Fatih Dinc

Bayram Cevdet Akdeniz

Ecda Erol

Dilara Gokay

Ezgi Tekgul

Ali Emre Pusane

Tuna Tugcu

Analytical Derivation of the Impulse Response for the Bounded 2-D Diffusion Channel

This letter focuses on the derivation of the hitting probabilities of diffusing particles absorbed by an agent in a bounded environment. In particular, we analogously consider the impulse response of a molecular communication channel in a 2-D and 3-D environment. In 2-D, the channel involves a point transmitter that releases molecules to a circular absorbing receiver that absorbs incoming molecules in an environment surrounded by a circular reflecting boundary. Considering this setup, the joint distribution of the molecules on the circular absorbing receiver with respect to time and angle is derived. Using this distribution, the channel characteristics are examined. Then, we extend this channel model to 3-D using a cylindrical receiver and investigate the channel properties. We also propose how to obtain an near-exact analytic estimate for the unbounded 2-D channel from our derived solutions, as no analytic derivation for this channel is present in the literature. Throughout the letter, we perform particle-based simulations to compare the analytic results and lay evidence for our findings.

Introduction

Molecular communication (MC) has recently gained much attention as a promising method for communication among nanodevices. A method of such communication is the diffusion of molecules in biological environments, where the messenger molecules are used to mediate signals between transmitters and receivers. Medical applications constitute a promising application field for such biocompatible nanodevices. Therefore, examining the response of molecular communication channels is an important task to determine communication characteristics and possible communication scenarios. Fortunately, Brownian motion has been explored extensively in the physics literature in the context of first passage processes [START_REF] Chevalier | First-passage quantities of brownian motion in a bounded domain with multiple targets: a unified approach[END_REF][START_REF] Mejía-Monasterio | First passages for a search by a swarm of independent random searchers[END_REF][START_REF] Majumdar | Hitting probability for anomalous diffusion processes[END_REF][START_REF] Condamin | First-passage times in complex scale-invariant media[END_REF][START_REF] Mattos | First passages in bounded domains: When is the mean first passage time meaningful?[END_REF]. As they describe the hitting probability in diffusive environments, the first-passage processes found many applications in the molecular communication literature to describe diffusion channels consisting of transmitters and receivers [START_REF] Redner | A Guide to First-passage Processes[END_REF][START_REF] Akyildiz | Nanonetworks: A new communication paradigm[END_REF][START_REF] Akyildiz | The internet of bio-nano things[END_REF].

Many different types of receivers have been extensively explored in the molecular communication literature [START_REF] Farsad | A comprehensive survey of recent advancements in molecular communication[END_REF][START_REF] Nakano | Molecular communication and networking: Opportunities and challenges[END_REF][START_REF] Malak | Molecular communication nanonetworks inside human body[END_REF][START_REF] Noel | Improving receiver performance of diffusive molecular communication with enzymes[END_REF]. Among those, two are more commonly utilized: absorbing receivers that consume the incoming molecules upon contact and observing receivers that track the number of molecules inside a volume without absorbing them. In the literature, impulse responses for both types of channel models have been investigated in great detail. In general, these channels can be categorized into two groups according to their environments as well. While some channels are placed in a free unbounded environment, others are placed in a bounded (and usually tubular) environment. For the first group, the impulse response for a 1-D channel is derived [START_REF] Eckford | Nanoscale communication with Brownian motion[END_REF], while in [START_REF] Yilmaz | Three-dimensional channel characteristics for molecular communications with an absorbing receiver[END_REF] the 3-D channel's impulse response is examined for a point transmitter and a spherical absorbing receiver. Nevertheless, the impulse response in a 2-D unbounded medium for a point transmitter and a circular absorbing receiver has not been derived, except for some special cases presented in [START_REF] Akdeniz | 2-D channel transfer function for molecular communication with an absorbing receiver[END_REF][START_REF] Dy | First-passage-time distribution for diffusion through a planar wedge[END_REF]. The channels with point transmitters, as well as the ones with spherical transmitters, are considered in a 3-D medium in [START_REF] Genc | Reception modeling of sphere-to-sphere molecular communication via diffusion[END_REF] and [START_REF] Noel | Channel impulse responses in diffusive molecular communication with spherical transmitters[END_REF].

As stated in [START_REF] Giné | Molecular communication options for long range nanonetworks[END_REF], vessel-like channels, one type of bounded channels, have beneficial effects for long-range molecular communication by preserving released molecules in a bounded range. Therefore, they have higher Figure 1: (a) General Channel Model for a point transmitter situated at r = r 0 , a cylindrical absorbing receiver with radius d 0 and height h surrounded by a larger cylindrical reflecting walls with radius R and height h. Note that the diffusion of the molecules is confined inside a finite annular volume depicted in (b) as the z-dependence is suppressed through symmetry arguments. (c) The receiver can be modified to count only the particles inside the angle range (-θ f , θ f ). power efficiency, which is one of the reasons why many biological systems evolved in this direction. Since the molecules are not dispersed too much compared to the case of unbounded environments and due to their possible practical use in health applications, bounded and particularly vessel-like channels gained much attention in the literature. In [START_REF] Turan | Channel model of molecular communication via diffusion in a vessel-like environment considering a partially covering receiver[END_REF], the 1-D and 3-D hitting location distributions of messenger molecules on a planar receiver are examined when there is no flow in the vessel-like environment. In [START_REF] Wicke | Modeling duct flow for molecular communication[END_REF], the impulse response of a 3-D vessel-like channel is obtained for a spherical observing receiver when there is a laminar flow in the environment. In [START_REF] Bicen | System-theoretic analysis and least-squares design of microfluidic channels for flow-induced molecular communication[END_REF], the flow models for microfluidic channels with different cross-sectional areas are presented, and the impulse response is derived by solving the 1-D diffusion-advection equation, which is only valid for some specific cases. Besides the channel impulse response, the capacity of the single-input single-output molecular communication channels with flow and drift is derived in [START_REF] Sun | Channel capacity modelling of blood capillary-based molecular communication with blood flow drift[END_REF].

In addition to vessel-like channels, diffusion processes that are bounded by membranes are also encountered commonly in nature. One such example is the transmission of messengers inside a spherical cell bounded by the cell membrane to an organelle, which can be modeled by a diffusion channel consisting of an absorbing spherical receiver and a reflecting/absorbing spherical boundary [START_REF] Dinc | Impulse response of the channel with a spherical absorbing receiver and a spherical reflecting boundary[END_REF][START_REF] Al-Zubi | Modeling of ligand-receptor protein interaction in biodegradable spherical bounded biological micro-environments[END_REF]. In general, the spherical model is not always accurate to describe the diffusion processes inside the cell. In some cases, a cylindrical cell model can be more accurate, as there are many cylindrical structures in living organisms, like oval cells in the liver or simple columnar epithelium. Hence, the impulse response of a bounded cylindrical environment can be useful to describe diffusion processes inside such environments more accurately.

In this letter, we first derive the SO(2) symmetric1 impulse response of a 2-D annular channel that consists of an absorbing circular receiver, a reflecting circular boundary and a point transmitter. Then, we find the channel characteristics of such a system and discuss many effects that arise due to the existence of the circular boundary. Afterwards, we derive the generalized angle-dependent impulse response for the annular channel, where the receiver only counts certain particles, which are absorbed inside the angle range [-θ f , θ f ]. Next, we use the impulse response of the 2-D annular channel to find an analytic estimate with an arbitrarily small error for a circular receiver located in a 2-D unbounded channel, an exact analytic result which does not exist in the literature. As the impulse response of the 2-D bounded channel can be simulated via particle-based simulations, we perform further analysis to illustrate the accuracy of the analytic estimate. Later, we consider the 3-D concentric cylindrical diffusion channel that involves a point transmitter, a cylindrical absorbing receiver, and a cylindrical reflecting boundary. We show that the 3-D cylindrical channel can be described by the impulse response of the 2-D annular channel under certain assumptions. Finally, we conclude with further remarks on our findings and future work.

System Models

The system models for various channels considered in this letter are depicted in Fig. 1. In this letter, we will simplify a 3-D coaxial cylindrical channel to a 2-D annular channel by certain assumptions and carry out the derivations for the impulse response of the 2-D channel. In the 3-D channel, shown in Fig. 1(a), a coaxial cylindrical absorbing receiver is placed at the center of the microfluidic channel whose boundary is reflecting and a point transmitter transmits messages by releasing molecules to the diffusion channel. Assuming that there is no flow in the environment, the propagation of the released molecules is modeled by Brownian Motion as ∆x ∼ N (0, 2D∆t) , (1a)

∆y ∼ N (0, 2D∆t) , (1b) ∆z ∼ N (0, 2D∆t) , ( 1c
)
where D is diffusion coefficient, ∆x, ∆y and ∆z are incremental step sizes in the three dimensions, ∆t is time step, and N (µ, σ 2 ) is the normal distribution with mean µ and variance σ 2 . The cylindrical receiver absorbs the molecules that come to the vicinity of its receptors and makes a decision by counting these absorbed molecules.

If the heights of the boundary h b and the receiver h are equal, this system can be reduced to a 2-D bounded environment that has a concentric absorbing receiver and reflecting boundary as shown in Fig. 1(b). In practice, the length of the receiver can be smaller than the length of the channel as in the case of oval cells. Therefore, we shall evaluate the required condition for the height of the receiver h and height of the position of the transmitter h 0 for justifying the reduction this system to 2-D. The distribution of the released molecules along the z-axis can be modeled using (1) as N (h 0 , 2Dt). Therefore, if the arrival probability of any molecule at either of the ends of the receiver is almost 0, then our approximation is still valid. Nonetheless, we stress that this is a simplifying assumption, as the diffusion along the z-axis is not independent of the absorption of molecules by the receiver. In fact, this assumption will lead to an over-estimate of the error bound, as we shall illustrate in Section 4.3. For now, we assume that we can reduce the microfluidic channel to 2-D with a coaxial cylindrical receiver if the following condition

P (z < 0) + P (z > h) < (2)
is satisfied. Then, using the distribution of z, we can write this condition explicitly as

Q h 0 √ 2Dt + Q h -h 0 √ 2Dt < , (3)
where t represents the maximum time of interest. Taking h 0 = h/2, one obtains

2Q h 2 √ 2Dt < . (4)
Therefore, with the simplifying assumption h 2 √ 2Dt, the system can be reduced to 2-D. We shall first discuss the 2-D annular channel and then generalize its applications to various other channels: the 2-D annular channel with a partially absorbing receiver (with an angle-dependent impulse response), as an estimate for the impulse response of the 2-D unbounded channel with an absorbing receiver, and finally the 3-D channel with a coaxial cylindrical receiver.

