G Croizat

A Kehren

H Roux De Bézieux

A Barakat

Influence of pulsatile blood flow on allometry of aortic wall shear stress

Shear stress plays an important role in the creation and evolution of atherosclerosis. A key element for in-vivo measurements and extrapolations is the dependence of shear stress on body mass. In the case of a Poiseuille modeling of the blood flow, P. Weinberg and C. Ethier [2] have shown that shear stress on the aortic endothelium varied like body mass to the power -3 8 , and was therefore 20-fold higher in mice than in men. However, by considering a more physiological oscillating Poiseuille -Womersley combinated flow in the aorta, we show that results differ notably: at larger masses (M > 10 kg) shear stress varies as body mass to the power -1 8 and modifies the man to mouse ratio to 1:8. The allometry and value of temporal gradient of shear stress also change: ∂τ /∂t varies as M -3/8 instead of M -5/8 at larger masses, and the 1:150 ratio from man to mouse becomes 1:61. Lastly, we show that the unsteady component of blood flow does not influence the constant allometry of peak velocity on body mass: umax ∝ M 0 . This work extends our knowledge on the dependence of hemodynamic parameters on body mass and paves the way for a more precise extrapolation of in-vivo measurements to humans and bigger mammals.

Introduction

The formation of atherosclerosis in arteries is a multifactorial process, still not fully understood in its mechanical factors ([12], ch. 23, p. 502). Yet, it has been shown that wall shear stress plays a key part in this phenomenon [START_REF] Cunningham | The role of shear stress in the pathogenesis of atherosclerosis[END_REF]. For theoretical analyses as well as for experimentation, it is very useful to know the dependence of wall shear stress τ in arteries, and especially in the aorta, on body mass M . It has long been assumed that wall shear stress was uniform accross species, thus independent of body mass. However, recent works have shown that this assumption was not correct: allometric relationships, expressed as τ ∼ M α , with α ∈ R, can be obtained from simple models of blood flow in the aorta. Note that ∝ shall be understood as "is proportional to". P. Weinberg and C. Ethier have modeled blood in the aorta as a Poiseuille flow and have deduced the following result: τ P oiseuillle ∼ M -3/8 [START_REF] Weinberg | Twenty-fold difference in hemodynamic wall shear stress between murine and human aortas[END_REF]. This result implies that wall shear stress in mice should be 20 times higher than in men. The objective of this article is to study the allometric relationship between body mass and wall shear stress and other key parameters with a more precise model of blood flow in the aorta. In order to be closer to physiology, we added a purely oscillatory unsteady profile to the steady Poiseuille one. This oscillating component is called Womersley flow. We will first detail the mathematical derivation of the Poiseuille and Womersley profiles, before exploring their influence on the allometry of wall shear stress (WSS), temporal gradient of wall shear stress (TGWSS) and peak velocity.

Mathematical modeling

In everything that follows, blood is considered as an incompressible Newtonian fluid of density ρ = 1060 kg.m -3 and dynamic viscosity µ = 3.0 10 -3 P a.s. The arteries are assumed to be rigid of radius a, and we adopt a no-slip condition on the arterial wall (u(r = a) = 0). We also assume the flow to be axisymmetric. Model-dependent hypotheses are described in the relevant subsections. See table 5 at the end of the document of all constants used in this work. All numeric simulations were done in Matlab R2016 a, The MathWorks, Inc., Natick, Massachusetts, United States. log stands for logarithm in base 10, and ln for natural logarithm.

Poiseuille flow

In the case of a Poiseuille flow, shear stress on the wall is worth:

τ = 2µ u max a (1)
Let us detail the allometric arguments of equation 1, applied to the aortic artery:

• the aortic diameter a is assumed to vary as M 0.375 , both from theoretical ( [START_REF] West | A general model for the origin of allometric scaling laws in biology[END_REF]) and experimental considerations ( [START_REF] Clark | Comparative physiology of the heart MacMillan[END_REF] and [START_REF] Holt | Geometric similarities of aortas, venae cavae and certain of their branches in mammals[END_REF])

• the dynamic viscosity of blood µ is assumed to be independent of the body mass

• the aortic velocity u max can be seen as the cardiac flow rate divided the aortic cross-section u max = Q πa 2 . The cardiac flow rate varies with M 0.75 , as described in [START_REF] West | A general model for the origin of allometric scaling laws in biology[END_REF] and [6]. Therefore, u max is independent of M .

We can conclude that in the case of a Poiseuille flow, wall shear stress scales as:

τ P oiseuille ∝ M -0.375 (2)
Additionally, we can deduce the allometry of temporal gradient of wall shear stress. Since ∂τ ∂t ∼ ωτ , ω ∼ M -0.25 being the cardiac frequency, we get:

∂τ ∂t P oiseuille ∼ M -0.625 b (3)
We obtain the following values for mice, rabbits and humans (tab. 1), as described in [START_REF] Weinberg | Twenty-fold difference in hemodynamic wall shear stress between murine and human aortas[END_REF], relatively to the human value:

Animal

Womersley flow

We consider a purely oscillating pressure gradient (eq. 5) in the same geometry, and derive the allometry of wall shear stress, temporal gradient of wall shear stress and peak velocity associated to this flow.

Ruling equations

In this context, the Navier-Stokes equation projected on z can be written as:

ρ ∂u ∂t = -∇P + µ∆u (4)
We inject the expression of the pressure gradient (with G 1 ∈ R constant) in 4:

-∇P = ℜ(G 1 exp(iωt)) (5)
as well as the expression of the velocity (note that A(r) ∈ C):

u(r, t) = ℜ(A(r) exp(iωt)) (6)
We obtain the following equation, ′ being the differentiation with respect to r.

