Be it enticing or not, Math is a crucial subject that is necessary to understand the working of the world better. While it is true that the concepts turn complex as a student passes through grades, one of the grounds may be the introduction of newer concepts. When one reaches high school, the concept of calculus is introduced. With the appearance of derivatives, limits, functions, and integrals, one can comprehend the inception of calculus in their academics.
That being the case, these lessons can be intriguing for some and arduous for others. Breaking down the obligatory sub-concepts and mastering them can be a finer solution. To comprehend the same in detail, we here, in this post, look into what calculus is all about, the concepts it encompasses, and finally, strategies to grasp them better.
Calculus- The math of changing entities!
Most of the concepts in math usually encompass the study and estimations of stationary objects. For instance, geometry is the study of shapes, and algebra is the study of operations, especially arithmetic. Calculus, inceptively known as infinitesimal Calculus, is the study of the movement of objects. Accordingly, it deals in areas like accumulation, convergence, sequences, and series, presenting the change in the form of expression.
Grasping calculus concepts- Evaluating the imperative traits in dyscalculics!
Calculus in general needs the learners to have grasping knowledge in concepts like Geometry, numbers, analytics, and verbal skills. To look meticulously into these traits, here we look into those areas that can make learning calculus facile.
1. Readiness to solve and Right attitude- Being ever ready!
Having trust in self and abilities would stimulate readiness to address the query with an appropriate attitude. Getting familiar with inventory, assessments, and course performance can be further aiding. Laura J, Pyzdrowki[1] made a quantitative analysis that showed that high school grade point average and the Calculus Readiness Assessment had significant positive correlations with course performance. Nonetheless, the strongest positive significant correlation was between attitude (Attitudes Toward Mathematics Inventory overall and confidence subscale) and course performance.
Dyscalculics often have compromises with readiness. Anne Neville[2] in her research outlined that dyscalculics can have readiness lag along with general learning difficulty. Due to some personal grounds, they often strive to ensure readiness to address queries, thereby obligating additional effort in notions like calculus.
2. Precision- Attention to minor details!
Math is one of those few subjects that require the learners to be precise. In concepts like calculus, it becomes obligatory, as a change in one symbol can lead to errors in the expressions later. Accordingly, Rebecca- Anne Dibbs[3] studied the importance of precision teaching in calculus. The results found that; although precision teaching requires more work for the instructor, students’ high perceived utility makes precision teaching a valuable method of undergraduate instruction because they claim to study more, understand the material better, and earn higher grades. Further, ensuring precision can also be ensured at a personal level as well.
Dyscalculics may face challenges with attention to detail. Sarit Askenazi[4] studied the attentional networks in developmental dyscalculia. The results remarked deficits in alerting networks, which have a major role in executional functions, thereby they often get slower to respond- implying attentional compromises.
3. Creativity- Where Dyscalculics Excel!
Math, often being an abstract set of notions, is actually implemented in real life. For employing these concepts, a certain level of innovation is necessary, as real-life challenges may come up with newer complications. Dennis B. Roble[5] studied the effect of using creative techniques to preach Calculus concepts like algebra in class. The results found that creative pedagogies revealed a positive effect on students’ achievement and enhanced the level of students’ mathematical creativity in terms of fluency, flexibility, and originality of their solutions. While we comprehend, dyscalculics are often more than creative, this trait can be their strength in grasping calculus concepts.
4. Mastery in underlying concepts
As the concepts for high schoolers and colleges, addressing calculus needs a gripping mastery in previously learned math notions like algebra, geometry, exponentials, logarithms, and trigonometry. Students with knowledge of these concepts can look at calculus with ease. Sometimes, the complications of calculus may arise due to compromising algebraic expressions.
- Algebra in Calculus
Usually, the expressions in Calculus are algebraic. Representation of a movement on paper is facile in the form of expressions. Putting instantaneous value can determine positions at that moment. Apart from BODMAS and substitution, linear algebra notions like matrix are also occasionally employed. Jack W. Rogers Jr.[6] in his research mentioned the role of linear algebra in calculus. He states that students are assumed to have mastered these techniques in their calculus courses and to be familiar with the problems to which they are applied. These problems can be used to motivate new ideas in linear algebra.