2-D Annular Channel

Channel Impulse Response

In this section, we start by finding an impulse response for a 2-D annular channel, as depicted in Fig. 1(b), while deriving the probability density function of the molecules. To describe the diffusion of a molecule inside the annular region, we shall find a solution to Fick's Law

D∇ 2 P (r, t|r 0 ) = ∂P (r, t|r 0 ) ∂t , (5)
where ∇ 2 is the Laplacian operator and P (r, t|r 0 ) is the probability density function (PDF) of the molecules inside the diffusion channel. The circular boundary at r = R is reflecting. Furthermore, the transmitter is assumed to be situated at a distance r = r 0 from the origin. In this section, we are interested in the absorption probability (an angle-independent quantity, see [START_REF] Yilmaz | Three-dimensional channel characteristics for molecular communications with an absorbing receiver[END_REF]) of the molecules by the receiver. Therefore, our calculations include an SO(2) symmetry. The physical meaning of this is the following: Instead of releasing molecules from a single point transmitter at a distance r 0 from the origin, we release them on a circle with radius r 0 according to a uniform angular distribution. Even though implementing this process could be physically more challenging, this assumption simplifies the theoretical problem significantly. As the receiver consumes molecules from all angles upon contact, both original and SO(2) symmetric problem will lead to the same absorption probability at a given time. In Section 4.1, the angle of absorption will be of interest as we discuss a partially absorbing receiver; hence, we will remove the SO(2) symmetry assumption there. In addition to the initial condition the probability distribution P (r, t|r 0 ) should be zero when the molecules hit the receiver, where we assume a perfect receiver due to simplicity. This results in the boundary conditions ∂P (r, t|r 0 ) ∂r r=R = 0, (6a)

P (r, t|r 0 ) r=d0 = 0, (6b)
P (r, 0|r 0 ) = 1 2πr δ(r -r 0 ). ( 6c
)
We recall that, since the boundary is described by a mixture of Neumann and Dirichlet boundary conditions, the Laplacian operator is guaranteed to have a unique solution by the uniqueness theorem for the 3-D diffusion/heat equation [START_REF] Guenther | Partial differential equations of mathematical physics and integral equations[END_REF]. Now, we shall start with the separation of variables Ansatz

P (r, t|r 0 ) = φ(r, θ)T (t), (7)
which leads to the equation

D ∇ 2 φ(r, θ) φ(r, θ) = T (t) T (t) = -µ 2 , (8)
from which we can easily deduce

T (t) = Ae -µ 2 t , (9)
where A and µ are constants. Afterwards, we arrive at the eigenvalue problem for the Laplacian operator

∇ 2 φ(r, θ) = - µ 2 D φ(r, θ). (10)
We note that the overall factor of the solution is lumped into the time portion, T (t). Therefore, we can arbitrarily normalize the space part, φ(r, t).

The eigenvalues µ 2 /D are non-negative and real as either Neumann or Dirichlet conditions impose the boundary conditions. Furthermore, eigenfunctions corresponding to distinct eigenvalues are orthogonal and form a basis for all possible solutions [START_REF] Grebenkov | Geometrical structure of Laplacian eigenfunctions[END_REF]. Here, we invoke the idea of SO(2) symmetry in our system. Due to angular symmetry, the position-dependent part of the Ansatz depends only on the distance from the origin and not the angle, i.e. φ(r, θ) = φ(r). This choice eliminates certain eigenvalues (and their corresponding eigenfunctions) from the solution.

Rewriting the eigenvalue equation in polar coordinates, we obtain

r 2 φ (r) + rφ (r) + µ 2 D r 2 φ(r) = 0, (11)
which has the general solution

φ(r) = J 0 µ √ D r + cY 0 µ √ D r , (12)
where J n and Y n are the Bessel functions of the first and second kind of nth order, respectively, and c is a constant which shall be determined using boundary conditions. We are now ready to shape our solution according to the boundary conditions given in [START_REF] Redner | A Guide to First-passage Processes[END_REF]. At this point, we shall stress that our solution will be a combination of both J 0 and Y 0 . This adds a layer of complexity to the problem, as opposed to many common cases where the origin is in the region of interest. If a solution should exist at the origin, one can immediately set c = 0, as Y 0 (x) diverges as x → 0. In our initial value problem, this is not the case and further computation will be required. For now, we shall postpone this computation and consider the special function η s 0 (x)

η s 0 (x) = J 0 (x) + c s Y 0 (x) , (13)
such that η s 0 (β s ) = 0 and η s 0 (αβ s ) = 0, where α = d 0 /R (from now on called the aspect ratio) and η s 0 (x) denotes the derivative of η s 0 (x) with respect to x. The construction of such a function and the computation method of the set {β s } are discussed in detail in Appendix A. We note that {β s }, called eigenvalues from now on, is an (increasingly) ordered, discrete, and infinite set.

It can be verified through straightforward algebra that the function η s 0 (β s r/R) satisfies the two boundary conditions and is a radial solution for the diffusion equation given in [START_REF] Mattos | First passages in bounded domains: When is the mean first passage time meaningful?[END_REF]. Moreover, the following orthogonality condition can be shown to hold for η s 0 (β s x):

1 α η s 0 (β s x)η l 0 (β l x)x dx = 1 2 (η s 0 ) 2 (β s ) -α 2 (η s 1 ) 2 (αβ s ) δ sl , (14)
where we simply replace J 0 and Y 0 by J 1 and Y 1 in ( 13) to find η s 1 (β s x). We find the probability distribution function to be of the form

P (r, t|r 0 ) = ∞ s=1 A s η s 0 β s r R e -β 2 s Dt R 2 , (15)
where we note that β 1 > 0 (see, for example, Table 1 in Appendix B), indicating that for t → ∞ the probability distribution of molecules vanishes everywhere in the space. Taking the orthogonality condition into account, we find the general normalization constants

A s A s = 1 πR 2 η s 0 β s r 0 R 1 ((η s 0 ) 2 (β s ) -α 2 (η s 1 ) 2 (αβ s )) , (16)
from which we find the solution

P (r, t|r 0 ) = ∞ s=1 η s 0 β s r0 R η s 0 β s r R πR 2 ((η s 0 ) 2 (β s ) -α 2 (η s 1 ) 2 (αβ s )) e -β 2 s Dt R 2 , (17)
where we recall that the set {β s } is defined such that η s 0 (αβ s ) = 0 and η s 0 (β s ) = -η s 1 (β s ) = 0 to satisfy the boundary conditions. Now that we have P (r, t|r 0 ), we can calculate the hitting rate2 as

n hit (t) = 2πd 0 D ∂P (r, t|r 0 ) ∂ r r=d0 , (18)
where D ∂P (r,t|r0) ∂r r=d0

represents the probability current into the absorbing receiver. Thus, we find the hitting rate

n hit (t) = -2D ∞ s=1 αβ s η s 0 β s r0 R R 2 ((η s 0 ) 2 (β s ) -α 2 (η s 1 ) 2 (αβ s )) η s 1 (β s α) e -β 2 s Dt R 2 . ( 19
)
We conclude this section by emphasizing that we are required to find different eigenvalues {β s } for each aspect ratio α to construct the special functions η s 0 (x) (See Appendix A).

Verification of Analytic Result and Channel Characteristics

Having found the analytic solution for the 2-D annular channel, we shall now focus on verifying our findings through comparison with particle-based simulations and then discuss the effects of a reflecting boundary on the channel response. In this section, we simulate the hitting rate n hit (t) for different aspect ratios α = d0 R and interpret certain channel characteristics.

In our simulations, we take the radius of the outer cylinder as R = 100µm, unless otherwise stated, and simulate the system for different receiver radii d 0 by changing α = d0 R . Changing the diffusion coefficient only re-scales time, which does not affect the correctness of the comparison. For simplicity, we set D = 80 µm 2 /s for our illustrations. Unless otherwise stated, simulations are performed with 10 6 particles, and the reflections are performed according to the rollback mechanism, which is discussed in [START_REF] Turan | Channel model of molecular communication via diffusion in a vessel-like environment considering a partially covering receiver[END_REF]. Now that we have both our analytic solution and the simulation framework, we shall compare the hitting rate, n hit (t), for different aspect ratios α, and different initial positions r 0 , in Figure 2. As can be seen from the figure, the simulation and the analytic results are in agreement for multiple scenarios, as expected. In these illustrations, we truncate the analytic result, given in [START_REF] Giné | Molecular communication options for long range nanonetworks[END_REF], after the 350th term. This is possible as the terms in the sum are exponentially suppressed for larger β s values. In fact, we can choose a parameter where t is the final time of interest and truncate the series afterward. In order to visualize the relative contribution of each term in the summation, we first define a time period of interest [∆t, 10t γ ], where ∆t is the smallest time possible to detect a signal and

β sc such that exp -β 2 sc Dt R 2 1, (20)
t γ = (r 0 -d 0 ) 2 D ( 21
)
is the scale-free time of the channel such that [∆t, 10t γ ] includes several factors of the characteristic time of the diffusion process for any set of r 0 , d 0 and D. Therefore, for each set of parameters, a new t γ is calculated. For the truncation analysis, we set ∆t = t γ /1000.

