A Pathway to a Student-Worded Definition of Limits at the Secondary-Tertiary Transition

Abstract

Over the last forty years of mathematics education research, a coherent body of knowledge has accumulated regarding the teaching of limits. On this basis, it remains a challenge to identify goals and design tasks compatible with ordinary teaching conditions. This paper reports on a teaching experiment carried out in France with year 12 students, which led to the formulation by the students of a correct formal definition of the infinite limit for sequences, with minimal background logical prerequisites and in the course of a 2-h session. On a more theoretical level, the teaching project was developed in the framework of didactic engineering, and provides opportunities to contribute to the ongoing work on its adaptation to the specific context of tertiary education. In the a priori analysis, we highlight the didactical potential of tasks of differentiation between neighboring concepts as a pathway to advanced mathematical concepts. In the a posteriori analysis, we focus on the nature and extent of teacher intervention in the shaping of a mathematical milieu that is conducive to the definition of an advanced mathematical concept.

Appendix. Candidate-definitions written by students – or pairs of students – at the beginning of phase 2 in the 2016 experiments

The candidate-definitions are numbered for the sake of clarity. We stayed as close as we could to the original French wording and to the layout of the original texts. The candidate-definitions selected for the collective discussion are in bold print.

2016–1.

• 1. For a sequence to tend to + ∞, we need to have∀ n ∈ ℕ, u_n +1> u_n.

• 2. The sequence (u_n) tends to + ∞ if and only if for all natural numbersn, (u_n) is increasing and not bounded above.

  ∀ n ∈ ℕ  u_n +1> u_nand (u_n) not bounded above

• 3. The sequence (u_n) tends to + ∞ when it is not bounded above.

• 4. The sequence (u_n) tends to + ∞: the sequence increases while getting ever closer to + ∞ but without ever reaching it.

• 5. A sequence (u_n) which tends to + ∞ is a sequence whose terms increase so that we cannot determine the last term of this sequence.

• 6. ∀M ∈ ℝ  ∃ n ∈ ℕ  u_n > M, moreover, the sequence (u_n) has to be bounded below.

• 7. To say that a sequence tends to +∞ means that there is no upper limit to the sequence.

• 8. A limit is a real number, unreachable for a given sequence.

• 9. A sequence (u_n) tends to +∞ if it is constant in +∞, or increasing without being bounded above.

  Sometimes some sequences tend to both +∞ and -∞

• 10. $\underset{n\to \infty }{\lim }=+\infty$

  In a sequence (u_n), there exists a n such that u_n =  + ∞

• 11. A sequence is said to “tend to +∞” when it is either constant at +∞, or constantly increasing. Exception: a sequence can also tend to +∞ and -∞, then it tends to +∞.

• 12. You never reach the limit, you can just come closer to it. By taking the greatest value of n that you please, you come really closer to the limit but you never go beyond it.

• 13. A sequence (u_n) which tends to +∞ is a sequence that is increasing and not bounded above by one of its terms.

• 14. The limit of a sequence is the value beyond which it cannot go any further

• 15. A sequence which tends to +∞ is a sequence which increases more than it decreases.

• 16. This sequence has no end. It is increasing. This increasing is endless, it is infinite. You always find an ordinate greater than the preceding one.

  u_n + 1 > u_n holds all in all for sequences in general, but, for instance, for (−1)ⁿ × n it does not, it becomes u_n + 2 > u_n.

2016–2

• 1. A sequence has limit + ∞ when it’s bounded below by a real number m .

• 2. Let (u_n) be any sequence. Then the sequence tends to + ∞ when (u_n) is strictly increasing

• 3. One can say that a sequence has limit + ∞ if all the terms of the sequence belong to (a number known or unknown, n ) or ( n , a number known …), this number of terms must not be finite.

• 4. The greatest value to which the sequence tends, whether it’s increasing or decreasing, is its limit.

• 5. A sequence has limit + ∞ ⇔ it’s increasing and not bounded above

• 6. A sequence (u_n) has limit + ∞ if and only if whenntends to + ∞, (u_n) tends to a unique number close to + ∞. If (u_n) is bounded above, the limit of (u_n) is the upper bound.

• 7. A limit is a value that a given sequence will never go beyond, whatever the value of its unknown.

  A sequence with limit + ∞ is an increasing sequence which has no limit.

• 8. A sequence has limit +∞ when ∀n, u_n + 2 > u_n and the sequence is bounded above by no real number.

• 9. There exists a unique limit to a sequence such that the values of that sequence are not greater than or equal to +∞

• 10. The limit of a sequence is a maximal value towards which the sequence tends.

• 11. A sequence (u_n) has limit k if and only if u_n tends towards k.

• 12. A sequence (u_n) has limit +∞ if and only if ∀n ∈ ℕ, (u_n) is not bounded and strictly increasing.

• 13. A sequence has limit +∞ if and only if it is strictly increasing and not bounded above and u_n can increase up to infinity

• 14. A sequence (u_n) has limit +∞ if and only if ∀n ∈ ℕ, the sequence (u_n) is increasing and not bounded above.