Teacher Tap

The Question of Context: How Real Is Real?

Jo Boaler

"When students try to memorize hundreds of methods, as students do in classes that use a passive approach, they find it extremely hard to use the methods in any new situations, often resulting in failure on exams as well as in life. The secret that good mathematics users know is that only a few methods need to be memorized, and that most mathematics problems can be tackled through the understanding of mathematical concepts and active problem solving" (What's Math Got to Do with It?, Jo Boaler, 2008, p. 41).

You're already a great teacher. Unfortunately, not all children share your enthusiasm. How can we increase interest and learning? How do we convince children that math is exciting? Technology can help.

This session won't focus on drill and practice, but this is something a computer does very well. Instead, the session will focus on the application of skills. Technology resources and tools can help in this area too.

Watch What is in a microleter (ul)? This short Prezi was designed to show why it's so important to understand what a microliter is.

bugBrainstorm another measurement that young people need to understand to make a math problem meaningful or understand the context of a problem. Can you provide a reference point? Young people are familiar with a 3-minute song, a 2-liter bottle of Mountain Dew, and a loaf of bread. Go to Prezi. Build a short presentation like Pop the Pop Myth. Compare Prezi to using a traditional presentation tool such as Google Present.

What do today's students really need to know about math? In his TED presentation Teaching Kids Real Math with Computers, Conrad Wolfram argues that people in technical jobs need math skills, we need math in everyday life, and it's important that educated people have the ability to think logically. However he notes that we spend 80% of our time on calculation when it's also important that people can pose the right questions, convert from the real world to mathematical formulation and convert from mathematical forumulation back to the real world.

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The key to authentic learning is making the experience meaningful for your learners.

Let's create a poll "on the fly" to collect data from this group.
Let's use Flisti. What should we ask?
Create your own poll and share it with someone else in the workshop.

This page is divided into sections. Explore each section:

Expert vs Novice Mathematicians and Scientists

Besides in-depth knowledge and skills, what does a math expert do that a novice doesn't? An expert has a high degree of proficiency, skill, and knowledge in a particular subject. Experts are able to effectively think about and solve information problems. They see patterns in information and are able to identify solutions. Moving from novice to expert involves much more than simply developing a set of generic skills and strategies. Experts develop extensive knowledge that impacts the way they identify problems, organize and interpret data, and formulate solutions. Their approach to reasoning and solving information problems is different than a novice.

Based on our knowledge of the differences between novices and experts, how do we help student information scientists develop the necessary repertoire of knowledge and range of skills and strategies? Consider some of the following key areas:

Excerpt from my web page Expert vs Novice Information Scientists.

Use screen recording software as a way for student to share their explanations. It's a fun alternative to writing out an explanation. Watch a few calculus examples: Student 1, Student 2, Student 3, Student 4.

Vi Hart is a mathemusician and enjoys math as a hobby. Watch Visual Multiplication: YouTube. Her videos are a fun way to think about math. Be sure to check out her Mathematical Doodling. Explore Vi Hart's YouTube channel or her website at Vi Hart.

Watch how teens are Using the Flip Video in a High School Math Class (YouTube): Part 1 and Part 2. Add pipecleaners (Youtube) for some fun with graphs. Check out Mr. Sladkey's video projects.

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Young mathematicians need to be aware of their thinking. Talking through a process can be helpful. Technology added an additional dimension to recording and sharing the thought process.

Go to Sketchcast. Sign-up. Click CREATE. Enter a Title. Click the box next to WITH VOICE. Record yourself drawing and talking about the angles of a geometric shape. When you PUBLISH your project, you'll be given a unique URL to share.

Use a FLIP camera to record a problem. Read the real classroom assignment.

There are other options for creating screen captures.

Try another screen casting tool such as screencast-o-matic which captures your movement on the screen and works well with any software you wish to use.

Download Jing, make a short video and upload instantly.

Or, download the screen recording software Camtasia (trial or subscription). Then, Screencast (you need to verify your email) to share your projects online.

Or, download open source choices such as CamStudio (Windows).

How do Experts Do STEM?

"One clear difference between the work of mathematicians and school children is that mathematicians work on long and complicated problems that involve combining many areas of mathematics. This stands in stark contrast to the short questions that fill the hours of math classes and that involve the repetition of isolated procedures." (Boaler, p. 24)

Professionals need math skills. However, doctors, geologists, and engineers don't sit around running calculations. Instead, they apply their skills to authentic situations and problems. This requires the ability to apply knowledge to new situations.

