**Inquiry Teaching: Models of Instruction**

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**What is Inquiry Teaching and how do you do it?**

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Inquiry teaching is teaching “exclusively oriented towards ‘enabling independent reasoning’”, and there are no strong “preconceived learning outcomes to be made explicit, but is dependent upon the students’ responses (Jaworski, p. 11). Children are responsible to learn mathematical concepts as dictated by the State Core Curriculum, and teachers are responsible to make sure that the students do so. Instead of “directly lecturing or modeling the desired [learning]” (Baer, 2010, p. 161), the teacher constructs a learning environment in which the concept is present, and assists students to explore that environment by designing experiences that assist in assimilating and accommodating what they learn to form new meaning and understanding. (Baer, 2010, p. 161).

These ideas promoted in inquiry teaching are based on the constructivist learning theory which states that “Knowledge is not passively received but actively built up by the cognizing subject”, and learners construct their own meaning and understanding based on their experiences (Jaworski, p. 16).

In order to describe what inquiry teaching is, it is helpful to list some of the characteristics. These characteristics include:

- Focusing on the process used to arrive at the answer, not the answer itself. It is through the process of figuring out the answer that the student constructs meaning for him or herself, and concepts are internalized.

- A balance between content and procedure. In Barbara Jaworski’s book Investigating Mathematics Teaching, it’s states, “ Mathematics is identified by the particular ways of thinking, conjecturing, searches for informal and formal contradictions etc., not by the specific “content”. Thus investigational work, through an emphasis on process, might prove to be an effective way of approaching the content of the mathematical curriculum” (Jaworski, p. 5).

- Investigating: Investigating means a teacher poses a problem dictated by the content or by the students’ questions, and then the students explore, or investigate, the processes and concepts within the problem (Jaworski, p. 6)

- Collaborative learning. When exploring the problem/activity, students often work in partners or small groups (and whole-class discussions towards the end of the lesson) to assist one another through the learning process. This enables the students to articulate their thinking, build upon ideas, and use one another as sounding boards as they work towards a solution.

- Allowing and expecting that more than one process will lead to the right answer. All children have their own problem-solving methods and strategies they use to arrive at conclusions, and all are legitimate in inquiry teaching (Bender, p. 72). For example, there is more than one strategy to solving 4 x 4. One student might add 4 four times. Another student might know that 4 + 4 = 8, and then double that to reach 16.

- The role of the teacher as a guide. In an inquiry lesson, the teacher does not give the students the answer directly. Rather, the teacher uses questions and prompts to guide student’s thinking towards the concept and help the students really visualize the problem (Bender, p. 72) allowing the students to arrive at the conclusion through their own problem-solving strategies. This requires the teacher to understand the concepts very well, and to also have a repertoire of ways in which to solve the problem at hand. This is because the different processes the students use dictate the questions to be asked.

- Questioning and monitoring. As the students are exploring and figuring out how to solve the task at hand, the teacher will go around and listen to their discussions, ask questions to assess the students’ understanding, and to identify possible misconceptions the students may have about the mathematical concept being explored (Baer, 2010, p. 163).

- Adaptive teaching. Teachers need to constantly reflect upon what they are observing, students’ responses, insights, misconceptions, etc., and adapting the lesson according to these observations and reflections (Barnett, 1998, p. 81).

- Giving tasks that have real-life, meaningful application, and “testing [students] ideas and hypotheses against relevant experience” (Jaworski, 12). For example, a teacher may pose the problem 3 x 4 like this: “Sally has 3 bags of candy. Each bag has 4 candies inside. How many candies does Sally have altogether?” (Bender, p. 72).

- Using mathematical tools, such as Cuisenaire rods, base ten blocks, ten frames, paper/pencil, etc. to assist the students in solving the problem. This helps visualize a problem, and can bridge the gap between concrete and abstract thinking and understanding (Bender, p. 73). It also assists in explaining and sharing problem-solving strategies.

