A Lazarus

Discrete dynamical stabilization of a naturally diverging mass in a harmonically time-varying potential

Keywords: Dynamical systems, Stability analysis, Modal analysis, Floquet theory, Standing waves

We numerically investigate the stability and linear oscillatory behavior of a naturally diverging mass whose potential energy is harmonically modulated. It is known that in the Kapitza limit, i.e. when the period of modulation is much smaller than the diverging time, the collapsing mass can be dynamically stabilized and behave like an effective classic harmonic oscillator. We find that in the regime where the period of modulation is larger than the collapsing time of the mass, dynamical stabilization is still possible but in a discrete fashion. Only almost-periodic vibrational modes, or Floquet forms (FFs), are allowed that are located in independent stability stripes in the modulation parameter space. Reducing the FFs to their periodic eigenfunctions, one can transform the original equation of motion to a dimensionless Schrödinger stationary wave equation with a harmonic potential. This transformation allows for an analytical prediction of the stability stripes and the modal shapes of the vibrating mass. These results shed new light on the stability of linear dynamical systems, analytical solutions of Mathieu equations and on the relations between Initial and Boundary Value Problems.

Introduction

Modal analysis is a linear perturbation method that allows to characterize the local oscillatory and stability behavior of stationary states of dynamical Email address: arnaud.lazarus@sorbonne-universite.fr (A. Lazarus) systems [START_REF] Guckenheimer | Nonlinear oscillations, dynamical systems, and bifurcations of vector fields[END_REF][START_REF] Strogatz | Nonlinear dynamics and chaos: with applications to physics, biology and chemistry[END_REF]. It is used in various area of physics, from molecular vibrational frequencies [START_REF] Nakamoto | Infrared and Raman spectra of inorganic and coordination compounds Part A[END_REF] to the stability of engineered structures [START_REF] Nayfeh | Linear and nonlinear structural mechanics[END_REF]. Reduced to a single dimension in space, this concept is modeled by the archetypal example of a mass moving in a quadratic potential energy whose governing equation is a linear homogenous Ordinary Differential Equation (ODE) with initial conditions. In the case of a perturbed equilibrium, i.e. for a constant potential in time, two qualitative behaviors exist: the mass is either neutrally stable, harmonically oscillating in a potential well (this case is the classic harmonic oscillator) or unstable, exponentially diverging on a potential hill.

A less constrained situation eventually occurs when the potential energy of the perturbed stationary state is free to periodically vary with time [START_REF] Bergé | Order within chaos[END_REF][START_REF] Bolotin | The dynamic stability of elastic systems[END_REF], i.e. when the motion of the mass is mathematically governed by a linear homogenous ODE with periodically time-varying coefficients. This generalized framework explains a broader class of physical problems from parametric oscillators [START_REF] Turner | [END_REF]8] to the emergence of Faraday waves [9,10,11] or the motion of the lunar perigee [12,13]. Again, two different scenarios should be considered whether the mass is naturally diverging or not. In a periodically time-varying potential well, the quasi-periodically oscillating particle eventually destabilizes for certain regions in the modulation parameters space, a.k.a. Mathieu's tongues [14,15]. On a modulated potential hill, there is an asymptotic limit for which the modulation parameters allow to stabilize the naturally diverging mass [16,17]. This last model is generally called a Kapitza oscillator since Kapitza was the first to explain the dynamical stabilization of a collapsing inverted pendulum in which the pivot point vibrates in the vertical direction [18,19].

Although simplistic, the aforementioned 1D oscillators allow for the modeling and understanding of an outstanding number of physical systems and their complete theoretical description is therefore a necessity. Yet, to the best of our knowledge, we believe the linear vibratory analysis of a 1D Kapitzalike oscillator has only been partially studied, so far. Here, we present a numerical and theoretical study of the modal analysis of a naturally diverging mass on a harmonically modulated potential hill. Unlike focusing on the Kapitza asymptotic regime where the modulation periodicity is much smaller than the collapsing time of the particle, we are interested in the regime where the modulation periodicity is larger than the diverging time. In Section 2, we present the system under study and the numerical tools we use to perform its modal and stability analysis. Section 3 highlights the original mechanical behavior numerically observed when the modulation is slow as compared to the natural diverging time. Section 4 shows the theoretical reduction process that allows to analytically characterize the numerical results obtained in Section 3. We conclude and discuss those results in Section 5.

We find that, opposite to the Kapitza limit, the dynamical stabilization of the naturally diverging mass consists in almost-periodic vibrational modes, or Floquet Forms (FFs), that are located in independent stability stripes of the modulation parameter space. Based on simple numerical observations, we show it is possible to reduce the harmonically time-varying governing differential equation under study to a dimensionless Schrödinger stationary wave equation with a harmonic potential. The latter can be analytically solved and is used to predict the location of the stability stripes as well as the periodic modal shapes of the stable FFs of the modulated oscillator. These results should help improving our understanding of periodically time varying systems and the relation between initial and boundary value problems, two notions that are notably at the heart of the recently unveiled discrete time crystals [20,21,22,23]. This work could also benefit to the modeling of hydraulic quantum analogs that consist in a bouncing droplet "walking" on a harmonically vertically vibrated bath [24,25,26,27,28,[START_REF] Fort | Proc. Natl. Acad. Sci. USA[END_REF][START_REF] Perrard | [END_REF]. Although we believe the mathematics leading to Schrödinger's like equations are fundamentally different, it is interesting to notice that both dynamical systems involve a naturally collapsing or bouncing mass in a time-varying potential energy and lead to discrete mechanisms.

(a) (b)

EQUATION OF MOTION AND FLOQUET THEORY

Figs. 1(a),(b) show the 1D periodically time-varying linear dynamical system under study. A particle of mass m locally moves with a position r(τ ) on a harmonically modulated, quadratic potential hill, so that its total kinetic plus potential energy reads E(τ ) = 1/2m ṙ(τ ) 2 + 1/2k(t)r(τ ) 2 where ṙ(t) = dr(τ )/dτ is the velocity of the particle. The linear equation of motion of m associated with E(τ ) is modeled by the 1 degree-of-freedom mass-spring system shown in Fig. 1(b). In this periodically conservative system, the particle experiences a parametric excitation F (τ ) = -k(τ )r(τ ) that derives from the modulated potential energy and that can be represented by a spring with a T -periodic varying stiffness k(τ ) = k(τ + T ) = -k 0 + k 1 cos(2πτ /T ) where -k 0 and k 1 are the fundamental and modulated stiffness, respectively. The only difference with a classic parametric oscillator is that the fundamental stiffness -k 0 is negative: if k 1 = 0 N/m, the system is linearly unstable and the mass exponentially diverges following r(τ ) ∝ e 2πτ /T 0 where T 0 = 2π/ k 0 /m is the natural diverging time.

According to Newton's second law and Lagrangian mechanics, the dimensionless Initial Value Problem (IVP) governing r(τ ) reads

d 2 r(t) dt 2 -r(t) + α cos(2πt/T )r(t) = 0 ( 1
)
where t = 2πτ /T 0 is the dimensionless time. In this fundamental Mathieu equation [START_REF] Whittaker | A course of modern analysis[END_REF][START_REF] Magnus | Hill's equation[END_REF], the natural diverging time is now 2π and the dimensionless modulation period of the potential energy is T = 2πT /T 0 . The two relevant modulation parameters are the time scale ratio T /2π and the stiffness ratio α = k 1 /k 0 . For modal and stability analysis of the linear differential Eq.( 1), one can use Floquet theory [START_REF] Floquet | [END_REF][START_REF] Whittaker | A course of modern analysis[END_REF] to express r(t) as a linear combination of two almost-periodic vibrational modes, or Floquet forms,

r(t) = c 1 Ψ(t)e st + c 2 Ψ * (t)e -st (2)
where c 1 and c 2 are constants determined upon initial position r(0) and velocity ṙ(0). Replacing the Floquet form Ψ(t)e st in Eq.( 1) leads to an eigenvalue problem that can be numerically solved for each set of parameters (α, T ) [34,35,36]. According to Floquet theory, s is a complex eigenvalue called a Floquet exponent and the computed eigenfunction Ψ(t) and its complex conjugate Ψ * (t) are periodic with a dimensionless modulation period T . Because of the dimension and the linearity of Eq.(1), only three qualitative stability behaviors can be observed in the (α, T ) space:

• If (s) = 0 and 0 ≤ (s) ≤ π/T , the particle is neutrally stable and r(t) is an almost-periodic oscillation about r(t) = 0.

• If (s) > 0 and (s) = 0, the position r(t) = 0 is dynamically unstable and r(t) is a T -periodic motion that exponentially diverges from r(t) = 0 with a growth rate (s).

• If (s) > 0 and (s) = π/T , the position r(t) = 0 is dynamically unstable and r(t) is a 2T -periodic motion that exponentially diverges from r(t) = 0 with a growth rate (s).

MODAL AND STABILITY ANALYSIS (T > 2π)

The oscillating system represented by the governing equation Eq.( 1) has been well studied in the asymptotic limit where the modulation period T is much smaller than the natural diverging time T 0 , i.e. for T << 2π [START_REF] Magnus | Hill's equation[END_REF]. In this regime, first understood by Kapitza thanks to averaging techniques when studying the inverted pendulum whose pivot point is vertically vibrated [18,16], it is possible to stabilize the naturally diverging mass if (αT ) 2 /8π 2 > 1. Here, we will focus on the dynamical behavior of the modulated oscillator of Fig. 1 for T > 2π that is the overlooked regime we are interested in.