A ′′ + 1 r A ′ - iωρ µ A = - G 1 µ (7)
where we recognize the sum of a constant particular solution and a Bessel differential equation. We get the following velocity field, with J 0 the 0-order Bessel function of first kind. We also introduce the Womersley number α = a ωρ µ . This dimensionless number corresponds to the ratio of inertial transient forces to viscous forces.

u(r, t) = ℜ( G 1 iωρ (1 - J 0 (i 3 2 α r a ) J 0 (i 3 2 α) )e iωt ) on u z (8) 1.2.

Derivation

Let us now calculate the flow rate Q:

Q(t) = a 0 u(r)2πrdr (9)
and the wall shear stress τ :

τ (t) = -µ ∂u ∂r | r=a (10)
The calculation of (10) goes down to ∂ ∂r

[J 0 (i 3 2 α r a )] |r=a (11)
Using

J ′ 0 = J 1 in C, we can conclude that τ (t) = ℜ( aG 1 i 3 2 α J 1 (αi 3 2 ) J 0 (αi 3 2 ) e iωt ) (12)
Concerning (9), the key point is

a 0 rJ 0 (i 3 2 α r a )dr (13)
Using

x n J n-1 (x) = d dx (x n J n (x))
for n in N* and x in C, we get

Q(t) = ℜ( πa 2 iG 1 ωρ (1 - 2J 1 (αi 3 2 ) αi 3 2 J 0 (αi 3 2 ) )e iωt ) (14)

Physiological pulse wave: Poiseuille + Womersley flow

In reality, the pressure gradient is neither steady, nor purely oscillatory. It was shown ([13], [START_REF] Womersley | Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is known[END_REF]) that the pressure gradient ∇P could be reasonably approximated by the first six harmonics of its Fourier decomposition. In our case, six harmonics are not needed, as we are only looking at allometric variations. Along with the allomerty of the Womersley flow (first harmonic), we will present the sum of Poiseuille and Womersley flows (fundamental + first harmonic), which we denote PW flow from here on:

u P W = u P oiseuille + u W omersley (15)
Consequently, by linearity of differentiation, wall shear stress and its temporal gradient are obtained by: τ P W = τ P oiseuille + τ W omersley and ∇τ P W = ∇τ P oiseuille + ∇τ W omersley . Concerning peak velocity however, calculations are done using the total velocity field u P W .

2 Results

Wall shear stress (WSS)

To evaluate the allometry of wall shear stress, we need to know the allometry of every variable present in its expression. Concerning a, α or ω, the allometry is known from [START_REF] Weinberg | Twenty-fold difference in hemodynamic wall shear stress between murine and human aortas[END_REF], as summed up in table 5. But the allometry of G 1 is unknown. To overcome this problem, we substitute the flow rate Q, which follows an experimentally determined allometry Q ∼ M 0.75 . This step can be discussed, since the Womersley flow model implies a zero average flow rate. Although < Q >, the time average value of Q, is always null, its effective value < Q 2 > is indeed an increasing function of body mass, which justifies the use of Q ∝ M 0.75 in the Womersley model. Combining ( 12) and ( 14), we obtain

τ = ωρJ 1 (x) πai(xJ 0 (x) -2J 1 (x)) Q (16)
with x = i 3 2 α and Q, τ the complex variables such that Q = ℜ(Q) and τ = ℜ(τ ). We introduce the following function f , such that τ = ωρ πa f (α)Q:

f (α) = iJ 1 (x) (2J 1 (x) -xJ 0 (x)) (17)
We plot |ℜ(f )| as a function of α in logarithmic scale on figure 1). Note that ℜ(f

) is negative, ie |ℜ(f )| = -ℜ(f ).
) ∝ α -1 for α > 5 • for α < 2, ℜ(f ) ∼ α -2 (r 2 = 0.98) • for α > 5, ℜ(f ) ∼ α -1 (r 2 = 0.99)
We can confirm these regressions using Taylor and asymptotic developments of J 0 and J 1 :

J 0 (x) = 1 -x 2 /4 + O 0 (x 4 ) (18)
and

J 1 (x) = x/2 -x 3 /16 + O 0 (x 5 ) (19)
using big O notation ( [START_REF] Arfken | Mathematical methods for physicists[END_REF], p. 687). Consequently, for α << 1, f (α) ≈ 4i/x 2 ≈ -4/α 2 , which confirms ℜ(f ) ∼ α -2 for α << 1.

Conversely, for |x| → +∞:

J 0 (x) ≈ 2 πx cos(x -π/4) (20)
and

J 1 (x) ≈ 2 πx sin(x -π/4) (21)
introducing complex sine and cosine ( [START_REF] Arfken | Mathematical methods for physicists[END_REF], p. 723).

In our case, ). We observe that Poiseuille shear stress is, in absolute value, much lower than Womersley stress (10 to 20 times lower as mass increases). At low masses (M < 1 kg), both flows follow the same allometry: τ P W ∝ M -0.375 . At higher masses however, (M > 10 kg), the Womersley stress accounts for the variations of total stress and thus τ P W ∝ M -0.125 . In practice, it means that the difference in wall shear stress between mice and men is greatly reduced by the influence of pulsatile flow in larger arteries, and shows the necessity of the unsteady modeling of blood flow for correct body mass allometries.

x = i 3/2 α = α exp(3iπ/4), therefore f (α) ≈ -itan(x) x = tanh(α √ 2/2) α exp(-3iπ/4) and finally ℜ(f (α)) ≈ √ 2 2α , which confirms ℜ(f ) ∼ α -1 for α → ∞. Knowing that α ∼ M 0.

