Stephanie Cambrea's mini-course
Hands-On Manipulatives in Mathematics
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Overview and Purpose
Many mathematics teachers at the middle and high school level have started to move away from the traditional instructional approaches of direct instruction followed by drill-and-kill practice towards designing lessons where students learn with understanding (Perkins & Blythe, 1994). When students learn with understanding, the focus is on comprehending the mathematic reasoning that allowed them to arrive at the solution instead of just memorizing a list of steps and procedures to get the correct answer. There is a difference between knowing what the correct solution should be and actually understanding what that solutions means and being able to apply it to other situations (Collins, 1996). An effective way for mathematics teachers to promote higher-levels of understanding in their classroom is through the use of hands-on manipulatives.
This mini-course will be discussing how the use of hands-on manipulatives is beneficial to supporting students understanding mathematical concepts. Hands-on manipulatives allow students to create a visual represent of a difficult abstract concept to make it more concrete. Hands-on manipulatives also gives students the opportunity to work through a problem on their own as they create evidence of their thinking and sense-making during the learning process. The purpose of this mini-course is to education middle and high school mathematics teachers about the benefits of hands-on manipulatives and effective strategies for utilizing them so they can successfully implement them in future lesson. This mini-course would also be informative for AIS Math teachers, special education teachers, and ENL teachers who also work to support students learning needs in mathematics.
Needs Assessment and Learner Analysis
Part 1: The 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.
Part 2: Gathering Information
The purpose of this needs assessment 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. 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. They 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.
Part 3: Learner Analysis
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.
Performance Objectives
- Participants will be able to understand the benefits of using hands-on manipulatives in mathematical instruction and how it enhances student learning.
- When given videos showing different types of manipulatives used in math learning, participants will be able to analyze how the use of hands-on manipulatives supports mathematical reasoning.
- Participants will be able to apply their learning to design multiple lesson plans that involves students using hands-on manipulatives in meaningful ways to promote greater conceptual understanding.
- Participants will be able to create their own instructional video that successfully incorporates the use of hands-on manipulatives to teach a mathematical topic.
Course Units
This mini-course includes the following units. Click the title of a unit to go to its page.
Unit 1: Research on Hands-on Manipulatives in Mathematics
In the first unit of the mini-course, participants will learn about the benefits of hands-on manipulatives in mathematics and reflect on why they should start using them in their classrooms. Various scholarly articles with educate participants about how hands-on manipulatives can improve instructional design by promoting learning for understanding and higher levels of mathematical reasoning. Participants will also learn about strategies to effectively help them implement hands-on manipulatives into their teaching for the first time. A scholarly article will inform participants how to best introduce hands-on manipulatives to students and model how to use them.
Unit 2: Examples of Hands-On Manipulatives in Algebra Topics
In the second unit of the mini-course, participants will explore how hands-on manipulatives can be used to strength understanding of various algebra topics, such as adding/subtracting integers, solving two-step questions, multiplying polynomials, and more. The unit ends with participants working in small groups to create a lesson plan for a given algebra topic that incorporates the use of hands-on manipulatives to support mathematical understanding to demonstrate their learning.
Unit 3: Examples of Hands-On Manipulatives in Geometry Topics
In the third unit of the mini-course, participants will discover how hands-on manipulatives can be used to enhance comprehension of various geometry topics, including surface area and volume of three-dimensional figures, different types of transformations, and more. The unit ends with participants working in small groups to create a lesson plan for a given geometry topic that incorporates the use of hands-on manipulatives to support mathematical understanding to demonstrate their learning.
Unit 4: Putting It Into Practice
In the final unit of the mini-course, participants will reflect on their learning as they create a 5-10 minute instructional video on a topic of their choosing. When participants share their instructional video with the rest of the class, they will have the opportunity to give and receive feedback on how the implementation of hands-on manipulatives was successful in their video and how it promoted greater levels of understanding from students.
Extended 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).
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.
Swan, P., & Marshall, L. (2010). Revisiting Mathematics Manipulative Materials. Australian Primary Mathematics Classroom, 15(2), 13–19.