## Abstract

Programming-based activities are becoming more widespread in curricula. Our theoretical and empirical investigation seeks to identify appropriate ways to connect computer programming and algorithmics to mathematical learning. We take the intermediate value theorem as our starting point, as it is covered by the French school curriculum, and because of its links with the bisection algorithm. We build upon the theory of mathematical working spaces, distinguishing between algorithmic and mathematical working spaces. Both working spaces are explored from the semiotic, instrumental, and discursive dimensions that support learning. Our two research questions focus on the suitable algorithmic and mathematical working spaces in which students develop an understanding of the intermediate value theorem, and the bisection algorithm. Our method starts at the reference level, with an epistemological and curricular analysis. Then, a series of tasks is designed for students working in adidacticity, and suitable working spaces are determined a priori. The tasks have been implemented in French classrooms with students aged 16–19. An analysis of their work supports an a posteriori examination of the working spaces. Our findings demonstrate that the students were able to make connections between algorithmics and mathematics in each of the three dimensions, semiotic, instrumental, and discursive, and point out the interplay between these dimensions.

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## Data availability

Data that support the findings of this study are available in Laval (2018) on the repository https://tel.archives-ouvertes.fr/ with the identifier tel-01943971.

## Notes

Fraillon et al. (2020a) reviewed curricula in thirteen countries over three continents, looking for “[an] emphasis on aspects of computational thinking” (p. 15). Of the twelve aspects the authors identified, four relate to programming: developing digital applications; evaluating code, programs or macros; writing code, programs or macros; and creating algorithms. Eight countries emphasize all four aspects, while two only mention creating algorithms. Other data are reported in Stephens (2018).

The IVT guarantees, for a continuous function

*f*defined on an interval [*a;b*] and for every*m*in the interval [*f(a); f(b)*], the existence of one or more values*c*in [*a;b*] such that*f(c)*=*m*. In this paper, we consider a lemma that guarantees, for a function*f*defined on an interval [*a;b*] the existence of one or more zeros under the following sufficient conditions:*f*is continuous and*f(a)*^{.}*f(b)*≤ 0. A corollary states that the zero is unique under the supplementary sufficient condition that the function is strictly monotonic. The general IVT can be deduced from the lemma, using a simple algebraic manipulation which is not considered here, due to our focus on real analysis.The French curriculum includes algorithm design as one of four overarching skills, but does not indicate any particular learning goals. It reflects the idea that teachers should avoid teaching content that is too different to ordinary mathematics, and the fact that specifying learning goals would not be easy given the lack of tradition and experience in this domain.

One could argue that algorithmics in mathematics education involves a single WS, with features that are associated with either algorithmic or mathematical thinking. This position would ignore the fundamental coherence of each WS. The metaphor of a professional with two specialties illustrates our point: A French elementary teacher teaches both mathematics and French in the same classroom; situations such as problem solving involve objectives in both domains, and the teacher has to act in two WSs (mathematics and French) that cannot be merged, but have to be coordinated.

Many notation systems use the

*equal to*sign. In this case, for example,*x*=*x*+ 1 is a valid instruction for incrementing a variable. In this paper, we use an arrow rather than the*equal to*sign:*x*←*x*+ 1.Throughout this paper, “suitable” refers to a WS level as understood in MWS theory, and not in the general sense of “appropriate.”.

See Table 1 for examples of iteration and iterative variables. An iteration starts with a specific marker (

*while*in the examples shown), followed by a block of instructions. An iterative variable is a variable whose value changes with each iteration. In the bisection algorithm,*u*and*v*are iterative variables, in contrast to*a*and*b*(constants), and*m*(a local variable). Unlike bisection, iteration in Nguyen and Bessot (2010) is a loop with a fixed number of repetitions:*repeat*<*n*>*,*<*block*> .In principle, identifying reference WSs would begin by reviewing previous research. However, we found no study that directly investigated the IVT and the bisection algorithm. Douady (1980) implicitly considers the use of the IVT by students aged 8–10 in the generation of decimal numbers to approach a solution. However, while the class is able to figure out new numbers by bisection, there is no systematization. The goal is to encourage them to construct decimal numbers, rather than to raise, and investigate questions relative to the IVT. Other studies on the IVT concern undergraduate students in relation to the completeness of the set of real numbers, and do not include the bisection algorithm. Furthermore, we were unable to identify any studies on binary search, as research on computer science education is in its infancy.

Unlike the bisection algorithm, Stevin’s algorithm divides each interval into ten steps, adding a new decimal place at each step.

Barany (2013) refers to “a method of approximating roots.” We do not discuss methods and algorithms here, as it is clear to us that the method was systematic, and can therefore be qualified as an algorithm.

Two sequences of real numbers (

*u*_{n}) and (*v*_{n}) are adjacent when (*u*_{n}) is increasing, (*v*_{n}) is decreasing, and the sequence (*u*_{n}*—v*_{n}) converges towards zero. The theorem states that two adjacent sequences converge towards a common limit. It is equivalent to the idea of the completeness of the set of real numbers, and is a trivial corollary of the nested closed intervals theorem.A function changes sign when it is negative at one boundary of the interval of definition, and positive at the other:

*f*defined on [*a;b*] and*f(a)*^{.}*f(b)*≤ 0.Without this constraint the algorithm may not terminate. For instance, searching for a hidden integer inside the interval [0;100], at the

*n*^{th}iteration*u*= 100*p*/2^{n}*; v*= 100*(p*+ 1*)*/2^{n},*p*is an integer. This can be proved by induction, and it follows that only 25, 50, and 75 can be found by a bisection algorithm that would overlook this constraint.See the definition in footnote 12.

Footnote 2 explains why this lemma is considered, and not the full IVT.

The relationship can be expressed in compact form, using an alternative statement inside a recurrence between ordered pairs, as follows:

(

*u*_{n+1,}*v*_{n+1}) = if*f(u*_{n}*f(v*_{n}*)*> 0: (*m*,*v*_{n}) else (*u*_{n,}*m*),*m*being the mean of (*u*_{n,}*v*_{n}).This means that our results for the three parts refer to different classes. The 11

^{th}graders had to be prepared for part 2 by completing part 1. Similarly, 12^{th}graders completed parts 1 and 2 in preparation for part 3 (Laval, 2018). We do not report here on this preparation. In the context of our framework, we assume that after this preparation, 11^{th}graders had adopted the components of suitable WSs covered in part 1, and that 12^{th}graders had adopted the components of suitable WSs covered in part 2 (Table 3).In particular, see pp. 278–345 for part 1, pp. 377–416 for part 2, and pp. 450–464 for part 3. The assignments and the transcripts of classroom discussion are translated from the thesis.

Being aware of the equivalence, the students identified the IVT and this lemma.

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Lagrange, JB., Laval, D. Connecting algorithmics to mathematics learning: a design study of the intermediate value theorem and the bisection algorithm.
*Educ Stud Math* **112**, 225–245 (2023). https://doi.org/10.1007/s10649-022-10192-y

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DOI: https://doi.org/10.1007/s10649-022-10192-y

### Keywords

- Algorithmic and mathematical working spaces
- Mathematics-computer science connections
- Bisection algorithm
- Intermediate value theorem
- Programming tasks
- Algorithmic thinking
- Adidacticity