Course Activity: Pre-planning Guide

In this activity you will complete and reflect on a pre-planning guide that will prepare you for more focused work on the four standards of authentic instruction.

1. 
   Consider a future lesson that you will be teaching and complete the following questions:
   

2. 
   In the space below, summarize your responses. Please note any key ideas that came up while you completed the questions. These questions will guide you as you develop your authentic lesson in this unit.
   

3. 
   Return to the course to continue.
   

Our world is making advancements in every area, every day. Students are expected to know more, teachers are expected to teach more, employees are expected to produce more, and employers are requiring more. Burkhardt et al. (2003 p. 1) states, "Most Americans agree that the workplace is changing and that the skills necessary for success in the 21st Century workplace are different from those needed in the 20th Century." When preparing students, teachers should implement more authentic ways to teach our students the skills necessary to guarantee achievement in the classroom as well as preparation for the real world. To help students produce authentic intellectual work (AIW), teaching should be grounded in three criteria: enabling students to construct meaning, knowledge through analysis, interpretation and evaluation, focus on limited topics while communicating conclusions, and apply their findings beyond school achievement (Newmann & Associates, 1996 p. 2). This high quality style of teaching specifically focuses on 3 main criteria: the construction of knowledge, disciplined inquiry (including in-depth understanding and elaborated communication), and connections beyond school (Newmann & Associates, 1996, p. 2).

This is especially important in secondary mathematics classrooms because the world is becoming more and more mathematically complex. Students will need to master the mathematical complexity in order to be successful adults in the workforce beyond school. Almost all adults will need to go beyond the manipulations of numbers to function well; it is integrating technology, analyzing data and statistics, programming, and much more. Mathematics classes need to include more than just numbers, counting, and memorizing basic facts. Instruction should include complex thinking strategies that enable students to go beyond the memorization of facts. LEARN, North Carolina (2007, p. 1) agrees with this statement by declaring, "Complex thinking goes beyond basic recall of facts, such as evaluation and invention, enabling students to retain information and to apply problem-solving solutions to real-world problems."

In order to reach these instructional standards, classrooms need to contain instruction that focuses on the use of higher order thinking skills. Thomas and Thorne (2007, p. 1) claims,

"Higher Order Thinking, or HOT for short, takes thinking to higher levels than just restating the facts. HOT requires that we do something with the facts. We must understand them, connect them to each other, categorize them, manipulate them, put them together in new or novel ways, and apply them as we seek new solutions to new problems."

When teachers implement higher order thinking skills and questions into their every day instruction, students are encouraged to go beyond the reproduction of knowledge to using, organizing, and manipulating knowledge as in activities such as hypothesizing, evaluating, analyzing, and creating in order to challenge students in their thinking and problem solving.

1. 
   Construction of Knowledge: Manipulate knowledge by analyzing, evaluating, and creating ideas to extend learning.
   
   1. 
      Analyze: When analyzing in mathematics, students should distinguish between different parts.
      
      1. 
         The students could be asked to differentiate, examine, outline, organize and test in order to produce charts, graphs, and/or spreadsheets.
         
      2. 
         The students could distinguish between relevant and irrelevant numbers in a mathematical word problem.
         
      3. 
         The students could determine a relationship between concepts in order to solve a problem.
         
   2. 
      Evaluation: When students evaluate in mathematics, they should be able to make judgments based on criteria and standards and then defend the idea with supporting evidence.
      
      1. 
         The students should be able to judge which methods are the best for solving the problem.
         
      2. 
         Given a set of occurrences, the student should be able to determine which outcome is the most likely.
         
   3. 
      Create: When creating in mathematics, the students should be able to put elements together to create a whole concept by generating, planning, and producing.
      
      1. 
         Given dimensions, students would need to determine if a site is feasible for a playground in a local park.
         
2. 
   Disciplined Inquiry: Gain detailed insight of controlled subject matter and communicate findings and conclusions in various ways.
   
   1. 
      Describe: The student should describe the process which he/she used to determine an answer.
      
   2. 
      Identify a rule: The student must state a rule to provide proof for a given answer.
      
3. 
   Value Beyond School: Students are able to make connections between mathematical concepts in order to gain a thorough understanding. It is also important to make connections with other content areas as well as with the real world. Students begin to see importance in what they are learning when they can connect to real-life situations, issues, and ideas.
   
