Central Bucks Schools Teaching Authentic Mathematics in the 21st Century

3. 
   Summarize your lesson enhancement ideas in the space provided. You will build these ideas into your next lesson. Be prepared to add the summary to your Learning Log.
   

4. 
   Enter a summary of your responses in your Learning Log by clicking on "Resources" and then "Learning Log." (Label your entry "Developing Substantive Conversation.")
   
5. 
   Close the Learning Log window to return to the course.
   

 As students enter the 21^st Century workforce, their ability to apply math strategies to their daily life will be crucial to their success as both 21^st Century workers and citizens. Thus, creating connections to the world beyond the math classroom must be the goal. According to the research of Dr. Fred Newmann, in order to achieve authentic intellectual work in the classroom, tasks must replicate work—which is more "complex socially or personally meaningful," produced by adults (2001, p. 14). It is work in which explicit attention needs to be given to the following points (p. 14):

The importance of...

• 
  Constructing knowledge instead of reproducing knowledge students have been given.
  
• 
  Constructing knowledge through disciplined inquiry that demands in-depth conceptual understanding and elaborated communication.
  
• 
  Selecting real-world content by focusing on problems, questions, and issues in the real world that can/must be answered (at least in part) by the application of mathematics.
  

Connecting to the world beyond the classroom necessitates more than creating or implementing an activity which in part could be repeated in the real world. Instead, the activity must replicate what happens in the real world and contain knowledge—which the student already possesses—and a problem to be solved. Research by Dr. Fred Newmann (2001) suggests that student work becomes authentic when the following occurs: knowledge is paired with complex intellectual work and real-life applications which are of value to others (p. 14). It is his contention that authentic student work must include the same components as adult work if students are to find success beyond the classroom (2001). Therefore, mathematics instruction must reach beyond "mere tables, symbols, formulas, and abstractions" (Orey, 1998, p. 2). Research has suggested that "the kind of math children are doing in the classroom is often of a different type from that found in the real world" (p. 2). Orey suggests that research findings point to the need for mathematics to be taught as a discipline rather than as school math (p. 2). Thus, the approach to teaching math must be amended for a 21^st Century workplace demanding workers and citizens be able to make "informed decisions (requiring) greater knowledge of concepts related to data, models, and computers" (Tinker, 2006, p.1).

A starting point would be to make the content personally interesting to students. Linda Starr suggests that "students need to be actively involved in the learning process" (2004, p. 1). To do so, the lessons need to be relevant and interesting. Students can invest in lessons personally when the lessons and activities show a desire to establish "common ground with the kids," which may "mean the difference between success and frustration" (2004, p. 1). For example, college-bound high school students studying algebra might be interested in how the cost of tuition is connected to specific post-college jobs. Introducing a lesson from PBS Teachers called "Are Colleges Still Affordable?" would interest the students. In the lesson, the students "investigate a mathematical model that compares the cost of education to potential earnings" (PBS, 2007). The students "compute loan repayments, compare salaries, and discuss related factors to be considered" (PBS, 2007). Using graphing software, students can further develop valuable 21^st Century technology skills by using technology to create a matrix (Nothwehr, 2007). From the manipulation of technology, students can make generalizations and create webpages to provide statistical information to other teens (Nothwehr, 2007). The lesson not only interests students personally, but it also meets authentic intellectual work criteria: students are constructing knowledge through disciplined inquiry with real-world content focusing on a relevant problem to be solved (Newmann, 2007).

Although the idea of affordable colleges may be related to the real world, it is important to remember that a global perspective must guide student learning. In an article in Time Magazine, CEO of UPS, Mike Eskew, spoke about "global trade literate" workers (Wallis & Steptoe, 2007, p. 2). However, in most U.S. schools the focus is on core knowledge rather than "portable skills"—critical thinking, making connections between ideas, and knowing how to keep learning—necessary for the global economy (p. 2). Thus, the approach means "students must collaborate and solve problems in small groups and apply what they've learned in the real world" (p. 6). Real-world issues in a global market involve problem solving. This can extend through compounding interest for credit cards, projecting the exponential growth of a population in an area, calculating savings for retirement, statistically determining the reliability of polling or advertising, deciding whether to lease or purchase a vehicle, converting metric recipe measurements, or applying geometry to home décor.

Each real-world issue applies mathematical concepts such as multiplication, statistics, geometry, and metric conversion, to data and models used in daily life. Therefore, Newmann's structure (construction of knowledge, disciplined in-depth inquiry, and problems related to real-world issues) is an essential guide when designing authentic intellectual work. Utilizing an authentic intellectual work approach, teachers will find that not only does student interest and participation in the classroom increase, but also the age old question, "When am I ever going to use this?" is answered.