Having defined the time of interest, we can find the relative contribution of each term in the summation as

f s = -2 αη s 0 (βs r 0 R ) βs((η s 0 ) 2 (βs)-α 2 (η s 1 ) 2 (αβs)) η s 1 (β s α) (e -β 2 s D∆t R 2 -e -β 2 s Dtγ R 2 ) tγ ∆t dt n hit (t) , (22)
such that ∞ s=1 f s = 1. Furthermore, we define the cumulative contribution as

F s = s r=1 f r . (23)
We show the relative contribution of each term in the summation and their cumulative behavior for the time period [∆t, 10t γ ] in Fig. 3. As can be seen from the figure, the initial position of the transmitter is of high importance for the most efficient truncation. This can be understood by the exponential suppression of higher index terms for later times, as exp -β 2 s Dtγ R 2 becomes more negligible for large t γ , which is the case for large r 0 values. This also justifies the choice of the upper-bound, 10t γ , for the time of interest, as later times can already be described by the terms, which one picks when describing earlier times. For many cases, the most efficient truncation (at term s = s c ) can be performed by considering the difference between the absolute relative contribution of the two consequent terms such that

||f sc+1 | -|f sc ||
, where is a small positive number. An equivalent method is to consider |1 -|F sc || and then to choose the cut-off s c accordingly, since F s is unit-normalized for s → ∞ by definition. For cases where computing {β s } becomes costly, the most efficient truncation method discussed here can be easily implemented. Fortunately, the channel impulse response discussed in this letter allows finding a large number of {β n } very efficiently. Hence, for the entirety of the letter, analytic results in each applicable figure are computed using more than 300 terms. , where l c = R -d 0 is the channel length. This is due to molecules reflecting from the boundary being dominant for the absorption. For closer initial release points r 0 , the peak time vs. initial distance scales as τ peak ∼ (r 0 -d 0 ) 2 in agreement with the 3-D spherical receiver point transmitter case [START_REF] Yilmaz | Three-dimensional channel characteristics for molecular communications with an absorbing receiver[END_REF]. The channel characteristics are calculated and illustrated for R = 500nm and D = 80µm 2 /s. Due to inherent space scaling symmetry, the shape of the curves are the same for micro-scales as well, within a scaling of time. We use more than 300 terms (different for each α) in the summation for the analytic result while computing the channel characteristic times. Now, we can focus on the channel performance, specifically effects caused by the existence of the reflecting boundary. First we define some useful concepts: Definition 1 (Peak time) The peak time τ peak is defined in [START_REF] Yilmaz | Three-dimensional channel characteristics for molecular communications with an absorbing receiver[END_REF] as the time such that the hitting rate is maximum, e.g. ∂n hit (t) ∂t

t=τ peak = 0. ( 24
)
Definition 2 (Average time) The average time τ average is defined as the expected value of the time where the hitting rate n hit (t) is taken to be the probability density function, i.e.,

τ average = ∞ 0 t n hit (t) dt. ( 25
)
Note that the hitting rate being the probability density function for time is a direct consequence of the continuity equation.

Definition 3 (Half time)

The half time τ half is defined as the time it takes for the molecule to be absorbed with a probability of 0.5, i.e.,

τ half 0 n hit (t) dt = 0.5. ( 26
)
Many of the 2-D annular channel characteristics and the effects of the boundary on the channel can be captured through the peak, average, and half-time values and their dependence on the initial release point r 0 . An illustration of these parameters is shown in Fig. 4.

The importance of peak time, τ peak , in communication aspects has been discussed in [START_REF] Yilmaz | Three-dimensional channel characteristics for molecular communications with an absorbing receiver[END_REF]. As an addition, we believe that both τ average and τ half can be useful when characterizing the channel. Most importantly, τ half could be used as a reliable measure when defining sampling time for communication purposes, as one requires to sample at least more than half of the messenger molecules to have a signal-to-inter-signal-interference ratio higher than 1. Unfortunately, determining τ half has high computational cost, and therefore a measure, which is easy to compute, is required. Fortunately, τ average has similar behavior and is within the same order of magnitude of τ half while having an easy-to-compute analytic expression. Moreover, comparing τ peak with τ half and τ average , we can qualitatively infer the importance of tail effects in a 2-D channel. The difference between these quantities shows the relative contribution that comes from the tail of n hit (t) for t ≥ τ peak with respect to earlier times. If most of the molecules arrive at the receiver at early times close to τ peak , we expect the quantity |τ average -τ peak |/τ peak to be of the order unity. Nonetheless, there is an order of magnitude difference between τ peak and τ average as can be seen from the Fig. 4(a). Hence, tail effects are highly dominant in a 2-D geometry, meaning that a communication process taking place in 2-D should include sampling of messenger molecules for times much larger than τ peak . Moreover, Fig. 4(a) illustrates that τ peak and τ average have similar behavior versus r 0 , where both measures are dominated by the effects due to boundary.

The effect of the reflecting boundary at early times, ∼ O(t γ ), can best be seen from the peak time τ peak , which is plotted for three different aspect ratios α in Fig. 4(b),(c),(d) for illustration purposes. As is apparent from the figure, the shape of the plot, τ peak vs r 0 , depends solely on the initial condition R 0 and is independent of α, hence d 0 and R. When the molecule is initially close to the receiver, the effect of the boundary at early times is negligible and we observe a square-law dependence between the distance and the peak-time τ peak , as was the case for a 3-D spherical receiver and a point transmitter [START_REF] Yilmaz | Three-dimensional channel characteristics for molecular communications with an absorbing receiver[END_REF]. As we shall see in section 4.2, the unbounded channel can be described by the analytic expression for the bounded channel for t ∈ [0, τ peak ] in this region. Defining the channel length l c = R -d 0 , we realize that the deviation from the square-law is apparent when the release distance is (r 0 -d 0 ) 2 3 l c . This (r 0 -d 0 ) is indeed the critical distance, after which the boundary effects become dominant at time scales t τ peak . The same transition is not as apparent with τ half and τ average due to the following reason: the existence of the reflecting boundary ensures that the molecules are absorbed earlier than in the unbounded case, hence the effect of the boundary on τ half and τ average is present even when the molecule is initially very far away from the boundary or close to the receiver.

Considering that there are infinitely many summed terms in the expression for n hit (t), it is not straightforward to obtain a formula for τ peak and τ half . Fortunately, one can find an analytic expression for the average time:

τ average = -2 ∞ s=1 αR 2 η s 0 β s r0 R η s 1 (αβ s ) Dβ 3 s ((η s 0 ) 2 (β s ) -α 2 (η s 1 ) 2 (αβ s )) . ( 27
)
While finding this expression we exchange the integral and the sum, therefore, we shall present a proof of concept for the convergence of this expression. In order to do so, we first define an estimate for τ average as where lim t→∞ τ est average (t) = τ average . We compare the estimate with the analytic expression in Fig. 5(a) for various final times t max versus r 0 /R. As can be seen from the figure, the estimate converges to the analytic result. This is expected, since in [START_REF] Grebenkov | Geometrical structure of Laplacian eigenfunctions[END_REF] the terms in the summation behave as ∼ 1/β 3 s for large indices and are suppressed accordingly.

τ est average (t) = t 0 n hit (τ )τ dτ, ( 28
)
We note that the similar behavior of τ half and τ average is not surprising at all. We already illustrated this similarity in Fig. 4(a), but this result can be considered more general due to the scale invariance of n hit (t) as long as tail effects are dominant. One way to see this scale invariance is to consider [START_REF] Giné | Molecular communication options for long range nanonetworks[END_REF], where n hit (t) depends dominantly on the parameter α and r 0 /R as long as t ≥ O(τ peak ) or equivalently t ≥ O(t γ ). Therefore, as long as times t ≥ t γ have more dominant effects on τ average , we can expect similar behavior for both τ average and τ half . Finally, we plot τ average and τ est average (t γ ) versus initial condition r 0 in Fig. 5(b) to show that earlier times, t ≤ t γ , have negligible contributions for τ average . Therefore, as t γ /τ peak is of the order unity, the tail effects are indeed dominant when determining τ average .

Applications to Various Channels

Having characterized the 2-D annular channel, we now perform comparisons with particle based simulations to illustrate that the impulse response can be used for different types of channels under certain conditions. In addition, we show that with a similar approach, we can also find the angle-dependent impulse response for the 2-D annular channel.

2-D Annular Channel: Angular Dependent Impulse Response

Inspired from the nature of diffusion, it has been shown that using a partially-counting receiver based on angular position has beneficial effects in molecular communication [START_REF] Akdeniz | Molecular signal modeling of a partially counting absorbing spherical receiver[END_REF]. Since molecules move slowly, it takes much higher expected time to move to the part of the receiver that is far from the transmitter. These parts can also be represented by the reception angle as shown in Fig. 1(c). In addition, angle-dependent channel impulse response can be used to improve the channel performance by reducing inter symbol interference as proposed in [START_REF] Akdeniz | Molecular signal modeling of a partially counting absorbing spherical receiver[END_REF]. The receiver (in Fig. 1(c)) absorbs all the molecules upon collision, but counts only those that arrive inside the angular interval [-θ f , θ f ] and disregards the rest. We can modify our previous calculations to find an analytic solution for this case as well, which is carried out in Appendix B.

For this channel, we define the hitting rate as the probability of a single released molecule to hit the receiver inside the angle range [-θ f , θ f ] between times t and t + dt

n hit (θ f , t) = θ f -θ f Dd 0 ∂ ∂r P (r, θ, t) r=d0 dθ. (29)
From (B.10), we find the hitting rate as (For definitions, see Appendix B)

n hit (θ f , t) = ∞ s=1 θ f Dαβ 0s πR 2 I 0s η s 0 β 0s r 0 R η s 0 β 0s d 0 R e -β 2 0s Dt R 2 + ∞ p=1 ∞ s=1 2Dαβ ps pπR 2 I ps sin(pθ f )η ps p β ps r 0 R η ps p β ps d 0 R e -β 2 ps Dt R 2 . (30)
A comparison of analytic and simulation results for this channel type is given in Fig. 6. For the remainder of this letter, we focus on the receiver type with θ f = π, for which (30) reduces to [START_REF] Giné | Molecular communication options for long range nanonetworks[END_REF].

