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Prove it! Engaging Teachers as Learners to Enhance Conceptual Understanding

Using a Journal Article as a Professional Development Experience


Title:             Prove it! Engaging Teachers as Learners to Enhance Conceptual Understanding
Authors:       Julie Sweetland and Meghann Fogarty
Journal:        Teaching Children Mathematics
Issue:           September 2008, Volume 15, Issue 2, pp. 68-73.


Rationale/Suggestions for Use 

  • The authors engage teachers as learners in a professional development experience, intended to heighten awareness of the difference between procedural knowledge and conceptual understanding. As teachers deepen their understanding of selected mathematical ideas for which they might have had only procedural knowledge, they develop strategies they can use to teach for understanding in their own classrooms.
  • The need for teachers to deepen their conceptual knowledge of mathematics so they in turn can help children learn with understanding is supported in the Teaching Principle (PSSM, 2000).  It states that “effective teaching requires knowing and understanding mathematics… and effective teaching requires continuing efforts to learn and improve. These efforts include learning about mathematics and pedagogy” (pp. 17, 19).


  • Facilitator will present a task (one of three described in the article) that can be solved by applying a procedure or using a known formula. In each case participants will be asked to “demonstrate their thinking behind the answer”, or as stated in the title of the article, “Prove It!”.  Teachers will reflect individually and in groups on their own thinking as to why the procedure used makes sense, and share strategies they can use to help their children learn with understanding.
    Note: Facilitator may use one of the three tasks shared in the article or ones of own choosing.  Participants may observe that tasks presented are not at a level they would use in their classroom. Explain that this is intentional.  The purpose is to have teachers work on problems beyond the mathematics level they are probably teaching children in their classrooms so that they can better reflect on their own understanding of mathematical concepts.
  • Invite participants to find the answer to 1/5 x 1/4.

    NOTE: This is the same question posed to teachers in the staff development workshop described in the article.
    Allow time to find the answer. Ask selected participants to share their solution and how it was found. Most will know automatically the answer is 1/20 and most will probably say you find the answer by multiplying the numerators and multiplying the denominators.
  • Ask participants, as the author suggested in the article, to “demonstrate the understanding behind the answer” or “show that this answer makes sense”.  
    Allow time for participants working in small groups to demonstrate their understanding of the procedure used to find the answer.
  • Ask selected groups to share their solutions with the whole group. Choose groups that used different solution strategies and reasoning. Highlight with the whole group that there are many different ways to think about the problem, and here, as in the classroom, we share different ways of finding a solution to highlight the diverse ways of thinking among any group of learners.

    NOTE:  If teachers experience difficulty in showing why 1/20 makes sense, you could suggest that 1/5 x 1/4 can be read as 1/5 of 1/4.  Also, if teachers didn’t use the area model as demonstrated in the article (page 69) you can share this with participants. Have manipulatives available for participant’s use, e.g., strips of paper for folding, commercial fraction strips, number lines, or counters.

Reflection and Discussion 

  • Ask participants what was learned while they demonstrated their thinking of the solution to 1/5 x 1/4 and why the rule multiply numerators, multiply denominators makes sense.
    • What was learned from listening to the thinking of others?
    • How was it learned?
    • Were some ways of finding solutions easier to understand?  For example, some may note that they were more comfortable using a concrete model than a drawing. This could lead to a discussion of the importance of recognizing multiple learning styles in the mathematics classroom.
  • Discuss implications the experience of “demonstrate your thinking” might have for teaching in their own classroom. Encourage teachers to share examples from the classroom

    NOTE: If time allows the other two tasks described in the article could also be explored with the participants.

Next Steps 

  • Working in small groups participants choose a concept that would be suitable for the children they are teaching.  Discuss the instructional strategies they could use to help children develop understanding of the concept and develop a plan for teaching the concept.  Participants are encouraged to use the developed plan in their classrooms, record observations, share experiences and what they noticed about children’s learning at the next session.
  • Provide copies of the article (referenced above) to participants and refer them to the following quote on page 71.

    "The teachers talked as they worked, and as the activity came to a close, the facilitator noted how much vocabulary was used during the lesson…. Participants agreed that this was the sort of atmosphere they would like to create in their own classrooms - an environment where mathematical terms come up as frequently as literary terms. Providing a context where students can acquire and use mathematical terminology is a crucial step toward helping students build mathematical communication skills; such skill is an important element of students’ overall mathematical proficiency”.
    • What vocabulary was used in your group as you demonstrated your thinking in making sense of the solution?
    • Share examples of how students use mathematical vocabulary as they share a solution process.
    • What strategies do you use in your classroom to help students acquire and use mathematical terminology while learning mathematics?
  • In the article the author states that: “Just as a student’s productive disposition toward mathematics is essential for mastering content and concepts, teachers  also must be given the chance to develop the sense that they are capable of learning and teaching mathematics” (p.71-72).

    “Productive disposition” is defined as the tendency to see sense in mathematics, to perceive it as both useful and worthwhile, to believe that steady effort in learning mathematics pays off, and to see oneself as an effective learner and doer of mathematics.  ( “Adding It Up: Helping Children Learn Mathematics”, Jeremy Kilpatrick, et al). Ask participants to:
    • Share instructional strategies they use to help children develop a productive disposition towards mathematics.
    • Discuss the quote above as it refers to teachers developing the sense they are capable of doing mathematics. 
      • What opportunities have you had in your own professional learning experiences to do this?
      • How do you think ‘demonstrating your thinking’ in the example explored at the beginning of this session (1/5 x 1/4) contributed to your productive disposition towards mathematics.
  • An excellent resource for teachers is the NCTM publication (2007), Mathematics Teaching Today, which addresses standards for the teaching and learning of mathematics. In the Learning Environment standard, it is stated:

    “A central focus of the classroom environment must be sense making (Hiebert et al, 1997)… Teachers should consistently expect students to explain their ideas, to justify their solutions, and to persevere when they encounter difficulties. Teachers must also help students learn to expect and ask for justifications and explanations from one another. A teacher’s own explanations must similarly focus on underlying meanings; something a teacher says is not necessarily true simply because he or she “said so.” (p. 40).

    NOTE: A copy of this excerpt can be handed out to participants.
  • Working in small groups, invite participants to reflect on the excerpt considering the following:
    • What you find interesting about it and why?
    • What challenges have you had in establishing a sense making environment in your classroom?
    • What supports do you need to help students understand that explaining ideas, justifying solutions, and persevering to find solutions are expected?
    • Share classroom examples where you intentionally focussed on sense making. Discuss both the successes and the challenges.

Connections to Other NCTM Publications 


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