Temporal gradient of wall shear stress (TGWSS)

Concerning the temporal gradient of shear stress, we get ∂τ ∂t ∼ ωτ , with ω ∼ M -0.25 [START_REF] Weinberg | Twenty-fold difference in hemodynamic wall shear stress between murine and human aortas[END_REF], and therefore: Poiseuille makes only 12% of total stress for a mouse (M = 0.025 kg), and this percentage goes down to 5% for a human (M = 75 kg). Concerning allometry, both processes induce τ P W ∝ M -0.375 at low masses (M < 1kg), but at higher masses (M > 10kg), the Womersley stress sets the allometry τ P W ∝ M -0.125 .

We compare the values of shear stress gradient between Womersley and Poiseuille in tab. 3. Poiseuille TGWSS is lower than Womersley TGWSS, and the allometric domination at high masses is also present. The graph of these functions (fig. 3) is very similar to the one on wall shear stress (fig. 2). At low masses (M < 1kg), both flows result in ∂τ ∂t P W ∝ M -0.625 allometry, and at higher masses (M > 10kg), Womersley stresses dominate the allometry:

∂τ ∂t P W ∝ M -0.

Peak velocity

And finally, we can express the maximum velocity. First we inject the expression of the flow rate in the Womersley velocity to absorb the time dependence (again using x = i 3/2 α) :

u W om (r) = Q πa 2 × J 0 (xr/a) -J 0 (x) xJ 0 (x) -2J 1 (x) (22)
We know that Q πa 2 is independent of α (table 5), therefore Poiseuille TGWSS is much smaller than Womersley TGWSS. At low masses (M < 1kg), ∂τ ∂t P W ∝ M -0.625 , while at higher masses (M > 10kg), the Womersley stress accounts for the body mass allometry ∂τ ∂t P W ∝ M -0.375 .

u W om max (α) ∝ max r∈[0,a] ℜ J 0 (xr/a) -J 0 (x) xJ 0 (x) -2J 1 (x) = max r∈[0,a] (g wom (α)) (23)
Recalling equivalents of J 0 and J 1 (eq. 18 to 21), we can find equivalents of g wom (α) at low and high α:

• for M << 1, g wom ∼ √ 2
2α ∼ M -0.25 , we observe a divergence:

lim M→0 u W om max = +∞
.

• for M >> 1, we also obtain g wom ∼ 1 α ∼ M -0.25 , and

lim M→+∞ u W om max = 0 .
We plot max r (g wom ), which represents the Womersley peak velocity only, as a function of M on figure 4. We also add the Poiseuille peak velocity. Concerning PW flow, calculations must be done considering the total velocity field: recalling the expression of u P ois as a function of Q and r: u P ois = 2Q πa 2 (1 -r a ), u P W max is:

u P W max (α) ∝ max r∈[0,a] ℜ 2(1 - r a ) + J 0 (xr/a) -J 0 (x) xJ 0 (x) -2J 1 (x) = max r∈[0,a] (g P W (α)) (24)
On figure 4, we can see that the Womersley flow alone gives rise to a peak velocity which vanishes with M (blue curve), while the PW flow (yellow curve) shows a constant peak velocity for reasonably high M . This proves that the constant peak velocity measured experimentally [START_REF] Schmidt-Nielsen | Scaling Why is Animal Size so Important[END_REF] is due to the steady Poiseuille component of the flow, rather than to the Womersley unsteady component. We also observe that the maximum velocity is systematically reached in the center of the artery (r = 0, data not shown), which is not the case for a pure Womersley flow at high Womersley number. This confirms the domination of the Poiseuille flow. Figure 4 further proves the necessity of a combined Poiseuille -Womersley flow to account for the variations of the different hemodynamic parameters.

Conclusion

In this study, we have recalled the derivation of wall shear stress, temporal gradient of wall shear stress and peak velocity, for a Poiseuille and subsequently Womersley modeling of the blood flow in the aorta. We have shown that a Poiseuille modeling alone (and equivalently, a Womersley modeling alone) was not able to account for the variations of the previously mentionned parameters. We showed that a combination of Poiseuille and Womersley flows, here called PW flow, was necessary to conduct an allometry study.

• Concerning wall shear stress, Poiseuille and Womersley flows agree on a M -0.375 at low Womersley numbers, while the Womersley component dominates at higher α (M -0.125 against M -0.375 ). This changes the "Twenty-fold difference in hemodynamic wall shear stress between murine and human aortas" advanced in [START_REF] Weinberg | Twenty-fold difference in hemodynamic wall shear stress between murine and human aortas[END_REF] to a lower 8.2 fold difference.

• About the temporal gradient of wall shear stress, the same phenomenon occurs: while M -0.625 is valid at low α, the unsteady component dominates at higher α (M -0.375 against M -0.625 ), and brings mice to men ratio from 150 : 1 to 61 : 1.

• For peak velocity on the contrary, the independance on α observed experimentally [START_REF] Schmidt-Nielsen | Scaling Why is Animal Size so Important[END_REF] is due to the Poiseuille component rather than to the Womersley one, which follows a vanishing u wom max ∝ M -0.25 .

As a consequence, only considering a Poiseuille flow to infer the wall shear stress allometry in mammals is not a precise method. On the contrary, one should take into account the two components of the blood flow profile (steady Poiseuille and unsteady Womersley), to account for their respective ranges of domination. This work gives a new insight on the allometry of 3 key hemodynamic parameters in atherosclerosis. It can also help extrapolate measurements from small mammals like mice, which are common in cardiovascular laboratories, to bigger ones and to humans. A comprehensive imaging study of aortic velocity profiles on heavy mammals could help support or reject this theory and greatly benefit the field.