For example, in a Pennsylvania classroom, Matt Davis teaches the Pythagorean Theorem to students each year (interview, 2007). Textbook practices ask students to solve problems, which are classroom specific; however, Davis uses a different approach. He asks how many students have a cell phone. Most hands go up. Then, he has them predict why calls are dropped or dead zones appear in the call area. Students make predictions. Davis encourages the students to construct new knowledge by using the Pythagorean Formula to check their hunches. Working in small groups, students continue to construct new knowledge in an in-depth inquiry, learning that the cell towers cover specific areas with a calling area. Then, to discover why calls are dropped or dead zones occur, students continue their inquiry by graphing cell towers and including physical barriers that alter the coverage area. Finally, Davis asks students how they might decide the reliability of "best coverage" advertisements for cell phone plans, which is a real-world application of mathematics (2007).

Students in Davis' classroom continue the lesson beyond the classroom by incorporating their findings into their blogs and personal webpages. They communicate their new knowledge to others beyond the confines of the classroom walls, and often, Davis reports, students make adjustments to their own cell phone plans based on their new knowledge (2007).

In the end, students benefit by investing in their own learning via a sustained and complex inquiry using core knowledge (Newmann, 2001). The teacher benefits by becoming a facilitator as students incorporate learning components into their work: prior knowledge, data collection, problem solving, and interpersonal skills—all valuable 21^st Century strategies (Davis, 2007).

Connection to 21^st Century Skills

In Assessment of 21^st Century Skills: The Current Landscape, the Partnership for 21^st Century Skills, a business advocacy organization, believes that necessary skills for the 21^st Century include information and communication skills, thinking and problem solving skills, as well as interpersonal and self-directional skills (2007, p. 4). In Davis' lesson, students collect and use information, both prior knowledge and data, to communicate ideas to one another and solve the mystery of dropped calls. Students develop interpersonal skills as they communicate ideas and self-directional skills as they stay focused on the task in the time limit of the class period. Most importantly, students critically analyze as global market consumers. Students in Davis' classroom are learning to "critically analyze information, comprehend new ideas, communicate, collaborate, solve problems, and make decisions, which are essential in today's work place" (Partnership, 2007, p.5).

The Partnership for 21^st Century Skills suggests guiding principles and strategies to implement mathematics instruction to adequately prepare students (2007, p.16):

• 
  To increase global awareness in the classroom - use 21^st Century skills to understand and address global issues, work collaboratively with individuals representing diverse cultures, religions, and lifestyles in a spirit of mutual respect and open dialogue, and promote the use of a universal, non-English language—such as math—to communicate ideas.
  
• 
  To increase information and communication learning skills - provide opportunities for accessing, analyzing, managing, integrating, evaluating, and creating information in a variety of forms and media, including technology.
  
• 
  To increase thinking, problem solving, interpersonal, and self-directional learning skills - provide opportunities for exploring one's ability to frame, analyze, and solve problems, as well as exercise sound reasoning in understanding and making complex choices.
  

• 
  Creativity and intellectual curiosity - Developing, implementing, and communicating new ideas to others, staying open and responsive to new and diverse perspectives.
  
• 
  Interpersonal collaborative skills - Demonstrating teamwork and leadership; adapting to varied roles and responsibilities; working productively with others; exercising empathy; respecting diverse perspectives.
  
• 
  Accountability and adaptability - Exercising personal responsibility and flexibility in personal, workplace, and community contexts; setting and meetinghigh standards and goals for one's self and others; tolerating ambiguity.
  
• 
  Social responsibility - Acting responsibly with the interests of the larger community in mind; demonstrating ethical behavior in personal, workplace and community contexts.
  
• 
  Intrapersonal and self-direction skills - Becoming more productive in accomplishing tasks and developing interest in improving one's own skills; monitoring one's own understanding and learning needs, locating appropriate resources, transferring learning from one domain to another.
  

Infusing math class work with real-world technology takes planning, but it can be done. The planning involves an objective, research, and organization. In a technology infused lesson by Brenda Dyck, students focused on collecting and analyzing data (2003, p.2). Dyck located web resources prior to student inquiry, which included articles explaining how numbers can be informative or misleading through online surveys. The information was obtained from Gallup Polls webpage, an online site that turns numbers into colorful graphics, and an array of topic related sites for a variety of student interests (Dyck, 2003, p. 2), providing students with information enabling them to communicate their ideas. In their inquiry, they strove to develop an understanding of "how numbers can lead or mislead, the usefulness of unbiased data, the art of creating a good survey question, and how to analyze data and present the results effectively" (Dyck, 2003, p. 2), developing thinking and problem solving skills. Students were told that their learning would be placed online, which increased student interest and accountability as students developed interpersonal and self-directional skills (Dyck, 2003, p. 2). And finally, once published on the web, the students became part of global awareness of the issues related to how data is collected and used.