2-D Unbounded Channel: An Analytic Approximation

The impulse response for the 2-D unbounded channel with an absorbing receiver has been missing from the literature, whereas the impulse responses for 1-D and 3-D channels are well-known [START_REF] Yilmaz | Three-dimensional channel characteristics for molecular communications with an absorbing receiver[END_REF][START_REF] Redner | A Guide to First-passage Processes[END_REF]. Fortunately, our derivations for the 2-D annular channel can be utilized to estimate the impulse response for the 2-D unbounded channel with an arbitrarily small error. The trade-off for obtaining an arbitrarily small error is the computational complexity, as more β s terms will be required in [START_REF] Giné | Molecular communication options for long range nanonetworks[END_REF] to describe the channels with smaller aspect ratios α. This is due to the fact that as α → 0, the eigenvalues β s become closer and closer. Therefore, it is not trivial to obtain an exact analytic result in the limit α → 0, since this would require integrating an integrand, which has combinations of various Bessel functions both in its numerator and denominator and has infinitely many poles. Therefore, we shall focus on making accurate estimations for the 2-D unbounded channel while confirming our observations regarding the effects of the boundary on τ peak from section 3.2.

In Fig. 7, simulation results for the 2-D unbounded channel and corresponding analytic estimates are illustrated. As is apparent from the figure, as long as the boundary R is sufficiently far away, the analytic estimate matches the simulations quite well. In fact, from our discussion on τ peak , we expect that the boundary effects on the peak-time will be apparent when (r 0 -d 0 ) 2 3 l c . This is certainly the case for Fig. 7(c), where the analytic estimate with R = 100µm diverges from the simulation around the peak position. In this case, simply increasing the boundary radius R → 1000µm results in an estimate that matches the simulations well. We can infer from the simulations that choosing the boundary radius R according to (r 0 -d 0 ) 0.2 results in a good-agreement between the simulation and the analytic estimate for the time of interest t ∈ [0, O(t γ )]. Fig. 7(b) illustrates the strong influence of the boundary on τ half and τ average , as the analytic estimate with R = 100µm does not match the tail of the simulation. Even though τ peak matches in this case, the increase in the tail results in molecules being absorbed earlier than in the unbounded case. This leads to the saturation effect we observe in Fig. 4(a) for τ half and τ average for larger r 0 /R values. Thus, we see once more that τ peak alone is not sufficient to describe the characteristics of the 2-D bounded channel.

3-D Channel: Comparison with Simulations

Now, we turn to estimating the impulse response for the 3-D channel with coaxial cylinders, as depicted in Fig. 1(a). In section 2, we discussed a simplified assumption such that the 3-D channel can be approximated by the 2-D annular channel. Now, we shall perform comparisons with particle based simulations to verify our findings.

For now, we will revisit (4) and re-write it, for simplicity, as follows

χ = 2Q h 2 √ 2Dt max =⇒ χ < , (31)
where is our crude over-estimate of the error-bound and t max is the maximum time of interest. For illustration purposes, we shall consider the most extreme case, where the boundary height h b → ∞ and the receiver height is h = 2h 0 . The transmitter is located in the middle as in Fig. 1. The comparison is shown in Fig. 8 for two different initial conditions r 0 = 20µm and r 0 = 70µm. As can be seen from the figure, χ is indeed an over-estimate, since the analytic estimate follows the simulations even for high χ values.

As can be seen from Fig. 8(b), the discrepancy between the analytic estimate and the simulations becomes more apparent for large r 0 /R values, as the tail effects become important at earlier times and more particles arrive at the receiver until the time of interest, amplifying the existing error correspondingly. For the usual time of interest, t ∈ [0, O(t γ )], the analytic estimate matches the simulations if h = 10(r 0 -d 0 ) very well. We can also see this from our crude estimate by setting t max = κt γ and h = 10(r 0 -d 0 ):

χ = 2Q h 2 2Dκt γ = 2Q 5 √ 2κ . ( 32
)
For example, this function yields 0.04 for κ = 3 and 0.15 for κ = 6. Both cases are acceptable since this is a crude overestimate of the upper-bound on the error.

Conclusion

In this work, we derived the impulse response of the 2-D annular channel first for SO(2) symmetric initial conditions, then broke the symmetry while offering a more rigorous and angle-dependent description for the impulse response inside the channel. In this pursuit, we defined a special function η s 0 (x) (or η ps p (x) in general), which is a combination of Bessel functions of the first and second kind. This function satisfies the necessary boundary conditions and is an exact solution to the radial part of the diffusion equation under separation of variable Ansatz. This method of obtaining an impulse response leads to an infinite number of terms, the sum of which converges for t > 0. It is shown that the infinite sum in the analytic solution can be truncated after a certain number of terms depending on the time interval one is interested in. Furthermore, we show the agreement between the Monte-Carlo simulations and the analytic solutions for different channel and receiver parameters, such as the aspect ratio α, the boundary radius R, and the receiver radius d 0 . This equivalence constitutes the evidence for the accuracy of our findings.

Having verified our findings, we explore the dependency of certain channel characteristics on the initial position of the transmitter. Regarding the peak time τ peak simulations, the effect of the boundary on the peak time is more apparent as the initial position of the transmitter is around 2l c /3, where l c = R -d 0 is the channel length. As r 0 > l c , there is an apparent shift in the peak time, caused by the boundary. Nonetheless, this trend shift is not apparent for the average and half time, τ average and τ half . The intuitive reason behind this phenomenon can be explained through the tail effect. The average and half time values are more dependent on the existence of the boundary as they rely not only on the peak of the hitting rate, but also on the behavior of the tail that follows the peak. As the transmitter is placed further from the receiver, the contribution from the tail surpasses greatly the contribution from the peak, hence smoothing out the distinct trend shift for r 2l c /3. Evidence for this phenomenon can be observed from the relatively large values of average and half times compared to lower values of the peak times, as the difference between τ peak and τ average /τ half depicted in Fig. 4 is approximately an order of magnitude. Moreover, we present an analytic expression for τ average , which can be extremely useful when choosing a sampling time for the receiver in a communication scenario.

Finally, we used our findings to describe the impulse response of various channels: the angular-dependent response of the 2-D annular channel, an analytic estimate for the 2-D unbounded channel and finally an analytic approximation for the 3-D diffusion channel consisting of a cylindrical receiver and reflecting boundary. Since no exact analytic result exists in the literature, our findings constitute a leap forward in understanding the impulse response of the 2-D unbounded channel. As a future work, we plan to explore the angular dependent impulse response of the 2-D annular channel and the corresponding channel characteristics, as well as possible applications to many transmitter communications.

B Derivation of Angle-Dependent Response

To describe the diffusion of the molecule inside the annular region under angle-dependent conditions, we shall find a solution to Fick's Law, satisfying the necessary boundary conditions ∂P (r, θ, t) ∂r r=R = 0, (B.1a)

P (r, θ, t) r=d0 = 0, (B.1b) P (r, θ, 0) = 1 r δ(r -r 0 )δ(θ -θ 0 ). (B.1c)
We shall start with the separation of variables ansatz

P (r, θ, t) = φ(r, θ)T (t), (B.2)
which leads to

T (t) = Ae -µ 2 t , (B.3) and R R + R rR + Θ r 2 Θ = - µ 2 D , (B.4)
where we define φ(r, θ) = R(r)Θ(θ) and A is the overall normalization factor to be determined by the initial conditions. R (r) denotes the derivative of R(r) with respect to r. This equation can be transformed into a Bessel differential equation for the radial part if we set

Θ = -p 2 Θ =⇒ Θ(θ) = A p cos(pθ) + B p sin(pθ). (B.5)
Then, the radial equation becomes

r 2 R (r) + rR (r) + µ 2 D r 2 -p 2 R(r) = 0. (B.6)
After proper re-scaling of the independent variable r, we can find the solution as (B.10) Some β ps values are given in Table 1 in the following page while we note that c ps values can be calculated easily by straightforward algebra.

Figure 2 :

 2 Figure 2: Simulations of hitting rate n hit (t) versus time for D = 80µm 2 /s, R = 100µm, d 0 = 1µm (a,b) and d 0 = 10µm (c,d), r 0 = 20µm (a,c) and r 0 = 70µm (b,d). Note the correspondence between the analytic solution and the simulation in each case. We use 350 terms when finding the analytic result for n hit (t).

Figure 3 :

 3 Figure 3: Relative contribution f s and cumulative contribution F s for finite number of terms in (19) for R = 100µm, d 0 = 10µm (a)-(c) and d 0 = 1µm (b)-(d).

Figure 4 : 2 3

 42 Figure 4: The channel characteristic times (τ average , τ half and τ peak ) for α = 0.01 (a), the peak time behavior versus initial condition, r 0 , for α = 0.01 (b), α = 0.1 (c) and α = 0.5 (d). Note the apparent trend change for τ peak once the initial release point (r 0 -d 0 ) 2 3 l c, where l c = R -d 0 is the channel length. This is due to molecules reflecting from the boundary being dominant for the absorption. For closer initial release points r 0 , the peak time vs. initial distance scales as τ peak ∼ (r 0 -d 0 ) 2 in agreement with the 3-D spherical receiver point transmitter case[START_REF] Yilmaz | Three-dimensional channel characteristics for molecular communications with an absorbing receiver[END_REF]. The channel characteristics are calculated and illustrated for R = 500nm and D = 80µm 2 /s. Due to inherent space scaling symmetry, the shape of the curves are the same for micro-scales as well, within a scaling of time. We use more than 300 terms (different for each α) in the summation for the analytic result while computing the channel characteristic times.

Figure 5 :

 5 Figure 5: Comparison of τ average and τ est average (t max ) for d 0 = 1µm, R = 100µm, D = 80µm 2 /s and various t max versus r 0 /R (a), for t max = t γ versus r 0 /R (b). All analytic results are calculated using 350 terms.

Figure 6 :

 6 Figure 6: Comparison of simulation and angular-dependent analytic result of n hit (t) for R = 100µm, d 0 = 10µm, r 0 = 20µm, D = 80µm 2 /s, θ f = π/2 (a) and θ f = π/6 (b). Comparing with Fig. 2(c), one can see that for r 0 = 20µm almost all particles hit inside the angle range [-π/2, π/2]. We use 4 × 10 6 terms when obtaining the angular-dependent analytic result.

Figure 7 :

 7 Figure 7: Comparison of Simulation and Analytic Estimations for unbounded channel for α = 0.1 and r 0 = 20µm (a), for α = 0.1&0.01 and r 0 = 50µm (b), for α = 0.1&0.01 and r 0 = 70µm (c), for α = 0.01 and r 0 = 200µm (d). For all simulations, d 0 = 10µm and D = 80µm 2 /s, whereas R changes in each subfigure according to α = d 0 /R.