Jo Boaler (2008) described the activities of a typical engineer designing a parking lot or building a wall.

  1. Interpret problems they are asked to solve.
  2. Form a simplified model.
  3. Select and adapt mathematical methods.
  4. Run calculations and create representations.
  5. Justify and communicate the result.

Help children identify mathematical practices such as justifying claims, using symbolic notation, and making generalizations. These are things that successful math learners do.

Examine The Correct Numbers Behind Cell Phone Usage to see how you can "correct" a graphic.

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Involve young people in examining the work of professionals. How are mathematics used in the lives of professionals?

Go to Twiddla and click GO. Click the WEBPAGE tab and to enter the URL of a webpage. Annotate the math that they see. Or, interpret an infographic such as Water, the poster How to Win Rock Paper Scissors, Road Traffic Accidents, or explore the math in the HowToons comic.

Authentic Collaborations

Leone Burton interviewed mathematicians and found that over half their professional papers were written collaboratively. Mathematicians enjoy learning from each other and sharing the excitement of shared problem-solving (The Practices of Mathematicians: What Do They Tell Us About Coming to Know Mathematics? Educational Studies in Mathematics, 37, Leone Burton, 1999, p. 36).

In the real-world, people work collaboratively to solve problems. Can students apply mathematical practices to real-world collaborative settings? If they can't, they don't really understand the concepts. Consider how to incorporate both individual and group assignments. Incorporate peer review. Create synergy by combining data.

Information graphics have become a popular way to visualize data. Watch David McCandless' TED program The Beauty of Data Visualization. Notice the variety of information graphics he uses in his presentation.

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Share your Twiddla URL with a peer. Ask this person to add other ideas and annotations to your page.

In addition to Twiddla, there are other tools with similar features such as Vyew, Dabbleboard

Joining a larger group can make young people part of something unique and special. Consider celebrating special days in your classroom:

World Statistics Day - US Resources and Lesson Ideas

Mathematical Deception

proofiness"Proofiness is the art of using bogus mathematical arguments to prove something that you know in your heart is true, even if it's not." (Proofiness: The Dark Art of Mathematical Deception, Charles Seife, 2010, p. 4)

"Truthful numbers tend to come from good measurement. And a good measurement should be reproducible.. and objective." (Seifer, p. 11).

In the book Proofiness, Charles Seife describes his concern about people who use precise numbers for things that are actually imprecise such as "normal body temperature." He calls this a "disestimate." He called disestimation "the act of taking a number too literally, understating or ignoring the uncertainties that surround it" (Seife, p. 23).

bugTry It
Did you know that your body temperature may not be 98.6 degrees?
Read about Carl Wunderlich and how the number 98.6 was decided.
Read A Critical Appraisal of 98.6 to find out why some people are choosing another number.
Take your own temperature with a disposable thermometer.
Let's measure 60 seconds with our Online Stopwatch.
Compare your temperature with someone else.
Share your results on the class AnswerGarden page.
Use the Mean Machine to figure out the mean temperature.
How does it compare to Wunderlich's number?
Convert Fahrenheit to Celsius. Check your work at Best Option (includes formulas), Option 1, Option 2, Option 3, Option 4.
How does this compare to Wunderlich's original work?

The Disconnect

Red Pyramid"One learns by doing," Zia said. "This is not school, Sadie.  You cannot learn magic by sitting at a desk and taking notes.  You can only learn magic by doing magic." - The Red Pyramid by Rick Riordan.

Sometimes there's a disconnect between skills learning and application. This seems to be a problem in both the real and fantasy worlds.

Have you seen any of the following disconnects in your classroom?

Do students know how to ask and refine their own questions?
Do students have a generic set of skills to break down a problem and figure out what math is needed?
Do students know where to go to find additional information?
Do students have the skills needed to actually solve the problem?

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Brainstorm the conditions when your students seem to "lose it."
Share these ideas on TodaysMeet:LoseIt.
Use the following ideas as a guide.

Where is the disconnect?

How do we solve the problem?

There are many interesting real-world examples, unfortunately they won't all appeal to every student. The key is providing enough variety that each youth will find a connection. For example, explore Science of NFL Football.

Although they are based on TV programs, they can be adapted for other situations. Check out We All Use Math Every Day.

Maps are a wonderful tool for real-world applications. Let's get to know Google Maps:

Math & the Range of Contexts

"Students want to know how different mathematical methods fit together and why they work. This is especially true for girls and women." (Boaler, p. 43).

