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Inquiry-based Approaches

"The students who learned through a problem-solving approach did not regard the mathematics of school and the world as different and talked with ease about their use of the school-learned math." (Boaler, p. 11)

butterflyBegin with (a)an interesting problem or situation that (b) requires thinking. Incorporate technology to match the inquiry need and time available.

Think of ways to make use of both the computer as well as hands-on classroom and outdoor situations. Read Math icebreaker and Results of Math Icebreaker for an example.

With computers, it's easy to find the answer. Give students the "answer," then focus on analyzing proof, evidence, and explanation. Grade the process, rather than the product. To learn more, read Shifting Assessment in a World with WolframAlpha.

Go to WolframAlpha's Knowledge Base for lots of great examples of why we must change our approach. If it's just calculation, students can simply enter their problem, find the solution, and choose what to copy.

With technology, it's easy to find answers. The old assignment is to "Find the ... and show your work. Instead we need students to focus on analyzing proof, evidence, and explanations. The new assignment is to "Prove that... Show your work." Then, grade the thinking process.

bugTry It
Examine Snake Oil? infographic. Ask students to show the math that proves some aspect of this infographic.

Ask Yourself: What can you do with the computer you can't do on paper :

bugTry It
Draw Interest - Examine the Vampire Power infograph.
Learn More - Read about Energy Vampires. Are energy vampires living at your house? Why or why not?
Connect to Real Event - It's Vampire Power Awareness Month! Go to Vampire Power website to explore the issue of energy use.
Infuse Controversy - Read Vampire Power: Fact versus Fiction.
Connect to Personal Life - Use the Energy Hog website to find out where energy might be wasted in your home. Do a study of your house. Use the Standby Power Summary Table and Standby Power Summary Power Chart to find the worst offender at your house.
Connect to STEM standards - Read the FAQs to find out how to convert the watts used into energy costs. How much can you save?

Inquiry-based Learning

graphic inquiry"Inquiry is a process that involves asking questions and searching for evidence that can be used to design arguments, make decisions, and draw conclusions" (Lamb & Callison, 2010).

It's an active process that addresses meaningful questions and results in practical solutions. It provides students with an opportunity to apply their math skills to authentic situations that cross disciplines. By applying the power of mathematics to real-world problems, students are able to see the relevance of these skills.

Establish opportunities to explore, apply, and learn. Dan Meyer in his session "Math Class Needs a Makeover" identified five indicators that there's a problem with reasoning in the classroom.

Which of these problems are reflected in your students?

View Meyer's TED video to learn more about his perspectives and approach to math inquiries. Dan Meyer recommends the WCYDWT (What Could You Do With This) approach. He proposes separating math problems into four layers: visual, question, structure, and steps. Meyer notes that working through the math reasoning process is essential.

bugTry It
Do a search at Illuminations for "real world data". Do these have elements that focus on inquiry?
Explore the Science project Prince William's Oily Mess: A Tale of Recovery - Working with Real Data

What elements make this an effective or ineffective inquiry-based learning experience?

Annette Lamb (1997) developed a model called the New Ws using everyday terms to describe the inquiry process. We don't just want students to be able fill in the blanks and apply the formula. We want them to be thinkers. We want them to be perplexed. We want them to create synergy through cross curricular connections.

Think of these Five Phases of Inquiry as a pinwheel that folds back on itself.

bugTry It
Explore the Five Phases of Inquiry and select a technology tool for each phase that you will build into an activity.

Phase 1 - Wonder

Wondering focuses on brainstorming options, discussing ideas, identifying problems, and developing questions.

The inquiry process begins with an open mind that observes the world and ponders the possibilities. Ask inquirers to become observers of their environment becoming in tune with the world around them from family needs to global concerns.

Use our senses and intuition to experience the world. Hook students through multimedia. Start with a photo, image, or video. The more compelling the starter, the more engaged the students. The media should be catchy enough that you can't help but be absorbed in questions. In other words, transform your traditional problem into an image that motivates students to formulate the problem and generate specific questions. Ask questions and phrase problems.

What is the question I’m trying to answer, the problem I’d liked to solve, or the key issue I need to resolve?

bugTry It: Trash Project
Ask yourself, "how do I encourage students to ask deep questions rather than surface level questions?"

Starter examples:
Images and Video - Images and Video - Turtle deformed by being trapped in a six-pack plastic ring
Comic - Deadly Dilemma of the 6 Pack Rings.

Generate a list of questions about a topic. Then, explore images, videos, audio, statistics, or text. Refine the questions. What's the impact of the visuals or audio on your ability to generate questions?

bugTry It: Smart Phone Project
Get started exploring the world of cell phone:
ViewGraph Jame Phone Charts. What are your questions about cell phones?

Use technology to inspire thinking:

Explore current trends. Try Google Trends and Google Flu Trends.