Figure 8 :

 8 Figure 8: Comparison of Simulation and Analytic Estimations for the 3-D channel with R = 100µm, d 0 = 10µm, D = 80µm 2 /s, the boundary height h b → ∞ and various receiver heights h for r 0 = 20µm (a) and for r 0 = 70µm (b). χ values are evaluated with t max = 3.75s for (a) and with t max = 135s for (b). 350 terms are used when obtaining the analytic estimate in each case.

  shall define the function η ps p (x) analogously to[START_REF] Eckford | Nanoscale communication with Brownian motion[END_REF] asη ps p (x) = J p (x) + c ps Y p (x) (B.8)such that η ps p (β ps ) = 0 and η ps p (αβ ps ) = 0, where α = d0 R as usual. Then, η ps p (β ps r R ) are indeed solutions to the radial equation with the boundary conditions satisfied, where β ps = Rµ √ D . In general, to find β ps , we shall solve a linear set of equations similar to what we have done for η s 0 (β s x). Before continuing, we shall give the normalization condition for the special function η p (β ps x) as1 α η ps p (β ps x)η p s p (β p s x)x dx = I ps δ pp δ ss ,(B.9)where we note that I ps can be written in terms of linear combinations of Bessel functions of the first and second kind. Without loss of generality, we can set θ 0 = 0 and find the probability density function for the molecules as P (r, θ, t) =

Table 1 :

 1 β ps values for α = 0.1 calculated by our algorithm to be used in (30). β 0s values can also be used for the impulse response given in[START_REF] Giné | Molecular communication options for long range nanonetworks[END_REF].

		s=1	s=2	s=3	s=4	s=5	s=6	s=7	s=8	s=9	s=10	s=11	s=12	s=13
	p=0	1.103

SO(2) symmetry in a

2-D coordinate system corresponds to the angular symmetry around the origin.

This quantity is also known as first-passage time probability density distribution in the literature. We occasionally use the term "hitting number" to describe this quantity as well.

Acknowledgements

Research at the Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development Canada, and by the Province of Ontario through the Ministry of Research and Innovation. Fatih Dinc would like to thank Leander Thiele for commenting on the manuscript and pointing towards important references in the diffusion literature. The work of A.E. Pusane and T. Tugcu was partially supported by the State Planning Organization (DPT) of Turkey under the project TAM (2007K120610).

APPENDIX

A Derivation of η s 0 (β s x)

When finding the impulse response of the 2-D channel, we have assumed that there exist functions η s 0 (β s x) such that η s 0 (β s ) = 0 and η s 0 (αβ s ) = 0. In this section, we shall discuss how to construct such functions and illustrate an algorithm to find the eigenvalues ({β s }).

To begin with, we can rearrange the radial solution slightly (ignoring the general normalization constant for now) as φ(r) = J 0 (ar) + cY 0 (ar) ,

where we define a ≡ µ √ D for simpler algebra. Using the boundary conditions (6a) and (6b), we find the following set of linear equations:

-aJ 1 (aR) -caY 1 (aR) = 0, (A.2a)

Rearranging the terms, we can obtain

where setting aR = β and α = d0 R , we look for the solutions of the equation

which we call the characteristic equation. There are infinitely many solutions for this equation, each of which corresponds to a distinct eigenvalue and an eigenfunction of the Laplacian operator. We also note that c is fully determined by the procedure above. Finally, we define the function

Without a given aspect ratio α, this is the most general function we can define. If α is given, then we can construct a code that finds the roots of the characteristic equation. This is feasible, because the roots become periodic as β → ∞. This can be shown by considering the large argument asymptotic behavior of the Bessel functions, where both J p (x) and Y p (x) behave as ∼ cos x -p π 2 -π 4 /

√

x and ∼ sin x -p π 2 -π 4 / √ x, respectively. Then, (A.4) becomes a trigonometric equation:

For small α, such that α < 0.5, the shift β → β + π leaves the left-hand-side (LHS) of this equation invariant while the RHS changes slowly. Then, in the second shift β → β + π, the RHS does not complete a periodic rotation over its range, as α2π < π, whereas the LHS completes the second periodic rotation. Hence, for α < 0.5, the consecutive roots are contained at least within the interval β n-1 -β n ∈ [π, 2π]. A similar argument can be made for cases 0.5 < α < 1. Hence, as β → ∞, the roots are contained within periodic intervals, an example of which can be seen in Table 1 for α = 0.1.

As {β s } depends solely on α, this computation needs to be carried out only once for each α value. Once the roots β s are found, we can construct the eigenfunctions η s 0 (β s x) given in ( 13) by finding c s 's.


Eva Arbona
email: eva.arbona@uv.es

María José Beltrán-Meneu

Ángel Gutiérrez
email: angel.gutierrez@uv.es

Strategies exhibited by good and average solvers of geometric pattern problems as source of traits of mathematical giftedness in grades 4-6

Keywords: Early algebra, geometric pattern problems, mathematical giftedness, primary school

We describe and analyse the strategies used by students in primary grades 4, 5 and 6 to solve linear and affine geometric pattern problems. Based on two problems posed in a teaching experiment, we have identified several profiles of strategies used by students to solve the problems. We consider the profiles of students who were good geometric pattern problem solvers as traits that may help identify mathematical giftedness. Our results show that average students used very often incorrect functional strategies and were consistent in using incorrect proportional strategies along the grades. On the other hand, good geometric pattern problem solvers tended to use correct functional strategies, although, when they had difficulties in identifying the structure of a pattern, they tended to switch to correct recursive strategies, because they are easier to apply and more reliable.

Introduction

Learning algebra is a very fruitful way to boost mathematical abilities of all primary school students, in particular mathematically gifted students (gifted students hereafter). Algebra is also an essential tool in secondary school, since it is needed to solve problems in the different areas of mathematics. However, most secondary school students have difficulties to understand and learn algebra, which hinder their mathematical progress. Some of those difficulties are understanding the meaning of letters and the equal sign, distinguishing between the notions of variable and unknown, and transforming word statements into algebraic expressions [START_REF] Banerjee | Evolution of a teaching approach for beginning algebra[END_REF][START_REF] Jupri | Improving grade 7 students' achievement in initial algebra through a technology-based intervention[END_REF].

In the early grades, algebra can be introduced through algebraic thinking that allows working and operating with variables and unknowns, avoiding the use of symbolic alphanumeric expressions (Radford, 2011a). The context of geometric pattern problems (gp problems hereafter) has proved to be successful in developing algebraic thinking in primary school [START_REF] García-Reche | La aprehensión cognitiva en problemas de generalización de patrones lineales[END_REF][START_REF] Rivera | Teaching and learning patterns in school mathematics[END_REF]. 1 and2) present a graphical representation of the first few terms of an increasing sequence of natural numbers and ask for values (V) or positions (n) of specific terms of the sequence. These problems are especially useful to facilitate the access of gifted students to basic algebraic concepts. Furthermore, gp problems are an adequate context to identify possible gifted students, since they have to use their abilities for generalization, abstraction and symbolization. Those abilities are important components of mathematical reasoning, more developed in gifted students than in the majority of students of the same age or the same school grade (we refer to them as average students) [START_REF] Krutetskii | The psychology of mathematical abilities in school-children[END_REF]. [START_REF] Amit | Rising to the challenge": using generalization in pattern problems to unearth the algebraic skills of talented pre-algebra students[END_REF] analysed strategies used by gifted students in grades 6 and 7 when solving gp problems, while [START_REF] Fritzlar | Continuing patterns as a component of algebraic thinking -An interview study with primary school students[END_REF] distinguished a set of five components of algebraic thinking by observing algebraic abilities of gifted and non-gifted primary school students when identifying and generalising patterns. However, there is little research reporting mathematically gifted students' behaviour when solving gp problems, so it is of interest to inform on traits of mathematical giftedness in the specific context of gp problems.

Gp problems (Figures

Teachers should provide all their pupils with opportunities to develop high order thinking. In particular, gifted students require more complex problems compared to average students; thus, it is important to provide teachers with tools to identify their gifted students. Students' solutions to gp problems may be very diverse, depending on the ways of reasoning and performing calculations used, since they reflect different levels of mathematical talent. Gifted students are unusually good problem solvers compared to average students, so a reliable way to identify gifted students, used by most researchers, is to observe their problem solving profiles.

In this context, we present results from a research project aimed to identify and analyse profiles of good gp problem solvers in grades 4, 5 and 6 when solving linear and affine gp problems, to characterise profiles that could differentiate gifted students from average students, and also gifted students in different school grades. The specific objectives of the part of the research presented here are:  To identify differences between profiles of good gp problem solvers and average students in their solutions of gp problems, as possible traits of mathematical giftedness.  To identify differences in students' strategies along 4, 5 and 6 primary school grades.

Theoretical framework

Gp problems present realistic contexts to help students give meaning to the pictorial representations of sequences and their numerical values and answer questions [START_REF] Billings | Algebraic thinking and pictorial growth patterns[END_REF]. Direct questions ask for values of immediate, near and far terms [START_REF] Stacey | Finding and using patterns in linear generalizing problems[END_REF], i.e., the number of pieces in the graphical representation of those terms, and also ask to write a general rule and an algebraic expression to calculate the value of any term of the sequence. Inverse questions ask for the position of the term with a given value [START_REF] Rivera | Teaching and learning patterns in school mathematics[END_REF]. Here we focus on answers to direct questions.

Gp problems could have several levels of complexity depending on their characteristics. [START_REF] Friel | A framework for analysing geometric pattern tasks[END_REF] analysed 18 different geometric patterns and ordered them from less to more complex, the most basic patterns being those based on linear relationships V=an (Figure 1). Patterns based on affine relationships V=an±b (Figure 2) increase their difficulty. Lastly, patterns based on quadratic relationships V=an 2 ±bn±c are more complex than the previous ones.

