Stephanie Cambrea's Portfolio Page

Navigation links: ETAP 623 Spring 2023 (Zhang) | Stephanie Cambrea: Hands-On Manipulatives in Mathematics

About Me

My name is Stephanie Cambrea and this will be my final semester at UAlbany. I graduated from Siena College in May 2020 with a certification in Mathematics Education grades 7-12 and an extension for grades 5-6. In September 2020, I started working at Farnsworth Middle School in the Guilderland School District as an AIS Math teacher. This year I am an eighth-grade math classroom teacher at FMS where I teach accelerated algebra and Math 8. I have been very grateful to use this year to improve my teaching practices. I am currently working toward my Master's in Curriculum Development and Instructional Technology.

When I am not teaching, I enjoy hiking and watching the Yankees. My goal is to go to all 30 Major League Baseball Stadiums. I have been to four so far and plan to go to at least two more this summer.

My Topic and Purpose

The topic of my mini-course is Hands-On Manipulatives in Mathematics. The goal is encourage the use of various types of hands-on manipulatives in mathematics instruction to promote higher levels of mathematics reasoning, sense-making, critical thinking, and over all enhancing the way that students learn. The purpose of this mini-course is to educate mathematics teachers about the benefits of using hands-on manipulatives to promote true understanding of concepts and mathematics reasoning rather than memorization. It will also inform teachers about the ways hands-on manipulatives can be incorporated into lesson plans so they feel more prepared to use them in their future instruction. More specific details about my topic and purpose can be found on my project page.

Scope of Learning Outcomes

The scope of learning outcomes will take place over the span of four units in my mini-course. In the first unit, participants will read scholarly articles about the advantages of hands-on manipulatives in mathematics and why it would be beneficial to include them in instructional design. The second and third unit focus on teaching participants how to use various types of hands-on manipulatives to support learning in both algebra and geometry topics. Participates will watch videos to learn how to use the hands-on manipulatives themselves so they can see first hand how it helps students increase their understanding and get inspiration for future lesson ideas involving the hands-on manipulatives. In the fourth and final unit, participants will continue to reflect on their learning as they apply their new knowledge to create an instructional video where they model the use of hands-on manipulatives to teach a concept of their choosing on which they will receive feedback from their classmates.

By the end of the mini-course, participants will be able to:

Acknowledge the benefits of using hands-on manipulatives to teach mathematics
Connect the use of hands-on manipulatives to learning with understanding
Explore different examples of hands-on manipulatives and how they work
Describe multiple ways that hands-on manipulatives can be used in algebra and geometry lessons
Identify the key strategies to successfully implementing hands-on manipulatives into mathematics instruction

Needs Assessment

Educational Problem

When mathematics instruction is procedural, students struggle to apply their learning to real-world situations. According to Sarwadi & Shahrill (2014), "teaching the students only procedural skills will impair learning in the classroom and will not equip students well with the necessary skills mathematically for the future" (p.2). It is not enough for students to be able to successful find a solution to a problem as they must also be able to truly understand where that solutions comes from. Students also need to understanding the mathematical reasoning behind concepts to develop their problem-solving skills instead of just memorizing the steps to solve a problem. After all, "it is partly true that students will be able to do computation if they are drilled but will not be able to do problem solving and application questions properly because the latter demands both procedural and conceptual understanding" (Sarwadi & Shahrill, 2014, p.2). While many math teachers are effective in supporting procedural understanding in students, they many struggle with the best way to promote conceptual understanding.

One of the best ways to teach students about the mathematical reasoning behind concepts is through the use of hands-on manipulatives. With this being said, "to be effective, however, simply placing one’s hands on the manipulative materials will not magically impart mathematical understanding. Without the appropriate discussion and teaching to make the links to the mathematics explicit, the very opposite may be true; children may end up with mathematical misconceptions" (Swan & Marshall, 2010, p.19). This means that simply having students use hands-on manipulatives during a lesson is not enough to increase their understanding. Teachers must be educated about how to effectively implement hands-on manipulatives into their instruction and use it to support student learning.

Learners/Participants

Middle and high school math teachers, as well as some special education teachers, were given a Google Form were they were asked to describe their experience and confidence levels about using hands-on manipulatives in their instruction. Fifteen Google Forms were distributed and twelve responses were collected.

Analysis of Gaps

The purpose of the Google form was to gather information about the prior knowledge that teachers already have regarding effectively using hands-on manipulatives in mathematics classrooms and discover the gaps in knowledge the exist. Teachers were asked to share examples of hands-on manipulatives that they were familiar with using and reflect on the impact the tools had on their students' learning. The two most common examples shared where number lines for teaching integer rules and base-ten blocks for teaching place value and multiplication. Some of the special education teachers stated that they were somewhat familiar with using hands-on manipulatives at the elementary level, but struggled to see how hands-on manipulatives related to higher-level math topics, such as solving equations. Participants were also asked to rate how confident they felt using hands-on manipulatives to promote conceptual understanding and mathematical reasoning, to which most responded "No Confidence" or "Little Confidence". When participants were asked to share specific topics where they start using hands-on manipulatives in their instruction, responses included but were no limited to volume/surface of three-dimensional figures, solving multi-step equations/inequalities, and various triangle theorems. Thus, the results of the needs assessment reflect that teachers would benefit from a mini-course on the use of hands-on manipulatives in mathematics. Teachers' responses to the Google Form were all taken into considered as the mini-course was designed.