We perform a numerical stability analysis of Eq.( 1) by analyzing the growth rate (s) and fundamental frequency (s) of the computed Floquet forms Ψ(t)e st and Ψ * (t)e -st in the modulation parameter space (α, T ). Fig. 2 shows the linear stability chart of the particle for 0 ≤ α ≤ 10 and 0. T /2π ≤ 5. Following the stability classification given at the end of Section 2, we find an alternation of T -unstable (red dots in Fig. 2) and 2T -unstable regions (yellow dots) which is typical of classic Mathieu equations [START_REF] Whittaker | A course of modern analysis[END_REF][START_REF] Magnus | Hill's equation[END_REF]. If the system is not modulated enough, i.e. for α < 1 (the potential energy is never positive), the mass cannot be dynamically stabilized. For T < 2π, a stability region opens whose instability lower limit corresponds to the classic frontier of Kapitza, α = √ 8π/T , shown in black line in Fig. 2. For T > 2π, i.e. when the period of modulation is slower than the naturally diverging time, Kapitza's theory does not hold but the dynamical stabilization of the mass is still possible in the (α, T ) modulation space: our goal is to get some physical and mathematical insights into this stabilization process.

According to Floquet theory, the T and 2T instability regions cannot merge in the (α, T ) space so each alternation of colors indicates a tiny stability domain that we denote l = 0, 1, 2, . . . as shown in Fig. 2 [37]. Those regions of stability form independent "stripes", whose finite width drastically decreases as T increases (we barely reach inside the stability regions above T /2π > 5 because of machine epsilon of our computational software). Figs. 3(a) and (b) show the evolution of the spectrum of the two Floquet forms Ψ(t)e st and Ψ * (t)e -st as a function of T /2π for α = 2.25. According to Floquet theory, the eigenfunction Ψ(t) is T -periodic with a fundamental frequency 2π/T . Therefore, by Fourier expansion, the spectrum of the two Floquet forms read s + h i2πh/T and -s + h i2πh/T , where h is an integer. By the superposition principle of Eq.( 2), the solution r(t) contains the sum of both spectrum in it. Fig. 3(a) shows the location of the frequency spectrum of the two vibrational modes ± (s)+ h h as a function of T /2π for α = 2.25 where we introduced (s) = (s)T /2π (only the positive part of the spectrum is shown as the latter is symmetric with respect to the x-axis). For most modulation period T , the frequency spectra of the two modes are locked, alternatively in 0 + h h2π/T (T -periodic spectrum) or in π/T + h h2π/T (2T -periodic spectrum). As shown in Fig. 3(b), this lock-in is associated with a finite growth rate (s) so that one of the periodically oscillating Floquet form is exponentially diverging ( (s) > 0) and one is damped ( (s) < 0). Between those locked unstable regions, tiny zones exist where the two Floquet modes are unlocked and where the associated growth rate (s) is zero for both modes, a situation that leads to the stability stripes in Fig. 2 which correspond to neutrally stable almost periodic solutions r(t). The insets in Figs. 3(a) and (b) display a zoom on the first stability region l = 0. In stability regions, the fundamental frequency (s) evolves continuously from 0 to π/T for tiny variations of T . Note that the green and blue vertical lines are the limits of the stability zones corresponding to neutrally stable T and 2T -periodic solutions with (s) = 0, respectively, and located at modulation periods T which are characteristic values of the so-called periodic Mathieu functions [START_REF] Whittaker | A course of modern analysis[END_REF].

We now focus on the neutrally stable vibrational motions of the mass. Figs. 4(a),(b) and (c) display three typical examples of such vibrational modes over three periods T for modulation parameters in the stability regions l = 0, 1 and 2, respectively [37]. In each panel, we show the position r(t) = Ψ(t)e i (s)t , velocity ṙ(t) and mechanical energy E(r(t)) = 1/2 ṙ(t) 2 + 1/2(-1 + α cos(2πt/T ))r(t) 2 of the computed vibrational modes. The modes are normalized so that N = T Ψ * (t)Ψ(t)dt = 1. Full lines represent the computed almost-periodic motion when dotted lines show the periodic eigenfunctions Ψ(t), Ψ(t) and E(Ψ(t)). The stable almost-periodic vibrations of the mass described in Fig. 4 are synchronized with the modulated potential energy and can be decomposed in a successive repetition of similar motions that are scaled copies of their T -periodic eigenfunctions (the scaling factors can take all the values between -1 and 1 and will be discussed in next section). The eigenfunctions can themselves be decomposed in three parts over one period T : i) when the curvature of the quadratic potential energy in r(t) = 0, κ(t) = (-1 + α cos(2πt/T )), is negative, the mass exponentially diverges, ii) when κ(t) > 0 the mass oscillates, iii) when κ(t) < 0 the mass exponentially converges to a state very almost identical to the previous period, close to r(t) = 0 [37]. The qualitative difference between vibrational motions with ascending order of l from Fig. 4(a) to (c) is that in the l th stability region, the mass is able to do (l + 1) oscillations during the time κ(t) is positive. Like in a classical oscillator, the velocity is zero when the position is at a local maximum. Finally, the mechanical energy E(r(t)) illustrates the "diverging/vibrating" binary behavior of the studied oscillator: over one period T , it seems null when κ(t) < 0 and positive when κ(t) > 0.

REDUCTION OF THE DYNAMICS TO Ψ(t)

4.1. Reduction to the neutrally stable periodic solutions r l (t) = Ψ l (t)

As already mentioned and illustrated in Fig. 4, the almost-periodic neutrally stable modes of the l th stability region, r l (t) = Ψ l (t)e i (s)t , can be decomposed in a succession of cycles that are scaled copies of the periodic eigenfunctions Ψ l (t) = Ψ l (t + T ). To highlight this time-translational property, we superpose, on the primitive time cell -T /2 < t ≤ T /2, the 20 first consecutive periods of the position r(t) of some computed vibrational modes. The result is shown with T /2π = 5 and α = 1.14905152323442 for modes in the l = 0 stability region in Fig. 5(a) where the chronological order is indicated by 20 gradual colors from cold blue to hot red. The dotted line represents the periodic eigenfunction Ψ 0 (t). Around T /2π = 5, the width of the stability region has already decreased to ≈ 10 -13 in the (α, T ) parameter space. Throughout this width , the eigenvalue or fundamental frequency (s) varies in the primitive spectral cell 0 ≤ (s) ≤ π/T (see insets of Fig. 3(a) and (b) to see the evolution of s in a stability region); unlike the periodic eigenfunction Ψ(t) that remains unchanged. As a consequence, unlike its periodic eigenfunction Ψ(t), the position of the stable vibrational mode, r(t) = Ψ(t)e i (s)t , is sensitive to the width of the stability regions (the same is true for ṙ(t) and Ψ(t) or E(r(t)) and E(Ψ(t))). This sensitivity property is highlighted in the top and bottom panels of Fig. 5(a) that show vibrational modes in the l = 0 stability region for T /2π = 5 ± /10, corresponding to (s) ≈ 0.98 × π/T and (s) ≈ 0.62 × π/T , respectively. From one panel to another, Ψ 0 (t) as well as the various curves of r(t) have similar shapes, the only change lies in the chronological order in which r(t) appears. The same numerical observations on sensitivity can be made for vibrational modes in higher regions of stability as illustrated for l = 1 in Fig. 5(b), where the top and bottom panel correspond to a fundamental frequency (s) ≈ 0.25 × π/T and (s) ≈ 0.04 × π/T .

To go further, one needs to specify the scaling factors that relate the position r(t) = Ψ(t)e i (s)t to the eigenfunction Ψ(t) for each successive periods. According to Floquet theory [START_REF] Whittaker | A course of modern analysis[END_REF][START_REF] Magnus | Hill's equation[END_REF], the almost-periodic solution r(t) theoretically verifies the periodic mapping

r(nT ) = cos( (s)nT ) × r(0) = cos( (s)nT ) × Ψ(0) ( 3
)
where n is a positive integer and T is the dimensionless modulation period. What we observe in the numerical results illustrated in Fig. 5 is stronger than Eq.( 3) since it is the whole functions r([nT -T, nT ]), i.e. r(t) for t ∈ [nT -T, nT ], that are related to the unique T -periodic function Ψ([nT -T, nT ]) = Ψ([0, T ]). From numerical observations of r(t), we are able to generalize the quasi-periodicity property of Eq.( 3) from a single time value t to a full period [t, t + T ] so that

r([nT -T, nT ]) = ρ n × Ψ([0, T ]) ( 4
)
where ρ n = cos( (s)(nT -T )) is a real between -1 and 1. In Fig. 6(a r l ([nT -T, nT ]) all collapse on their respective eigenfunctions Ψ l ([0, T ]). If we increase the number of periods, the scaling factors ρ n take a very large number of different values between -1 and 1 and all the r l ([nT -T, nT ]) would collapse on Ψ l ([0, T ]).

The property of Eq.( 4), altogether with the aforementioned sensitivity property of thin regions of stability when T > 2π, offers a reduction opportunity. One could choose to neglect the width of the l th stability regions and reduce the latter to widthless stripes, i.e. to represent a neutrally stable mode of vibration r l (t) = Ψ l (t)e i (s)t by its sole T -periodic eigenfunction Ψ l (t) in the primitive periodic cell -T /2 < t ≤ T /2 by taking (s) = 0. By doing so, we loss informations on the vibrational modes r l (t) whose Floquet exponents actually span 0 ≤ (s) ≤ π/T , but we will gain analytical insights on the oscillator governed by Eq.( 1), notably on Ψ l (t) and their location in the (α, T ) modulation parameter space.