Several authors [START_REF] García-Reche | La aprehensión cognitiva en problemas de generalización de patrones lineales[END_REF][START_REF] Radford | Embodiment, perception and symbols in the development of early algebraic thinking[END_REF][START_REF] Stacey | Finding and using patterns in linear generalizing problems[END_REF] have identified different strategies used by students to solve direct questions in linear or affine gp problems. As some strategies are labelled differently by those authors, we have merged them into a single categorization of students' answers: the counting strategy is based on drawing the graphical representation of the term asked and counting its pieces. The recursive strategy uses the constant difference between the values of two consecutive terms to calculate the value of another term, by adding this difference to the value of a known term as many times as necessary. The proportional strategy assumes that there is a proportional relationship between the positions of a known term (n) and the asked term (m) and their values, V(n) and V(m): if m=a n, with a , then V(m)=a V(n). The functional strategy consists of calculating the value of a term by using an arithmetic or algebraic expression based only on the position of the term in the sequence, and not on the value of a known term. The counting, recursive, and functional strategies give the correct answer if applied properly, while the proportional strategy is correct for linear problems but it is wrong for affine problems.

Gifted students are those who show "a unique aggregate of mathematical abilities that opens up the possibility of successful performance in mathematical activity" (Krutetskii, 1976, p. 77), their problem solving abilities being higher than those of average students. [START_REF] Greenes | Identifying the gifted student in mathematics[END_REF], [START_REF] Krutetskii | The psychology of mathematical abilities in school-children[END_REF], and [START_REF] Miller | Discovering mathematical talent[END_REF] detailed several characteristics that gifted students can present, some of them being particularly important to solve gp problems, like identifying patterns and relationships among elements, generalising and transferring mathematical ideas or knowledge between numeric and algebraic contexts, locating the key of problems, abbreviating solution processes, or inverting mental procedures in mathematical reasoning.

The experimental part of our research took place in ordinary classrooms, where, quite often, there are gifted students who have not been identified by their teachers. We did not have any external way to identify gifted students in the sample of our research (e.g., their IQ), so we looked for good gp problem solvers. By comparing the solutions provided by the good gp problem solvers and the other students in their classrooms, we aim to identify specific good gp problem solvers' solution profiles that could be considered as traits of mathematical giftedness and used, together with other procedures, to identify gifted students in ordinary classrooms.

Research methodology

We present results from a study based on grades 4, 5 and 6. These are the last grades in Spanish primary schools, just before students start learning algebra, in grade 7. Two classroom groups in each grade followed an experimental teaching unit for early algebra based on gp problems. There were 43, 34, and 41 students in grades 4, 5, and 6, respectively.

The teaching unit took place during three 45-minutes ordinary mathematics classes. It was aimed to work on: i) the generalisation of functional relationships from the geometric representations, ii) the meaning and use of basic algebraic concepts and symbols, like letter notations and the translation of verbal expressions into algebraic ones, and iii) the reinforcement of the algebraic contents previously learned. The teaching unit was based on linear and affine gp problems similar to those shown in Figures 1 and2. Students solved individually about 3 problems in each session, depending on their ability and quickness. The teacher (the first author) provided some guidelines to the students; after they had solved each problem, she collected the students' answers, encouraging them to share on the blackboard their different strategies and debate whether they were correct or wrong. Finally, the teacher explained why the wrong strategies did not work. In this paper, we analyse two problems (Figures 1 and2) posed in the third session of the experiment.

The two problems are based on the same well-known context of tables and chairs. Both have the same wording (Figure 1), with 2 direct questions (a, b), an algebraic generalisation question (c), another direct question (d) aimed to check their generalisation, and an inverse question (e). However, they have a difference: the first problem (Figure 1) presents tables (T) with chairs (C) only at the sides, so the sequence is linear (C=2T); the second problem (Figure 2) presents tables with chairs both at the sides and the ends, so the sequence is affine (C=2T+2). The problems were posed to induce students to generalise the relationships, write an algebraic expression, and use it.

María is organising her birthday party with her friends and relatives. She wants to calculate how many tables and chairs she needs to sit people as in the pictures: After analysing the answers to the direct questions in the problems (a, b, and d), we have identified different profiles of students' solutions, depending on which strategies were used. For example, the recursive-functional profile identifies those students who used a recursive strategy in the linear gp problem and a functional strategy in the affine one.

Analysis of students' answers

We have analysed the types of students' strategies in the direct questions in both gp problems. We have considered as good gp problem solvers those students who solved correctly, or with minor errors, all the gp problems in the teaching unit. We identified as good gp problem solvers three students in grade 4, four in grade 5, and three in grade 6. Table 1 shows the number of good gp problem solvers and average students in each grade that used each profile to answer the direct questions. In the average students' profile functional-functional, we use brackets to show the number of their correct answers.

In grade 4, two out of the three good gp problem solvers based their solutions to both problems on functional strategies. On the other hand, 23 average students in grade 4 used functional strategies in both problems, but only 7 students solved them correctly. The other students used a diversity of combinations of functional and proportional strategies, all incorrect.

Linear problem

Student 5B-13: 6×2 = 12 Because you must multiply the table x2.

Affine problem

Student 5B-13: 14 Because I've been adding. In grade 5, the good gp problem solvers provided solutions based on functional and recursive strategies, half of them changing from using the functional strategy in the linear problem to the less complex recursive strategy in the affine problem (Figure 3), or vice versa. On the other hand, average students mostly used the functional strategy in both problems, but only 8 students produced correct answers in both problems. The change of strategy made by the good gp problem solvers seems illogical, but it allowed them to succeed in solving the problem. On the contrary, the average students used the same strategy in both problems, maybe because they were comfortable using it or considered that both problems were analogous, but many average students produced incorrect answers (Table 1). In grade 6, all good gp problem solvers and most average students used functional strategies in both problems, with a few average students using such strategies incorrectly. As in the other grades, some average students used proportional strategies. The fact that all good gp problem solvers in grade 6 used correct functional strategies, while a part of the average students used wrong proportional and functional strategies, points to traits of mathematical giftedness in the context of gp problems, namely identification of patterns and relationships among different elements, and generalising and transferring mathematical ideas from a numeric context to an algebraic one.

Profiles of solutions

A profile typical of students when they start solving gp problems is that they tend to move from a strategy to another in the consecutive questions of the same problem [START_REF] Gutiérrez | The cognitive demand of a gifted student's answers to geometric pattern problems[END_REF]. However, as the problems we are analysing were posed in the third class session of the teaching experiment, the students had already learned that the final aim of the gp problems was to state a generalisation. Then, all students but one in the sample used the same strategy for all direct questions in each problem, although some students used different strategies for the linear and affine problems, showing diverse profiles in their solutions. Students had also learned that recursive strategies are efficient only for the immediate or near terms.

After comparing the data in Table 1 for the different grades, we get the following conclusions:

i) The good gp problem solvers used, in each problem, a strategy with which they felt confident and that they were sure it was correct, even using a different strategy in each problem (Figure 3). They were more successful than average students in using simpler recursive strategies, and avoided the proportional strategy even in the linear problems, were it provides a correct answer.

The good gp problem solvers also became more efficient along the grades using functional strategies.

ii) Average students in all grades used mostly functional strategies, with an increase of the percentage of correct answers along the grades, but we do not observe a reduction in the use of (incorrect) proportional strategies in the affine problem.

Due to the relevant number of average students using the same strategy in the linear and affine gp problems, we have analysed the errors caused by this profile. We have identified three types:

Constancy of proportional relationship

Some average students used a (correct) proportional strategy in the linear problem and they used it again in the affine problem, although now it was wrong. Students did not analyse the whole pattern, but only one term: they calculated proportionally the value of the term requested, considering only the number of guests sitting around one table in the first or the second term of the pattern. Figure 4 shows the written answers of an average student who only considered the number of chairs around the table in the first term.

Linear problem

Student 5A-6: 50×2 = 100 Because there are twice as many guests as tables.

Affine problem

Student 5A-6: 50×4 = 200 Because, if there are 50 tables, there are 4 guests for each table.

Constancy of recursive relationship

Some average students identified the difference between the values of two consecutive terms and used it in a repeated addition or a multiplication. In Figure 5, the student did not pay attention to the chairs at the ends of the tables in the affine problem and used the increment of 2 chairs ("two more chairs") as proportional ratio.

Error of analysis of diagrams

Some average students did not analyse correctly the parts of the patterns. They identified a wrong difference between the linear and the affine patterns and used it to create a wrong formula. In Figure 6, the student did not identify correctly the chairs at the ends of the tables and, furthermore, he did not use correctly the number of chairs around the first

Conclusions

We have presented a comparative analysis of strategies of solution used by a sample of good gp problem solvers and average students in grades 4 to 6 when solving direct questions in a linear (V=an) and an affine (V=an+b) gp problems posed as part of an experimental teaching unit. The analysis of students' answers to those problems shows some significant and original findings: most students used mainly functional strategies, although good gp problem solvers showed a tendency to follow profiles using functional and recursive strategies, which were less complex but more successful than the profiles followed by average students, based on functional and proportional strategies. There is a tendency of average students in all grades to use wrong proportional strategies. According to [START_REF] Van Dooren | Not everything is proportional: effects of problem type and age on propensities for overgeneralization[END_REF], students are prone to apply proportional strategies when they should not be applied. The frequency of such strategies seems to be higher when students start learning proportional reasoning, which, in Spain, usually happens in grade 4.

The analysis of the data collected suggests a difference between the profiles of good gp problem solvers and average students in the use of recursive strategies of solution in the (more difficult) affine problem. Data also seem to show a clear difference between the profiles of both types of students in the use of proportional strategies, even when they were correct. Hence, a contribution of this research is the suggestion that a trait of giftedness in solving gp problems seems to be the use of correct recursive and functional strategies and the absence of proportional strategies to solve gp problems.

Respect to the differences between grades, we have observed an increment in the amount of solutions based on functional strategies, and an increment in the percentage of correct functional strategies. However, it is not apparent a (expected) reduction in the use of incorrect proportional strategies.

  many people will sit around 6 tables? How do you know it? b) How many people will sit around 50 tables? How do you know it? c) Explain to a friend how she can calculate how many people will sit depending on the number of tables. Write down the formula you have mentioned.d) Use the previous formula to find how many people will sit around 15 tables.e) If there are 22 people sitting, how many tables will be? How do you know it?