Use guiding questions to facilitate inquiry:

Exploring leads back to questioning. Questions may be refined, restated, or new queries may emerge. Encourage inquirers to be risk-takers. Ask:

Phase 2 - Wiggle

Wiggling involves evaluating information, along with twisting and turning data looking for clues, ideas, and perspectives.

Explore and structure our thinking. Provide some structure to think about the problem (i.e., drawings, charts, graphs). Talk about ho the problem might be rephrased. Identify strategies to address problem. Break the problem into smaller pieces or elements.

Data is located and relevant resources are organized into meaningful clusters. One piece of information may lead to new questions and areas of interest. Students assimilate information. Make connections among ideas and applications. Then, they consider, select, and justify methods of approaching the problem.

Exploring leads back to questioning. Questions may be refined, restated, or new queries may emerge.

Assimilation involves processing, associating, and integrating new ideas with already available knowledge in the human mind. This can be the toughest phase for young people because they may be uncertain about what they've found and where they're going.

bugTry It: Trash Project
Encourage young people to explore related topics.
Introduce the book Tracking Trash by Loree Giffin Burns. Provide quality resources to begin the investigation including statistics about the problem. Explore Hi-Cone Ring Recycling - Vital Statistics. Encourage young people to explore different perspectives on a particular topic. As students if they agree or disagree with the following article: Should you cut up six-pack rings so they don't choke sea birds?

bugTry It: Smart Phone Project
Encourage young people to explore related topics.
Ask students to think about how they might use the cell phone and design specific problems to solve using the ATT Data Calculator to estimate monthly usage. For instance, how might some months of the year be different than others?

Compare online data plans such as the three levels of ATT and cell phone plans.

Work in small groups to build questions based on data plan information.
If I plan to use 500 minutes per month, which plan is the best value?
A cell phone plan consists of a fixed cost (monthly fee) and a variable cost (charge for extra minutes). Find a linear function that gives the total cost of each plan when x extra minutes are used.
Using these costs, determine the cost of using 600 total minutes and 1000 total minutes with each plan. What's the best deal?
For other ideas, use the MarkItRealLearning examples.

Use a data collection tool to collect information about how what types of plans students in the school have.

Use the Make It Real Learning Sampler for ideas on how to turn online statistics into effective learning activities.

Use technology to structure exploration and collect data:

Use guiding questions to facilitate inquiry:

  1. What does this problem involve?
  2. What information do I have?
  3. What information is not needed or useful? Why?
  4. What additional information is needed? Where can it be found?
  5. What are the facts of the situation? How are these facts connected?
  6. How have I tackled similar problems in the past?
  7. How can I break down the problem into smaller pieces, fewer numbers, or chunks?
  8. Can I use a chart, graph, number line, drawing, or other visual to help visualize and organize thinking?
  9. What strategies will I use? What's my plan?
  10. What tools will I use? Calculator, online tools?
  11. What are my guesses? What's the range of solutions? What's the wrong answer? What guesses am I rejecting?
  12. What information do I need to solve this problem?
  13. How do I know what I know?
  14. What structure do we need to visualize our thinking? Would a concept map, chart, graph, help me visualized?
  15. How do I simplify and attack a complex problem?
  16. What's the relevant and irrelevant data?

Phase 3 - Weave

Weaving consists of organizing ideas, calculating numbers, creating models, and formulating plans. It focuses on the application, analysis, and synthesis of information.

Synthesize information, make calculations, and draw inferences. Apply evidence and take action. As students weigh evidence, they may go back and collect additional information to support their inferences. This process of assimilation and inference reoccurs as young people accumulate information.

Although assimilation and inferential thinking occur deep within our brain, we can use visual activities to build these associations. Marzano, Pickering and Pollock (1997) identified six graphic organizers that correspond to six common information organization patterns:

As students explore, look for unique aspects of at least 3 pieces of evidence and make comparisons. Apply the Ds of Evidence to a problem:

bugTry It: Trash Project
Select an infographic such as Trashonomics and The Most Dangerous Species in the Mediterranean. Or ask students to evaluate the quality of an online research project that represents the main idea of an investigation. Apply the Ds of Evidence to this image. What do you think about the quality of the evidence? Is it good or bad? Do you need additional information?

Check out calculators to help you explore data related to trash such as the Ecoconsumer calculator, Conversionator, Trash Cost Calculator.

bugTry It: Smart Phone Project
Get started exploring the world of cell phone through infographics:
Afterlife of a Cell Phone
Say I Am Green
Are you obsessed with your cell phone?
Cell Phones & Airplanes
Should You Use Your Cell Phone on the Road?
Less Bars in More Places
Ultimate Cell Phone Comparison
Cell Phones: The 7th Mass Media
Evolution of the Cell Phone
Do You Care About Privacy?
Things You Do With Your Phone
Real Figures Behind Cell Phone Usage

Use technology to scaffold thinking.

Go to WolframAlpha's Knowledge Base for lots of great examples and tools.