• 
  Incorporating online quizzes to challenge students with problems to be solved
  
• 
  Utilizing online data bases for research during in-depth inquiries
  
• 
  Developing webpages to support classroom learning outside the classroom with links to real-world resources, such as interest calculators, loan calculators, and graphing programs to encourage in-depth inquiry and replicate authentic intellectual work done by adults
  
• 
  Encouraging email communication for specific inquiries, feedback, and sharing
  
• 
  Utilizing Smartboards, digital cameras, laptops, and computer software to enhance communication of newly constructed knowledge and elaborate communication of learning
  
• 
  Implementing classroom blogs to encourage student communication and feedback
  
• 
  Creating student-generated portfolios and blogs to communicate conceptual understanding and elaborated communication
  
• 
  Organizing and editing newly constructed learning on a wiki, a website which can be easily edited, as students continue and expand in-depth inquiry and communicate new solutions to problems
  
• 
  Using podcasts to provide additional content lectures, peer tutoring, and/or sharing of projects related to authentic intellectual work
  
• 
  Joining Taking It Global to collaborate with other teachers and classrooms worldwide, to select globally thematic units, and to search activities databases, as well as to create content
  
• 
  Related projects for students involving urban sustainability or an action plan for AIDS
  

In the end, research and current practices, specifically the research by Dr. Newmann on authentic intellectual work, suggest that real-world problem solving belongs in the math classroom. The goal of the problem solving is not simply to encourage higher level thinking, but also to develop a global awareness and knowledge of technology to create learners who will be ready to enter the 21^st Century workforce as valuable assets in the global community.

Davis, M. (2007). Personal interview.

Dyck, B. (2003). When technology intergration goes to math class. Meridian: A Middle School Computer Technologies Journal. Retrieved May 11th, 2007 from http://www.ncsu.edu/meridian/sum2003/math/print.html

Newmann, F. (2007). Personal interview.

Newmann, F., Bryk, A, & Nagaoka, J. (2001). Authentic intellectual work and standardized tests: Conflict or coexistence? Consortium on Chicago School Research.

Nothwehr, S. & Boxer, F. (2007). Beanie babies basics. Iste Nets: Connecting Curriculum and Technology. Retrieved May 5th, 2007 from http://cnets.iste.org/students/pf/pf_beanie_babies.html

Orey, D. (1998) In my opinion: Mathematics for the 21^st Century. Teaching Children Mathematics. Retrieved May 9th, 2007 from http://www.csus.edu/indiv/o/oreyd/papers/math21.html

Partnership for 21^st Century Skills. (2005). Assessment of 21^st Century skills: The current landscape. Retrieved on May 11th from

Starr, L. (2004). Sports math scores points with students and teachers. Education World. Retrieved May 6th, 2007 from http://www.education-world.com/a_curr/curr106.shtml

Tinker, R., & Collison, G. (2006). What is 21^st Century mathematics? The Concord Consortium. Retrieved May 5th, 2007 from http://www.concord.org/publications/newsletter/2006-fall/math.html

Wallis, C. and Steptoe, S. (2007). How to bring out schools out of the 20^th Century. Time Magazine. Retrieved May 9th, 2007 from http://www.time.com/time/printout/o,8816,1568480,00.html www.takingitglobal.org. (2007).

© Copyright 2007 Learning Sciences International.
All Rights Reserved.

Web Research Activity: Making the Connection Beyond the Classroom

This activity helps you to find and explore information on making the connection beyond school.

1. 
   Click to open a search engine that will display a list of Web sites that relate to connecting class content to the world beyond the classroom.
   
2. 
   Identify three Web sites that will help you to better make this connection:
   
   1. 
       
      
   2. 
       
      
   3. 
       
      
3. 
   Use the following questions to guide your thoughts on how to better connect content to the world beyond the classroom. Summarize your findings in the space below.
   
   1. 
      Identify three concepts or ideas that will help you to make the connection. Please explain specifically what you found and how you will implement them into your selected lesson. Please include at least one idea for higher level technology infusion:
      
      • 
         
        

4. 
   Close the browser window.
   
5. 
   Return to the course.
   

Complete an activity in which you analyze and improve the authenticity of a sample lesson plan based upon standards and scoring criteria.

1. 
   Below,Review the lesson plan for evidence and opportunities for authentic instruction.
   
2. 
   Use the following table to document evidence of and opportunities for authentic instruction. Please be specific in your recommendations.
   

 Measures of the Interior and Exterior Angles of Regular Polygons

• 
  To define regular polygons
  
• 
  To determine how to find the measure of the interior and exterior angles of regular polygons
  
• 
  To create a regular polygon using Geometer's Sketchpad
  
• 
  Design a regular polygon given the measure of its interior or exterior angles