Figure 1 :

 1 Figure 1: Graph of log|ℜ(f )| as a function of α, logarithmic scale. We can clearly identify two power laws: ℜ(f ) ∝ α -2 for α < 2, and ℜ(f ) ∝ α -1 for α > 5

Figure 2 :

 2 Figure 2: Compared wall shear stresses for Poiseuille, Womersley and PW flows as a function of body masslogarithmic scale. We observe that Poiseuille stress is globally dominated, in absolute value, by Womersley stress:Poiseuille makes only 12% of total stress for a mouse (M = 0.025 kg), and this percentage goes down to 5% for a human (M = 75 kg). Concerning allometry, both processes induce τ P W ∝ M -0.375 at low masses (M < 1kg), but at higher masses (M > 10kg), the Womersley stress sets the allometry τ P W ∝ M -0.125 .

Figure 3 :

 3 Figure 3: Compared temporal gradient of wall shear stresses for Poiseuille, Womersley and PW flows as a function of body mass -logarithmic scale. Poiseuille TGWSS is much smaller than Womersley TGWSS. At low masses (M < 1kg), ∂τ∂t P W ∝ M -0.625 , while at higher masses (M > 10kg), the Womersley stress accounts for the body mass allometry ∂τ ∂t P W ∝ M -0.375 .

Figure 4 :

 4 Figure 4: Peak velocity of Poiseuille, Womersley and PW flows as a function of mass, logarithmic sclae: the constant peak velocity observed experimentally ([18]) is due to the Poiseuille component rather than the Womersley one which vanishes at high M

Table 1 :

 1 Relative flow characteristics for various species -Poiseuille flow

		Mass Shear stress Gradient of shear stress Peak velocity
	Human 75 kg	1	1	1
	Rabbit 3.5 kg	3.0	6.8	1
	Mouse	25 g	20.2	148.8	1

Table 2 :

 2 25 , and that ωρQ πa ∼ M 0.125 (see table 5 at the end of the document), we obtain:

	Human 75 kg	13	1	1	1
	Rabbit 3.5 kg	6	3.0	1.7	1.8
	Mouse	25 g	1.8	20.2	7.6	8.2

τ womersley ∼ M -0.375 for M < 1 kg and τ womersley ∼ M -0.125 for M > 10 kg which differs notably from τ poiseuille ∼ M -0.375 for all M We can complete the previous chart of relative shear stress for different animals (tab. 2). We also plot the compared shear stress induced by a Poiseuille, Womersley or physiological PW flows as a function Animal Mass α aorta Shear stress (Poiseuille) Shear stress (Womersley) Shear stress (PW) Relative shear stress for various species -Poiseuille, Womersley, and Poiseuille + Womersley (PW) flows of mass on figure

2

. First of all, we can see that despite several allometric approximations (flow rate, heart rate, aortic radius, Womersley number), the absolute value of wall shear stress τ ∈ [0.1; 10] P a is within the range of measured values ([12], ch. , p.502

Table 3 :

 3 375 . Here again, we observe a reduced difference between men and mice due to the influence of unsteady flow at high Womersley numbers. Relative shear stress for various species -Poiseuille vs Womersley flow

	Animal Shear stress gradient (Poiseuille) Shear stress gradient (Womersley) Shear stress gradient (PW)
	Human	1	1	1
	Rabbit	6.8	3.8	3.9
	Mouse	148.8	55.8	60.5

Table 4 :

 4 We summarize the different allometries obtained throughout this work in table 4. Summary of body mass allometry of different hemodynamic parameters

	Flow modeling	Poiseuille	Womersley	PW
	Hemodynamic parameter		M < 1 kg M > 10 kg M < 1 kg M > 10 kg
	WSS	-0.375	-0.375 -0.125	-0.375 -0.125
	TGWSS	-0.625	-0.625 -0.375	-0.625 -0.375
	Peak velocity	0	-0.25	0

Appendix


Øyvind Jacobsen Bjørkås

Kjetil B Halvorsen
email: kjetil.b.halvorsen@nord.no

Dag Oskar Madsen
email: dag.o.madsen@nord.no

Informal statistical inference and permutation tests

Keywords: Mathematics curriculum, statistical inference, generalization

Background

In the Norwegian Curriculum for the Common Core subject of Mathematics, there are competence aims after the second, fourth, seventh and tenth years in primary and lower secondary education. In the subject area of Statistics, there seems to be little progression through the various grades, in our opinion. The relevant competence aim after Year 4 is "collect, sort, note and illustrate data using tally marks, tables and bar graphs, with and without the use of digital tools, and converse about the process and what the illustrations tell us about the data" (Utdanningsdirektoratet, n. d.).

The competence aims after Year 10 read as follows.

carry out investigations and use databases to search for and analyse statistical data and critically assess sources; order and group data, find and discuss and elaborate on the median, mode, average and spread, and present data with and without digital tools, and discuss and elaborate on different ways of presenting data and what impressions these can give. (Utdanningsdirektoratet, n. d.) The competence aims in Statistics after Year 10 and are in textbooks often dealt with in grade 8. In the textbook Faktor for grade 8 (Hjardar & Pedersen, 2014, p. 110), the following problem is marked as difficult. A table that shows the population for each county in Norway is given with the task: "Find the top three counties with the highest population". In terms of difficulty, there is very little progress to see from tasks given based on the competence aims after grade 4. Even if the competence aims ("analyse statistical data") and the related document Mathematics: Characteristics of goal achievement seem to support a more exploratory approach leading for instance towards informal statistical inference, our impression is that the subject area of Statistics is treated superficially and poorly in textbooks.

