Course Activity: Student Focus Group Protocol

In this activity you will choose a cross section of students for your student focus group.

1. 
   Select two students who you think are "low performers."
   

   1.

   2.

2. 
   Select two students who you think are "average performers."
   

   1.

   2.

3. 
   Select two students who you think are "high performers."
   

   1.

   2.

4. 
   As you progress through the course, you will utilize these six students throughout various activities in this course. As directed, record your results of these activities.
   
5. 
   Enter a summary of the responses in your Learning Log by clicking on "Resources" and then "Learning Log." (Label your entry "Student Focus Group Protocol.")
   
6. 
   Close the Learning Log window to return to the course.
   

In this activity you will continue to evaluate your classroom instruction by interviewing the student focus group. Interview questions will be based on Dr. Fred Newmann's four standards of authentic instruction. The student perspective will be critical in your self analysis.

You may also conduct the student interviews online. Look for free online survey resources to use or consult your technology coordinator for more directions. If you use the online tools, be sure to include open-ended questions and not just yes/no questions and be sure the tool allows space for students to write adequately.

1. 
   Summarize the focus group's responses to each question in the space provided. Be prepared to add the summary to your Learning Log.
   

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  Do you use technology in this class to find answers, evaluate, explain, and/or create something?
  

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  Do you develop arguments in this class or explanations for the way you feel, with which other people might not agree? How do you share this information?
  

Depth of Knowledge:

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  What topics do you focus on in this class? How do you apply these concepts to real situations?
  

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  Do you feel comfortable taking risks when sharing ideas or answering questions in class?
  

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  How do you use technology in class to extend your thinking on major class topics?
  

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  Describe your use of technology in the classroom.
  

Substantive Conversation:

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  How do you feel about the discussions that occur in this class, both between students and between the teacher and students? Do these discussions stay focused on the subject matter?
  

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  Do these discussions extend beyond the class day, e.g., is there a class blog, wiki, or email exchange that allows students to reflect on the day's learning?
  

Connecting to the World Beyond the Classroom:

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  What connection do you see between what you are learning in class and the world outside of school?
  

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  What have you learned about professions or jobs that are related to this subject?
  

2. 
   Enter a summary of the responses in your Learning Log by clicking on "Resources" and then "Learning Log." (Label your entry "Student Interviews")
   
3. 
   Close the Learning Log window to return to the course.
   

In one high school classroom, the teacher tells students, "Look at the relationships in the following drawing. Your job is to fill in the missing lengths and angles of the sides of objects." He models how to find two of the missing lengths and angles, and then asks students to find the next one. When most have an answer, he again shows the step-by-step process on the board so students can check their work. The homework assignment provides practice in the procedure.

The room erupts into conversation. Students know from television that this is a real-life mathematical application. They draw on prior knowledge about proportionality, ratios, angles, and geometric formulas to form their positions. The all-class discussion helps students clarify and improve their understanding and methodology. Later, the teacher gives an extension problem. What if the window had been open and there was no bullet hole? From which windows could the shot then have been fired? This gives students who struggled with the first problem a chance to apply what they learned from the ensuing discussion of solutions.

In the first classroom, the teacher is relying on the recitation method—providing knowledge and chances for student learning through repetition. In the second classroom the teacher is asking the students to engage in authentic intellectual work—and the students in that class are far more likely to grasp and retain how to work with angles and proportionality.

What is Authentic Intellectual Work in Mathematics?

Authentic intellectual work engages students in "complex thinking and elaborated communication about issues important in students' lives" (Newman, 2001, p. 10). Teachers work to connect content to real-life issues to which the students will need to apply mathematics to find solutions. The second classroom involves authentic intellectual work because students are involved in (Newman et. al, 2001):

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  Construction of knowledge. Students are organizing, interpreting, evaluating, or synthesizing information to address a mathematical concept, problem or issue. Here, the second teacher built on what students knew about angles to help them discover methodologies that had not been previously given to them; they had to construct the knowledge of determining trajectories.
  
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  Disciplined inquiry. Class discussions and assignments ask students to elaborate on their understanding, explanations or conclusions. The students needed to go beyond applying procedural understanding. They had to determine what mathematical knowledge was relevant and form a logical argument. Through dialogue and, later, written explanation and justification of the individual word problems the students created, they engaged in elaborated communication about mathematics.
  
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  Value beyond school. Assignments ask students to address concepts, problems or issues that are similar to ones they have encountered or are likely to encounter in daily life outside of school. These could involve workplace tasks, personal finances, or participating in evaluating political or environmental programs. In the above example, the second teacher helped his students see and then construct real-life applications of the foundations of trigonometry. A final assessment might ask them to find the height of a building with a few simple tools, design a new roof with different pitches for an old home, or plan a low-ropes course, using existing equipment at a local playground as much as possible.
  