Reduction to a dimensionless Schrödinger stationary wave equation with

a harmonic potential Reducing the regions of stable vibrational modes to the limit branches of periodic Mathieu functions r l (t) = Ψ l (t) (that correspond to the stability frontiers s = 0 of the red regions in Fig. 2), Ψ l (t) are solutions of the initial value problem (IVP), Eq.( 1), so that we can write

d 2 Ψ(t) dt 2 -Ψ(t) + α cos(2πt/T )Ψ(t) = 0. (5)
Unlike the general Eq.(1), Eq.( 5) is periodic and can therefore be reduced to a single representative period T . The complete mathematical analysis of Eq.( 5) is beyond the scope of this paper but one could first resolve it thanks to numerical observations. By construction and for sufficiently thin stability regions with << 1, the normalized periodic solutions Ψ l (t) has a compact support over a period T and can be decomposed in two exponential functions, mirrored with respect to a central vertical axis, connected by a function with l + 1 extrema (see throughout Figs. 23456). Consequently, the intensity of the normalized Ψ l (t) is localized in the center of the period and almost zero elsewhere (this is especially pronounced as the width of the stability region decreases). So, multiplying Ψ l (t) by a T -periodic function F (t) would localized the intensity of F (t) at the center of the period. Choosing the primitive periodic range -T /2 < t ≤ T /2 as the representative cell for the whole solution Ψ(t), Ψ(t) cos(2πt/T ) would cancel the intensity of cos(2πt/T ) away from the origin t = 0 so that, assuming Ψ(t) is localized enough, the Taylor series approximation

Ψ(t) cos(2πt/T ) ≈ Ψ(t)(1 -2π 2 t 2 /T 2 ) (6)
would be legitimate. Replacing Eq.( 6) in Eq.( 5) and upon the change of variable χ = vt with v = (2π 2 α/T 2 ) 1/4 , Eq.( 5) governing Ψ l (t) can be rewritten in the form of a Boundary Value Problem (BVP) on the dual primitive cell 1) (green lines) are compared to the analytical results from the BVP Eq.( 7) (black lines). Each solution E l = l + 1/2 of Eq.( 8) corresponds to a branch of stability, s = 0. (b) In blue line, absolute square of the first four computed eigenfunctions Ψ l (t) of Eq.( 1) on the dual primitive cell -vT /2 < χ ≤ vT /2 for T /2π = 5 and v = (2π 2 α/T 2 ) 1/4 = 0.389351968371210, 0.416745399049464, 0.444673895477715, 0.472863276327510 from bottom to top. The eigenfunctions are normalized so that N = T |Ψ(t)| 2 dt = 1. Black lines are analytically obtained from Eq.( 9).

-vT /2 < χ ≤ vT /2

EΨ(χ) = 1 2 χ 2 Ψ(χ) - 1 2
d 2 Ψ(χ) dχ 2 (7)
where E = (α -1) / √ 2α × T /2π. The linear eigenvalue problem with variable coefficient in Eq.( 7) is well-known as it is the dimensionless form of a stationary Schrödinger equation with a harmonic potential, usually predicting the total energy E and wavefunction Ψ(χ) of a 1D quantum harmonic oscillator [START_REF] Messiah | Quantum Mechanics[END_REF]. For Ψ(-∞) = Ψ(+∞) = 0, the discrete set of eigenvalues E and eigenfunctions Ψ(χ) take the form

E l = (α -1) / √ 2α × T /2π = l + 1/2 (8)
and

Ψ l (χ) = H l (χ)e (-χ 2 /2) /(π 1/4 √ 2 l l!) (9)
where H l (χ) are Hermite polynomials and l = 0, 1, 2, . . . The analytical results of Eqs.( 8)-( 9) allow us to predict the shape of the T -periodic Mathieu functions Ψ l (t) as well as their location in the modulation parameter space (α, T ). Fig. 7(a) shows that each analytical E l of Eq.( 8) accurately predict the T -periodic limits of the l th numerical stability regions of Eq.(1) in the (α, T ) space. As expected from numerical observations, since the width of the l th stability region and the errors in the approximation Eq.( 6) decrease as T increases, the analytical prediction Eq.( 8) for each mode l becomes better with T as shown in Fig. 7(a). Fig. 7(b) illustrates, for T /2π = 5, that the analytical eigenfunctions Ψ l (χ) of Eq.( 9) are in good agreement with the periodic eigenfunctions Ψ l (t) that were numerically computed from Eq.( 1) and shown in Figs. 4(a), (b) and (c) for a α in the l = 0, l = 1 and l = 2 stability regions, respectively. Thus, in a certain asymptotic limit that will need to be rigorously defined in future work, the stationary wave equation Eq.( 7) allows to compute, in the (α, T ) modulation parameter space, the eigenfunctions Ψ l (t) of the stable vibrational motion of the oscillator. The actual position of the mass of the oscillator, r l (t) is not predicted by Eqs.( 8)-( 9), but it can be infer from the previous subsection that it will be a successive repetition of scaled periodic eigenfunctions Ψ l (t).

Conclusions and discussions

In summary, we presented a numerical and theoretical study of a fundamental vibratory mechanism: the discrete dynamical stabilization of a har-monically modulated 1D linear oscillator whose mass is naturally diverging. We have shown an intriguing asymptotic limit of a Kapitza like oscillator governed by a Mathieu equation in which the stable vibrational modes, or Floquet forms, get restricted to thin stability stripes of the modulation space. The multiple initial value problems (IVP) governing the motion of the neutrally stable particle in those modulation regions can be recast in a single boundary value problem (BVP) on a primitive periodic cell that is a dimensionless Schrödinger stationary wave equation with a harmonic potential. This stationary wave equation allows to analytically predict the modulation parameters leading to dynamical stabilization as well as the periodic modal shapes (which are the periodic Mathieu functions of the system) of the stable vibrational modes, albeit with a certain ignorance on the actual motion of the mass.

This paper is a first study, mainly numerical, that helps to shed new light on periodically time-varying oscillators governed by Floquet theory that are at the heart of many problems in physics. In future work, a more rigorous mathematical approach would be needed, notably to characterize the asymptotic limit where the original equation of motion can be approximated by a dimensionless Schrödinger's equation. Another mathematical task would be to determine whether the presented reduction from IVP to BVP is unique, or generalizable to a broader class of ordinary differential equations with periodic coefficients. Meanwhile, this theoretical work raises a practical question: is it possible to experimentally validate the described mechanism? A trivial answer would be to set up an experimental system whose linearized equation of motion is in the form of Eq.( 1). The simplest realization could be an inverted pendulum whose pivot point is vertically vibrated which is the archetypal example of a mass that naturally diverges under periodically modulated gravity. The major problem with such a basic experiment would lie in the theoretical width of the stability regions that we uncovered. The latter are intrinsically so small that any additional experimental noise would probably made the pendulum unstable. A possibility could be to add some damping to the oscillator that would decrease the growth rate (s) and widen the instability regions for any given modulation parameters. The drawback of his approach is that the eigenfunctions Ψ(t) as well as the actual positions r(t) would loose their symmetry in the primitive periodic cell. Another solution would be to start the experiment from modulation parameters for which the stability regions are reasonably large and strive to progressively explore the regions as it shrink. In any case, a study of the dependence of an additional noise on the stability of the motion in a nonlinear framework would be needed to get some physical insights in the transitions between stable regions.

Last but not least are the mathematical analogies of the presented harmonicallyvarying 1D oscillator with the 1D quantum harmonic oscillator [START_REF] Messiah | Quantum Mechanics[END_REF]. Quantum analogs have regained considerable attention due to the recent experimental discovery of a mechanism that consists in a bouncing fluid droplet "walking" on a vertically vibrated bath [24,25,26,27]. By varying the experimental setup, this new framework coined "dual walkers" has already allowed to describe quantum tunneling [28], quantization of classical orbits [START_REF] Fort | Proc. Natl. Acad. Sci. USA[END_REF] and the quantum harmonic oscillator [START_REF] Perrard | [END_REF] in an analogous fashion. Although powerful, the mathematical modeling underlying this hydraulic quantum analog is based on nonlinear interactions between a wave and a particle such as chaotic behavior and time-delay phenomena, which are rather elaborate ingredients when Schrödinger's equation could reduce to a 1D linear ODE in the case of the quantum harmonic oscillator [START_REF] Messiah | Quantum Mechanics[END_REF]. Since the existence of dual walkers is deeply related to Faraday instabilities that are parametric instabilities explained by Floquet theory, it is a possibility that the presented periodically time-varying oscillator framework could help in the physical understanding and modeling of quantization phenomena observed in macroscopic wave-particle interactions.

Figure 1 :

 1 Figure 1: The periodically time-varying oscillator under study. (a) Mass on a local quadratic potential hill, harmonically modulated over a period T . (b) Associated linear mass-spring system with a time-varying stiffness k(τ ) = -k 0 + k 1 cos(2πτ /T ).

Figure 2 :

 2 Figure 2: Numerical stability chart of the harmonically modulated collapsing mass for 0 ≤ α ≤ 10 and 0.2 ≤ T /2π ≤ 5. Red and yellow regions show T and 2T -periodic unstable solutions, respectively. Between those regions are tiny zones, denoted by l = 0, 1, 2, . . . in ascending order of T , where r(t) is a neutrally stable almost-periodic solution. The black line represents the Kapitza limit α = √ 8π/T .

2 ≤

 2

Figure 3 :

 3 Figure 3: Spectrum of the solution r(t) as a function of T /2π for α = 2.25. (a) Location of the frequency spectrum (s) + h. Green and blue vertical lines show the neutrally stable T and 2T -periodic solutions, respectively. Inset: Zoom on the first stability region l = 0 where the spectrum of the two Floquet forms are unlocked. (b) Evolution of the growth rate of the two vibrational modes Ψ(t)e st and Ψ * (t)e -st . Inset: Zoom on the first stability region l = 0 where the growth rate of both modes is (s) = 0.