Intent Statement

The goal of this mini-course is to expose students to various types of hands-on manipulatives that can be used in a mathematics classroom to strengthen students understanding. Participants will become more familiar with how to use hands-on manipulatives so they can incorporate them into their future lesson plans to improve their instruction. Participates will have multiple opportunities throughout the mini-course to analyze the use of hands-on manipulatives in videos, create their own lesson plans that use hands-on manipulatives to promote mathematical reasoning, and reflect on their learning.

Analysis of the Learner and Context

There are two main groups of learners who will participate in this mini-course. The first group is middle and high school math teachers, including AIS math teachers. The second group are middle and high school special education teachers who serve as co-teachers in mathematics classrooms. All of the participants have solid math content knowledge with at least five years of experience, but have little to no experience with incorporating hands-on manipulatives into their instruction. The participants have expressed interest in wanting to engage in this mini-course to enhance their knowledge of hands-on manipulatives to improve their future teaching. Some participants have background knowledge with some examples of hands-on manipulatives, such as number lines and base-ten blocks, but are still looking for ways to expand their knowledge. The mini-course will serve as a professional development opportunity as participants build on their experience with hands-on manipulatives to introduce new manipulatives and resources that can be used to strengthen students mathematical reasoning.

The context of this mini-course supports teachers learning how to successfully implement hands-on manipulatives into their mathematics lessons. After learning about why hands-on manipulatives are beneficial to student learning in the first unit, participants will learn about how to use hands-on manipulatives to promote greater levels of understanding in various algebra and geometry topics in the second and third unit. During these units, participants will have the opportunity to collaborate with each other in small groups to design their own lesson plans that incorporate the use of hands-on manipulatives in a meaningful way. Ideally, participants will share their ideas as they learn from each other's strengths as math teachers and special education teachers will be grouped together. This will continue in the final unit when participants create an instructional video involving hands-on manipulatives and receive feedback from their peers. The overall goal is for participants to become more familiar and confident with how to enhance students' conceptual understanding in mathematics through the use of hands-on manipulatives and give them inspiration for how to improve their existing lesson plans with the new ideas learning throughout the mini-course.

Performance-Based Objectives

After completing this mini-course, learners will be able to:

Understand the benefits of using hands-on manipulatives in mathematical instruction and how to introduce them to students.
Analyze given videos that demonstrate the use of hands-on manipulatives and explain how they enhance student learning.
Apply learning to design lesson plans that involves students using hands-on manipulatives to promote conceptual understanding
Create an instructional video that successfully incorporates the use of hands-on manipulatives to teach a mathematical topic
Reflect on learning experiences throughout the course and evaluate classmates' instructional videos to provide feedback

More specific performance objectives can be found on my project page.

Task Analysis

Unit 1: Research on Hands-On Manipulatives in Mathematics

At the end of this unit, participants will be able to:

Understand the advantages of using hands-on manipulatives to enhance students' mathematical understanding
Identify key strategies for implementing hands-on manipulatives in a math lesson for the first time
Interpret scholar articles and explain the ways the research supports the use of hands-on manipulatives in mathematics instruction
Reflect on the reasons why they should implement hands-on manipulatives in their teaching and how they can do it successfully

Unit 2: Examples of Hands-On Manipulatives in Algebra Topics

At the end of this unit, participants will be able to:

Identify hands-on manipulatives that specifically apply to algebra topics
Analyze and explain how the use of hands-on manipulatives is beneficial to student learning and adds value to the lesson
Reflect on their own teaching experiences and how to improve their future mathematics instruction
Create a lesson plan on a given algebra topic that effectively incorporates the use of hands-on manipulatives to support student learning

Unit 3: Examples of Hands-On Manipulatives in Geometry Topics

At the end of this unit, participants will be able to:

Identify hands-on manipulatives for multiple geometry topics
Analyze and explain how the use of hands-on manipulatives is beneficial to student learning and adds value to the lesson
Reflect on their own teaching experiences and how to improve their future mathematics instruction
Design a lesson plan on a given geometry topic that successfully utilize the use of hands-on manipulatives to mathematical reasoning

Unit 4: Putting It Into Practice

At the end of this unit, participants will be able to:

Create an instructional video that involves hands-on manipulatives to model any mathematics topic
Analyze their classmates' instructional videos presentation and provide them with helpful feedback
Reflect on their major takeaways from the course and how it will impact their teaching moving forward

Curriculum Map

References and Resources

Collins, A. (1996). Design issues for learning environments. In S. Vosniadou, E. De Court, R. Glaser, & H. Mandl (Eds.), International perspectives on the design of technology-supported learning environments (pp. 347-361).

Garzón, J. A., & Bautista, J. (2018). Virtual Algebra Tiles: A pedagogical tool to teach and learn algebra through geometry. Journal of Computer Assisted Learning, 34(6), 876–883.

Lehrer, R., & Schauble, L. (2000). Developing Model-Based Reasoning in Mathematics and Science. Journal of Applied Developmental Psychology, 21(1), 39–48.

Perkins, D. L., & Blythe, T. (1994). Putting Understanding Up Front. Educational Leadership, 51(5), 1–5.

Sarwadi, H., & Shahrill, M. (2014). Understanding Students’ Mathematical Errors and Misconceptions: The Case of Year 11 Repeating Students. Mathematics Education Trends and Research, 2014, 1–10.

Starr, J. (2022, March 7). Manipulatives in Math: Hands-On Strategies to Scaffold Instruction. Teaching with Jillian Starr. https://jillianstarrteaching.com/manipulatives-in-math/

Swan, P., & Marshall, L. (2010). Revisiting Mathematics Manipulative Materials. Australian Primary Mathematics Classroom, 15(2), 13–19.