Go to Google Search Features for many types of data sources.

Evidence is necessary to support a claim, justify change, or make an informed decision. Students must learn to identify, process, and judge evidence. This begins with looking for patterns of evidence. Ask:

Arguments provide evidence to support a claim. To develop useful arguments, inquirers must evaluate evidence, examine different points of view, and determine the most logical approach or meaningful conclusion. Ask:

Use guiding questions to facilitate inquiry:

  1. What evidence have I collected?
  2. What are the patterns, relationships, connections, sequences, or causes/effects?
  3. How do I handle ambiguity?
  4. How does this new evidence match my prior knowledge?
  5. How does this relate to...?
  6. What ideas have we learned that I can apply in this situation?
  7. Can I give examples and nonexamples?
  8. How and why is this happening
  9. What inferences and be drawn?
  10. What additional information is needed?
  11. How can this data be synthesized?
  12. How do I know what formula or concept is most useful in applying to this situation?

Phase 4 - Wave

Waving involves creating and packaging ideas and solutions. Why is this important? Who needs to know about it? How can I effectively convey my ideas? Waving also consists of communicating ideas to others through presenting, publishing, and sharing. How will I market my ideas and who will I ask for feedback?

As students identify solutions, draw conclusions, make decisions and solve problems, they think about the process and consider how to share their conclusions and plan for future inquiries. How can ideas be represented in meaningful ways? How can I communication the solution and explain the calculations?

In most academic situations, inquiry involves accumulating evidence that supports inferences that seem reasonable, logical, and persuasive. Students ask:

Another approach is to ask students to turn facts into a visually convincing argument that can be shared. For example, show me why you think the penny should or shouldn't be discontinued. Or, show me that this computer game is or isn't an accurate reflection of history.

When designing persuasive messages, ask:

With each inquiry cycle, inquirers must revisit questions with an open mind.

Use technology to create and share solutions.

tessellationsUse guiding questions to facilitate inquiry:

  1. Is there a single solution or alternate solutions?
  2. Does the solution make sense? Is it reasonable? Why or why not?
  3. What evidence supports my conclusion?
  4. How does this result compare to my original guess?
  5. How will I explain this conclusions to others or take action?
  6. Is my conclusion correct or valid?
  7. How can the result be visualized?
  8. How can the result be shared?
  9. What can I or others use this information? How can this be applied to other problems?
  10. What are the sources of errors or problems in the solution?
  11. How is my answer like and unlike others?
  12. Do you want to share with people in your class or the world?
  13. Do you want to share temporarily or permanently?
  14. Could you write a letter or email?
  15. Could you make a sign for the kitchen, hallway, or area business?
  16. Could you make a group decision and create a "Math Squad" shared announcement?

bugTry It
Use the Flip camera to create your own short movie. Or, use it to pose a real-world problem.

bugTry It
Try a creation tool such as a comic generator like Make BeliefsComix.

Create a sample product for your classroom that you could use as an example.

Explore the Glogster Cell Phone Student Survey. Explore other examples of Glogs for math: angles, geometry, Greeks, Pythagorean Theorem, Constructions, trigonometry, trig, trigo-tecture

Phase 5 - Wish

Wishing involves assessing, evaluating, and reflecting on the process and product of inquiry. Was the project a success? What will I do next?

Reflect, stretch, and imagine. After rounds of questioning and exploring, assimilating and inferring, ask students to revisit the questions and goals of their inquiry. How did the project evolve?

lemonsEncourage products that build in metacognitive aspects and opportunities for reflection. Examples:

Rather than just copying from Wikipedia, I thought about what a tourist would really want to know about the desert.

I’ve created both a family timeline and a Civil Right Movement timeline so we can talk about how each member of the family might have been impacted by what was happening nationally.

My exploration of music from the 1850s lead me to songs about fashion. I create a song in GarageBand that’s a parody of the fashion industry.

Inquiries may go in different directions depending on the questions. While some inquiries look for answers, others seek solutions. The goal may not be apparent in the first round of the cycle. By encouraging inquirers to reflect throughout the process, inquiry becomes a cycle building deep understandings. Ask:

Use technology to reflect.

Use guiding questions to facilitate inquiry:

  1. How did I move through the process?
  2. How does this inquiry connect to possible future inquiries?
  3. What are my new questions?
  4. Do I see patterns in my work?
  5. What if things were changed? How would the solution be different?
  6. What have I learned? How could I apply this to a future situation?
  7. What are the biggest ideas from the inquiry?

bugTry It
What elements do you think are important in the reflective aspect of inquiry? Why?

Inquiry-based Learning Assessment

What might this type of activity look like?


Criteria for evaluating inquiry-based activities

Assessment at Each Phase

Assessment Checklist

bugTry It
What are the biggest frustrations in student assessment in inquiry-based learning environment? How can these be addressed?

Use the links on the left to move through this online workshop.

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