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Making Mathematical Arguments in the Primary Grades: The Importance of Explaining and Justifying Ideas

Using a Journal Article as a Professional Development Experience
Reasoning & Proof 

PDF 

Title:             Making Mathematical Arguments in the Primary Grades: The Importance of Explaining and Justifying Ideas
Author:         Joy Whitenack and Erna Yackel
Journal:        Teaching Children Mathematics
Issue:           May 2002, Volume 8, Issue 9, pp. 524-527

Rationale/Suggestions for Use 

Principles and Standards for School Mathematics (2000) states that “instructional programs should enable students to develop and evaluate mathematical arguments and proofs” (p. 56). Furthermore the Reasoning and Proof Standard for Grades 3 -5 states that “mathematical reasoning develops in classrooms where students are encouraged to put forth their own ideas for examination. Teachers and student should be open to questions, reactions, and elaborations from others in the classroom.  Students need to explain and justify their thinking and learn how to detect fallacies and critique others thinking.” (NCTM, 2000, p. 188).

This article explores how teachers might encourage students to explain and justify their reasoning during class discussions and as the author states to look at the “complex and important roles that both explanations and justification play as students develop arguments during discussion” (p. 524). Participants, while exploring the views of the author related to explaining and justifying reasoning at the classroom level, will reflect on implications for their own practice.

Procedures 

  1. Introduction to Reasoning and Proof: Provide a copy of the Introduction to the Reasoning and Proof Standard for Grades PreK– 2 and 3 -5 from Principles and Standards for School Mathematics (NCTM, 2000), i.e. Page 122 and Page 188. 
    • Divide participants into two groups.  Assign the first group to read the Introduction to the Reasoning and Proof Standard for Grades PreK – 2 (p. 122) and the second group to read the Introduction to the Reasoning and Proof Standard for Grades 3 - 5 (p. 188). Ask them to consider how the information in these introductory sections compares to their own understanding of and experiences with reasoning and proof at these grade levels.
    • Facilitate a whole group discussion on reasoning and proof in PreK- 5, asking participants how the information read connects to or expands their own knowledge of reasoning and proof in the mathematics classroom.
     
  2. Provide copies of the article, Making Mathematical Arguments in the Primary Grades: the Importance of Explaining and Justifying Idea to all participants.  In table groups invite participants to read the introduction to the article on page 524 to the end of the first column on page 524.
    • Ask participants, still working in table group, to discuss the following (these discussion questions could be prepared beforehand and a copy placed on the table):
      • What do you see as the difference between explaining and justifying your reasoning?
      • What aspects of the dialogue among the students in Ms. Jones class showed the difference between explaining and justifying?
      • Are there some other questions you as a teacher would ask of Casey to encourage him to explain and justify his reasoning further and to make it clearer for the other students in the class?
      • How did the class discussion on the solution to the problem discussed help further children’s mathematical understanding?
       
    • Facilitate  a whole group discussion by asking participants to share ideas about the authors statement at the end of the section page 525, “ In sum, when students make mathematical arguments, they do not simply share their answers; instead, they explain and justify the ideas that they had as they thought about and solved the problem” Encourage participants
      • To relate their discussion to the classroom conversation among Ms. Jones and her students, especially distinguishing between explaining and justifying ideas.
      • To identify challenges presented in their own practice when children are asked to explain and justify solutions.
       
     
  3. Allow time for participants to read the section titled, Developing Arguments during Mathematics Instruction, pages 525 -526. 
    • In table groups discuss the following :
      • One idea in this section that most challenged your thinking.
      • Strategies used in your own practice to create an environment where children feel safe to explain and justify their reasoning.
      • The author refers to the teacher initiating discussions in which the class ‘talks about talking about mathematics’.  What do you see as the benefits and challenges of doing this in your own practice?
      • One idea in this section that you would like to know more about.
       
    • Facilitate a whole group discussion asking table groups to share ideas on how to develop, foster, and promote a mathematical environment in which students explain and justify their ideas.
     
  4. Summarize the session by asking participants to respond to the question “After our discussions on the article, what are the implications for your own practice?” Allow time for participants to think about the question, and then share their ideas, first with a partner or table and then with the whole group. Facilitator can chart ideas shared to refer to at a future session. 

Next Steps 

  1. Ask participants to record and reflect on children’s conversations as they explain and justify their solution to an assigned problem of the participants’ choice. If there is an opportunity for a follow-up session ask participants to share ‘aha’ experiences from student’s conversations and also the challenges they met in facilitating the conversations.
  2. Give each participant a copy of the following quotes taken from the Reasoning and Proof Standards in Principles and Standards for School Mathematics (NCTM, 2000). Pair participants, preferably so that there is one whose main experience is Grades Prek-2 and the other Grades 3-5.
    • Ask each pair to discuss the quotes below and select two that best represents the ideas expressed by the author in the article, Making Mathematical Arguments in the Primary Grades: the Importance of Explaining and Justifying Ideas.  Facilitator needs to provide a copy of the quotes to all participants.
      • “From children’s earliest experiences with mathematics, it is important to help them understand that assertions should always have reasons. Questions such as ‘why do you think it is true?’ And ‘does anyone think the answer is different, and why do you think so?’ help student see that statements need to be supported or refuted by evidence” (p. 56)
      • “Beginning in the earliest years, teachers can help students learn to make conjectures by asking questions: what do you think will happen next? What is the pattern? Is this true always? Sometimes?” (p. 57)
      • “Teachers should prompt students to make and investigate mathematical conjectures by asking questions that encourage them to build on what they already know” (p. 125)
      • “When students make a discovery or determine a fact, rather than tell them whether it holds for all numbers or if it is correct, the teacher should help students make determination themselves. Teachers should ask such questions as “How do you know it is true?” and should also model ways that students can verify or disprove their conjectures.” (p. 126)
      • “Mathematical reasoning develops in classrooms where students are encouraged to put forth their own ideas for examination. Teachers and student should be open to questions, reactions, and elaborations from others in the classroom.  Students need to explain and justify their thinking and learn how to detect fallacies and critique others thinking.” (p. 188)
      • “During grades 3 -5, students should move toward reasoning that depends on relationships and properties. Students need to be challenged with questions, such as, What if I gave you twenty more problems like this to do – would they all work the same way?  How do you know?” (p. 190)
      • “Even students who seem to have developed a clear argument about a mathematical relationship need to be questioned and challenged when they are ready to encounter new aspects of the relationship” (p. 192)
      • “ Being able to explain one’s thinking by stating reasons is an important skill for formal reasoning that begins at this level” (p. 123)
      • “Mathematical reasoning develops in classrooms where students are encouraged to put forth their own ideas for examination…. Students need to explain and justify their thinking and learn how to detect fallacies and critique others thinking They need to have ample opportunity to apply their reasoning skills and justify their thinking in mathematics discussions” (p. 188)
       
    • Facilitate a whole group discussion by asking each pair to share the quote they selected and why they selected it. Facilitator should chart ideas shared.
     

Connections to Other NCTM Publications 

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