Theoretical perspectives

There is already an extensive research literature on teaching experiments introducing statistical inference in middle school (see e. g. [START_REF] Watson | Exploring Beginning Inference with Novice Grade 7 Students[END_REF]. One common approach is to use graphical representations of datasets, often with the help of digital tools such as TinkerPlots [START_REF] Konold | TinkerPlots: Dynamic data exploration[END_REF], to make comparisons between samples and informal inferences about differences of populations. We intend to go one step further and introduce to middle school students test procedures that have theoretical justification in permutation methods [START_REF] Ernst | Permutation methods: A basis for exact inference[END_REF]. The focus will be on the significance (perhaps without using this technical word) of difference of means, not just comparing means. Rossmann (2008, p. 17) suggests that "simulation of randomization tests provides an informal and effective way to introduce students to the logic of statistical inference". Zieffler, Garfield, Delmas & Reading (2008, p. 44) define informal inferential reasoning as "the way in which students use their informal statistical knowledge to make arguments to support inferences about unknown populations based on observed samples".

Developing a teaching plan

We want to develop a teaching plan aimed at grade 8 or 9 about statistical inference focusing on experiments with realistic problems, perhaps using a design-based research methodology. As mentioned we want to include permutation tests, but they will be presented to the students an informal way. There should be no instruction about distributions or formal methods, only an appeal to the underlying logic that we hope can be intuitively acceptable for students. This last point should be one focus point of our research. We have tried out one experiment with teacher students about comparing the weight of apples and oranges in samples and then repeatedly and randomly reassign which fruits are "apples" and which are "oranges". How does the real difference in weight compare with the randomized differences? If there is no significant difference in weight between apples and oranges, then the real difference should not be among the most extreme in the simulation. This is an example of the logic behind permutation tests. But is it informal inferential reasoning? This is a question we should have in mind when developing the plan.

The details of the teaching plan are still very much in development. One idea we have is to offer the permutation tests to the students late in a sequence of lectures, where earlier in the sequence more student-driven activities, perhaps involving digital tools and graphical representation of data sets, are planned and where students are encouraged to use their own reasoning. We have done the activity with apples and oranges three times with teacher students and gotten valuable feedback. We want to develop this into a larger project involving real classroom settings.


Martin Andre
email: m.andre@tsn.at

Zsolt Lavicza
email: lavicza@gmail.com

Theodosia Prodromou
email: theodosia.prodromou@une.edu.au

Formalizing students' informal statistical reasoning on real data: Using Gapminder to follow the cycle of inquiry and visual analyses

Keywords: Statistics education, statistical literacy, visual data analysis, statistical reasoning, intuitions

) by using the software Gapminder for data visualization in classrooms. In this exploratory study 19 students aged 14-15 were guided through two lessons of 1 hour and 40 minutes, each, using the Gapminder tool and materials to explore and visualize sets of Open Data while working on several worksheets and test items developed by the research team. Findings of this initial study are being used to further conceptualize and design successive investigations.

Theoretical background

This study is primarily based upon the cycle of inquiry and visual analysis [START_REF] Prodromou | Drawing Inference from Data Visualisations[END_REF], complemented by works on students' statistical literacy [START_REF] Prodromou | Data Visualisation and Statistics Education in the Future[END_REF], their development of the big ideas of statistics [START_REF] Garfield | Developing students' statistical reasoning: Connecting research and teaching practice[END_REF]) and various research studies on the use of technology in the era of Open Data with a special focus on the visualization of context-based data. Acknowledging that graphical representations play a fundamental role in intuitively understanding of abstract mathematical or scientific concepts [START_REF] Fischbein | Intuition in science and mathematics: An educational approach[END_REF], theories on intuition [START_REF] Fischbein | Intuition in science and mathematics: An educational approach[END_REF], heuristics and biases (Tversky & Kahnemann, 1974) are considered. Tversky and Kahnemann (1974), [START_REF] Fischbein | Intuition in science and mathematics: An educational approach[END_REF] stress the influence of connate heuristics, biases and misconceptions on making decisions or judgements under uncertainty (Tversky & Kahnemann, 1974). Studies have examined specific heuristics, biases and misconceptions within the learning processes of statistics and probability theory [START_REF] Garfield | Developing students' statistical reasoning: Connecting research and teaching practice[END_REF]. Thus, it is a central challenge in statistics education, to address students' intuitions in order to identify their biases and misconceptions and support students to overcome their misconceptions. In fact, misconceptions are strongly bound to context because students think differently about stochastic concepts in various contexts. [START_REF] Makar | The reasoning behind informal statistical inference[END_REF], as well as other researchers in Statistics Education, emphasize the importance of context and the use of real data in statistical reasoning. [START_REF] Ben-Zvi | Children's wonder how to wander between data and context[END_REF] demonstrate students' transition between context and data and also report on students' growing understanding of the ways to combine these contexts and data. Although there is little research on the question what makes data "real", we can presume that students perceive data to be real when they get personally affected by the data. Therefore, Open Data on topics of general interest may be considered as real data easily accessible for teachers and students.