Figure 4 :

 4 Figure 4: Time evolution position r(t), velocity ṙ(t) and energy E(r(t)) of neutrally stable vibrational modes Ψ(t)e i (s)t with N = T |Ψ(t)| 2 dt = 1. Periodic eigenfunctions Ψ(t), Ψ(t) and E(Ψ(t)) are shown in dotted lines. T -periodic cells are delimited by thin vertical lines. We fixed T /2π = 5. (a) Mode in the l = 0 stability regions for α = 1.14905152323442. (b) Mode in the l = 1 stability regions for α = 1.5081802859941. (c) Mode in the l = 2 stability regions for α = 1.9549540063305.

Figure 5 :

 5 Figure 5: Sensitivity analysis of the stable vibrational modes r(t) = Ψ(t)e i (s)t for T /2π = 5. (a) For a mode in the l = 0 stability region (α = 1.14905152323442), the 20 first consecutive periods of r(t), chronologically ordered from cold blue to hot red, are superposed on the primitive time cell -T /2 < t ≤ T /2. Dotted lines represent the periodic eigenfunction Ψ 0 (t). Top and bottom panel have slightly different T /2π corresponding to (s) ≈ 0.98 × π/T and (s) ≈ 0.62 × π/T , respectively. (b) Same as (a) but for a mode in the stable zone l = 1 (α = 1.9549540063305). Top and bottom panel have slightly different T /2π corresponding to (s) ≈ 0.25 × π/T and (s) ≈ 0.04 × π/T , respectively.

  ), we show for T /2π = 5 the evolution of cos( (s)(t -T )) as well as the scaling factors ρ n = cos( (s)(nT -T )) represented by color dots. From top to bottom, we varied (s) in the primitive spectral cell 0 ≤ (s) ≤ π/T by taking (s) ≈ {0.98, 0.62, 0.25, 0.04}×π/T . Those values of (s) are the ones we obtained in Fig.5from top to bottom. In Fig.6(b) and (c), we divide each position function r([nT -T, nT ]) of Fig. 5(a) and (b) by the corresponding scaling factor ρ n of Fig. 6(a): the 20 first periods of the various computed

Figure 6 :

 6 Figure 6: Periodic mapping and reduction of the neutrally stable vibrational modes to their periodic eigenfunctions for T /2π = 5. (a) Evolution of the n = 20 first scaling factors ρ n = cos( (s)(nT -T )) as a function of number of periods represented by color dots from cold blue to hot red. Black lines show the functions cos( (s)(t-T )). From top to bottom: (s) ≈ 0.98 × π/T , (s) ≈ 0.62 × π/T , (s) ≈ 0.25 × π/T and (s) ≈ 0.04 × π/T . (b) Reduction of the positions r([nT -T, nT ]) of Fig. 5(a) on the eigenfunction Ψ([0, T ]) by ρ n scaling. (c) Same as (b) but for the eigenfunction Ψ 1 (t) of Fig. 5(b).

Figure 7 :

 7 Figure 7: Analytical predictions from a dimensionless Schrödinger equation with a harmonic potential. (a) Limits of T -periodic instability regions of Fig. 2(a) in the (α, T ) space for 1 ≤ α ≤ 2 and 1 ≤ T /2π ≤ 20. Numerical outcomes from the IVP Eq.(1) (green lines) are compared to the analytical results from the BVP Eq.(7) (black lines). Each solution E l = l + 1/2 of Eq.(8) corresponds to a branch of stability, s = 0. (b) In blue line, absolute square of the first four computed eigenfunctions Ψ l (t) of Eq.(1) on the dual primitive cell -vT /2 < χ ≤ vT /2 for T /2π = 5 and v = (2π 2 α/T 2 ) 1/4 = 0.389351968371210, 0.416745399049464, 0.444673895477715, 0.472863276327510 from bottom to top. The eigenfunctions are normalized so that N = T |Ψ(t)| 2 dt = 1. Black lines are analytically obtained from Eq.(9).


domain: math shs

text:
Cássia Aline Schuck
email: cassia.schuck@blumenau.ifc.edu.br

Cláudia Regina Flores
email: claudia.flores@ufsc.br

Being in the diversity and going through the infinity

Keywords: Teaching education, multidisciplinary approach, critical theory

Introduction

The debate on visual culture raises questions that have been stressed in the educational research, such as the role of the image in the process of subject-formation and the visuality as a form of knowledge. Visual Culture is an interdisciplinary field that combines arts, philosophy, anthropology, and cultural studies. The main point of this field is the question of visuality, which means focusing on the relationship between the seen and the seer. In short, the target of Visual Culture Studies is not an object at all, but it "consists of things we can see or whose existence is motivated by their visibility; things that have a particular visuality or visual quality that addresses the social constituencies interacting with them" [START_REF] Bal | Visual essentialism and the object of visual culture[END_REF]. Thus, the term of visuality has become an important keyword for this field [START_REF] Mirzoeff | On visuality[END_REF], because it refers to the vision and visuality are both socially and historically constructed.

With regards to mathematics education research, [START_REF] Flores | Visuality and mathematical visualization: Seeking new frontiers[END_REF] has proposed this perspective of visuality to investigate how images, particularly paintings, affect and are affected by Cartesian perspectivalism visual practices, as well as to grasp the fabrication of a certain mathematical rationality to look and to think within the realms of culture. The author has considered the term 'visuality' instead of 'visualization', because the former leads to a deconstruction of the founding principles of sense of vision and perception. In contrast, visualization is understood as a process of construction and transformation of mental images, whereas visuality is the sum of discourses that inform how we see. Thus, while the latter is concerned with learning geometry's concepts and visual skills, visuality discusses visual practices in the context of history and culture. (p. 7060). Thus, at stake is a kind of mathematical thinking presents in contemporary school practices and that is provoked notably itself through images of arts [START_REF] Flores | Descaminhos: Potencialidades da arte com a educação matemática[END_REF].

Going through infinity

In response to this, we have been examining historically the artistic practices of infinity considering, particularly, those involving the discussions of perspective in renaissance painting. Furthermore, we have analysed modern paintings exhibiting characteristics of a classical system of visuality, which means using concepts of harmony, symmetry, parallelism and perspective. On a whole, we have found not only that our way of looking is shaped within an already built field of techniques and discourses, but also that mathematics ideas play an important role in the constitution of how we think.

Being in the diversity

In the methodological path of cartography, [START_REF] Schuck | Cartografar na diferença: Entre imagens, olhares ao infinito e pensamento matemático (Dissertação de Mestrado)[END_REF] has developed an intervention plan with alcohol and drugs dependents in treatment on a Psychosocial Care Center in Florianópolis, Brazil, in 2014. Four workshops were conducted and each one was centered on images suggesting the infinity idea. By focusing on discourse analyses, interpreting multi-faceted narrative, looks and affections by the subjects involved, it was possible to highlight the effects of looking at infinity rather than 'truth' per se. For instance, seen from the perspective of the people involved, the ways of looking at infinity deal with reflections situated between the mystical and the emotional experiences such as nothingness, emptiness, freedom. In this regard, we call attention to two points: "mathematical mindset" is not the sole result of a schooling regime; and subject's visual subjectivities emerge in the entanglement of the individual in discursive formations. In order to discuss this, we, in the poster, displayed some information concerning on the visualization and visuality, the cartography as research methodology, the workshops itself, and the discourses about infinity staged by the people involved in the intervention mentioned above. Following Foucault's assumptions, we consider that by a better understanding of how a certain kind of mathematical visualization has been constituted within the sociocultural practices might contribute not only teacher education but also mathematics learning practices. In sum, we bring under our attention that both mathematical rationality and visualization are staged and configured within the diverse sociocultural practices.

Proceedings of CERME10

Acknowledgment

We gratefully acknowledge the Brazilian agencies CNPq and CAPES for the financial support.


text:
Rachel-Ann Friesen
email: rachel-ann.friesen@tu-dresden.de

Collaborative learning in multigrade mathematics education at primary level

Keywords: Multigrade classes, collaborative learning, teacher training

Background

In Saxony, predominantly in rural areas, social and political challenges are being posed by a declining number of pupils. To prevent especially younger pupils from having to travel long distances to school, organisational and curricular alterations are being considered. One suggestion is the merging of grades to establish multigraded classes. Multigrade education is also being discussed for its pedagogical advantages; for example, it can make the first phase of primary school more flexible by allowing pupils to remain within the same multigraded class for either one, two or three years. However, even in learning groups which are homogeneous in age, the differences between pupils' preconditions for learning can be up to four years [START_REF] Hirt | Lernumgebungen im Mathematikunterricht. Natürliche differenzierung für Rechenschwache und Hochbegabte[END_REF]). Yet, when making learning groups heterogeneous in age, the diversity of learning preconditions increases, and with it also the necessity for differentiation through diverse tasks with various levels of difficulty. For a subject like mathematics, which is strongly guided by a systematic course, this often means extremely individualised and separated learning [START_REF] Nührenbörger | Unterrichtsgespräche zwischen Schülern und Lehrkräften in jahrgangsgemischten Kleingruppen[END_REF]. However, interaction is seen as foundational constituent of learning in early years [START_REF] Miller | Kollektive Lernprozesse. Studien zur Grundlegung einer soziologischen Lerntheorie[END_REF]Schütte, 2009) and should not be neglected. More specifically, for mathematical learning to occur it is of great importance for the pupils to participate increasingly in collective reasoning within classroom interaction [START_REF] Krummheuer | Representation of the notion "learning-as-participation'' in everyday situations of mathematics classes[END_REF]. To enable individualised learning to take place in cooperation with others, substantial learning environments hold great potential both for mathematics education in general and specifically in multigrade mathematics education because they offer the opportunity for natural differentiation, which means that learners all work on the same task and the differentiation is not predetermined by the teacher but chosen by the pupils themselves [START_REF] Scherer | Natural differentiation in teaching of mathematics for school beginners[END_REF]. The possibility for students to learn with and from others by communicating and helping each other is also seen as one of the pedagogical arguments for purposeful mixed-age grouping [START_REF] Wagener | Potenziale der Jahrgangsmischung[END_REF].

