# Thought Process to Discover Knowledge : unique proposition of nubtrek

Books and education websites treat knowledge as “matter-of-fact”. In comparison to those, nubtrek provides a thought process to discover the knowledge. In this post, the two are is explained for differential calculus.

### matter-of-fact knowledge

Derivative as slope of curve : Reference Khan Academy

Formal definition of derivatives : Reference Khan Academy

In these the knowledge is presented as “matter-of-fact” for students to learn.

### Thought Process to Discover Knowledge from nubtrek

Nubtrek starts with rate-of-change in numerical problems.

- If a car travels a distance of 20m in 2 seconds, the speed of the car is 10m/sec. The speed is the rate of change of distance traveled.

Students are lead to understand that the quantities can be function of variables.

- if a car is moving at 10m/sec, the distance covered in t sec is 10t meter. The expression 10t is a function of variable t.
- If a car’s position of displacement is given as a function of a variable, How would one go about finding the speed? This is the problem we try to solve in differentiation. Car’s displacement is given as s=3t×t+2.

nubtrek takes the students in a thought-process to discover knowledge.

- Person A does the following: At t=2 sec, the distance traveled is 3×2×2+2 = 14m. So the speed should be 14/2 = 7 m/sec
- Person B does the following: At t=1 sec, the distance traveled is 3×1+2 = 5m. So the speed should be 5/1 = 5 m/sec

Why these two are different? What is being calculated by Person A and B? These students are calculating the average speed as a numerical value.

Students are lead to realise that displacement is a function of variable (s=3t×t+2), the speed is also a function of variable.

nubtrek continues to take students in a thought process to discover knowledge.

- Knowing that speed has to be computed as a function of time…
- Person A calculated rate of change for 1 sec interval as [s(t+1) — s(t)] / 1 = 6t+3
- Person B calculated rate of change for 2 sec interval as [s(t+2) — s(t)] / 2 = 6t+6
- Person C calculated rate of change for 0.5 sec interval as [s(t+.5) — s(t)] / .5 = 6t+1.5

Students are lead to understand what instantaneous speed is.

- A speedometer measures instantaneous speed and we would like to calculate that as a function of given variable.
- Instantaneous speed is when the rate of change is calculated for a time difference of 0

nubtrek challenges the young brains with questions to discover using limits.

- considering the time interval as δ, the speed is [s(t+δ) — s(t)] / δ
- For δ=0, the speed becomes [s(t+0) — s(t)] / 0 = 0/0
- What would be the value of function evaluating to 0/0?
- Students should shout USE LIMITS, if they have learnt the limits in nubtrek
- Thus the differentiation in first principle is discovered as limit δ tending to 0 of [s(t+δ) — s(t)] / δ

The thought process in nubtrek is designed to be

- start with something students know
- create situations or problems that challenges their current knowledge
- lead the students to figure out the answers
- connect the right elements of knowledge to make them discover the fundamentals

After all this, the graphical meaning of derivatives is introduced as slope of the tangent at the point and that too is presented in a thought-process to discover.

Reference: nubtrek

Thanks for reading