Advances of new technology with numerous opportunities for data visualization prompted [START_REF] Prodromou | Data Visualisation and Statistics Education in the Future[END_REF] to argue for the profound importance of using Open Data in statistics education. Analyzing Open Data with the software TinkerPlots in classroom, [START_REF] Watson | Open Data in Australian schools: Taking statistical literacy and the practice of statistics across the curriculum[END_REF] describes the chances of learning statistics in a transdisciplinary way, integrating statistics education in non-mathematical topics, such as to Health and Physical Education, Social Science or History. [START_REF] Garfield | Developing students' statistical reasoning: Connecting research and teaching practice[END_REF] outline various research studies on students' development of different big ideas of statistics, which they identified as crucial for statistical reasoning: a) data, including the nature of data and the diverse types and sources of data b) statistical models, such as regression and the Normal Distribution as statistical models, c) distribution, involving the ideas of shape, centre and spread, d) centre, covering the difference between median and mean, e) variability, including the measuring of the spread of a distribution, f) comparing groups by centre and spread, g) sampling, containing, e.g., the effect of sample size, h) statistical inference as to testing hypotheses or confidence intervals, and i) covariation with scatterplots, correlation and linear regression. The theory-based recommendations given with the presented activities for the development of the big ideas of statistics comprise learning these ideas explicitly, addressed in a constructive way and facilitated by examining real data sets. Furthermore, [START_REF] Garfield | Developing students' statistical reasoning: Connecting research and teaching practice[END_REF] also suggest learning the big ideas of statistics beginning with students' intuitions and preconceptions, taking their informal notions of the single ideas and turning them into formal notions. Teaching must start with introducing students' intuitions and preconceptions, identifying intuitions and preconceptions and rectifying them using different techniques including the use of digital technologies and graphical representations of data to display the summaries and visualizations of data.

Visualizing data is one of the most intuitive ways of understanding underlying concepts from data and "graphical excellence is that which gives to the viewer the greatest number of ideas in the shortest time with the least ink in the smallest space." (Tufte, 1983, p. 51). Moreover, presenting information with diverse graphical representations is a highly important skill for many scientific disciplines aiming to make data meaningful [START_REF] Tufte | The Visual Display of quantitative information[END_REF]. Therefore, new types and the diversity of data in our age also require new ways of structuring, organizing, and visualizing data. Traditional, static visualization techniques such as bar charts, boxplots or distributions for single variables often fail to meet requirements of current data sources and analyses. Thus, processing and visualizing data with new techniques using specific, sophisticated technology plays an important role in students' developing understanding of statistics.

Using TinkerPlots for exploratory data analysis in the classroom, Ben-Zvi and Ben-Arush (2014) focus on the instrumentation process of transforming an artifact (i.e., a software with no meaning for the students) into an instrument (i.e., a tool, that is meaningful and useful to the learners). Based upon their observations, the authors suggest three types of instrumentation processes: a) unsystematic: students playing and experimenting with the software in a non-intentional way, b) systematic: students being focussed on the tool rather than on the task, but applying the software purposefully, and c) expanding: students using the software fluently and focussing on the task. [START_REF] Prodromou | Drawing Inference from Data Visualisations[END_REF] presents a project concluding that the students were able to flexibly use the software Gapminder to build visual structures that highlight information relevant to their analysis task. In addition, various studies concerned with the implementation of Open Data in school education (e.g., in [START_REF] Prodromou | Data Visualization and Statistical Literacy for Open and Big Data[END_REF]Engel et al., 2016) and the technologies used for this purpose (e.g., [START_REF] Forbes | Use of data visualisation in the teaching of statistics: A New Zealand perspective[END_REF] show promising possibilities to beneficially integrate Open Data in statistic courses by operating with visual methods.

To meet the needs of exploring and analysing data particularly by visual methods, [START_REF] Prodromou | Drawing Inference from Data Visualisations[END_REF] presents the cycle of inquiry and visual analysis, especially elaborated for that purpose. The steps of this cycle are: a) identifying the task, b) foraging for data, c) searching for visual structure and implementing visualizations, d) developing insight through interaction with the resulting data visualizations, and e) acting with regards to work further on any step to develop deeper insight or ending the cycle. All these stages are connected to each other and the process of visual analysis includes continuous interactions between the stages. Furthermore, basic statistical literacy is a prior condition for exploring data by visual methods, being deepened with these investigative processes. Therefore, [START_REF] Prodromou | Data Visualisation and Statistics Education in the Future[END_REF] have introduced a framework for constructing statistical literacy in schools, incorporating new possibilities arising in the age of Open Data and strongly bound to the diverse graphical representations of data.

In the teaching of basic statistics courses, students' intuitions and preconceptions have to be addressed and we also need to consider their biases and misconceptions. Working with graphical representations assists students to access their intuitive knowledge [START_REF] Fischbein | Intuition in science and mathematics: An educational approach[END_REF]. According to researchers (e.g., [START_REF] Makar | The reasoning behind informal statistical inference[END_REF], it is crucial to work with context-based data that are meaningful and important for students. Thus, meaningful data-processing needs visualizing real data, includes entire cycles of inquiry, and particularly addresses the big ideas of statistics [START_REF] Garfield | Developing students' statistical reasoning: Connecting research and teaching practice[END_REF]. Furthermore, an updated concept of statistical literacy required for our age of data society [START_REF] Prodromou | Data Visualisation and Statistics Education in the Future[END_REF] should address students' perceptions and assessment of large sets of data. Finally, using technology autonomously and fluently, allowing focus on the task rather than on the software, is a fundamental factor of students' constructing their statistical knowledge and skills (Ben-Zvi & Ben-Arush, 2014). Therefore, the aim of this study is to develop new ways of teaching basic statistics, integrating students' intuitions addressed by the visualizations of Open Data with the software Gapminder. In our exploratory pilot study, we aimed at identifying important patterns and fundamental limitations regarding students' intuitions on statistical concepts, their statistical literacy, and their instrumentation processes to further conceptualize this investigation.

