Essential Questions
How can using manipulatives make mathematics more accessible to students?
Learning Goals
Participants will experience mathematical concepts by using manipulatives to solve non-routine tasks.
Participants will explain how these experiences can make mathematical ideas more accessible to all students and develop students' conceptual foundations about mathematics.
Participants will create a lesson or a non-routine task using manipulatives to make the concept more accessible and foster students' conceptual understanding.
Materials List
"Mathematical Statements" for the Magnetic Statements
"Math Manipulative Centers" note sheet
Activity and Manipulative Handouts found in the ATTACHMENTS
Manipulatives and Activity handouts for "Probability Centers"
Manipulatives and Activity handouts for the LEARN lesson, "Pythagor-eatin' Theorem"
Notecards (four for each participant)
"SCORE Reflection" note sheet
Engage
Begin with slide two, displaying the professional development title. Welcome participants and briefly introduce yourself and the session. Objectives and goals will be addressed after the opening activity.
Transition to slide three, introducing the "Mathematical Statements" for the Magnetic Statements strategy. Participants will read the statements on slide four and identify the one that they are either most attracted to or most repelled by, then move to stand next to that statement. (Note: in a small session, participants may remain seated and select the statement they feel most attracted to or repelled by. These can be shared out individually to the whole group or they may turn and talk with an Elbow Partner.) Once participants understand the objective, display the statements and allow them time to find a statement to stand by. After groups have formed around the statements, instruct participants to share within their group why they chose that particular statement. One person will summarize what was discussed to the whole group.
Ask participants to return to their seats. Then briefly share the essential question and learning objectives for the session, found on slides five and six. Participants should keep these in mind as they explore each activity using manipulatives. Transition to slide seven, reading the quote. Explain that we will begin exploring different mathematical activities that use manipulatives, but not all activities can foster deep thinking about mathematical concepts. It is important to be intentional with the tasks selected. Tasks and activities must open and provoke deep thinking and questioning about mathematical ideas, concepts, and relationships.
Explore
Provide three to four activity and manipulative centers for the participant groups to rotate between. Each small group will explore each center for about 10 minutes and then use 5 minutes to reflect and answer the two prompts on the "Math Manipulative Centers" note sheet. Groups rotate after 15 minutes (time must be monitored and notify groups when the first 10 minutes have passed, and reflections should begin).
Transition to slide eight, introducing each of the mathematical activities and manipulatives that each group will be exploring during the rotations. Also, provide instruction regarding the "Math Manipulative Centers" note sheet. At each center, use this note sheet to record the following (shown on slide nine and on the top of the note sheet):
How does each manipulative develop and foster students’ conceptual understanding and thinking about mathematical ideas?
How does each activity open mathematics, making it accessible to all students?
After the instructions have been given, have participants explore each activity in timed rotations. (Note: see attached "Activity and Manipulative Handouts" for all optional rotations, resources, and instructions.)
Explain
After participants complete the assigned centers, display slide 10, introducing the 3-2-1 strategy. Invite participants to individually reflect on the manipulatives and activities they explored, recording their thoughts on the designated "3-2-1" handout. After a few minutes of personal reflection, invite participants to share (either in small groups or as a whole) how manipulatives, in general, can make mathematics more accessible and foster students' conceptual mathematical thinking and mindset.
Extend
After participants have reflected and shared the benefits of using manipulatives, transition to slide 11. Participants will now select a topic they plan to teach in the next two to three weeks and begin to develop a lesson outline/plan or a non-routine task that incorporates manipulatives. The lesson/task should utilize manipulatives to make mathematical ideas and relationships more accessible to all and fosters students’ conceptual foundation. This is protected time to apply new knowledge into their practice.
While groups work, display slide 12. Participants will summarize their lesson/task in preparation to share with other participants. Encourage participants to think deeply about numbers three and four as they will expand and connect to those ideas during the Evaluate portion of the session.
If participants need guidance, ask them some of the questions below:
Which standards and mathematical ideas will be addressed by the lesson/task?
How much time can you allot? Be realistic, if the lesson/task takes a whole week and addresses one standard then it won’t be worth it. However, if it addresses multiple standards and takes a week, ask yourself, "How deep are students learning the mathematical ideas?" It may be worthwhile if mathematical ideas are being explored deeper.
How will manipulatives open mathematical ideas to all students, and how does the lesson/task develop and support their mathematical conceptual foundation?
How might you structure and prepare your class for this lesson/task? Should the seating/student groups be rearranged? Do students need to co-create norms for using manipulatives?
After participants have had at least 15 to 20 minutes of protected planning time, draw their attention to the notecards on the tables and change to slide 13. Participants will summarize their lesson/task using the details highlighted on slide 13 in preparation to share with other participants. (Note: each participant will record the exact same thing on four notecards. These notecards will be traded with others during the following activity.)
Transition to slide 14. Have participants mix and mingle around the room, locating three other participants with whom they will briefly describe their lesson/task and then exchange one copy of their notecards. From this activity, participants will gain three additional lesson/task ideas that use manipulatives to make mathematics more accessible to students and develop students' conceptual understanding.
