# Manipulatives: The Missing Link in High School Math

*The thrill of achievement comes from overcoming adversity in the accomplishment of an important goal ~ Unknown*

Algebra, Geometry and Trigonometry can be challenging and complex subjects for many high school students in the United States. These subjects are usually taught using textbooks, workbooks and examinations. While these resources are essential in developing learning skills in math, it doesn’t encourage problem-solving skills and retention. It is not surprising to learn many students view math as boring, difficult and irrelevant, rather than fun and interesting. The scope of what it means to be successful in mathematics expands beyond procedural knowledge and skill-acquisition to include sense-making and conceptual understanding (Buckley 2004). An integrated approach is needed to motivate students to learn math in a more relaxed environment, thus eliminating persistent math anxiety, the lack of math application skills and poor standardized test performance. The Third International Mathematics and Science Study (TIMSS) tested the students of 41 nations. Children in the United States were among the leaders in the fourth grade assessments, but by high school graduation, they were almost last. High school math should be taught in a way in which students can grasp and understand skills.* *Manipulatives are visual objects that help illustrate mathematical relationships and applications. These resources are used primarily in elementary schools and somewhat in middle schools. Manipulatives are valuable resources for accelerating and deepening students understanding of math, yet its use is almost non-existent in high school. Marilyn Burns, Creator of Math Solutions has “used manipulative materials at all levels for 30 years.” In every decade since 1940, the National Council of Teachers of Mathematics (NCTM) has encouraged the use of manipulatives at all grade levels (Bellonio), yet many high school teachers are reluctant to use this type of resource.

Almost all mathematics-teaching activities take place at the abstract level (Sharma, 1997). According to Sharma, students have a tendency to forget when taught only at the abstract stage. Thus students become frustrated because mastery was never fully attained. Students will begin to have difficulty in learning mathematics. The results in failure will cause many students to develop a fear of mathematics (Sharma, 1997). Students’ attitudes towards any curriculum area can be related to their achievement in ways that reinforce higher or lower performance (timss.org). Full 88% of Bill and Melinda Gates Foundations’ survey respondents said they had passing grades in high school. Asked to name the reasons they had left school, more respondents named boredom than struggles with coursework. Over 1 million students drop out of school each year. That includes nearly half of all African Americans, Hispanics, and Native Americans who fail to graduate from public high school with their class. Leaving many of them with a host of poor outcomes to follow, from low lifetime earnings to high incarceration rates to a high likelihood that their children will drop out of high school thus eliminating the cycle (Thornburg, 2006).

Sharma (1997) feels there are six levels of mastery of mathematical concepts: intuitive, concrete, representation (pictorial), abstract, applications and communication. According to Sharma, ideally, each mathematics concept should be introduced beginning at the communication level. Manipulatives will teach concrete understanding to the abstract math process, especially when the student may not understand the concept behind the skill. When students develop a concrete understanding of math skills, then they are more likely to perform that math skill and understand math concepts at the abstract level. Manipulatives can make math concepts come alive. According to Spikell (1993), most learners whether adults or children, will master mathematical concepts and skills more readily, if they are presented first in concrete, pictorial and symbols. By using manipulatives, pictures and symbols to model or represent abstract ideas, the stage is set for learners to understand the abstractions they represent (Spikell, 1993). Students will be able to practice and demonstrate mastery using concrete objects. Manipulatives appeal to the learning style of kinesthetic learners because they actually touch the objects. Pictures appeal visually for visual/spatial learners. “Visualization is the natural way one begins to think. Before words, images emerge” (Sharma, 1987). “Almost every mathematics idea, except simple arithmetic facts, consists of three components: linguistic, conceptual and skill/procedural” (Sharma, 1987). Therefore manipulatives offer benefits to a variety of learning styles. They also provide a change from the textbook for mathematical/logical learners.

Games with manipulatives are also valuable with helping students to apply what they learned to the real world, as well as provide a means to improve their math skills interactively. Using board games and card games along with cooperative learning are ways that students can become involved in a positive mathematical environment. Games are highly motivational to students and can be used effectively to practice specific skills. “Using games in the classroom and at home will maximize students’ problem-solving competence, ability to communicate and reason mathematically, perception of the value of mathematics, and self-confidence in their ability to apply mathematical knowledge to new situations.” Cooperative groups provide students a chance to exchange ideas, to ask questions freely, to explain to one another, to clarify ideas in meaningful ways and to express feelings about their learning. These skills acquired at an early age will be greatly beneficial throughout their adult working life.

