Last Updated on October 7, 2023 by Editorial Team

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Mathematics is not only about whole numbers, fractions, and decimals. There are certain values that are infinitesimally small and denote quantities such as a change in area or change in speed, etc. Calculus is the branch of maths that primarily deals with the changes and also helps determine the rate at which the changes take place.

It enters our lives in high school and almost looks like a concept from an alien world. The fact remains true that many students feel anxious about transitioning to calculus. Thankfully, there are manipulatives available that can help replace anxiety with curiosity and make the introductory phase pleasing for calculus beginners.

In this post, let’s explore the engaging manipulatives that calculus beginners can use to grasp concepts like integration, differentiation, etc. Before jumping on to the list, let’s talk about the reasons manipulatives have risen to glory among calculus teachers and learners.

**Using manipulatives: How it is changing calculus class environment**

Teaching calculus to beginners at the high school level feels challenging. Students find it difficult to relate the concepts to real-life situations around them. Also, they may feel lost in the technical jargon used in calculus. It is where the manipulatives prove their worth and make the classroom session more learning-oriented. Positive outcomes delivered are:

**Multisensory learning possible to achieve:**Manipulatives offer practical training through stimulating multiple senses. A study^{[1]}conducted on teaching integral through construction paper and scissors shows that the students build in themselves comfort with the terms. They grasp the practical meaning associated with the basic properties of the integral.**Active learning enhances the ingraining of concepts:**While using manipulatives, students can grasp the tangible aspect of the calculus concepts. Since they learn by doing, they are not burdened by the idea of memorizing concepts. In fact, they are driven to investigate^{[2]}to learn relationships and functions in an in-depth manner.**Allow learning by practical application:**The students can understand what all those signs of integration and differentiation implicate. By way of doing activities and trials, manipulatives drive critical thinking in learners and promote ease with the practical application of concepts.

So, why wait further? We have annotated here the best manipulatives to impart knowledge of basic Calculus concepts to beginners. Let’s take a look:

**Engaging manipulatives to impart calculus training to beginners**

**1. Taped Bowls with Paper-mache or plaster of Paris surfaces**

Aaron Wangberg of Winona State University came forth with the idea of creating dry-erase surfaces to use as manipulatives. He explains in the interview how he found these surfaces quite useful and practical to apply in introducing multivariable calculus to students. These surfaces proved effective in learning gradients and partial derivatives.

Usually, the students can employ a formula to find the gradient vector, but they cannot manipulate where it actually points. This hands-on manipulative allowed the students to experiment with different values and they were astounded to find that the gradient vector points in the same direction irrespective of the changes in data. Thus, this manipulative helped fill the gap in the practical application of the concept. Wangberg evolved this idea further and finally created more sophisticated models that are now used in various institutions.

**2. AP Calculus Flash Cards**

AP Calculus Flashcards offer a dependable resource for improving test-taking skills. About 425 flashcards added to this product allow students to brush up on all basic and advanced calculus facts. The users can employ these flashcards to create quizzes and practice tests for each other.

Trying to learn or memorize calculus facts alone can prove to be quite overwhelming at times. This flashcard set promotes collaborative learning and students can come together and share their knowledge. Also, there is a ring provided to arrange the flashcards so that you don’t waste time searching for the required one. All flashcards are arranged topic-wise for easier access and quicker viewing.

**3. Movable XY Axis Pegboard**

Calculus involves solving lots of problems based on functions. A pegboard with a movable X-Y axis helps model various graphing functions. Calculus beginners can shrug away the hesitation around learning functions with this engaging manipulative. By adjusting the X-Y axis, and experimenting with various variables, the visual learners can be conversant with the calculus functions and various terminologies.

Bring the formulas to form with this pegboard and solve various calculus equations without getting entangled in calculation hassles. There will be a lot of ‘Ahaa’ moments while studying when you find the answer from calculations and from adjusting the axes on the pegboard to be the same. It encourages to try different variable combinations and indulge in more practice.

**4. Bouncing Ball**

How a bouncing ball can be used to teach calculus? No matter how unusual it sounds, the bouncing ball proves to be a hands-on manipulative to teach Calculus concepts. The ball bounces uncountable times and creates a situation similar to infinite series.

A yardstick is placed against the wall to measure the bounce height. You will find that the bounce height reduces gradually till the ball comes to a stop. This activity using the Bouncing Ball was mentioned in the study^{[3]} by Melissa K. Taft and showed how Calculus can be learned from real-life objects and simple activities. The infinite series formula **∑∞n=Nf(n)** ∑ n = N ∞ f ( n ) becomes easier to visualize with the activity using the Bouncing Ball.

**5. Accordion Figures**

Accordion figures are nothing but two cardboard-based joined by accordion-design-based tissue paper. The study^{[4]} on how to use manipulatives to teach Calculus and Trigonometry functions used these Accordion figures to understand their impact on teaching volumes of solids.

The Accordion figures can be transformed into various figures such as cones, spheres, etc. Using these figures, the students could convincingly agree to the fact that the integral they used was indeed helpful in finding the volumes of solids. Thus, integration and its use in calculating volumes of solids become easier to learn with manipulatives.

**Summing up,**

Manipulatives enter our lives at the pre-primary level of education and refuse to lose their relevance at all advanced stages. While the counting manipulatives and those used to teach math operations to little learners are mostly toy-based or have a gamified look, calculus students can find easy learning support in things of daily use as well.

We have included both the pre-designed and things of common use as manipulatives in our list to help you grow your thinking. If falling short of any gamified manipulative, just look at the world around and you may find a teaching tool such as balls, accordions, Cardboard boxes, etc. All it will take is a little thinking to conceive the use of these objects as calculus manipulatives.

**References**

- Benjamin Thirey benjamin.thirey@usma.edu & Robert Wooster (2013) The Touchy-Feely Integral: Using Manipulatives to Teach the Basic Properties of Integration, PRIMUS, 23:7, 605-616, DOI: 10.1080/10511970.2013.796576
- Kapor-Mater, A. (2016, June 22).
*A study in the use of maniplatives to teach topics in differential and integral calculus*. https://dash.harvard.edu/handle/1/33797357 - Taft, M. K. (1994).
*Using manipulatives in trigonometry and calculus : an honors thesis (HONRS 499)*. https://cardinalscholar.bsu.edu/items/24879ee7-c028-4caf-9f1f-d40e6902152a - Kapor-Mater, A. (2016, June 22).
*A study in the use of maniplatives to teach topics in differential and integral calculus*. https://dash.harvard.edu/handle/1/33797357

I am Pratiksha Bhatt, Bachelor of Life Science, and Masters in Management Studies. I have done certification courses in early education counseling. I am a writer, a mother of a child with spelling difficulties which drove me to alternative resources of education like manipulatives and participatory activities. My areas of expertise are learning difficulties, alternative learning methods, and activity-based learning.