The manner in which students learn mathematics influences how well they understand its concepts, principles, and practices. Many researchers have argued that to promote learning with understanding, mathematics educators must consider the tasks, problem-solving situations, and tools used to represent mathematical ideas. Mathematical tools foster learning at many levels–namely, the learning of facts, procedures, and concepts. Tools can also provide concrete models of abstract ideas, or, when dealing with complex problems, they can enable students to manipulate and think about ideas, thereby making mathematics accessible and more deeply understood. [1]
Below are some examples to better explain what mathematical tools are available.
Learning Tools
A learning tool can be as simple as an image or as complex as a computer-based environment designed to improve mathematical understanding. The key characteristic of a learning tool is that it supports learners in some manner. For example, a tool can aid memory, help students to review their problem-solving processes, or allow students to compare their performance with that of others, thereby supporting self-assessment. Learning tools can represent mathematical ideas in multiple ways, providing flexible alternatives for individuals who differ in terms of learner characteristics. For example, learners who have difficulty understanding the statistical ideas of arithmetic mean (center) and variance (spread) may be assisted through interactive displays that change as data points are manipulated by the learner. A mathematical learning tool can scaffold the learner by performing computations, providing more time for students to test mathematical hypotheses that require reasoning. In the statistics example, learners can focus on why changes to certain parameters affect data–and in what ways, rather than spending all their time calculating measures. [4]
Traditional Tools
Traditional tools are best suited for facilitating students’ learning of basic knowledge and skills. Objects that can be manipulated, such as cubes, reduce the abstract nature of concepts, such as numbers, thereby making them real and tangible, particularly for younger children. Such tools support the development of children’s understanding of arithmetic by serving as a foundation for learning more complex concepts. Visualization tools, such as graphs, can support data interpretation, while paper-and-pencil tools that provide practice of computational skills can support memory for procedures and an ability to manipulate symbols. [4]
Dice
Dice games combine a mathematical skill or concept with elements of problem solving, reasoning and communicating and are engaging for children. Games support children to become mathematically literate, which means they are able to understand, use and justify their mathematics. [7]
Ruler, Compass, Protractor and Divider
We are surrounded by all kinds of shapes around us. Circular, spherical, triangular, cylindrical are just some of the various types of shapes that we encounter in our day-to-day life. Mathematics is a vast field of study which is branched into several other fields which often correlate to each other. The reason that we are surrounded by these shapes is that they are an integral part in many aspects which makes our lives easier. The study of these shapes of various sizes and dimensions is known as geometry. Geometry is one of the most widely studied branches of mathematics which has been igniting curiosity in human beings since ancient times. To draw the shapes of the practical geometry we require tools like a ruler, a compass, a protractor, set squares, roller scales, and a divider, known as geometrical tools. [8]
Technological Tools
Technological tools are most effective in facilitating students’ understanding of complex concepts and principles. Computations and graphs can be produced quickly, giving students more time to consider why a particular result was obtained. This support allows students to think more deeply about the mathematics they are learning. Electronic tools are necessary in mathematics because they support the following processes: (a) conjectures–which provide access to more examples and representational formats than is possible by hand; (b) visual reasoning–which provides access to powerful visual models that students often do not create for themselves; (c) conceptualization and modeling–which provide quick and efficient execution of procedures; and (d) flexible thinking–which support the presentation of multiple perspectives. [4]
Desmos
A free suite of math tools that help people represent their ideas mathematically, connect different representations dynamically, make conjectures, and then develop entirely new ideas. [3] Online tools on Desmos include:
Calculators (Graphing & Non-graphing)
Calculator.net‘s sole focus is to provide fast, comprehensive, convenient, free online calculators in a plethora of areas. Currently, we have around 200 calculators to help you “do the math” quickly in areas such as finance, fitness, health, math, and others, and we are still developing more. Our goal is to become the one-stop, go-to site for people who need to make quick calculations. Additionally, we believe the internet should be a source of free information. Therefore, all of our tools and services are completely free, with no registration required. [5]
While tablets and phones may have some of the computational features math students need, their use is not generally permitted during testing. This is where it’s essential to be familiar with a graphing calculator if you are a student. But the usefulness doesn’t stop when you exit the classroom. Professionals in many fields benefit from using a graphing calculator in their everyday work. What can they do? Well, a standard model can handle simultaneous equations, plotting graphs, and more. [6]
Mathematical Instruments
Most instruments are used within the field of geometry, including the ruler, dividers, protractor, set square, compass, ellipsograph, T-square and opisometer. Others are used in arithmetic (for example the abacus, slide rule and calculator) or in algebra (the integraph). [9]
French Curve
A French curve is a template usually made from metal, wood or plastic composed of many different segments of the Euler spiral (aka the clothoid curve). It is used in manual drafting and in fashion design to draw smooth curves of varying radii. The curve is placed on the drawing material, and a pencil, knife or other implement is traced around its curves to produce the desired result. They were invented by the German mathematician Ludwig Burmester and are also known as Burmester (curve) set.