- Trigonometry in Calculus
A science of triangles and their measuring angles that can be applied in any shape like the circle is encompassed in trigonometry. While the expressions are algebraic, and figures need geometry. Measurement of angles and related operations in calculus fall under trigonometry. USask made a report on trigonometry review in Calculus, elucidating where these are implied and how.
Dyscalculia is characterized by number compromises. Daniela Lucangeli in his book Understanding Dyscalculia outlined that Dyscalculics can have a wide range of persistent challenges in mathematical skills, The same is the case with abstract reasoning skills like algebra, trigonometry, and geometry. These observations denote the need for special care and strategies to ameliorate skills in calculus
Learn calculus finer- Strategies for dyscalculics to revamp
Apprehending what abilities are obligatory to grasp Calculus, one may be bewildered how these can be ensured. The above-mentioned traits can be learned over time; the strategies mentioned below can speed up by boosting the aspirant.
1. Start with basics:
Incepting with basic concepts rather than looking at calculus, on the whole, can mitigate the chances of anxiety, enduring ameliorate performance. Looking into mastering basic concepts of math like algebra and then discerning obligatory theorems and limit regulations can prove to be a sequential approach and thereby effortless calculus.
2. Make use of apps and video sources:
Online learning has been turning popular due to edges like interactive UI and multi-sensory pedagogies where needed. Consequently, having relevant apps and learning from them can be a preferable choice. As most of these are free to use and are easily available in app stores, learners can get adorned with an additional educational source. Further, pupils can also traverse through training videos on websites like youtube and save them for later as well.
3. Ensure precision and ask for doubts:
Precision, as depicted in the previous lines, may be stipulated in the classroom. Organically, the practice of relevant sums and notions can take learners closer to being precise. Pupils must be ready to ask for doubts from their mentors, who are always ready to address. Further, they can also try online forums to discern valuable discussions and get a few doubts clear.
4. Make use of your tools:
Number-related compromises can be effectively assessed by the use of some tools. Say, a graphing tool can be employed to speed up graph learning. Employing math fact cards (or flashcards) can make learning formulae and regulations. Further, text to speech strategy can be preferred over paper notes to recall concepts later. These tools can also aid in achieving precision.
5. Adaptive learning technology:
The level of understanding and the needs of each learner can be distinct. To address the same, classroom teaching can be transformed with adaptive learning technology, where the unique needs of individuals are taken care of through just-in-time feedback with pathways and resources. Jennifer B. Daines[7] made one pilot study of adaptive learning applied to campus-based courses, which covered a year of data that preliminarily indicates positive results for student success. These results showed a noteworthy difference in performances of students in trigonometry and calculus. Further, the drop rate was also reduced drastically.
Concluding thoughts- Dealing with calculus
Some notions of Math can be strenuous for learners. Even if there is a mathematical compromise, being disciplined to learn and choosing befitting strategies along with practice can manage to compete. While you can mull over the above-suggested strategies, you may need to choose and fabricate a custom-tailored plan of actions to ensure mastery in Calculus. Evidently, coordinating with instructors and mentors can be beneficial to acquire regular insights.
References:
- READINESS AND ATTITUDES AS INDICATORS FOR SUCCESS IN COLLEGE CALCULUS. (2012, July). Laura J. Pyzdrowski.
- Dyscalculia/specific learning difficulty in mathematics: Identification and intervention in irish primary schools. (2012). Anne Neville.
- Students’ Perceived Utility of Precision Taught Calculus. (2013). Rebecca- Anne Dibbs.
- Attentional networks in developmental dyscalculia. (2010, January). Sarit Askenazi.
- Communication and valuing students’ productive struggle and creativity in calculus. (2020). Dennis Roble.
- Applications of linear algebra in calculus. (2018). Jack W. Rogers Jr.
- Colorado Technical University. (2016, June). Improving performance in trigonometry and Pre-Calculus by incorporating adaptive learning technology into blended models on campus. Jennifer B. Daines.
An engineer, Maths expert, Online Tutor and animal rights activist. In more than 5+ years of my online teaching experience, I closely worked with many students struggling with dyscalculia and dyslexia. With the years passing, I learned that not much effort being put into the awareness of this learning disorder. Students with dyscalculia often misunderstood for having just a simple math fear. This is still an underresearched and understudied subject. I am also the founder of Smartynote -‘The notepad app for dyslexia’,