Summary
By Fiona Campbell
Introduction
Students with dyscalculia and other mathematical learning difficulties often face significant challenges in transferring information to their long-term memory. Additionally, students who exhibit traits of Specific Learning Difficulties (SpLD), but lack a formal diagnosis, may encounter similar obstacles. This results in difficulties when learning mathematical formulas, as the abstract nature of symbols and the seemingly meaningless relationships between variables can be particularly problematic.
To support these students, mathematics must be taught for deep conceptual understanding. Students who understand the relationship between variables in formulas are less likely to forget them, particularly if teaching has used multisensory methods. Furthermore, these students may have strengths in other areas such as logic and reasoning. Hence, even if they forget formulas, they can reconstruct them based on their understanding of the relationship between variables. This approach is demonstrated in this article.
The article describes three experiments that students can undertake in math lessons to understand what pi (p) is and how the formulas for the circumference and area of a circle are derived.
Experiment 1. What is Pi (ฯ)? [1]
[1] Before undertaking this experiment, students should have a solid understanding of key terms and how to change the subject of a formula. If students do not, consider pairing weaker students with stronger students and displaying posters prominently in the classroom as external memory aids or handing out prompt sheets.
All circle calculations use the constant p (pi) or 3.14.ย But what is p and where does it come from?
Materials
- Tape measure.
- Ball of string.
- Several assorted circular and cylindrical objects.
- A table to record the results.
- Pencil or pen.
- Calculator.
Figure 1. Materials for experiment 1.
[1] Before undertaking this experiment, students should have a solid understanding of key terms and how to change the subject of a formula. If students do not, consider pairing weaker students with stronger students and displaying posters prominently in the classroom as external memory aids or handing out prompt sheets.
Class Activity
- Measure. Select five objects and measure their circumference and diameter using a string and a ruler.
- Record. Record the results in the table below.
- Calculate. For each object, divide the circumference of the circle by its diameter.
- Discuss. Discuss the results with your partner. What do you notice
Class Discussion
- What did you find?
- pย is the ratio of the circumference of a circle to the length of its diameter.
- For all circles, the circumference divided by the diameter equals approximately 3.14 or p.
- Pi (p) is a constant.
- p is an irrational number, which means it has an infinite number of decimal places without repeating. You can listen to it here.
- Some people have attempted to memorise pi to as many decimal places as possible. The current world record holder isโฆ (ask students to research).
- How do you think they managed to memorise such a large number?
- What techniques can you use to help you remember information? Letโs try it now to remember that pi is the ratio between the circumference of a circle and its diameter.
Experiment 2. Where Does the Formula for the Circumference of a Circle Come From?
In mathematics, we often need to calculate the circumference of a circle. If p is the circumference of a circle divided by its diameter, we can rearrange the formula to find the circumference.
Materials
- Whiteboards.
- Erasable pens and rubbers.
Class Activity
- Discuss. How do you rearrange the formula to make the circumference the subject? Discuss with your partner.
- Demonstrate. Select pupils to demonstrate on the board how they rearranged the formula to make the circumference the subject. Choose pupils with different approaches. Ask the class which method they prefer.
- Do. On your whiteboards, rearrange the formula to make the circumference of the circle the new subject.
- Discuss. What if we had the radius of the circle instead of the diameter? What effect would that have on the formula? Discuss with your partner.
Class Discussion
The circumference of a circle is 2pr and this experiment has shown us where this formula comes from. We can also watch this video.
Experiment 3. Where does the formula for the area of a circle come from?[1]
[1] Students who undertake this experiment should be confident calculating the area of a rectangle.
Theย formulaย for finding the area of a circle is ๐จ = ฯ๐ยฒ, where ฯ is aย constantย and ๐ is theย radiusย of the circle. But where does the formula come from?
Materials
- A circle printed on paper with dotted lines to indicate sections.
- Scissors.
- Ruler.
- Whiteboard and pen.
Class Activity
- Cut. Cut a circle into sectors.
- Rearrange. Rearrange the sectors from top to tail on a whiteboard. Draw a line around it.
- Discuss. What shape do they make? Discuss with a partner. Is it like a rectangle? The more sectors you can cut the circle into, the better the rectangular shape will be.
- Look. Examine the height of the rectangle. Consider the original circle. What part of the original circle gives the height of the rectangle? Now think about the length of the rectangle. What part of the original circle gives the length of the rectangle?
- Calculate. Use these measurements to calculate the area of the rectangle. What is it also the area of? The circle.
For an overview of these steps, see Figure 2.
Figure 2. A step-by-step visual guide to understanding the formula for the area of a circle.
Class Discussion
The height of the rectangle is the radius of the circle. The length of the rectangle is half of the circumference or pr. So the area of the rectangle is pr x r or p
. Watch how it is demonstrated in this video. Discuss how the class will remember this relationship and what methods they will use.
Conclusion
To think like mathematicians, students must grasp the relationship between variables. An overreliance on memorising formulas hinders students from becoming true mathematicians and disproportionately disadvantages those with dyscalculia and other mathematical learning difficulties. These students often face significant challenges in transferring information to their long-term memory. By teaching mathematics with a focus on deep conceptual understanding, we enable students to develop their logical reasoning skills, bypassing the need to memorise abstract and often meaningless formulas. For more ideas and resources for teaching maths to neurodiverse students, visit The Maths Hub.
You also might find our series, ‘Fixit with Karen’ available on our YouTube channel here- https://www.youtube.com/@thedyscalculianetwork2548 and on the blog section of our website here- https://dyscalculianetwork.com/insights-events/