- Communicating. Communicating includes students identifying and expressing problems for investigation, expressing their own ideas, and developing those ideas in solving problems” through speaking and through representing thought processes visually (Jaworski, p. 12). This also includes “rationally defending their own ideas and conclusions and submitting the ideas of others to a reasoned criticism (Jaworski, p. 12). Because learning is facilitated through collaborative efforts of all the students, it is essential that the students are able to do so (Jaworski, p. 8).

There are many websites online with information about how to plan an inquiry-based lesson. For more information or variations, see the Lesson Plan Websites listed below. For our intents and purposes, we chose one inquiry lesson template to discuss. In this model, there are three main parts to an inquiry-based lesson. They are:

** Launch**: The launch stage is the first portion of the lesson lasting usually no more than 5-10 minutes. This portion of the lesson is to orient the students and prepare them for the main task that will be the focus for the rest of the lesson. It also is an opportunity for the teacher to (1) introduce vocabulary (i.e. “what does division mean?”), (2) review previously learned concepts, (3) explain the task ahead in regards to management and expectations (like how to retrieve the math tools and materials, how to work in groups, what they would be doing as their task) etc.), and/or (4) give the students an activity that will help connect the students to what they’ve learned before, and prepare them to take their learning one step further. (Baer, 2010, p. 162-163).

** Explore**: The explore stage is the portion of the lesson devoted to exploring and solving the task given by the teacher in the launch. The explore stage is longer, lasting between 10-30 minutes. During this stage of the lesson, the students work together in partners or small groups to solve their task and representing their thinking using the manipulatives and paper and pencil. The teacher will walk around, listening, questioning, and evaluating, making note of misconceptions, and in all ways preparing for the third and final stage of the lesson. (Baer, 2010, p. 163-164).

** Summarize**: The summarize stage of the lesson is when students come together in a whole-class forum to discuss and summarize what they learned during the explore stage. This is not a time for each student to share his or her own work. Rather, it is a time in which to identify the concepts discovered, the methods used, and the conclusions drawn. The teacher’s role is to facilitate this class discussion by dictating what is shared first, helping students to articulate their thinking for the benefit of the entire class, etc. The order in which concepts and procedures are shared plays a very important role in this portion of the lesson. Usually processes and concepts that most of the class used or arrived at are shared first (whether or not theirs is a misconception). Then, the teacher invites students to share more sophisticated strategies that focus on the main idea. The teacher should be careful not to have students share things that detract from or confuse the big ideas that are the focus of the lesson, or are too advanced for the majority of students to connect with. As a general rule of thumb, move from the most concrete to the most abstract. If any algorithms were used, share them last of all. It is during this summarizing that the big ideas begin to surface, the new vocabulary learned is incorporated, and more sophisticated problem-solving methods are developed. The summarizing portion should last between 10-30 minutes. During this time, students should be given roles such as clarifier, questioner, and evaluator to keep them engaged and actively participating (Baer, 2010, p. 164-165).

**Why Teach Math Using Inquiry?**

“To compete in the global economy, you must know math. Therefore it is more important than ever that our students receive solid math instruction in the early grades to prepare them to take and pass Algebra and other challenging courses in middle school and high school.”

– U.S. Secretary of Education Margaret Spellings

(http://www.ed.gov/about/inits/ed/competitiveness/math-now.html)

“To understand is to invent.”

– Jean Piaget (1896-1980)

http://math.ucsb.edu/department/cmi/IBL_History.html

“Inquiry-based learning provides natural points of integration in problem solving across the curriculum between mathematics, language arts, science and other areas.”

(www.ed.gov/programs/**math**sci/reportbrookings.doc)

“Set a student to work on an addition problem without telling him what to do. He will discover what is to be done, and invent a way to do it. Let him perform several in his own way, and then suggest some method a little different from his, and nearer to the common method. If he readily comprehends it, he will be pleased with it, and adopt it. If he does not, his mind is not yet prepared for it, and should be allowed to continue his own way longer, and then it should be suggested again.”

