Central Bucks Schools Teaching Authentic Mathematics in the 21st Century

Covey, S., Merrill, A., & Merrill, R. (1994). First things first. New York: Simon and Schuster.

Principles and standards for school mathematics. (2000). National Council of Teachers of Mathematics.

De Lange, J. (2003). Quantitative literacy: Why numeracy matters for schools and colleges. Mathematical Association of America.

Webb, N. L. (2002). Depth-of-knowledge levels for four content areas. Retrieved August 5, 2007 from http://facstaff.wcer.wisc.edu/normw.

In this activity you will prepare for an online discussion that will help you to focus your attention on promoting depth of knowledge in key content areas during your instruction.

1. 
   Think about a lesson that you recently taught or will be teaching in the near future. Once you have identified the lesson, answer the following questions to provide a basis for your discussion input.
   
   1. 
      How do you delineate between what Fred Newmann refers to as knowledge that is deep or thick versus knowledge that is superficial or thin?
      

2. 
   In your lesson, what are some examples of "deep" knowledge and what would be examples of "superficial" knowledge?
   

3. 
   Around what core ideas or concepts is your lesson built?
   

4. 
   How or what opportunities are your students given to demonstrate a "depth of knowledge" in the core concepts and ideas?
   

5. 
   Considering the "Range of Instructional Practice" chart and the "Technology: Productivity Use vs. Higher-Level Thinking Use" chart, what meaningful ways could you use technology to provide depth of knowledge?
   

6. 
   What opportunities in your lesson exist for the following:
   

2. 
   For the purpose of the online discussion, summarize your responses and main ideas to the chart in the space provided.
   

3. 
   Fill in the "L" and "D" columns of your "Developing Authentic Instruction K-L-D Chart."
   
4. 
   Return to the course and advance to the next screen in order to receive further instructions to share your summary online.
   

A high school teacher, Mr. Cooper, gives his 10^th grade students the following problem:

Mr. Cooper knows that some of his students will fail to take into account whether final outcomes are dependent on previous outcomes. Instead of explaining a straightforward approach to solve the problem, he asks students to work in groups to decide on a method. After observing the diagrams the groups have put together as they work to solve the problem, he asks for explanations. Consider a discussion that might sound like this.

In substantive classroom conversations (Newmann, F. M., Secada, W. G., & Wehlage, G. G. 1995) discussion goes beyond facts and experiences to interaction about the ideas of a topic. To determine whether your students are carrying on substantive conversations, look for the following characteristics:

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  Students compare and analyze ideas, raise questions to clarify or further understanding, synthesize information, or in other ways use higher order thinking.
  
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  Students frequently interact directly with each other, rather than a teacher-student-teacher-student pattern of response. They might justify their ideas, ask for clarification, or restate another person's thoughts.
  
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  Students give sustained attention to a topic, staying with the issue or question until it is sufficiently solved or clarified.
  
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  The teacher refrains from correcting student reasoning or answers directly, instead encouraging students to support their answers with textual evidence or reasoning.
  
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  Conversations build on previous comments, working toward collective understanding of a theme or topic.
  

In substantive conversations, students decrease their dependency on teachers as the only source of knowledge. Instead they learn that they can question, correct their own understandings, learn from each other, and discover concepts. Talking about concepts helps students clarify their own thinking. The teacher is still directly involved in facilitating the process; in fact, substantive conversation may take as much or more preparation than traditional lectures. Teachers must think in advance about the misunderstandings students might have and how these might surface in conversation. They need to prepare prompts, examples, and questions that will direct student conversation toward the goals of the lesson while still allowing for construction of knowledge. They need to quickly identify which work examples will take the class discussion in the right direction. Further, they need to listen carefully for reasoning and justification as well as for correctness—and be prepared when students propose a method or solution that had not occurred to them. The teacher can participate as both a stimulator to move to succeeding productive steps and also as a learner.