   1. 
      Interpret: Interpretation of graphs
      
   2. 
      Utilize: Use of the world wide web
      
   3. 
      Problem Solve: Focus on different real life situations such as banking, construction projects, grocery shopping, etc.
      

Students need to leave school armed with the skills necessary to survive in the workplace of the 21st Century. Burkhardt et al. (2003, p.15) agrees, "To achieve success in the 21st Century, students also need to attain proficiency in science, technology, and culture, as well as gain a thorough understanding of information in all its forms." In order to be proficient in science and technology, students need to have a strong foundation of mathematical skills. The mathematics teacher is responsible for teaching concepts, implanting higher order thinking skills, and allowing students to explore problem solving with the use of technology by focusing on the skills that will lead our students to success in the workplace. Pink (as cited in Burkhardt, 2003, p. 2) states, "Schools must prepare students for a different workplace—one that values innovation, imagination, creativity, communication, and emotional intelligence." Technology plays a huge part of our education process. Students must learn to use modern day technology in order to be prepared for various jobs. Business Week (2006, p. 1) claims, "The work is moving into a new age of numbers. Partnerships between mathematicians and computer scientists are bulling into whole new domains of business and imposing the efficiencies of math." An example for incorporating technology into the classroom to promote Higher Order Thinking is through Podcasting. Students create podcasts describing how to solve mathematical problems or how to understand a mathematical concept or theory. As the Podcast is an auditory tool, students would have to describe in words or sounds the problem or concept, requiring the use of verbal council and not visual imagery. Having the students teach others using only verbal elements will stretch the students to develop communication skills while engaging in Higher Order Thinking.

Goldenberg (2000, p. 1) declares, "One of the strongest forces in the contemporary growth and evolution of mathematics and math teaching is the power of new technologies." Students can use technology in a wide variety of ways within the mathematics classroom. McClain, Cobb, and Gravemeijer (2001, p. 174) implemented a study where computer tools were used to supplement instructions. They conclude,

"In the twenty-first century, access to information will continue to be enhanced by new technologies. This fact highlights the importance of providing students with opportunities to develop and critique data-based arguments in situations that are facilitated by technologies. It is therefore imperative that learning opportunities of this type in which students can develop deep understandings of important statistical ideas become central aspects of school curricula."

By providing students with a wide range of technology experiences, they will be able to reach new innovative ways of thinking and attacking problems. It is our job to prepare our students for all occupational experiences beyond the school walls. McClain, Cobb, and Gravemeijer (2001, p. 176), claim, "Classroom experiences in which students use computer-based tools to help them think and reason about problem situations serve to prepare them for the twenty-first century." Implementing technology in the classroom can be as simple as using mathematics software programs, general calculators, graphing calculators, Power Point, Excel, etc. McClain, Cobb, and Gravemeijer's (2001) study lists a multitude of technological activities that teachers can incorporate into their mathematics instruction.

1. 
   Computer software enables students to construct graphs so that they will be able to portray their understanding of mathematical concepts using data (p. 175).
   
2. 
   The use of a computer projection system allows students to clarify, validate, and defend mathematical reasoning during presentations and/or discussions.
   
3. 
   Students can use computer programs to organize and structure data so that they can make sound educational decisions.
   
4. 
   Word processing enables students to prepare written explanations. Students are able to provide support by including visual examples such as graphs, equations, and charts.
   

1. 
   http://www.aimsedu.org/puzzle/index.html: This website individualizes puzzles for various learners. Students can work on solving the puzzles and exploring the mathematics behind them.
   
2. 
   http://mathforum.org/pow/: This website introduces students to a mathematics problem of the week. It includes word problems in five different categories such as: Mathematics fundamentals, Pre-algebra, Algebra, Geometry, and Pre-Calculus.
   
3. 
   http://www.mathcounts.org/webarticles/anmviewer.asp?a=196&z=60: This website offers a special section for students where they are able to work on a problem a week dealing with news-related word problems.
   

"Computers can provide interactive 'virtual manipulatives where physical devices do not exist. As always, the value of a tool depends on how it is used. If physical or electronic manipulatives are well designed and well used, they can increase the variety of problems that students can think about and solve."

