic experience with the listening, seeing and writing.

6. Define thermodynamics, conduction, convection, and radiation. After defining each, ask students for an example of this form of heat travel from their data page. Note who investigated each type.

Observations

Most students dove into some investigation using the array of materials provided. One or two pairs had trouble starting; they said they thought the question was too broad. During the investigation we noticed that very few groups were measuring temperature (evidence that heat was traveling), so we asked every group, “How do you know heat is traveling unless you measure it?” One group asked for food coloring, which we found. They wanted to observe how water moves at different temperatures.

During the next day’s lecture, one pair described how heat moves upwards and in a conical shape. This was an incredibly useful observation because it allowed us to think three dimensionally. Textbooks state that warm air rises and cool air sinks because of convection currents, using diagrams such as:

But our students drew more accurate diagrams, as their classmate explained that heat does not travel upwards two dimensionally; some heat spreads out while rising. Weeks later, students were able to describe convection using that investigation as evidence.

More Lesson Planning

A Mathematics Curriculum Framework benchmark for Geometry in 7-8 grades states that students will:

Demonstrate an understanding of the concepts and apply formulas and procedures for determining measures, including those of area and perimeter/circumference of parallelograms, trapezoids, and circles. Given the formulas, determine the surface area and volume of rectangular prisms, cylinders, and spheres. Use technology as appropriate. (MA DOE 2000, 50)

Corresponding to this benchmark, the Department of Education suggests the following problem to apply volume formulas:

At the end of every second mile of the Boston Marathon, a typical marathon runner takes a four-ounce cup of water. Instead of drinking all of the water, the runner sips some of it and then throws the rest on his or her head or body to cool off.

1. Assuming the typical runner drinks half of the water in the cup, how many ounces of water would an average runner drink during an entire 26.2-mile marathon? Explain how you found your answer.
2. Suppose that all of the runners in the Boston Marathon behaved like the "typical" marathon runner described above. About how many gallons of water would the 40,700 runners in the 1996 Boston Marathon have used? Record each step that you used to find your answer. (MA DOE 2000, 52)

We felt this problem would give students the opportunity to practice using the formula for volume, and it would allow students to practice skills in number calculation, but it does not require students to show their understanding of volume. We believed that if students first understood the concept of volume is, they would understand and remember a procedure for finding the volume of any shape. We wanted our students to develop a personal understanding of volume, one that they could later apply to other problems.

Volume Investigation

As when planning the Heat Transfer Investigation, in planning the investigation of volume we also made some deliberate decisions to ensure that our students felt responsible and engaged in the process of understanding the concept.

Below is the lesson we designed on volume. Again, in bold are our instructions to students. In plain text are directions to teachers. In the gray boxes we’ve included some of our reasoning behind the design of the procedure.

Lesson Plan: Volume Investigation

Objectives:

1. Students will begin to understand the concept of “volume”
2. Students will improve skills of investigation, communication, and problem solving
3. Students will begin to understand the basis and application of volume formulas

Procedure:

1. Place a large cylinder filled with navy beans at the front of the class. Place several beans at each table for student observation. Write on the board: Estimate the number of beans that you think it would take to fill the cylinder.

   We wanted students to begin to understand the concept of a unit of volume. In this activity, the beans represented a standard unit of volume.

2. Begin the class after all students have made a guess. Collect and record estimates… Tell students: Your task today is to find a way to count the number of beans in this can. Ask: What are some things you notice about the can itself? About the beans? Allow them to brainstorm some initial observations about the container and the filling material…This may help them begin. Tell students: You may use any and all methods you can think of in order to gain the most accurate count possible. You should work with your partner to document your procedure and record your final count total on the data sheet provided (see table below). After you complete your investigation, be ready to share your procedure and total count with the class.

   We wanted every student to have an answer and want to prove that he/she was right. We knew that there would be a very large discrepancy in the numbers and this initial sharing of guesses would begin to spark interest in finding the number of beans in the can. During this activity we knew there would be many different approaches and we wanted all students to value the diversity of approaches, share ideas and communicate results.

   Procedure
   Data
   Conclusion

   We wanted to encourage students to keep careful notes on a data sheet about their procedure because it would be these procedures that would later help them understand how the volume formulas were derived and applied.

3. Allow students time (approx. 45 minutes) to thoroughly plan and conduct their investigation. Be sure to encourage all solutions to the problem and push students to think of more than one way. Ask students questions such as: You’ve got a good method; are there any variations? I saw another group do this (describe the procedure). Could you improve upon it? As students are following through with their ideas, remind them to clearly document the procedure they are using. Also, ask clarifying questions to begin to assess their understanding of volume.

4. Once students have generated procedures and calculated the amount of beans in the can it is essential to share ideas and process with the class. Record the different methods on the board. Then, ask students to look for and share similarities and differences in the methods.

5. Define Volume for the class. Ask: How does this apply to the investigation you have just completed? There are many prompting questions possible here. Students should begin to recognize that volume is measured in standard units (in this case beans), and those units can easily change.

   We wanted students to be able to make the link between this activity and the concept of volume.