FigureFigure 2 :

 2 Figure 1: The linear gp problem

Figure 3 :

 3 Figure 3: Functional and recursive strategies in a good gp problem solver's answers to question a

Figure 4 :

 4 Figure 4: Constancy of proportional relationship in an average student's answers to question b

Figure 6 :

 6 Figure 6: Error of analysis of diagrams in an average student's answer to question b

Table 1 : Profiles of solutions to the direct questions (a, b, and d) in both problems

 1

			Good gp problem solvers		Average students
	Linear	Affine	4th	5th	6th	4th	5th	6th
	Recursive	Functional	0	1	0	0	0	0
	Functional	Recursive	0	1	0	0	0	0
	Functional	Functional	2	2	3	23 (7*) 20 (8*) 27 (22*)
	Functional	Proportional	0	0	0	6	2	0
	Proportional	Proportional	0	0	0	3	2	6
	Other solutions	1	0	0	8	6	5
	Total	3	4	3	40	30	38
	(*) Number of average students in this profile producing correct solutions to both problems.

Constancy of recursive relationship in an average student's answers to question b

  table (4).

	Linear problem	Affine problem
	Student 5A-11: 50×2 = 100	Student 5A-11: 50×2 = 100
	Because each time there are 2	I multiply the number of tables
	[more] chairs and 50 tables	by 2 more chairs.
	[multiplied] by 2 chairs too.
	Linear problem	Affine problem
	Student 5A-1: 50×2 = 100	Student 5A-1: 50×2 = 100; 100+3=103
	Because, each time, 2 chairs are	Because there are 4 guests in 1
	added with one table.	table, but 2 guests are added
		each time.

Figure 5:

Acknowledgments

The results presented are partially funded by the research projects EDU2017-84377-R (AEI/ERDF, EU) and GVPROMETEO2016-143, and by the predoctoral aid FPU16/04513 (Spanish Ministry of Science, Innovation and Universities), aimed to train new university teachers.


James Gray
email: james.gray@oslomet.no

Bodil Kleve
email: bodil.kleve@oslomet.no

Helga Kufaas Tellefsen
email: helgate@oslomet.no

Students' expected engagement with algebra based on an analysis of exams in Norway from 1995 till 2018

Keywords: Algebraic thinking, Assessment, Specialised and non-specialised mathematics discourses

Using the definition of algebra from the TIMSS framework, algebra questions were identified on a sample of Norwegian mathematics examinations for 10 th grade students from 1995 to 2018. These questions were then analysed using a discursive framework to look at the specialisation of the mathematical language. The analysis revealed consistency in the density of the language, a decreasing expectation of the depth of engagement with real world contexts, and a recent tendency to processes that direct the students in solving problems. It is argued that this gives an indication of the expected engagement with and implementation of the algebra curriculum in Norwegian schools.

Background and context

An ongoing issue in debates about Norwegian students' performance in mathematics is related to performance on algebraic tasks. On TIMSS tests since 2011, Norwegian 5 th grade students were among the top performers in mathematics in Europe [START_REF] Bergem | Hovedresultater i matematikk[END_REF]. They performed significantly higher than students in the other Nordic countries. However, the performance of 9 th grade students on the most recent TIMSS test in 2015 was at an intermediate level in a European perspective. Overall this was an improvement on previous years, but it was "particularly due to low scores in Algebra" (Bergem, 2016 p.22), which have decreased since 2011 and were at a very low level in 2015.

In Norway, the only externally set high-stakes written examination based on the mathematics common curriculum occurs at the end of grade 10. We consider tasks on these mathematics examinations to be important with regard to how students are expected to engage with the mathematical discourse. Like [START_REF] Morgan | To what extent are students expected to participate in specialised mathematical discourse? Change over time in school mathematics in England[END_REF], we assume that mathematics examinations mirror how mathematics curricula are implemented in the classroom [START_REF] Broadfoot | Education, assessment and society[END_REF]. From a discursive perspective, [START_REF] Morgan | To what extent are students expected to participate in specialised mathematical discourse? Change over time in school mathematics in England[END_REF] investigated differences in the language used in mathematics questions on examinations in England and Wales. The identified changes are discussed in the light of debates about 'standards', highlighting engagement with algebraic manipulations and independent problem solving. It is interesting to see if similar changes have occurred in other countries, to which our study may contribute.

Within this context, we investigated how algebra questions on the 10 th grade examinations have developed since 1995. During this period, there have been several curricula (M87, L97 and LK 06 introduced in 1987L97 and LK 06 introduced in , 1997L97 and LK 06 introduced in and 2006 respectively) respectively), which differ substantially in their treatment of algebra and contextualisation. In M87 (Mønsterplan for grunnskolen: M87, 1987) "Algebra and Functions" was one topic and "Mathematics in everyday life" was integrated into all topics. L97 (1999) had the following as separate topics: "Algebra and Number"; "Graphs and Functions"; and "Mathematics in everyday life". In the current curriculum LK 06 (Kunnskapsdepartementet, 2006), there is an emphasis on basic skills. In contrast to M87 and L97, in which the goals were knowledge oriented, each main area in LK 06 is formulated in terms of competence goals. As in L97, "Numbers and Algebra" and "Functions" are still separate topics. Mathematics in everyday life is no longer a separate topic, but integrated into all topics, as it was in the M87.

In light of this, we will investigate the following research question:

How has the nature of students' expected application of algebra in both mathematical and nonmathematical contexts developed since 1995 in 10 th grade school mathematics in Norway?

Theoretical framework

As analytical framework, we have drawn on the work of Morgan, Sfard, and Tang (Morgan & Sfard, 2016;[START_REF] Morgan | To what extent are students expected to participate in specialised mathematical discourse? Change over time in school mathematics in England[END_REF]. Related to their project The Evolution of the Discourse of School Mathematics (EDSM), they developed a substantial conceptual framework for analysing examinations. This framework has a theoretical foundation in social semiotics and Sfard's communicational theory [START_REF] Sfard | Thinking as Communicating. Human Development, the Growth of Discourses, and Mathematizing[END_REF]. For the purposes of this paper, it is sufficient to say that mathematics being a human activity is defined in terms of the discourse that one engages in when doing mathematics. [START_REF] Morgan | To what extent are students expected to participate in specialised mathematical discourse? Change over time in school mathematics in England[END_REF] argue that the language used on the exams is crucial and that the language use in examinations may change the nature of the mathematics students are expected to engage with. [START_REF] Dowling | Reading mathematics texts[END_REF] emphasised how the language used in school mathematics may both enable and deny access to mathematics. By analysing the language students used when telling stories about their own learning in mathematics [START_REF] Kleve | Learning subjects in School-being outsiders or insiders in the disciplinary discourse of mathematics and Language 1[END_REF] found that students who were successful in mathematics, used the subject specific language to a greater extent than those less successful. That is, the successful students were inside the mathematical discourse.

In our project, we aim to apply the whole EDSM framework to our selection of algebra questions. This paper will report on just one section of the framework. It is intended that the framework can be used in this way, and we are following the work of [START_REF] Morgan | To what extent are students expected to participate in specialised mathematical discourse? Change over time in school mathematics in England[END_REF] in focusing on the part of the framework that deals with specialisation of the discourse.

To analyse the algebra questions, nouns and verbs were allocated as SM (specialised to the discourse of school mathematics) or NSM (non-specialised). Specialised to the discourse of mathematics means that they were "used in accordance with mathematical definitions or in ways distinctive to school mathematics" (Morgan & Tang, 2016, p. 145). Then the whole questions could be classified as (entirely) SM, (entirely) NSM or Mixed (both SM and NSM). Although NSM was not included as a question code in the EDSM framework it was revealed during our analysis. Questions classified as SM were termed Abstract. Whole Questions classified as NSM or Mixed were further classified according to the depth of engagement with the context in terms of Ritual (repetitive exercise or following patterns widely used in text-books), Mundane (containing some unfamiliar elements, however requiring minimal engagement) and Deep (demanding more extensive engagement to determine how specialised mathematical discourse should be used). Illustrative examples of these categories appear in [START_REF] Morgan | To what extent are students expected to participate in specialised mathematical discourse? Change over time in school mathematics in England[END_REF]. With regard to vocabulary, we identified objects in terms of nouns, and processes in terms of verbs, coding each word as either SM or NSM.

Method

To explore our research question we needed a representative sample of examinations. We took examinations every five years from 1995 together with the most recent examination from 2018. This yielded a sample that was manageable within the remit of our project, had an even coverage of the study period, and that included an early and a late examination in each curriculum cycle. The sampled examinations and the curricula reforms are illustrated on the timeline below. To identify the algebra questions from each paper the authors made an initial selection individually. The selections were compared and the selection criteria discussed. It is now common to distinguish between algebra as cultural artefact and algebraic thinking as a human activity [START_REF] Hodgen | Algebraic thinking[END_REF]. Ideally, we would be able to pick out the questions that the majority of students would be expected to use algebraic thinking. However, we do not have access the thoughts of the candidates and examination setters so we could not identify questions by this criteria. We considered the curricula, but these did not provide us with an adequate definition because algebra has been grouped with different topics in the different curricula (see Background and context). Our project is partially motivated by the poor results in algebra in TIMSS (see Background and context), so we decided to use their definition of algebra to guide our selection ("TIMSS 2019 Mathematics Framework," 2017). For example, the questions asking students to use the Pythagorean theorem to work out the length of one side of a triangle were categorized as geometry and not algebra (as they are in the published TIMSS questions). An example of an algebra question involving geometry is given at the end of the Findings and analysis section. Using the definition from TIMSS, the authors adjusted their initial selection and reached a consensus.

Questions were coded as SM, mixed or NSM, and then mixed and NSM questions were coded as ritual, mundane or deep (see Theoretical framework). Working individually, one of the authors coded all the questions in the sample of examinations and the other two shared the sample. Revision and recoding of all the data set was then done by two authors in consultation with one another. Similarly at the level of specialised vocabulary. One of the authors coded the process (verbs) as SM or NSM and the other two coded the object (nouns) as SM or NSM. The authors consulted each other during this process so that difficulties could be resolved by a consensus decision. Inter-rater reliability was not calculated because the purpose of the project is to compare examinations and for each coding category each data set was coded completely by one of the authors.