Methodology and implementation

The present pilot study is embedded in a larger study investigating students evolve their intuitions and preconceptions from informal perceptions of statistical concepts to a higher and more formal level by using different visualization tools. The research question concerning the present study is: How do activities related to analyzing real data by visual methods contribute to formalize students' statistical reasoning? The entire study follows the methodology of design-based research [START_REF] Cobb | Design Experiments in Educational Research[END_REF], utilizing an iterative design of the implementation of a learning environment and approach based on analysis of variety of data sources at the subsequent stages. We analyze students' conceptualization of the big ideas of statistics inherent in the applied worksheets and tests while examining data with the software Gapminder. The worksheets were used to get a broad range of tasks and corresponding answers; the test items included both closed and open-ended questions. Results of the worksheets and tests were anonymized during transcription and translation by using pseudonyms. Transcripts were analyzed qualitatively using MAXQDA qualitative analytics software by categorizing the answers in response to various statistical ideas addressed. Although it was planned to visit classes and conduct interviews with students, researchers were not allowed to collect any additional data from students beyond the worksheets and tests. But, interviews with the teacher before and after the lessons offered further information on the circumstances and settings of lessons and insights into students' work.

In total, 19 students in the 8 th grade participated in the study. They had not worked with the software Gapminder previously nor had they received any prior instruction on distributions and variability in their school. However, their pre-knowledge, which could have an influence on solving tasks, includes calculating the mean and graphical representations of functional relationships in non-statistical, linear contents. The participating teacher was introduced by the researcher to the software and the instructional items of the study. Learning activities took place in a computer lab with a PC for each student and had duration of one week with two lessons of 1 hour and 40 minutes. The topic of the lessons was to find out more about poverty in the world.

Using two worksheets, the first lesson aimed at introducing students to the software as well as to ideas of investigating and exploring data sets. In the second lesson, using a third worksheet, students were asked to explore autonomously visualisations of various datasets. Between the different learning sequences, some test items were introduced to find out more about the students' statistical preconceptions and the conceptions they gained. The test questions were partly openended and partly closed, a structure familiar to the students from Austrian national standardized tests. The content of the first worksheet is about eight sets of interactive slides on human development trends, provided by the Gapminder foundation as instructional material (https://www.gapminder.org/downloads/human-development-trends-2005/). These slides show different visualisations (see figure 3) of the variables income and child mortality rate (health). Visualisations explained and described the distribution of income of all countries in the world and the relationship between income and health over time. This worksheet contains questions on statistical ideas that address central value, variety, distribution's shape, and correlation between variables based on the topic. As the tasks were open-ended and students did not get any instruction before, they were required to describe their ideas on the tasks intuitively. Connected to the first worksheet, the first test items were developed to directly address the students' intuitions of centre, distribution, and variability by giving them two diagrams of different income distributions (see figure 1). The students were asked to reason about the shape of the second distribution and to compare it to the first distribution regarding the centres of these distributions. Additionally, students were asked to explain why the median income in 2015 most probably belongs to a person from a middle-income country and to describe the income of Chinese people in 1987 (light green top of the left peak). The second worksheet was a step-by-step guideline to follow the cycle of visual analysis [START_REF] Prodromou | Drawing Inference from Data Visualisations[END_REF] with different variables using the Gapminder software. There was no data collected on this second worksheet. The third worksheet contained some general instructions and suggestions, which variables the students may use for their autonomous investigation of the topic, e.g., income and literacy rate. The students participating were asked to document the findings of their investigations.

A second test, held at the end of the second lesson, included questions on a bubble chart (see figure 2) showing the variables income and sanitation. The focus of the test was on the ideas of correlation, centre and distribution. Students were asked to describe the bubble chart depending on the income levels and got a set of five statements addressing the various statistical concepts (e.g., variety, centre, or spread) to choose the right ones. With exception of the investigational learning sequences, the students were asked to work independently. The teacher supported the students' learning process by assisting them while working with the software and the topic. When the students solved the tests independently, they did not receive any assistance.

Initial results and discussion

In this section we present and discuss some of the main results of the pilot study to pinpoint directions for further investigation. Results presented focus on some of the big ideas of statistics [START_REF] Garfield | Developing students' statistical reasoning: Connecting research and teaching practice[END_REF] connected to the students' intuitive approaches and their uses of technology. In the study, the main statistical model addressed was the model of regression. In several tasks students were asked to intuitively reason about the relationship between different variables (e.g., income and health) regarding various countries or a time series of one or more countries. All students recognized a correlation between income and health, building a model of regression. For example, one student, Richard, stated: "The higher the income in a country is, the better is the health rate of children." (Richard, worksheet 1) Students also interpreted the positive slope of the regression correctly: "That means that the countries constantly developin other words the GDP per person is growingand the percentage of surviving children is heightening." (Eva, worksheet 1) Still, students did not compare countries' development by recognizing and building two or more models of regression in one graph (see figure 3) with one exception: "That means that poor people in India are less healthy than the rich ones whereas in Namibia the poor ones have nearly the same rate of health as the rich ones." (Vincent, worksheet 1). Anyway, most students independently used a model of regression during their autonomous investigations: "The poorer the countries are the worse are the numbers [of other variables] or the more children they have." (Paul, Peter and Anna, worksheet 3).

Besides using regressions, some models of probabilities were detected in the data as well. The students were asked to decide whether the person with the median income most probably comes from a low, middle or high-income country. For example, Vincent marked the height of the income of middle-income countries in the graphic of the distribution and connected the height to its probability. But some misconceptions were also observed with students building a model for this probability while maybe having a linear (regression) model in mind, as exemplified by these remarks: "Poor income country, because 6-7 $ is very little" (Julia, test 1) or "middle income because the number is in the middle" (Ruth, test 1).