Project

The question this research project wants to address is how collaborative learning takes place in multigrade mathematics education at primary level. Therefore, firstly student teachers at the TU Dresden are asked about their experiences with multigrade education by using a questionnaire. Then, an interview study with individual teachers who teach in multigraded classes in Saxony will be conducted in order to identify their attitudes and concepts concerning multigrade mathematics education. Based on these empirical results, concepts and learning environments for multigraded learning within mathematics education for grades 1-6 will be developed in cooperation with teachers and student teachers in order to prepare them for the future challenges they will face when teaching mathematics in multigraded classes. Later these learning environments will be applied with pupils in practice to test the suitability of the tasks. The collaborative processes the pupils use to solve the tasks will be filmed, transcribed and then analysed from an interactionist perspective [START_REF] Krummheuer | Representation of the notion "learning-as-participation'' in everyday situations of mathematics classes[END_REF] to reconstruct constituent characteristics of successful multigrade mathematics education. This process will be accompanied by the development of seminars for student teacher concerning collaborative learning in multigrade mathematics education.

Initial results

The initial results of a questionnaire with student teachers at the TU Dresden show that less then 40% have experienced multigrade education in some form (e.g. during an internship, while being a student themselves). Even though they are all able to reflect about possible advantages and disadvantages of multigrade education, many of them say that they would feel overwhelmed, unsure or not prepared to teach in a multigraded class during their traineeship. These results need to be assessed in more detail but they emphasize the necessity of training teachers more specifically for multigrade education.

Proceedings of CERME10

Acknowledgement

The project "Collaborative Learning in Multigrade Mathematics Education at Primary Level" is part of a greater project called TUD-Sylber (Synergistic teacher education in an excellent framework) at the Technical University of Dresden, which is funded by the German Federal Ministry of Education and Research in the framework of the joint state and federal "Teacher Education Quality Offensive".


text:
David Swanson
email: david.swanson@manchester.ac.uk

Laura Black
email: laura.black@manchester.ac.uk

Integrating critical theory and practice in mathematics education

Keywords:

Recent years have seen many valid and important critiques of mathematics, mathematics education and mathematics education research (M, ME & MER). However, we also discern in some of these critiques a tendency toward one-sidedness and passivity. Unrelenting stress on the negatives of M, ME and MER can lead to a dismissal of the possibility of improving ME and a dismissal of those who attempt to do so. The separation of critical theory from critical practice which follows is then in danger of rendering critique sterile, becoming a mere pseudo-radicalism. As an alternative, we explore here the mutual relation between critical pedagogy and critiques of society, and the relationship between reform and more radical change, in wider society and ME. We argue that this analysis encourages a stress on joint activity, between individuals and organisations with a wide range of perspectives on what change is needed, to tackle the problems a critical perspective raises.

Introduction

A growing body of research in mathematics education has explored critically the socio-political function of school mathematics in terms of its role in the mobilisation and production of capitalism (e.g., [START_REF] Kollosche | Mathematics and power: An alliance in the foundations of mathematics and its teaching[END_REF][START_REF] Pais | An ideology critique of the use-value of mathematics[END_REF][START_REF] Pais | Economy: The absent centre of mathematics education[END_REF][START_REF] Swanson | Expressions of the commodity form: Alienation and mathematics education[END_REF][START_REF] Williams | Use and exchange value in mathematics education: Contemporary CHAT meets Bourdieu's sociology[END_REF] leading to the reproduction of inequalities in education along the lines of social class, gender and ethnicity (e.g., Jorgensen & Niesche, 2008;[START_REF] Solomon | Mathematical literacy: Developing identities of inclusion[END_REF][START_REF] Black | Teacher-pupil talk in whole class discussions and processes of social positioning within the primary school classroom[END_REF]Noyes, 2007). This work suggests that school mathematics (and associated qualifications) serves as a 'gatekeeper' in that it enables society and its dominant institutions (e.g., universities, employers etc.) to select and sort individuals under the rationale that 'mathematical ability' is a valued source of human capital. This critique is highly relevant to the current situation of mathematics education in England, where a new, more challenging curriculum has come into play since 2014 which serves to further substantiate the elitist position of mathematics in schools. At the same time, there is widespread concern in policy and in the mathematics education research community about achievement gaps (i.e. between rich and poor, or the most and least deprived)a concern which has been said to further produce social inequalities - [START_REF] Gutiérrez | A "gap-gazing" fetish in mathematics education?: Problematizing research on the achievement gap[END_REF] refers to this as "gap gazing".

Whilst such critiques of mathematics and mathematics education are important and necessary to challenge dominant ideologies (including those pertaining to education more broadly), at the same time, we argue, there is a need to propose an ideologically grounded alternative. This paper presents a case for an alternative way forward by first looking at the potential mutual relation between critical pedagogy and critiques of society. We then examine the relationship between reform (i.e. improvement whilst remaining within the same overall framework) and more radical change, in terms of wider society, education and mathematics education. What follows from this exploration, we argue, is the need for activity which tackles the problems a critical perspective raises. In particular, it suggests joint activity between those with a wide range of perspectives on what, and how far, things need to change. We then look briefly at the Stand Up for Education campaign in the UK, which brings together trade unionists, teachers, academics and parents, to show that what we outline here is not a purely abstract or ideal position. Spaces for the much required interrelation of critical theory and research with practice can and do exist, and we conclude by discussing why that matters for critical researchers.

What might a critical perspective on education look like?

We begin by looking at the relationship between critiques of education or society and critical mathematics pedagogy1 . Arguably the most radical perspective here is to imagine and work towards a society beyond capitalism (e.g. [START_REF] Bowles | Schooling in capitalist America[END_REF][START_REF] Counts | Dare the school build a new social order?[END_REF][START_REF] Freire | Pedagogy of the oppressed[END_REF] i.e. a change that involves a complete transformation of society. Discussions of what form education would take in such a society face certain limitations however. For instance, if we assume, as we should, that moving beyond capitalism entails the democratic collective control of society by the majority, then we who are so shaped by, and operate in, capitalist society are not best placed to either decide or predict what may happen. Nevertheless, we can speculate on how an alternative future education might work by taking the reverse of the features of capitalist education which are seen to lead to negative consequences today, for example the individual-competitive exam system which produces 'losers' who internalise their failure as objective qualities of themselves (for others, see [START_REF] Swanson | Expressions of the commodity form: Alienation and mathematics education[END_REF]. Then, we can combine these with the aspects of education that others have fought for (e.g. in school student strikes against corporal punishment, oppressive uniform policies or privatisation, see [START_REF] Lavalette | Schools out! The hidden history of Britain's school student strikes[END_REF] and at times, implemented (e.g., the banning of homework and exams in revolutionary Russia, see [START_REF] Karp | Soviet mathematics education between 1918 and 1931: A time of radical reforms[END_REF].

Among the features we might expect to see are i) democratic collective control of education by teachers, students and other education workers, within the wider framework of its democratic shaping for society's needs; ii) much greater control by individuals over their own learning within that, but with an emphasis on social rather than individually competitive learning; iii) an end to exams and their production and reproduction of societal inequalities; iv) an equivalent end to the performativity culture of continual measurement to judge teachers and other education workers, v) an increase in societal resources (such that, for example, class sizes reduce to the levels seen only within private education in this society), and vi) a closer integration of education with wider life, reducing both the formal detachment of schooling from the world outside, and the artificial separation of subjects from one another.

The perspective above can loosely be termed the revolutionary perspective in education. We would define reformist perspectives, and these are far more common than revolutionary perspectives 2 , as those which may agree with some, many or even all of the elements above but which are accepting of greater limits as to how much things can change, for example limiting the possible changes to within one classroom, or to what is possible within capitalism. Although this covers an enormous range of possible beliefs (e.g., the free schools movement in the U.S., see [START_REF] Miller | Free schools, free people: Education and democracy after the 1960s[END_REF], or on a small scale, Boaler's work on reform pedagogy in the US, [START_REF] Boaler | Creating mathematical futures through an equitable teaching approach: The case of Railside School[END_REF] and in Australia [START_REF] Sullivan | Transposing reform pedagogy into new contexts: Complex instruction in remote Australia[END_REF], in general we view such perspectives as radical and important. Fighting for fewer exams or less influence of exams on education clearly overlaps with fighting for no exams. We explore the general relationship between reform and revolution in a later section, but first we look at the relationship between the radical perspectives discussed so far and a particular type of reform, that of improving mathematics pedagogy.