Many teachers feel as though they do not know how to teach using manipulatives and therefore, hesitate to use them in the classroom. Many math teachers, who attended college more than ten years ago, were usually taught on the abstract level, textbook, pencil and paper, all through their school days. There are classes and workshops for teachers to learn how to teach using manipulatives. The companies that make the manipulatives also provide books and pamphlets on ways the material can be used. There are articles on using manipulatives in mathematics teaching journal such as the National Council of Teachers of Mathematics’ Arithmetic Teacher magazine. Manipulatives help relieve boredom in children allowing them to explore and use their imagination.

Many manipulatives are inexpensive and can be everyday objects. Money, 2 –color counters, calculators, rulers, dominoes, playing cards, button and number cubes are a few of the commonly available manipulatives that can successfully be used in the classroom. These manipulatives can be used to teach such concepts such as angles, area, decimals, factoring, estimation, fractions, measurement, counting, percent, prime numbers, probability, geometry and whole numbers.

There are companies, which specialize in manipulatives that can be ordered from a catalog or online. No matter where a school is located, materials can be made available through the mail. These companies also make available to teachers such manipulatives as tangrams, pattern blocks, fraction towers, geoboards, algebra tiles, Cuisenaire rods, miras and polyhedral models.

When I initially introduced pattern blocks to my Geometry students, I was met with opposition and disbelief. It was a consensus that these students had not used pattern blocks since elementary school. However, many students actually began to explore and build figures with these blocks. I gave them this little bit of play time before taking on the task for that particular lesson, which was angle measurement of polygons. We must remember, everyone likes to play games even adults, look at our various hobbies, such as bowling, tennis, golf, art, race cars, chess, playing cards, etc. This holds true for high school students, they still enjoy playing games. Manipulatives require a great deal of prior planning and organization. But considering the many benefits that manipulatives offer, it is well worth the effort. Students will find math more attainable, fun and exciting, hopefully returning them to the prior successes of their early math experiences.

### Research Evidence

In recent years, a growing body of research evidence and case studies has shed light on the effectiveness of using manipulatives in high school math education. One notable study, the Third International Mathematics and Science Study (TIMSS), highlighted the stark decline in mathematical performance among high school students in the United States compared to their peers in other countries. This decline has sparked interest in alternative teaching methods, including the incorporation of manipulatives.

Numerous educators and researchers have conducted studies to explore the impact of manipulatives on student learning outcomes. For example, Sharma (1997) emphasizes the importance of concrete understanding in mathematics education and proposes a six-level mastery framework, with manipulatives playing a crucial role in facilitating concrete understanding.

Additionally, Marilyn Burns, the creator of Math Solutions, has advocated for the use of manipulatives at all grade levels for over three decades, citing their ability to deepen students’ understanding of mathematical concepts. Moreover, Thornburg (2006) highlights the correlation between student engagement and dropout rates, underscoring the importance of implementing engaging teaching methods like manipulatives to combat student disengagement.

These research findings collectively suggest that the use of manipulatives in high school math classrooms can lead to improved student engagement, deeper conceptual understanding, and ultimately, enhanced academic performance.

**References**

Buckley, L. A. (2004, Oct) Course Taking and Equity: The Efforts of One High School Mathematics Department *Paper presented at the annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Delta Chelsea Hotel, Toronto, Ontario, Canada* Online.

Burns, M. How to make the most of math manipulatives – A fresh look at getting students’ heads — and hands! — around math concepts.

Sharma, M. (1987). How to take a child from concrete to abstract. Math Note book, 5, 8-10

Sharma, M. (1997, July). Improving mathematics instruction for all. Fourth Lecture in Colloquium “Improving Schools from Within: Your Role. Pp. 2-12

Spikell, M. (1993). Teaching mathematics with manipulatives: A resource of activities for the K-12 teacher. New York: Allyn and Bacon.

Third International Mathematics and Science Study (TIMSS), Institute of Educational Sciences, United States Department of Education

Thornburg, N. (2006), April 7), Dropout Nation, Time Magazine

TIMSS 1999 International Mathematics Report. Findings from IEA’s Repeat of the Third International Mathematics and Science Study at the Eighth Grade. Ina V.S. Mullis, Michael O. Martin, Eugenio J. Gonzalez, Kelvin D. Gregory, Robert A. Garden, Kathleen M. O’Connor, Steven J. Chrostowski, Teresa A. Smith.

Yale-New Haven Teachers Institute Home Multi-Sensory Manipulatives in Mathematics: Linking the Abstract to the Concrete by Judith L. Bellonio Contents of Curriculum Unit 01.06.12: Narrative Math Anxiety Types of …

Marilyn Curtain-Phillips teaches high school mathematics and is the author of several books on math education.