6. Share with the class the method and formula for finding the volume of the cylinder. Show them that in order to find volume of a cylinder you can find the area of the bottom and simply multiply this by the height of the can. Ask: Does this work with beans? Why or why not? Give the class plenty of time to discuss and then ask them if anyone actually used this method. Often times there are students who count the number of beans that fit on the bottom of the can, and estimate the number of bean layers that could fit in the can, and multiply. Illustrate to them that this is how the mathematical formula was derived, except that standard units of measurement (inches, centimeters, miles, etc.) are substituted for beans. Allow plenty of time for discussion and questions.

7. Ask students: Could you find the volume of any object? How?

8. Tell them the actual number of beans!

Observations

Every student pair had a method to solve the problem and calculate the number of beans. Students were very clear about their methods, and the data table was an excellent organizer and focusing tool. At the end of the lesson all students had a means by which they were beginning to understand the concept of volume as a three dimensional measurement. All of the students in class felt extremely successful and confident in their methods and as a result were eager to share their results with the class. The extension questions allowed them to begin to apply their new knowledge in different ways; they were inspired to find the volume of fish in our class fish tank and later find the volume of boats they designed.

Conclusion

The final step in an investigation is to draw some conclusions from the data and develop new questions for further study. Our goals in our experiment in open-ended questions were to provide students with opportunities to practice identifying a problem, take first steps, stay focused when confused, connect to their experiences, experiment, observe, and communicate their solutions effectively, all so ultimately they act like adult mathematicians and scientists. We worked toward those goals by informing students (up front) of our intent and goals (the big picture), having students practice the method, experience success, and enjoy themselves. We were also very careful about the questions we asked during the lesson; we posed questions that allowed students to begin and continue working, even if they were confused about the final answer.

Over the course of the school year, we noticed important changes in our classes. In September, students were more tentative, less likely to try something that they weren’t sure of. We worked hard in our class to develop a culture in which trying is more important than simply finding the right answer. By the end of the year, it was easier to introduce an open-ended activity because students had practiced them and knew what was expected. Moreover, students told us they enjoyed math and science at Parker more than they did at their “old school.” We gave unit evaluations, and in them students noted that in class they got a lot of individual attention, and their ideas were valued. They also stated that they preferred the work with partners (part of the open-ended approach) to lecture/discussions with the whole class.

Open-ended questions are designed so each student can draw on his or her experience and come to some solution. The approach helps assure that students experience success, and that success helps students have a more positive attitude about their math and science ability. By the end of the school year, students had expanded their perspectives; they’d “bought into” class. They were able to work with a limited amount of direction, trusting themselves to take risks and be successful (or not, and that was O.K. too). We knew that they’d learned to appreciate that there were multiple ways to arrive at an answer or another question.

We want to make sure the questions we utilize in class are rich enough and phrased in such as way as to allow students to explore a topic deeply. The two lessons we described in this paper were successful, but there were many more that weren’t all we hoped for. Perhaps the question was too broad, too specific, or just not the “right” question. We also want to improve how students use their results and connect their explorations with vocabulary, thereby constructing a complete picture of a concept. At the end of any good open-ended investigation (such as our developing practice), there should be next step questions. Ours will guide us to work to improve the effectiveness of the questions we use in class.

Our foray into utilizing open-ended questions worked for us this year. From our research and experience, we’ve come to achieve a better balance between lecture, practice, and investigation, and we’ve come a long way in developing our open-ended approach. Our research and application are unique to our school. Our goal, in documenting our work, is to increase awareness of open-ended questions as methodology. In this document, rather than provide a foolproof recipe, we sought to share how the approach can be used. Using open-ended questions has transformed our class and allowed us to improve our teaching practice, making student learning real. We wholeheartedly believe that these benefits are not unique to us and sincerely hope that more teachers will use this method, as open-ended questions are a tool with which to engage and prepare students for the world of math and science.

Contact Information

Daniel “Diesel” Noel and Gena “Annette” Merliss

Francis Parker Charter School

49 Antietam St.

Devens, MA 01432

978-772-3293

dan@parker.org

genam@parker.org

References

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Becker, Jerry and Shigeru Shimada, eds. 1997. The Open Ended Approach. Reston, VA: National Council of Teachers of Mathematics.

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Lambert, M. 1990. When the Problem is not the Question and the Solution is not the Answer: Mathematical Knowing and Teaching. American Educational Research Journal 27, no. 1:29-63.

Massachusetts Department of Education (MA DOE). November 2000. Mathematics Curriculum Framework. Retrieved 16 December 2003 from the MA DOE Web site: http://www.doe.mass.edu/frameworks/current.html.

Massachusetts Department of Education (MA DOE). May 2001. Science and Technology/Engineering Curriculum Framework. Retrieved 16 December 2003 from the MA DOE Web site: http://www.doe.mass.edu/frameworks/current.html.

National Council of Teachers of Mathematics (NCTM). 2000. Principles and Standards for School Mathematics. Reston, VA: NCTM.

National Council of Teachers of Mathematics (NCTM). 1989. Curriculum and Evaluation Standard for School Mathematics. Reston, VA: NCTM.

Sheffield, L., ed. 1999. Developing Mathematically Promising Students. Reston, VA: National Council of Teaching Mathematics.