Evaluate
Finally, change to slide 15, instruct participants to revisit the "Mathematical Statements" introduced at the beginning of the session. Participants will now move to the statement they believe will be most supported by or associated with their lesson/task (or a lesson/task they received from another participant). Have participants stand by the chosen statement. (Note: in a smaller session, participants may remain seated and select the statement they feel most strongly towards.)
Instruct participants to share their reasoning for choosing this statement within the statement groups. Ask one person from each group to summarize what was discussed to the whole group, connecting back to why these lessons and tasks support the selected statement. (Note: if participants struggle to connect to statement, ask them to specifically look at numbers three and four from the notecards. If they continue to struggle, ask them to consider how they might modify the lesson/task so at least one statement is better supported.)
Follow-up Activities
Display slide 17 as participants filter into the meeting. Once you and the participants are ready to begin the meeting, change to slide 18. Instruct participants to use the "SCORE Reflection Note Sheet" to record reflective notes about their own experience implementing any lessons and tasks that used the manipulatives explored from the professional development session.
After participants have reflected and recorded their thoughts, facilitate a small group share-out/discussion providing each attendee time to highlight the manipulatives they used and their students' overall experience. The questions on slide 18 mirror the "SCORE Reflection Note Sheet" and should be used to guide the discussion.
Encourage attendees to use another manipulative and continue to follow up with each participant, cultivating a safe environment for accountability and growth.
Research Rationale
Boaler (2016) suggests that one thing teachers must do in order to support and develop a mathematical mindset within students is to show them how mathematics is a visual and beautiful subject. Naturally, the human brain wants to think visually about mathematics. Therefore, instruction, opportunities to learn and explain mathematics visually must be presented to and explored by students. Complex mathematical ideas are abstract, but all mathematics originates from the concrete. Teachers must allow students time to construct meaning with the visual and concrete world of mathematics and manipulatives are one way for students to explore mathematics. We often move too quickly from the concrete to the abstract. Children need time to play, manipulate, and represent their thinking and understanding in order to fully process and engage in meaningful mathematics. Students should have time to create and discover mathematical patterns and relationships which connect to a larger mathematical idea or concept. Many researchers (Boaler, 2016; Boaler, 2014; Smith & Stein, 2018; Wheatley,1991) discuss how rich mathematical tasks, when properly supported by teacher facilitation and a carefully cultivated classroom environment which permits learning to be constructed in a natural, authentic process that is shared by students and teachers equally (Dewey, 1956, Freire; 1970, 2005; Boaler, 2016; Davis, 1997), can empower students who typically struggle in the traditional mathematics classroom. Furthermore, these tasks can change students’ perspectives and mindsets about mathematics. Pairing manipulatives with rich mathematical tasks encourage engagement and make mathematics more accessible to all students.
Resources
Boaler, J. (2014). Research suggests that timed tests cause math anxiety. Teaching Children Mathematics, 20(8), 469-474.
Boaler, J. (2015). Mathematical mindsets: Unleashing students' potential through creative math, inspiring messages and innovative teaching. John Wiley & Sons.
Davis, B. (1997). Listening for differences: An evolving conception of mathematics teaching. Journal for Research in Mathematics Education, 28(3), 355-376.
Dewey, J. (1956). The child and the curriculum and the school and society (Combined ed.).
Freire, P. (1970, 2005). Pedagogy of the oppressed 30th-anniversary edition. Translated by Myra Bergman Ramos. New York: Continuum.
K20 Center. (n.d.). 3-2-1. Strategies. Retrieved from https://learn.k20center.ou.edu/strategy/d9908066f654727934df7bf4f5059a7b
K20 Center. (n.d.). Elbow partners. Strategies. Retrieved from https://learn.k20center.ou.edu/strategy/ccc07ea2d6099763c2dbc9d05b00c4b4
K20 Center. (n.d.). Magnetic statements. Strategies. Retrieved from https://learn.k20center.ou.edu/strategy/d9908066f654727934df7bf4f50761bf
K20 Center. (n.d.). Pythagor-eatin' theorem. Constructing Pythagorean Theorem. Retrieved from https://learn.k20center.ou.edu/lesson/bb01792b8b7ae172f01b1c728a02b5ba
Smith, M. S., and Stein, M. K. (2018). 5 practices for orchestrating productive mathematics discussions. Reston, VA: National Council of Teachers of Mathematics.
Stein, M. K., Smith, M. S., Henningsen, M. A., & Edward, A. Silver. 2000. Implementing Standards-Based Mathematics Instruction: A Casebook for Professional Development.
Wheatley, G. H. (1991). Constructivist perspectives on science and mathematics learning. Science Education, 75(1), 9-21.
Youcubed. (n.d). Positive classroom norms poster. Ideas & impact. Growth mindset. Standford Graduate School of Education. Retrieved from https://bhi61nm2cr3mkdgk1dtaov18-wpengine.netdna-ssl.com/wp-content/uploads/2017/03/Norms-Poster-2015.pdf OR https://www.youcubed.org/resource/growth-mindset/