The relationship between critical mathematics pedagogy and (active) critiques of society

We can see within critical mathematics pedagogy (in the broad sense of the term critical) parallels of many of the more general demands of radical educationalists. For example, we see pedagogies which aim to promote: a more active role for students in learning through open problem solving (e.g. [START_REF] Barron | Doing with understanding: Lessons from research on problem-and project-based learning[END_REF]; teaching for understanding rather than for grades (e.g., [START_REF] Schoenfeld | When good teaching leads to bad results: The disasters of 'well-taught' mathematics courses[END_REF]; an emphasis on dialogue and social learning (e.g., [START_REF] Lerman | Intersubjectivity in mathematics learning: A challenge to the radical constructivist paradigm[END_REF]; and a more meaningful mathematics connected to the world outside of school and student experiences and concerns (e.g., [START_REF] Gainsburg | Real-world connections in secondary mathematics teaching[END_REF]. In doing this, pedagogy acts to counter some of the worst effects of capitalist education, even if it cannot overcome them fully. Here we argue, perhaps contra to some perspectives in critical MER, that it is worthwhile to subvert spaces, such as the classroom, as much as one can in these directions. Various forms of critical mathematics education which attempt to provide curricula and pedagogies which offer 'use value' to low status, disadvantaged or 'poor' learners and communities (e.g., [START_REF] Skovsmose | Opening the cage: Critique and politics of mathematics education[END_REF][START_REF] Gutstein | Reading and writing the world with mathematics: Toward a pedagogy for social justice[END_REF] have much to offer. They can potentially challenge the 'gatekeeper' role of mathematics described above and can maybe transform the function of education, that is, rather than the learner serving the school/education system, education can begin to serve the community/learner [START_REF] Williams | Mathematics capital in the educational field: Bourdieu and beyond[END_REF]. Perhaps more importantly, attempts at developing critical thinking within the mathematics classroom have the potential to be generalised and transferred to other aspects of life, for example, to a pupil's future life in the workplace (see [START_REF] Black | Developing a 'leading identity': The relationship between students' mathematical identities and their career and higher education aspirations[END_REF]. The experience of critical thinking, of challenging everything, of weighing up arguments can assist in developing the confidence to do so elsewhere.

The possible connections between critical pedagogies and critical perspectives on society can work in the other direction too. The real limitations which schooling imposes on such pedagogy means that it is difficult to sustain critical educational activity if it is solely limited to the individual classroom. Teachers attempting to develop or sustain attempts within their classroom will come up against obstacles. For example, a head of department on a professional development course led by one of the authors was instructed by management to reverse pedagogical changes because students were now talking too much in class. However, arguably, the experience of these obstacles can make teachers open to looking beyond their immediate situation to help them achieve the changes they want. If teachers are connected to networks which challenge how schooling is generally organised and which also show sympathy for progressive forms of pedagogy, they may potentially move towards engaging in critical activity outside the classroom, whether still directly related to pedagogy or beyond that. Such networks can also give teachers the confidence to persist with their efforts in their own classroom. (e.g., Volosinov, 1976, on the relationship between an individual's critical ideas and collective agency in such circumstances).

Taken together these points mean: Firstly, that it is in the interests of those who are critical of society to encourage meaningful activity in the classroom and to work alongside others who wish to do this, and, secondly, it is in the interests of those who want more meaningful activity in the classroom to work with those who have a critical perspective on society, precisely because they bring an understanding of the obstacles, and, usually, experience in organising networks to overcome these obstacles. A central task therefore for those who are critical of society and who work within mathematics education, is to help create, develop and shape organisational forms which encompass both these components.

Reform, revolution and the united front

The relationship between the particular reform, of developing more meaningful pedagogy in a classroom, and wider social struggles, rehearses similar arguments to that which can be made about the general relationship of reform to revolution. In general, reform and revolution are clearly different perspectives. As [START_REF] Luxemburg | Reform or revolution[END_REF] puts it:

[T]hose who pronounce themselves in favour of the method of legislative reform in place of, and in contrast to, the conquest of political power and social revolution, do not really choose a more tranquil, calmer and slower road to the same goal, but a different goal. Instead of taking a stand for the establishment of a new society, they take a stand for the minor modification of the old society. (p. 56) However, many of the elements key to a revolutionary strategy-for example, maximising active involvement and democratic control of movements; overcoming the division between purely economic and political struggles; attempting to connect up and generalise different struggles; developing an understanding of the interrelated nature of societal problems; and an emphasis that change comes from below, are not necessarily alien to those holding reformist ideas when they are engaged in struggles for particular demands (see, e.g., an account of the 2012 Chicago teachers' strike in [START_REF] Gutstein | The rebirth of the Chicago Teachers Union and possibilities for a counter-hegemonic education movement[END_REF]. At the same time, revolutionaries are also in favour of reforms. First because they improve immediate circumstances, but also because it is through the struggle for reforms that people develop the consciousness and confidence required to transform society: "The struggle for reforms is its means; the social revolution, its goal" (Luxemburg, 1986, p. 5). This overlap in immediate situational objectives, and the potentially shared belief in activity to achieve them, can provide a basis for joint activity. This joint activity between those who seek reform and those who aim for more fundamental change is central to a revolutionary approach and is termed the united front strategy. The strategy was explicitly formulated by the third congress of the communist international in 1922 as capitalism restabilised following the revolutionary wave around the end of the First World War. However, it has its roots in earlier practice. For example, during the Russian revolution of 1917, the unity of revolutionaries and reformists in repulsing Kornilov's attempted coup was central to the development of the revolution, and the key organisational form of the revolution, workers councils or soviets, can be viewed similarly as a united front (see [START_REF] Trotsky | Fascism, Stalinism and the United Front[END_REF].

For revolutionaries, there are two key aims of the united front strategy. The first is simply to increase the likelihood of success of the particular struggle through uniting the maximum number of people and organizations. Secondly, it aims to convince those involved in reform activity of the need for more radical change through i) joint experience of the benefits of revolutionary methods, ii) joint frustration at the limitations of reformist strategies, and iii) exposure to revolutionary ideas in ongoing dialogue. For these strategies to work, the unity and dialogue must be genuine of course, with the possibility of reformists winning revolutionaries to their strategy instead (see [START_REF] Trotsky | Fascism, Stalinism and the United Front[END_REF]. Although often from an alternative perspective, many with a reformist outlook on change equally see the importance of working together with others who hold different ideas to help achieve particular aims.

Implications for critical mathematics educators

Taken together, the arguments outlined so far imply the need for forms of organization which bring together various groups in mathematics education such as teachers, teacher educators, critical academics, parents, students and other education workers in common activity (a united front). This includes those who are particularly concerned with teaching and learning and those who are trade unionists; those who want to transform the world completely and those who just want to make things a little better. Through such activity radical mathematics educators can both assist in improving immediate circumstances in schools, classrooms etc., and also increase the numbers of those who see the necessity of more radical change (e.g., [START_REF] Gutstein | The rebirth of the Chicago Teachers Union and possibilities for a counter-hegemonic education movement[END_REF]. We speculate that such an organization in relation to mathematics education is more likely to arise as part of, or emerging through, more general forms developed for the field of education as a whole. To illustrate that organisational forms such as this can exist, we now briefly describe the emergence of a network, local to the authors, which brings together the various forces described above.

Stand up for Education

The Stand up for Education campaign by National union of teachers (NUT) (2014) first emerged as a campaign launched by the National Union of Teachers (the largest teaching union in the UK), in the build up to the 2015 UK general election, to influence educational policy discussions and mobilise NUT members and others. Through that campaign, a network of academics supportive of the NUT's aims was formed called Reclaiming Schools. Together Reclaiming Schools and the NUT jointly published a collection of short articles from academics and researchers in support of the campaign (see NUT, 2015). The Reclaiming Schools network continued, with a website devoted to putting research in accessible form for teachers and others campaigning to improve education, and with occasional meetings in local areas which bring academics and teacher activists together and promote the website's activities. At one such meeting in Manchester, partly inspired by recent parent campaigns to remove their children from standardised testing (the Let Our Kids be Kids campaign, see https://letthekidsbekids.wordpress.com), the idea emerged for a local conference to be held which could pull together wider forces. NUT activists organised a follow up meeting, which included some parent groups and academics, to plan the conference. The primary aim of the conference was to share and develop understandings of key issues affecting schooling; to develop and expand the different networks involved (parents, teachers, teacher educators and other academics), and to bring those networks together to promote mutual activity and campaigning. The conference (see https://www.facebook.com/standupforeducationmcr) united precisely the range of people that this paper has outlined, and, importantly, it discussed questions of organised activity, political issues and pedagogy. In future work we will discuss this movement in more detail, and in particular explore its potential in relation to critical mathematics pedagogy in particular. But we describe it briefly here to show that such networks can and do exist and are not merely an abstract desire of the authors.

Critical theory and critical practice

So far we have argued that i) critiques of ME and MER also require ideologically grounded alternatives; ii) both 'revolutionary' and 'reformist' alternatives exist; iii) critical pedagogy in mathematics (a particular reform) can be an integral part of both perspectives; and iv) this interrelationship between reform and revolution is a general one. These last two together entail v) the importance, and possibility, of united front activity and organization within the field of mathematics education for all those who are to any extent critical of how things currently are, whether their initial motivation is teacher wages and conditions, less stressful exams for children, or more meaningful activity in the classroom.

We conclude with the particular relevance of the above for critical mathematics educational researchers. Marx argued that "Practice without theory is blind. Theory without practice is sterile", and this point is relevant for those who wish to criticise the world of mathematics education without attempting to change it. Arguably though, theory and practice always form an interrelated unity. No practice is uninformed by theory, (it may be unconscious of course). And no theory is unshaped by practice. The question for educational researchers is which practice shapes their theoryacademic practice with its demands of publication and superficial novelty, or genuine critical practice and the needs of those trying to transform education. Critical theory detached from critical practice may provide useful insights, but ultimately its quality and usefulness will suffer from the separation. Uniting critical theory and critical practice, on the other hand, can enrich theory and research, and contribute to the development of the critical practice which can transform education.

Proceedings of CERME10

We use pedagogy here and throughout to mean teaching and learning combined due to the lack of an adequate single word in English.