Why would teachers put so much effort into teaching an algorithm as basic as computing trajectories and angles? Because research shows that neither students nor adults remember long-term how to solve mathematical tasks when they've been taught procedural knowledge through conventional drills. Further, they struggle to apply that knowledge correctly unless they've worked with authentic word problems that help them grasp the underlying concepts (Ma, 1999).

In contrast, students who are taught through authentic instruction showed significantly higher gains on tests of basic skills, the very learning that conventional wisdom tries to convey through procedural drills. A Chicago study comparing classrooms with high-quality assignments made learning gains in mathematics 20 percent greater than the Chicago average yearly gain on a national test while those with low intellectual demand assignments gained 22 percent less than the national average (Newmann et. al. 2001). The researchers summarized:

As students study a topic in some depth, the concepts that they learn are less likely to remain as disconnected skills and facts, and more likely to be integrated within a larger cognitive schema that connects new bits of information to one another and to students' prior knowledge. This cognitively integrated knowledge is more likely to be owned or internalized by students…participation in authentic intellectual activity helps to motivate and sustain students in the hard work that learning requires (p. 30).

In the real world beyond school, mathematical problems rarely present themselves as "naked" number sentences to be solved. Nor have many of us ever had to calculate when two trains leaving opposite stations and traveling at different speeds will meet. Authentic instruction benefits students by:

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  From the start, helping students see how they can put to use what they are learning because the teacher uses contexts they can understand.
  
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  Engaging their attention and sustaining their effort through meaningful real-world connections. Students grasp why they are learning the concepts.
  
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  Improving their higher-order thinking skills and communication skills. Students learn to reason and express their ideas both verbally and through written communication. They build on each other's ideas and learn to question in ways that develop everyone's understanding.
  
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  Helping them understand mathematics content through mathematical thinking that allow them to transfer their understandings of content and methods from one type of problems to new situations and procedures. In the above example, these students have learned that looking for patterns is a problem-solving strategy they can use in multiple, seemingly very different situations.
  

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  Using the framework to design rigorous assignments. By gauging whether students are using higher-order thinking and building conceptual understanding, they can avoid mislabeling tasks that are procedurally complex but lacking these characteristics as rigorous.
  
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  Being able to assess students' grasp of concepts in multiple ways. With regular paper/pencil drills, wrong answers can come from careless mistakes or from multiple misunderstandings, providing little information to inform instruction. As students work to justify their reasoning, teachers can hear those misunderstandings and provide examples designed to correct them.
  
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  Providing differentiation via multiple ways of assessment or choices in real-life connections. Authentic instruction allows for oral and written formative and summative assessments, adjusting instruction quickly through simple tools such as substituting simpler numbers in a problem for a student or group that is struggling, allowing students to generate examples connected with their own interests, and giving students more than one real-life issue to choose from in solving real problems.
  

Authentic intellectual work prepares students for meeting the complex intellectual demands that society will place on them personally, in the world of work, and as citizens. Instead of viewing mathematics as a series of procedures, students learn to think like mathematicians: describing the world in numbers, looking for patterns, searching for efficient strategies, justifying approaches and solutions, and solving problems in multiple ways. These abilities allow them to take their mathematical knowledge from the classroom to the real world.

For example, students with a deep grasp of the mathematics they have learned know how to transfer that knowledge to scientific and economic literacy because classroom work involved those applications. They can understand how to interpret statistical results, mathematical models, and problem-solving rationale across disciplines. They grasp how seemingly similar research studies can produce widely varying claims—and know how to delve into the studies to find the reasons.

The process is also directly related to the kind of inventive thinking required in the 21^st century workplace. From production lines to science laboratories to management teams to hospital emergency rooms, procedural knowledge is seldom sufficient for high-quality work. Students who learn to solve problems are learning to think for themselves and put forth the kinds of creative ideas, approaches and solutions that workplaces reward.

Finally, authentic instruction leads to enhanced communication skills. It requires teachers to move from assessments with one right answer to significant written or oral discussions that ask students to explain their mathematical reasoning, justify their solutions, and make generalizations or develop strategies they can use in the future. The processes ensure that students learn to communicate in ways that others can understand as well as ask questions that further their own understanding.

Content specific lesson example

In mathematics, authentic intellectual tasks are characterized by Newmann (2005):

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  Construction of knowledge. "The task's dominant expectation is for students to interpret, analyze, synthesize, or evaluate information, rather than merely to reproduce information (p. 1)
  
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  Elaborated written mathematical communication. "The task requires the student to show his/her solution path, and to explain the solution path with evidence such as models or examples" (p. 1)
  
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  Disciplinary concepts. Students develop understanding of concepts, apply them and explain or justify their methods.
  