– Warren Colburn (1793-1833)

(http://math.ucsb.edu/department/cmi/IBL_History.html)

**Outside the US.** “Although differences in the culture of teaching makes it difficult to compare US approaches with those outside the US, there is important evidence of the success of IBL approaches outside the US. For example, mathematics education in Japan has received a great deal of attention in the US recently, largely because of their high scores of international assessments (TIMSS, for example). The TIMSS contained an extensive video analysis of randomly selected 8th grade mathematics lessons from the US, Germany and Japan, which has provided baseline data on teaching practices in these three countries (this was coordinated by Jim Stigler of UCLA). One finds practices incorporating the most inquiry on the part of students in Japan. According to Stigler, the typical classroom lesson in Japan unfolds according to a different script than do US lessons. A typical Japanese lesson begins by recalling a concept discussed earlier, followed by posing a problem for students to investigate. After students work on the problem, multiple solution methods are examined and some closure is attained through a teacher-led discussion. In the typical US lesson the teacher demonstrates a procedure and students then practice what they have been shown, often completing 25 exercises in the same time Japanese students work on a single thought-provoking question. The analysis of the TIMSS lessons by mathematicians found that a striking difference in expectations of student work in class: in Japanese classrooms students spent more than 40% of class time engaged in tasks classified as “invent or think”, while in Germany and the US the time was less than 5%^{4}. An excellent sourcebook of Japanese lessons is the book, The Open Ended Approach: A New Proposal for Mathematics Teaching by Jerry Becker and Shigeru Shimada (NCTM 1997). It contains a collection of problems assigned daily and a landscape of anticipated student responses. Study of this book clarifies the Japanese approach, where students engage in inquiry about open-ended problems and then teachers are expected to pull the big ideas out of the collective ideas generated by the class.”

http://math.ucsb.edu/department/cmi/IBL_History.html

Lesson Plan Web Sites

Akron Math Community. Retrieved December 15, 2008, from

http://www.math.uakron.edu/amc/ibl.html.

Concept to Classroom (2004). Retrieved December 15, 2008, from

http://www.thirteen.org/edonline/concept2class/inquiry/demo_sub1.html.

Emints National Center (2004). Teaching Tips: Inquiry-Based Learning.

Retrieved December 15, 2008, from

http://www.emints.org/ethemes/resources/S00000902.shtml.

Teachnology: The Online Teacher Resource. Retrieved December 15, 2008

from http://www.teach-nology.com/currenttrends/inquiry/.

**References**

Bahr, D., and L. A. de Garcia. (2010). Elementary Mathematics is Anything

but Elementary. Boston, MA: Cengage Learning.

Barnett, C. (1998). *Mathematics Teaching Cases as a Catalyst for
Information Inquiry. *Teaching and Teacher Education, 14(1).

Retrieved December 13, 2008, from

http://www.sciencedirect.com/science.

Bender, W.R. *Differentiating Math Instruction: Strategies That Work for
K-8 Classrooms! * Retrieved from http://books.google.com/books?id=S29qq7WMXeAC&pg =PA70&lpg=PA70&dq=guided+inquiry,+mathematics+instruction&source=web&ots=h-C0_Fn9ro&sig=8uSLPsYw65Tnog8vdgFks_msWAw&hl=en&sa=X&oi=book_result&resnum=1&ct=result#PPP1,M1

*.*

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Jaworski, Barbara. *Investigating Mathematics Teaching.* Retrieved from

http://books.google.com/books? fl=en&lr=&id=c1gnm2Ig5ewC&oi=fnd&pg=PR5&dq=inquiry+math&ots=T7gbbPElmO&sig=Z60n9g9r4se_h-wjle-SOtqxkps#PPR7,M1.

Loveless, T., Henriques, A., & Kelly, A. (2005). *Mathematics and Science Partnership (MSP) Program: Descriptive Analysis of Winning Proposals. *Retrieved December 15, 2008, from www.ed.gov/programs/mathsci/reportbrookings.doc

*Math Now: Advancing Math Education in Elementary and Middle School*. Retrieved December 15, 2008, from http://www.ed.gov/about/inits/ed/competitiveness/math-now.html

*UCSB Department of Mathematics Center for Mathematical Inquiry: The History and Promise of IBL in Math Education*. Retrieved December 15, 2008, from http://math.ucsb.edu/department/cmi/IBL_History.html