The real-world contexts are simply a way to get to the more important issue of "why we need math". Not everyone needs to know "why." However, research shows that the "why" question is particularly important for girls. A context helps youth understand why learning math is important and how it connects to the everyday world outside school.

Context is more important for some learners than others. Think about the range of situations where students do math going from abstract to concrete experiences.

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As you think about your "time on task" with children and math during a class or school year, what's the distribution of contexts? Create a pie chart using the following tools. Then, compare the pie chart tools:
Create a Graph
Graph Jam
Math Warehouse
Mr. Nussbaum
Pie Online

Real-World or Pseudocontext?

Read "It's Own Context?" to better understand context.
Read "An Argument Against the Real World".

In What's Math Got To Do With It? Jo Boaler writes about the idea of "pseudocontext." She states that "Instead of giving students realistic situations that they could analyze, textbook authors began to fill books with make-believe contexts - situations that students were meant to believe but for which they should not use any of their real-world knowledge." It's close to reality, but students aren't allowed to use real information and experiences. For instance, you're told to buy pepperoni pizza because "all kids like pizza." However what if "I'm a vegetarian", "I'm allergic to wheat", or "I just don't like pepperoni"?

According to Boaler (p. 53), pseudocontexts "should only be used when they are realistic and when the contexts offer something to the students, such as increasing their interest or modeling a mathematical concept. A realistic use of context is one where students are given real situations that need mathematical analysis, for which they do need to consider (rather than ignore) the variables."

There's nothing wrong with pseudocontext, but some people students need opportunities for messy math that's more real-world. However pseudocontext can lead to misconceptions and unrealistic ideas about the application of math. The biggest problem with pseudocontext is that students can see through fake problems and question to need for the underlying math. When math comes naturally out of a genuine situation, students see the value in math. When using pseudocontext it's important to talk about the conditions that have been established or adjusted for the purpose of the problem.

Learn more at an excellent online discussion on pseudocontext.

Real world, means "real world" where children make choices and decisions. It's fine to add parameters and problems, but make it real such as a real concert, event, guitar, bike... CONTEXT is the key. Try to avoid the "fake" real-world by using real-world issues, circumstances, and problems.

Provide models, examples, and practice, but then give students an opportunity to do something real.

It's easier to find an authentic or relevant context in particular areas of mathematics. As you explore ways to build inquiry-based activities into your classroom, start with those areas that have logical connections.

Watch the following YouTube videos for ideas: Real Life Math

Go to Viralheat: Social Trends. Explore social media data for popular topics.

Meaningful Work

The 2008 report titled "Charting the Path from Engagement to Achievement: A Report on the 2009 High School Survey of Student Engagement" found that many students are bored and don't see the value in school. Two of three students (66%) indicated they were bored at least every day. They found the following activities engaging:

Students enjoy activities with no-clear cut answers (65 percent) and students would welcome the opportunity to be more creative in school (82 percent). Service learning was indicated to be of more interest than classroom learning by 75% of students. Students (61%) indicated they would be more engaged if classes incorporated video and online activities.

Educational LeadershipThe September 2010 issue of Educational Leadership focused on the importance of "Meaningful Work."

In the article Seven Essentials for Project-Based Learning, John Larmer and John R. Mergendoller identify seven important elements of effective projects:

  1. A Need to Know - a news video showing "Beach Closed: Contaminated Water"
  2. A Driving Question - How can we ensure good water quality?
  3. Student Voice and Choice - individual and group elements
  4. 21st Century Skills - team product development
  5. Inquiry and Innovation - authentic questions and resources
  6. Feedback and Revision - peer critiques
  7. A Publicly Presented Product - exhibition night

A 2008 meta-analysis by Patall, Cooper, and Robinson, 2008) of 41 studies reported a strong link between giving students choices and their intrinsic motivation for doing a task, their overall performance on the task, and their willingness to accept challenging tasks. However, the researchers also found diminishing returns when students had too many choices: Giving more than five options produced less benefit than offering just three to five.

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Go back and explore one of the technologies from this page. How could you use it with students to promote meaningful work in a project-based environment?

Explore Illuminations activities: K-3, 3-6, 6-8, 9-12. Or, revisit one of the following:

Flisti - Quick Poll
Sketchcast - Draw and Record
Twiddla - Annotate
AnswerGarden - Collect
Mean Machine - Calculate
TodaysMeet - Brainstorm
Create a Graph - Create
Interactivate - Create
Math Warehouse - Create
Mr. Nussbaum - Create
Pie Online - Create

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

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