This is true even when reformist political organisations are weak or non-existent. However recent years have seen an important revival in reformist organisation with new parties such as Podemos in Spain, or around individuals in existing organisations, such as Corbyn in the UK and Sanders in the U.S.Thematic Working Group 10Proceedings of CERME10


text:
Helena Roos
email: helena.roos@lnu.se

Diversity in an inclusive mathematics classroom: A student perspective

Keywords: Inclusion, diversity, equality, access to mathematics

This paper reports on a study exploring inclusion in mathematics education from a student perspective. The theoretical and analytical approach in the study is discourse analysis. The results presented in this paper are based on 8 interviews with students from lower secondary school and 4 observations of mathematics lessons. The teachers describe the students as students in special needs in mathematics (SEM). The results show that, from a student perspective, the teaching and learning of mathematics in an inclusive classroom is complex and diverse. At the same time, as these students are similar in that they are SEM-students, they are different when it comes to how they themselves want to be included in the mathematics. These differences regard both the organization and the content. Thus, diversity among students demands diversity in the mathematics education.

Introduction

A growing body of research in mathematics education is focusing on access and equity. This can be seen in some of the research books that have been published in recent years, for instance Diversity in mathematics education: Towards inclusive practices [START_REF] Bishop | Diversity in mathematics education: Towards inclusive practices[END_REF], Towards equity in mathematics education [START_REF] Forgasz | Towards equity in mathematics education: Gender, culture and diversity[END_REF], and Mathematical literacy: Developing identities of inclusion [START_REF] Solomon | Mathematical literacy: Developing identities of inclusion[END_REF]. In addition to this, at school-level, the issue of the need to meet every student's needs in the mathematics classroom according to the preconditions and needs of each and every one has been recognized [START_REF] Roos | Inclusion in mathematics in primary school: What can it be?[END_REF]. This task, to be able to meet all students' needs and create opportunities to learn, is not at all an easy task. Some teachers even say that it is impossible to meet every student's needs in an inclusive classroom because of the diversity. On the other hand, one could look upon teaching diversity as [START_REF] Frederickson | Special educational needs, inclusion and diversity[END_REF] do, claiming that teaching is interesting because of the diversity among students, but it is only possible because of the similarities among students. This implies that even if there is diversity among students in our mathematics classroom the teaching does not have to be different for each and every student, but by being aware of the diversity as a teacher you can develop a sensitivity towards equality in the teaching. In that sense, you put the students' needs at the forefront in the explanations and tasks given.

In mathematics education one of the motivations to strive for inclusion and access to mathematics for all is that it is a human right to know mathematics to achieve participation in society [START_REF] D'ambrosio | Ethnomathematics: A response to the changing role of mathematics in society[END_REF]. Most often inclusion is discussed and researched from this ideological perspective even though it is often used as a method in schools. Hence, there is a need for studies investigating inclusion in practice, which this research aims at. This leads to the research questions of this paper, which are: how do students experience an inclusive mathematics classroom and when do they express having optimal opportunities to learn mathematics?

Diversity and Inclusion

What does diversity and inclusion in mathematics mean? It seems that when scholars talk about diversity in mathematics, they almost always speak of inclusion in the same breath (e.g., [START_REF] Bishop | Diversity in mathematics education: Towards inclusive practices[END_REF], indicating the two notions are closely related. When investigating further, inclusion, if used as a tool in classrooms, can be seen as a way of meeting diversity, supporting all learners within a local community [START_REF] Booth | Developing inclusive teacher education?[END_REF]. Diversity on the other hand is not often defined, but used together with gender and culture (e.g., [START_REF] Forgasz | Towards equity in mathematics education: Gender, culture and diversity[END_REF]. It is also used together with specific subject area in mathematics and students' performances within this area (e.g., [START_REF] Hopkins | Capturing diversity in the classroom: Uncovering patterns of difficulty with simple addition[END_REF]; hence diversity is here connected to some kind of ability in mathematics. Accordingly, diversity can be connected to different things. In this paper diversity is connected to inclusion on the level of optimising students' performance in mathematics.

An inclusive classroom is in this paper defined as a classroom that is not grouped by ability but instead as a classroom in which students struggling with mathematics as well as students in need of more challenges in mathematics are taught working with similar tasks and the same mathematical content. Hence, diversity from an ability perspective is prioritized. This puts demands on the teacher, being aware of the diversity [START_REF] Solomon | Mathematical literacy: Developing identities of inclusion[END_REF], to have equitable instructional quality. This quality can be seen in the teachers' mathematical knowledge and their preparation for the teaching of mathematics as well as their beliefs about and skills in teaching diverse students [START_REF] Allexsaht-Snider | Mathematics for all": How do we get there? Theory into Practice[END_REF]. So, in having equitable instructional quality, students' opportunities to learn mathematics might increase. But according to [START_REF] Rousseau | Understanding the significance of context: A framework to examine equity and reform in secondary mathematics education[END_REF], there are factors that can work as barriers for increased opportunities to learn mathematics: large class sizes, high-stakestandardized tests, absenteeism and mobility of students and a lack of a high-quality curriculum. All these issues arise from a teaching or organisational perspective. Hence, it becomes important to also listen to the students' voices, enabling teachers to understand processes of exclusion and inclusion in the mathematics classroom like [START_REF] Solomon | Mathematical literacy: Developing identities of inclusion[END_REF] highlights.

One important issue arising when talking about the ability of individuals is labelling. There are many teachers that claim that in order to be able to meet the diversity of students they have to "label" them in some way. Though, [START_REF] Askew | Diversity, inclusion and equity in mathematics classrooms: From individual problems to collective possibility[END_REF] claims that meeting diversity does not imply that we have to label the students, because labelling might perpetuate exclusion instead of promoting inclusion. One way to meet diversity and create an inclusive classroom is to support cooperative learning [START_REF] Askew | Diversity, inclusion and equity in mathematics classrooms: From individual problems to collective possibility[END_REF] by building a sense of belonging and safety where diversity is valued [START_REF] Reid | The discursive practice of learning disability: Implications for instruction and parent-school relations[END_REF]. This implies that the teacher and the pedagogy the teacher uses in the classroom are really important to create this learning community. Then, as [START_REF] Liasidou | Inclusive education, politics and policymaking[END_REF] points out, pedagogy is an important dimension of inclusion. All this implies that diversity and inclusion are intertwined, and if striving for inclusion one has to respect diversity [START_REF] Booth | Developing inclusive teacher education?[END_REF]. Accordingly, in this study the objective is to investigate diverse students' experience of an inclusive mathematics classroom.

The current study

The focus in this research is on students in special educational needs in mathematics (SEMstudents). Special educational needs are here defined as a need of another teaching then the regular mathematics teaching. This is not unproblematic, since it signals a labelling of students in the research. To be vigilant in terms of that issue, the teachers in the study have made the selection of students. In this research, inclusion in mathematics is investigated from two different SEMdirections: students struggling with mathematics, and students in need of more challenges in mathematics. This kind of selection is information-oriented and used to "obtain information on unusual cases which can be especially problematic or especially good in a more closely defined sense" (Flyvbjerg, 2006, p. 230).

An ethnographic approach is used, meaning the researcher participates directly in the social setting collecting data without meaning being imposed on the participants [START_REF] Brewer | Ethnography[END_REF]. The ethnographic approach also offers in-depth study [START_REF] Hammersley | Ethnography: Principles in practice[END_REF], which can be used to follow a process in a particular case, such as to be included in mathematics teaching and learning. An ethnographic study usually investigates people's actions and accounts in an everyday context. In this study, a Swedish lower secondary school (students are 13 to 16 years old) that sets out to work inclusively is observed. That is, the school strives to have all students in the classroom, even the students assumed to be in special educational need of any kind. To be able to meet all students' needs, they strive to have at least two teachers in the classroom for each lesson. The school is an urban school with about 500 students, located in the outskirts of the city.

One grade 7 and one grade 8 class were observed. The mathematics teachers in a discussion with the researcher selected the classes and the students. Grade 7 was selected based on the criteria that working inclusively might be new for them. Grade 8 was selected since they have been working inclusively for a year. Students from grade 8 and grade 7 that struggled with mathematics as well as students that needed more challenges were chosen and interviewed several times. Ethical considerations were made both before and during the process. Both the students and their guardians gave written consent. As a researcher, I reflected on what ways I affected the students and the research. Another aspect was that the students in the classes would be able to handle a researcher in the classroom.

Methodology

The approach used in this research is Discourse Analysis (DA) and the data consists of observations from two classes.

Discourse Analysis

Discourse Analysis is chosen as approach because of the power of DA to focus on language in interaction and language above or beyond the sentences (Gee, 2014a) and its explanatory power of social contexts and meaning making. The focus of DA is on language and text, what we actually can see, hear and read. In this study, ethnography was applied together with DA in order to make students' expressions of mathematics teaching and learning visible. DA and the ethnographic approach complement each other in this research; DA provides theoretical and analytical notions, while ethnography provides a way to conduct research.

In this paper, DA is used from the perspective of Gee (2014a[START_REF] Gee | How to do discourse analysis: A toolkit[END_REF], since this focus is descriptive and I intend to describe how students want to participate in an inclusive mathematics classroom to be able to have optimal opportunities to learn. From Gee's perspective, DA covers all forms of interaction, both spoken and written, and he provides a toolkit for analysing this interaction. These tools put focus on the communication and ask questions of the text. Hence, in this research, the toolkit is used as a methodological tool. Gee (2014a[START_REF] Gee | How to do discourse analysis: A toolkit[END_REF]) also provides theoretical notions, such as big and small discourses (henceforth referred to as Discourse with capital D and discourse with lowercase d), where Discourse is looking at a wider context, social and political. Discourses are always embedded in many various social institutions at the same time, involving various sorts of properties and objects. For example, a Discourse can be "assessment in mathematics." Discourses are always language plus "other stuff" (Gee, 2014a, p. 52). This other stuff compromises actions, interactions, values, beliefs, symbols, objects, tools and places. Small d discourse is focused on language in use, what stretches of languages we can see in the conversations or stories we investigate (Gee, 2014a). In this research, big and small discourses will be the theoretical perspective. Hence, DA is used both as a theory and a tool and provides a set of methodological and theoretical lenses.