Studies have found that the skills needed for substantive conversation—justifying, reasoning, and making connections—also lead to significantly higher performance on standardized tests (Chapin, 2003, Fosnot & Dolk, 2002). Specific benefits to students include:

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  Growth in mathematical understanding from hearing multiple ideas and strategies presented.
  
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  Improvement in logical reasoning through practice in presenting ideas so that others can understand.
  
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  Socialization in the fact that problem solving takes sustained effort and that effort increases mathematical ability. This counters many students' belief that they either are or are not smart or that speed equates with intelligence (Resnick, 1999).
  
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  Practice and modeling of applying prior knowledge to new situations. Students benefit from hearing others' connections and applications.
  

Substantive conversations also provide for differentiation, allowing students to help one another as they take a cooperative approach to learning. Students who grasp the concepts quickly often still need practice in explaining them so that others can understand. Substantive conversations help the teacher facilitate these different levels of learning.

Teachers also find that students are motivated through "taking an interest in their peers' claims and positions within a discussion" (Chapin, p. xii). The motivational factor helps students remain engaged in problem-solving situations that stretch their mathematical abilities.

How Does Substantive Conversation Connect To 21^st Century Skills?

NCREL and Metiri Group (2003) identified four skill clusters as essential to success in the 21^st Century workplace by looking at literature reviews, workforce trends, data from educators, and input from businesses and other constituencies. The four skill clusters are digital-age literacy, inventive thinking, effective communication, and high productivity.

As with the nature of the educational system, students are constantly being asked to go beyond the knowledge needed for content and standards-based tests to these higher intellectual demands. Substantive conversations apprentice students in communicating with others to develop knowledge beyond given procedures and facts to the ability to apply that knowledge in changing situations. They gain significant practice in the higher order thinking skills that inventive thinking requires and in the risks involved in sharing ideas and listening to feedback. Students also build skills in collaborating and interpersonal skills, listening, learning to talk about ideas without attacking the person who shared an idea, and building upon contributions of different participants to solve a problem. As such, substantive conversation is one important element of authentic intellectual work.

What Teaching Strategies Can Promote Substantive Conversation?

A first step for teachers in implementing the use of substantive conversations is evaluating their comfort level in shifting from the role of expert to facilitator. Teachers who have relied on recitation may struggle with planning for the less predictable student responses in substantive conversations. In contrast, teachers who embrace the more facilitative role may fail to plan enough for the desired results to arise. To implement the strategy well takes both preparation and practice. Because students will also be learning new roles and techniques, the teacher may need to both actively teach and persistently employ substantive conversations for several weeks before seeing a significant change in students' mathematical reasoning. The following steps may help:

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  Anticipate the misconceptions students may have as prior knowledge as well as the mistaken conclusions they might draw as they work on the problem.
  
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  Have several problems or examples ready for students to work with that will provide evidence to students to correct misunderstandings.
  
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  Prepare open-ended questions or "teacher moves" when incorrect reasoning or answers are provided.
  

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  Restating what a student has said. Repeating their words helps the student consider if they were clear, and it gives the teacher time to think of another appropriate question. Later, the teacher will use the same move with students, asking them to restate each other's ideas to check for understanding.
  
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  Waiting. If students do not respond within a few seconds, teachers traditionally step in with another question or clarification. Or, if some students have their hands up right away, the teacher allows them to answer before others have formed their own ideas. Practice giving students individual time to form an answer, waiting ten to twenty seconds, but work to develop a class norm that everyone should be ready to participate after the wait time. Also, provide wait time after calling on an individual student, saying, "This is a tough one but you can take time to put your thoughts together." If he really struggles, come back to him. Later, the student might be asked to restate another's answer or whether they agree or disagree and why. Eventually, students will also learn that it is okay to share in-process thoughts, such as, "I am thinking that maybe a data table would be helpful here, but I have only got one entry calculated."
  