Findings and analysis

Proportion of algebra questions and degree on contextualisation

Figure 2 shows the percentage of algebra questions on each of the examination papers. In the examinations in 2000 and 2005, candidates were given a choice of six questions from 22 (covering a range of topics) in the last part of the examination. In each of these years, it was possible to avoid, or choose almost exclusively, algebra questions in this part. These two extremes are shown in Table 1, and in Figure 2 by two data points for each of these years. We do not have any data about how many algebra questions a student would typically choose. We note that candidates had the opportunity to make a significant reduction in the proportion of algebra questions in these years, and from the authors' experience, it is very unlikely that many chose the highest proportion possible. LK06 was revised in 2013 and more algebra was included [START_REF] Utdanningsdirektoratet | Eksamensveiledning MAT0010 Matematikk[END_REF]. The proportion of algebra questions on the examinations in our sample has also increased significantly since 2013. For comparison, the target proportion for algebra in the TIMSS assessment is 30%. The examination in 1995 was approximately at this level. The examinations in 2005 and 2010 were below this level, whereas the examinations in 2015 and 2018 were above this level. Figure 3 shows the number of questions coded as abstract, ritual, mundane and deep, and these as a proportion of all of the algebra questions. In the years 2000 and 2005, we have included all the algebra questions that the students could choose in the last part of the examination. (In the year 2000, candidates could not do more than twelve algebra questions, but the figure shows the results for all fourteen algebra questions.) If we exclude the last part of these examinations, there were six abstract and one ritual questions in 2000, and eight abstract and one mundane questions in 2005. (There were fewer questions in total on the examinations under the curriculum L97, but the time given to the candidates has remained constant for all the examinations in the sample.) Thus, there was only one compulsory contextualised algebra question for candidates in each of these years.

There were no questions coded as deep in the examinations after 2005 in our sample. Moreover, the questions coded as deep in 2005 were all in the last part where the candidates had a choice of 0% 10% 20% 30% 40% questions. Thus, the examination in 1995 was the only one in our sample where candidates were expected to engage in 'deep' algebraic context.

The proportion and number of ritual questions has increased significantly in the examinations in our sample that occurred after the introduction of the present curriculum (2010, 2015 and 2018). In addition, the proportion of abstract questions has decreased. This indicates a trend where there are more contextualised algebra questions, but that the degree of depth with which the students are expected to engage with the context has fallen. We noted earlier that the proportion of algebra questions had increased significantly in the last two years of our sample. If the motivation for doing this was to raise standards in algebra, then our data indicates that the quality as well as the quantity of algebra questions needs to be addressed. This may be related to the emphasis placed on algebra in the curricula referred to above. However, it is beyond the scope of this paper to go into this further.

Figure 3. The degree of contextualisation

Specialised and non-specialised processes and objects

The first two rows of Tables 2 and3 shows the frequency and rate per question of SM processes (verbs) and objects (nouns) respectively. The rate of SM processes and objects per question is lower in 2005 which will be discussed more below. In the other years, the rate of SM processes is similar over the sample. There is more variation in the percentage of SM objects, and the rate of SM objects per question. These appear to increase after each curriculum reform and then decrease, but a larger sample would be required to confirm this. The number distinct SM processes and SM objects are shown in the third rows of Tables 2 and3 respectively. This measure gives an indication of richness of the discourse, and the data indicates that this has remained constant over the sample. The number of SM symbolic objects as a percentage of the SM objects is an indicator of the mathematical density of the discourse. This has also remained constant over the sample period apart from the year 2005. The examinations in the sample under the present curriculum (2010-2018) contained a larger total number of objects, and a larger number of objects per question. However, the number of distinct object words has not increased proportionately to this. This indicates a more lengthy written style and not a richer expected discourse. A recurring feature in these examinations is supplementary information in the text with for example historical notes about mathematicians. This may be a consequence of the greater emphasis on basic skills (in this case reading) in LK 06 (Kunnskapsdepartmentet, 2006).

In 2005, the rate of SM processes and objects per question is lower than the other years. A closer inspection of the data revealed that the rate of SM processes was similar with other years on abstract questions. However, the only SM processes on a mixed question was find which occurred just one time. In addition, there were only six SM objects in total on the mixed questions. Thus the algebra questions in 2005 were either almost entirely SM or NSM. This may be a consequence of the curriculum having "Mathematics in everyday life" as a separate topic and not integrated into other topics (L97, 1999). The lack of specialisation in the language on mixed questions meant that candidates were required to engage with the content, so all of these questions were coded as either mundane or deep (see Figure 4). However, as noted earlier, students could choose not to do these questions and so these questions may have had less impact on teaching and learning.

Sorting the data by year revealed that, of the 18 distinct SM processes in 1995 and 17 in 2018, there were just seven of these that occurred in both years. This change in language was the result of incremental changes over the sample period. There were, for example, no significant introductions of new processes with the curricular reforms. For objects, there were 17 common objects out of the 24 distinct SM objects in 1995 and 30 in 2018. When the objects were sorted by topic and the largest category was geometry with a total of 25 objects. Nine of these occurred in the exams under the previous curricula, whereas 20 of these occurred in the exams under the current curriculum. This shows the increased use of geometry as a context for algebra questions. The frequencies of the most commonly occurring SM processes are shown in Table 4. Find and calculate are the most commonly occurring instructions in 1995. These have not been used so frequently in recent years. The current examination guidelines [START_REF] Utdanningsdirektoratet | Eksamensveiledning MAT0010 Matematikk[END_REF] " (2015 paper, p.11).) The data appears to show a tendency away from find and calculate (regn ut) over to solve and determine (bestemme) in algebra questions. The word for determine in Norwegian (bestemme) is also used to mean decide, and thus we would argue that determine belongs to a less specialised discourse than the other instructions.

The verb use has occurred significantly more often since the last curriculum reform. This is often in formulations such as "Use the [given] formula to calculate Monica's maximum pulse rate" (Question 6, 2018 examination, our translation), and "Use a graph tool to…". Such formulations give more explicit instructions to candidates, and may also reduce candidates' role in decision making (Morgan & Tang, 2016, p. 53). Another consequence is that the question becomes an instruction to complete a process, rather than solve a problem. That is, that the discourse becomes one that is about performing actions rather than engaging in mathematics. This reflects the formulation of the current curriculum which is formulated in terms of competency goals and has an emphasis on basic skills (see Background and context). ICT is one of the basic skills (Kunnskapsdepartementet, 2006), so explicit instructions to use ICT tools have been included on the examination [START_REF] Utdanningsdirektoratet | Eksamensveiledning MAT0010 Matematikk[END_REF].

An illustrative example

The following question appeared in the 2018 examination. It contained a picture of Archimedes (not reproduced here), the diagram shown in Figure 4, and each sentence was on a separate line.

A sphere has diameter equal to . A cone and a cylinder both have height equal to . Archimedes showed that the volume of the cylinder is equal to the combined volume of the sphere and the cone. Use the formulas below, and show that this holds. (2018 examination, our translation) This question was coded as abstract: there is a context for algebra, but it is within the SM discourse. Our analysis suggests that geometry contexts have been more prevalent under the current curriculum. The last sentence directs the student as to how to solve the problem. This is further emphasised by the use of the plus sign and equals sign in Figure 4. These prompts reduce the candidates expected engagement with the discourse of the question. The mention of Archimedes is not essential to the question and his picture may be a distraction, especially for weaker candidates [START_REF] Dowling | Reading mathematics texts[END_REF].

This example shows that our analysis gives us insight into some of the changes in the expected mathematical discourse of 10 th grade students in Norway. However, we see the importance of other factors such as the use of diagrams, and the role of human actors in the discourse. These factors can be analysed by other parts of the EDSM framework, and will be part of our further research.

Concluding remarks

Our sample indicates that the proportion of algebra questions on 10 th grade examinations in Norway has increased since the last curriculum reform in 2006. The quantitative analysis at the question level suggests that the expected engagement in real-world contexts has become more ritual. At the level of objects and processes, the analysis suggests that the density of the discourse has remained constant over the sample period. The examination questions have contained more objects under the present curriculum, and the processes have become more directive. This indicates a discourse that reduces the need for decision making on the part of the student. These factors may contribute to the ongoing underperformance of Norwegian students in algebra on international test, but an inspection of the examination questions reveal many more factors that are in play. These need to be researched and synergised with our results to gain a better understanding of Norwegian students' algebra discourse.

Figure 1 :

 1 Figure 1: Timeline showing sampled examination years (dots), and curricular reforms

Figure 2 :

 2 Figure 2: The percentage of algebra questions in each examination in the sample.

Figure 4 :

 4 Figure 4: Diagram from question 19 on the 2018 examination

Table 1 : The number of questions and algebra in each examination in the sample

 1

Table 2 : Specialised processes

 2

Table 3 : Specialised objects

 3

Table 4 : Frequency of the most commonly occurring SM processes

 4

		1995	2000	2005	2010	2015	2018
	SM (% of all objects)	68 (64%)	70 (80%)	39 (60%)	118 (90%)	140 (83%)	109 (57%)
	Rate of SM objects per question	2.5	3.1	1.8	4.6	4.4	3.5
	Distinct SM non-symbolic object	24 (1.3)	25 (1.8)	14 (1.3)	30 (1.9)	35 (1.7)	30 (1.5)
	words (rate per question)
	SM objects in symbols (% of all	23 (34%)	26 (37%)	19 (49%)	45 (38%)	48 (34%)	39 (36%)
	SM objects)
	Total objects (rate per question)	106 (5.9)	88 (6.3)	65 (5.9)	131 (8.2)	169 (8.1)	190 (9.5)

  state that candidates can choose their own solution method when answering questions that are formulated with find, solve or determine. Questions that are formulated with calculate require the candidate to use algebraic methods. (Although calculate was used as an instruction seven times in 2015, it was only used twice to indicate symbolic manipulation. The other five times were to evaluate expressions, e.g. "Calculate which Pythagorean triple you get if

Acknowledgment

Thanks to Candia Morgan for email correspondence.