Procedure

During one semester (January to June 2016) I observed the two classes at the chosen lower secondary school. I was present at least one mathematics lesson each week for each class doing observations. After observations, I conducted interviews with the selected students. Since I had both ethical and organisational issues to take into consideration, the interviews were done when the students wanted to and had time, and the teachers allowed it (they did not want me to interview them when they had their ordinary lessons). The interviews took place in a room next to the classroom when the students had "class time" once a week. The interviews were based on the close in time observations; hence, they were situated and narrative. I asked questions about situations and tasks and showed photos of tasks on the blackboard. We also looked at tasks in their textbooks. The first and the last interview were based on a questionnaire about their mathematics education.

Data analysis

In this paper, eight interviews and four observations have been used in the analysis, two interviews with a student in grade 7 named Billy and two interviews each with three students in grade 8, Edward, Ronaldo and Jeff. The teachers perceive Ronaldo and Jeff as students in struggle with mathematics and Billy and Edward as students in need of more challenges in mathematics. In the interviews the students got questions about what they wanted from the teaching in mathematics, how they learned mathematics best and also got questions arising from the previous mathematics lesson, which were the four observed lessons. The observations were used as contextualisation for the interviews as well as for supporting identification of big Discourses. When analysing the data by asking questions to the text, both small and big discourses appeared. That is, while examining the text, I used Gee's toolkit by asking specific questions. Depending on the type of text, different questions were asked. For example, when using the subject tool, I asked, "What are they talking about here, and why?" When using the deictic tool, I asked, "What is pointed out in the text, and what is the listener assumed to already know?" When applying the fill-in tool, I asked, "What needs to be filled in to achieve clarity? What is not being said overtly, but is assumed to be known or inferable?" Then, stretches of language(s) appeared when finding answers to the questions, which signalled for small discourses. When adding analysis of the data from the observations, such as text on the blackboard and the actions of the teachers, big Discourses could be identified.

Result and analysis

In the analysed data, three themes, or using Gee's (2014b) terminology, three "stretches of languages" emerged. The first theme was about how students wanted to participate from an organisational perspective. The second theme was about tasks they did or did not like and the third theme was how the student wants a mathematics lesson to be like for an optimal learning opportunity.

Organisational aspects

In the interviews, stretches of languages about organisational aspects were showing. In the first interview Ronaldo says, "We are starting to go outside [the classroom] into small groups, like we did not do before, and it feels much better now. I am concentrating a lot better [in a small group] and like that." He also explained why it felt better to be in a small group: "It feels better actually, you get peace and quiet and then… like me… if they talk a little there [inside the classroom] I lose concentration right away and listen to what they [the other students] are saying; when it's smaller groups I am able to concentrate better and learn more." In Jeff's first interview he also highlights the possibility of being able to go outside the classroom: "… if it's a test or something I would rather be outside [the classroom] since I am more focused then." Edward also talks about the organisation, but within the classroom when he is discussing cooperation. "It is not very easy, since I have often come a long way, so I always explain to them, it never gets to a discussion for me… I mean, with somebody else, that we discuss and so on […]." When the researcher asks Edward if it is hard to discuss with everybody he says, "It depends on whom I am sitting next to." He also expresses that he does not sit next to someone who can challenge him, and says that he would like to do that more. "I think I would get more out of it."

Tasks

Other stretches of languages showed talk about tasks. Billy explicitly talks about his need of more challenging tasks in the classroom. "I like those [tasks] which are harder, those that challenge you." "[I would like to have] more challenges […] at the lessons." He explicitly talks about problem solving tasks as something challenging: "I like it when we have problem solving. You get to think for yourself and then talk to friends [about it]." Ronaldo also talks about problem solving but in a rather different way. "I hate problem solving tasks more than anything! I just cannot do it." He also describes why he does not like it: "It is hard with reading comprehension and like that, and to connect it with like the task and the text, it gets too much. It is often that kind of task I fail at on the tests." Jeff talks about tasks that he likes. These are tasks he knows "how to calculate and tasks that I understand." He mentions geometry tasks as tasks he likes. The type of tasks Edward likes the most is Algebra tasks. "To be able to find out all the variables, it is fun to figure out what it is." Here we can notice a difference to how Ronaldo thinks about algebra tasks when he states that he thinks that it is hard with "like all this with X and Y and everything … it is terrible."

A good mathematics lesson

A third stretch of language is the talk about what the students want from a mathematics lesson in order to be able to learn mathematics the best. Jeff states that what is most important for him in a mathematics lesson is "if they [the mathematics teachers] explain good … and thorough, if they write step by step." He also says if he knows what to do it "feels good, I know what to do and I get on with it right away." Meaning, for him the thoroughness and structure in the instructions is important. Ronaldo likes to have a lesson when you "first work a little [by yourself] and then some 'going through'1 and then you work a little by yourself and then you do some group work with those you sit with […]. It is more fun when you are in a group and cooperate." Edward on the other hand thinks that it is hard to cooperate and "extremely hard to get something from the others in the class." Instead, he like lessons "with variation so that you don't get tired." He also likes "a going through or something like that, gladly a game or something. You should [also] count a little by yourself I think." Billy thinks that the lessons are best "when you get to explain to others [students] how you have done it." He likes when "we have like problem solving." He also stresses that he wants "more challenges at the lessons."

Identified discourses and Discourse

The three themes appearing in the dataorganisational aspects, tasks and a good mathematics lessoncan be interpreted as small discourses (Gee, 2014a) in the students' talk of their mathematics education. There are aspects in this talk from the different students that overlap, such as wanting to be in a small group sometimes, which both Jeff and Ronaldo stressed. Another aspect that overlaps is collaboration. Both Billy and Ronaldo highlighted collaboration, but Edward on the other hand felt that his peers did not challenge him in the discussions, but he said that he thought he would get more out of it (meaning the discussions) if he sat by a peer that had the possibility to challenge him. There are aspects that diverge, such as what type of task they want to have. Billy says he wants more problem solving and Ronaldo "hates" problem solving. Edward likes algebra tasks and Jeff likes geometry; hence there was no consensus on types of tasks or mathematical content they prefer. Another aspect that appears in the students' talk was the organisation of the lesson. To attend a structured lesson, with both explanations by the teacher and work by yourself or together with others seemed to fit them all. Jeff is more explicit about his wishes for structured lessons, indicating a need of structure to explain what to do. Even though all these small discourses contain diversities, they together indicate a big Discourse. When looking at the observation notes from the lessons you can identify support to this big Discourse, in terms of talking about tasks, organisational aspects such as "talk to your neighbour" or when the special teacher attending the classroom walks outside the classroom with some students; also, the type of task being addressed at the blackboard, and the way the teachers structure the explanations at the blackboard and the talk of what to include in an explanation. This big Discourse can be named "mathematics in school": what mathematics in school is or means for the students and what they want from the mathematics to be able to learn best.

Summary of results

The result shows both overlapping and diverging issues regarding how students experience an inclusive mathematics classroom and having optimal opportunities to learn mathematics. Regarding organisational aspects, Jeff and Ronaldo stressed the need of being in a small group outside the classroom from time to time even though the school promotes physical inclusion, and Billy and Edward highlighted collaboration in order to have opportunities to learn. Diverging aspects are type of tasks; for example, where Billy wants challenging tasks in the form of problem solving, Ronaldo hates problem solving. How the students want a mathematics lesson both diverges and overlaps: thoroughness and structure in the instruction; work a little by yourself and then in a group; lessons with variation and explain to others were expressed as good ways of learning.

Discussion

A diverse picture of how students want to participate in an inclusive mathematics classroom to be able to get optimal learning opportunities appears in this research. Although this is not unexpected, it is important to highlight this diversity and address the question of how the organisation and the education can support this diversity in order for students to be included in mathematics. This is not just only spatially included, but is also included in the teaching and learning of mathematics. The diversity of how students want to participate in the mathematics classroom that appears in this research stresses the need for diversity in mathematics education at school. One thing that is striking is that students expressed a need of being in a small group outside the classroom from time to time. The school promotes physical inclusion, and it did not seem to always benefit these students. You might say that diversity among students demands diversity in mathematics education. But, as Fredickson and Cline (2009) stress, even though students are different, teaching is only possible because students are similar in some ways. This research supports this, because even if the results showed diversity among the students (both within and between students that are perceived as students in need of more challenges in mathematics and students that are perceived as struggling students in mathematics), the results also showed similarities between the students. These similarities are something the teaching can take advantage of, in the organisation and planning of the mathematics education in order to get equitable instructional quality. However, barrier factors such as large class sizes, high-stake-standardized tests [START_REF] Rousseau | Understanding the significance of context: A framework to examine equity and reform in secondary mathematics education[END_REF], etc. can be prohibitive to the work of equitable instructional quality. If the organisation is responsive to the diversity among students and is aware of barrier factors it might be dynamic and adjust accordingly to improve access to mathematics and an increased inclusion in mathematics.

Proceedings of CERME10

Going through is "genomgång" in Swedish, which is when the teacher is explaining something on the blackboard.Thematic Working Group 10Proceedings of CERME10