French curves are used in fashion design and sewing alongside hip curves, straight edges and right-angle rulers. Commercial clothing patterns can be personalized for fit by using French curves to draw neckline, sleeve, bust and waist variations. [2]
Tape Measure
A tape measure is useful tool, so it’s important to know how to correctly read a tape measure to ensure you can transfer measurements from plans and get accurate measurements throughout a project. [10] Use a tape measure to teach fractions by teaching students HOW to read and use a tape measure.
Slide Rule
The slide rule (also known colloquially in the United States as a slipstick) is a mechanical analog computer which is used primarily for multiplication and division and for functions such as exponents, roots, logarithms, and trigonometry. It is not typically designed for addition or subtraction, which is usually performed using other methods, like using an abacus. Maximum accuracy for standard linear slide rules is about three decimal significant digits, while scientific notation is used to keep track of the order of magnitude of results.
Theodolite
TBS
References
[1] “Mathematics Learning – Learning Tools”. 2023. education.stateuniversity.com. https://education.stateuniversity.com/pages/2202/Mathematics-Learning-LEARNING-TOOLS.html.
[2] “French Curve – Wikipedia”. 2023. en.wikipedia.org. https://en.wikipedia.org/wiki/French_curve.
[3] “Desmos | Let’s Learn Together.”. 2023. desmos.com. https://www.desmos.com/.
[4] “Mathematics Learning – Learning Tools”. 2023. education.stateuniversity.com. https://education.stateuniversity.com/pages/2202/Mathematics-Learning-LEARNING-TOOLS.html.
[5] “Calculator.net: Free Online Calculators – Math, Fitness, Finance, Science”. 2023. calculator.net. https://www.calculator.net/.
[6] “Top 7 Uses For A Graphing Calculator | HP® Tech Takes”. 2023. hp.com. https://www.hp.com/us-en/shop/tech-takes/top-7-uses-for-graphing-calculator.
[7] “Using Dice Games in the Classroom”. 2023. nrich.maths.org. https://nrich.maths.org/8414.
[8] “What are Geometric Tools? – Ruler, Compass, Protractor and Divider”. 2023. Vendantu. https://www.vedantu.com/maths/geometric-tools.
[9] “Mathematical Instrument – Wikipedia”. 2023. en.wikipedia.org. https://en.wikipedia.org/wiki/Mathematical_instrument.
[10] “How to Read a Tape Measure in Inches”. 2023. Pine And Poplar. https://pineandpoplar.com/how-to-read-tape-measure/.
Additional Reading
“10 Best Math Tools for High School Students”. 2023. Common Sense Education. https://www.commonsense.org/education/lists/10-best-math-tools-for-high-school-students.
“6 Basic Benefits Of Game-Based Learning”. 2013. teachthought. https://www.teachthought.com/technology/6-basic-benefits-of-game-based-learning/.
“How To Read A Tape Measure – Simple Tutorial & Free Cheat Sheet”. 2019. Joyful Derivatives. https://joyfulderivatives.com/how-to-read-a-tape-measure-simple-tutorial/.
“Math Apps | The Math Learning Center”. 2023. mathlearningcenter.org. https://www.mathlearningcenter.org/apps.
“The #1 Thing to Teach Kids About Measurement”. 2023. therecoveringtraditionalist.com. https://www.therecoveringtraditionalist.com/teaching-measurement/.
Before we get into it, we are talking about measuring tools that we use here in the United States which deal a lot with fractions that involve 1/2, 1/4, 1/8, 1/16, 1/32 and I know that in other countries, you are metric systems, so you don’t have to deal with those. But as we go through these ideas, I want you to think about how they relate to the measuring tools you have because what we find here in the United States is that kids end up leaving school, not being able to use and read those tools. In this video we discuss 3 key points:
#1 Reading a Tape Measure Builds Fraction Understanding
#2 We use measuring tools all the time in our real-lives
#3 Don’t wait until you THINK kids are ready to use a tape measure
“The Geometry of Weird-Shaped Dice”. 2017. Skulls In The Stars. https://skullsinthestars.com/2017/03/09/the-geometry-of-weird-shaped-dice/.
The variety of dice got me wondering: how does one design dice with a weird number of faces? What mathematical strategies does one use to make them? What other types of dice are possible? And, perhaps most important: are these dice “fair”?
“Types Of Dice: From Six-Sided To D20 – 2023 Ultimate Guide”. 2023. Boards Cards Dice. https://boardscardsdice.com/types-of-dice/.
Whether you’re an experienced gamer or a novice, you’ve most likely used dice. But you might be wondering about the different types of dice. Why do some dice have so many sides? When do you use special dice? How many different dice are there, and does it make a difference which types you use? In this article, I’ve answered those questions and more. You’ll learn the different types of dice, what they’re called, and their purpose. You’ll also find some interesting information about dice in gaming.
“Unit 1 – Tape Measure Math”. 2023. unl.edu. https://www.unl.edu/synergycurriculum/Documents/Prevocational-Instructors/Teacher%20Module%202,%20Unit%201.pdf.
The featured image on this page is from the freepik website.
Mathematical Mysteries 