Instructional practices that incorporate students using technological tools will enhance their thinking, reasoning, and problem-solving skills as well as prepare them for the advancements of the 21st Century (McCain, Cobb, Gravemeijer, 200 p. 186). Another area where teachers can implement authentic intellectual work and actively engage students is in providing real world problems that they can relate with and bring meaning to their work. Newmann, Bryk, and Nagaoka (2001, p. 30) states, "Participation in authentic intellectual activity helps to motivate and sustain students in the hard work that learning requires." These types of activities enable students to examine areas that interest them in the world beyond school. There are various assignments in which teachers can incorporate real world problems and real data sets with mathematics instruction (Starr, 2005):

1. 
   Car payments: Students can go through the entire process of choosing a car and then determining which method is the best for making payments (Christopher, 2007, p. 1).
   
2. 
   Stock Market: Students are able to experience the stock market. The students are able to calculate commissions and odds, compare costs and ratios, use fractions and decimals, and read and interpret graphs.
   
3. 
   Finances: Students are able to discover the relationship of learning and earning by setting a budget based on income and spending. (Northwestern Mutual Foundation, 2007)
   
4. 
   Sports Statistics: Students can calculate sports statistics in basketball, baseball, football, etc. (American Statistical Association, 2007)
   

American Statistical Association. (2007). Sports statistics on the web. Retrieved May 28, 2007 from the American Statistical Association Website: http://www.amstat.org/sections/sis/sports.html.

Baker, Stephen. (2006). Math will rock your world [Electronic version]. Business Week, 1-6.

Burkhardt, G., Monsour, M., Valdez, G., Gunn, C., Dawson, M., Lemke, C. et al. (2003). EnGage 21st Century Skills: Literacy in the Digital Age. NCREL, 1-88.

Goldenberg, E. (2000). Thinking (and talking) about technology in math classrooms. Education development center, Inc., 1-8.

LEARN North Carolina (n.d.). Higher order thinking. Retrieved May 1, 2007, from LEARN North Carolina Web Site: http://www.learnnc.org/glossary/higher+order+thinking.

LEARN North Carolina (n.d). Examples of activities that promote higher order thinking. Retrieved May 1, 2007 from Faculty Center for Teaching and e-learning Website: http://www.fctel.uncc.edu/pedagogy/enhancinglearning/HigherOrderActivites.html

McClain, K., Cobb, P., & Gravemeijer, K. (2000). 12 Supporting students' ways of reasoning about data. Learning Mathematics for a New Century, 2000 Yearbook, 175-187.

Newmann, F.M. & Associates (1996). Authentic achievement: Restructuring schools for intellectual quality. San Fransisco: Jossey-Bass.

Newmann, F. & Wehlage, G. (1991). Five standards of authentic instruction. Authentic Learning, 50(7), 8-12.

Newmann, F., Bryk, A., Nagaoka, J. (2001). Authentic intellectual work and standardized tests: Conflict or coexistence? Consortium on Chicago Research, 1-48.

Northwestern Mutual Foundation. (2007). The Mint. Retrieved May 28, 2007 from the Mint Website: wwww.themint.org.

Starr, L. (2005). Get real: Math in everyday life. Retrieved May 28, 2007 from Education World Website: http://www.education-world.com/a_curr/curr148.shtml.

Sutton, J., & Krueger, A. (Eds.). (2002). EDThoughts: What we know about mathematics teaching and learning. Aurora, CO: Mid-continent Research for Education and Learning.

Thomas, A, & Thorne, G. (n.d.). Higher order thinking—It's HOT! Retrieved May 1, 2007, from Center for Development & Learning Website: http://www.cdl/resource-library/articles/higherorderthinking.php.

1. 
   Simplify each side of the equation by combining like terms.
   

2. 
   Use the addition property of equality to get the variable on one side of the equation and the numbers on the other side of the equation.
   

3. 
   Use the division property of equality to solve for the variable.
   

Experimental probability is the ratio of the number of times an event actually occurs to the total number of trials or times that the activity is performed. For example, in flipping a coin 50 times, the result will approach the ratio of 1:2, but will likely be different, say 29:50 or 19:50. The more times a coin is flipped, the closer the ratio will come to the theoretical probability ratio of 1:2.