As students gain proficiency with substantive conversations, teachers can use various forms of technology to enrich it. For all-class discussions, a teacher might use classroom performance systems, the handheld devices that allow for students to select their response, with the responses for the whole class tallied and shown on a projection screen.

Interactive whiteboards allow students to share representations and alter them as their understanding grows. Teachers can also use them to visually link ideas and conversation threads as students share.

Blogs are another common way to facilitate substantive conversation through technology in the classroom. Students have the ability to post problems and ask others to solve their problem, analyze, discuss, and interpret their reasoning.

Once students are proficient at substantive conversations, teachers might ask them to respond to each other via online conversation tools, wikis, blogs, or other electronic forms of communication.

A Sample Lesson Plan That Includes Substantive Conversation

Teachers can employ substantive conversations in multiple settings where concepts and processes rather than procedures and correct answers will be the focus. Examples and sources are listed below. Once an appropriate question or problem is chosen, careful planning is needed to ensure that the targeted learning goals are met through substantive conversation. Below is a sample problem along with a possible lesson plan. Such lesson plans could also include checklists for students to use in monitoring their own conversations, such as:

Are we

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  Sticking with one issue at a time until resolved?
  
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  Asking questions about the relevance of the information introduced?
  
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  Responding carefully to previous speaker using phrasing such as
  
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    I agree and…
    
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    I think you said that…
    
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    Could you explain that another way?
    
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    I understand your reasoning and would add…
    
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    Could you clarify…?
    
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    I disagree with…because…
    
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    I believe we have a solution and should move on to…
    
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    Who else has something to add?
    

You are moving into an upstairs apartment in an older home. The stairwell is rather small (3' by 6'). What is the maximum size mattress that will fit through the doorway and up the stairs? King, queen, full, or will you need to stick with twin mattresses? (Buhl, et. al.)

To help students grow in their understanding of optimization problems, especially regarding the real-world application of calculus. They will be able to explore the uses of The Geometer's Sketchpad, recognizing the process of solving the problem (identifying the variables) is at this point more important than actually laying out the entire solution.

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  Students may assume that the maximum height is measured when the mattress corner is at the base of the door when the mattress can in fact be rotated to lessen the height. The width of the stairwell needs to be considered as well as the height.
  
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  They may fail to identify all of the possible variables.
  
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  They may fail to model how the "clearance" is decreased as the mattress tilts and the clearance from the floor increases.
  

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  Start with a two-dimensional approach to better understand the movements of the mattress, i.e., assume it cannot be rotated but must be kept perpendicular to the stairs.
  
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  Work with manipulatives to identify the variables as it moves up the stairs.
  
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  Define a clearance function for the mattress's vertical progress up the stairs as a starting place (The more advanced students might do this step).
  
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  Use "trial and error"—plug in variables and see what Sketchpad diagrams.
  

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  Can you repeat the explanation that group gave, using their diagram?
  
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  How did you know to use that strategy?
  
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  Who has a different strategy?
  
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  Does this drawing match the mathematics they have used?
  
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  Are all variables accounted for? What else might change?
  
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  Can you restate the strategy your classmate used?
  
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  Does everyone agree that they have found the maximum height of the mattress?
  
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  Will the strategies you devised work in any staircase, or are they limited only to the one staircase considered here?
  

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  Model using the Geometer's Sketchpad and how it can reveal a misconception.
  
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  Let students work in groups. Do a process check after groups have identified variables, leading an all-class discussion to clarify all the variables to be considered.
  
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  Use an interactive whiteboard so students can easily revise their models.
  
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  Stop and make sure the entire class understands how the clearance value goes (possible other word choices here-"becomes a" or "changes to a") negative if students make the faulty assumption that the maximum height is when the corner of the mattress is on the floor.
  
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  Prepare in advance some diagrams for the whiteboard of what Sketchpad would show if students miss a variable or set up an equation incorrectly, and correctly, to aid in clarification.