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  Connection to student lives. "The question, issue, or problem clearly resembles one that students have encountered or are likely to encounter in their lives" (p. 1)
  

Or, a teacher might provide students with two different, commonly used maps of the world. The size and shape of countries vary significantly between the two maps. Students use what they have learned about measuring and estimating area to approximate the area of several countries that attract their interest. They compare and contrast their estimations of the representations of the area of several countries on the traditional Mercator projection and compare them to data in atlases about the square mileage of their countries. They then do the same for the Peters projection (Gutstein, 2005). For the final assessment, students might construct a map that would, without lying, portray one country as "larger" than another when it really isn't, perhaps to justify more resources for border security or health care. The grading rubric would emphasize justification and reasoning.

One way teachers may incorporate technology with teaching authentically in the mathematics classroom can be achieved through the use of a Wiki. Students are able to create a project based on a mathematical theory using words and pictures through the Wiki to describe how the project operates. The students encourage participation from other students within their class and reach out to mathematicians through universities and mathematics sites to ask for critiques and feedback on their projects.

Tasks like these help students realize the importance of competency with mathematical reasoning, build skills to which they will have lifelong access, and prepare them to compete in the world of work.

Many mathematics texts that meet the guidelines for instruction set forth by the National Council of Teachers of Mathematics contain tasks that meet the criteria for authentic instruction. Sometimes, though, teachers may wish to alter the context or instructional suggestions to increase higher-order thinking, classroom discussion, or the relevance of the context to students.

Many schools and universities have archived examples; type "examples of authentic teacher assignments" into an internet search engine. Other sources for examples are included in the "References and Resources."

References and resources

Chapin, S. H., O'Connor, C., & Anderson, N. C. (2003). Classroom discussions: Using math talk to help students learn. Sausalito, CA: Math Solutions Publications.

Gutstein, E. & Peterson, B. (2005). Rethinking mathematics: Teaching social justice by the numbers. Milwaukee: Rethinking Schools Ltd.

Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in China and the United States. Mahwah, New Jersey: Lawrence Erlbaum Associates.

Mueller, J. (2006). "High Rise Danger," Authentic assessment toolbox, http://jonathan.mueller.faculty.noctrl.edu/toolbox/index.htm

National Council of Teachers of Mathematics, www.nctm.org/resources

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

Newmann, F. M. (2005). "Standards and Scoring Criteria for Mathematics Tasks." Retrieved from http://qhs.humble.k12.tx.us/resources/5.2.1.htm on April 25, 2007.

www.urbanacademy.org, a small laboratory high school, has examples of student work for various discipline areas.

© Copyright 2007 Learning Sciences International.

The purpose of this activity is to critique examples of authentic, discipline specific instruction using the provided checklist. Then, use reflection to further extend your learning.

1. 
   On the following pages, read "Examples of Authentic Mathematics Instruction"
   
2. 
   Using the chart below, please analyze the authentic examples provided to you. The chart will help you identify if an authentic standard is present, whether technology is being utilized effectively, and describe how the standard has been implemented.
    
   

3. 
   For the purpose of the online discussion, summarize your responses in the space provided.
   
4. 
   Fill in the "L" and "D" columns of your "Authentic Teaching K-L-D Chart."
   
5. 
   Return to the course and advance to the next screen in order to receive further instructions to share your summary online.
   

 Examples of Authentic Mathematics Instruction

 

Example 1: A Different Sort of Math Book

Developed by Kelly Muzzy and Lori Schramm

In this project, you will choose to create either an original short story or original comic book that deals with mathematics. The following are requirements for this project:

COVERS

1. 
   Both the front and the back cover must be made either 9"x12" tag board or construction paper.
   
2. 
   The front cover must contain the title of your book, the author's name (you) and an appropriate illustration of some sort.
   
3. 
   The back cover must contain your created ISBN number, the price and publisher.
   

4. 
   The first page should be your title page. It should include the title, author's name, copyright date, and publisher.
   
5. 
   The second page should include any acknowledgements you care to make and a statement concerning what age and at what grade level you feel it could best be used.
   

6. 
   Your story is to be original and must deal with mathematics.
   
7. 
   Each page must be numbered.
   
8. 
   The text of your story must be typed or neatly printed.
   
9. 
   There must be at least one illustration on each page or on the page opposite the print.
   

10. 
    Your project must meet all of the above guidelines and will be evaluated using the attached rubric.
    
11. 
    Other points your project will be evaluated on will include: neatness, creativity, accuracy, mathematical content, originality and overall theme.
    
12. 
    You must submit and have me sign your Topic Approval by _________.