These outcomes show that students had an intuitive access to the basic ideas of regression and correlation. The attempts to apply the model of regression to the probability theory may be rooted in the proximity of tasks. Furthermore, we suggest addressing context, change and causality as a part of the new framework for statistical literacy [START_REF] Prodromou | Data Visualisation and Statistics Education in the Future[END_REF] in that connection, as some students intuitively described the relationship between the variables with a focus on causality.

A second statistical idea to be mentioned here is the idea of distribution. With the learning materials on worksheet 1, students were briefly introduced to the idea of statistical distributions. All students recognized a change in the parameters of the income distribution over time and interpreted this change correctly, e.g., "Heavily concerned [with poverty] were South and East Asia (1970). Since then, Africa got poorer" (Anna, worksheet 1). Moreover, some students gave a correct interpretation of the old-fashioned term "3 rd world (in 1987)" regarding the graphic in figure 1. As they did not get any instruction on regular or irregular shapes of distributions, students were not wondering about the different shape of the income distributions of 1987 and 2015.

When asked for the central value of the income distributions in 2015 and 1987, all students gave the right number for the "regularly shaped" distribution of 2015, but around half of the students wrongly identified the higher, more distinctive peak in 1987 (see figure 1) as the centre (in terms of the median) of the distribution, although the poverty line above this peak is marked with "47%". In our data, we have detected two answers of students autonomously differing between median and mean, but most students did not differentiate between these two. In sum, we can conclude that the ideas of distribution and the centre are intuitively accessible, but certainly should be addressed more explicitly to avoid or eliminate the misconceptions. Especially a more detailed focus on the idea of distribution regarding their shape must be explored.

The collected data show that all students adequately applied the idea of variability in different contexts. They intuitively defined and applied the idea of "variability", e.g., in terms of "constant appearing dispensation, spread of something; […] In Namibia there are some people, who earn very little and some who earn very much. Therefore, the spread is as large." (Anna, worksheet 1), "the diverse levels of economy and rate of child mortality in different countries" (Sarah, worksheet 1), or "continuously occurring distribution/spread" (Clemens, worksheet 1). Moreover, all students used the idea of variability in their autonomous investigation to describe different variables. Therefore, we believe that the basic concept of variability is relatively easy to access for students who have adequate pre-conceptions of spread or range.

Relying on the reports of the teacher, the instrumentation processes (Ben-Zvi & Ben-Arush, 2014) of students can be described as unsystematic in the beginning and partly systematic while they are getting familiar with the tool. This does not seem surprising, especially since the students did not spend a long time working with the software. As students were given hints which variables to use in their autonomous investigation, the development of the instrumentation processes from unsystematic to systematic may have been supported. More detailed observation on students' use of the Gapminder tool will be necessary to identify helpful and hindering conditions for their instrumentation processes.

Summary

By addressing students' pre-conceptions and intuitions on various statistical concepts with visual methods, the initial results of the study suggest that some of the basic ideas, such as models of regression, centre or variability, tend to be intuitively accessible for students, whereas especially the idea of distribution seems to demand a more extensive development. In addition, several opportunities to focus on students' statistical literacy, related to explicating inferences within data or context, change and causality [START_REF] Prodromou | Data Visualisation and Statistics Education in the Future[END_REF], were detected. Offering a selection of adequate variables to explore, students' instrumentation process using the Gapminder software was apparently supported. Results of this pilot study will lay the basis to redesign this research project and to generalize findings.

Figure 1 :

 1 Figure 1: Visualizations of income distributions used in the first test

Figure 2 :

 2 Figure 2: Bubble chart of the variables income and access to improved sanitation used in the second test

Figure 3 :

 3 Figure 3: Examples of slides from the Gapminder learning materials


Dani Ben-Zvi
email: dbenzvi@univ.haifa.ac.il

Koeno Gravemeijer

Janet Ainley

Design of statistics learning environments 1

Keywords: Learning environment, learning and teaching of statistics, statistics education

The goal of this article is to draw attention to the need to think about learning environments and their design as a way of considering how sustainable change in the learning and teaching of statistics can be supported. The goal is not to advocate one particular approach to the design of learning environments, but rather to raise awareness of the need to consider this lens in statistics education. We first present the rationale for the importance of a focus on learning environments for statistics education. We consider six design considerations for statistics learning environments. Finally, we discuss implications and emerging directions and goals for further implementation and research.

Introduction

Many of the research studies in the learning and teaching of statistics (e.g., Ben-Zvi, Makar, & Garfield, 2018) suggest innovative approaches that differ from the traditional classroom practices through which most current statistics teachers learned this subject themselves. However, innovation which addresses only one aspect of the pedagogical context, for example introducing technological tools in teaching when assessment practices remain unchanged, is likely to have only limited impact. This article offers starting points of design for deep learning of statistics to develop students' statistical reasoning. To do this, we use a learning environment perspective to provide a dynamic, holistic, integrated and multi-dimensional framework for sustainable educational change in statistics. The goal of this article is to draw attention to the need to think about learning environments and their design as a way of considering how sustainable change in the learning and teaching of statistics can be supported.

A learning environment is a complex and dynamic educational system, composed of multiple factors: key statistical ideas and competencies (content), engaging tasks, real or realistic data sets, technological tools, classroom culture including modes of discourse and argumentation amongst students and between students and teachers, norms and emotional aspects of engagement and assessment methods. Integrating all these factors in order to reform the way statistics is learnt and taught is a challenging endeavor. The focus in this article is on general characteristics of statistics learning environments that need to be examined and integrated in light of the new developments in mathematics, statistics and science education, and more generally in the learning sciences. Our specific objectives are to first present the rationale for the importance of a focus on learning environments for statistics education; we provide a potential framework for considering aspects of statistics learning environments. We then develop six design considerations for statistics learning environments. Finally we discuss implications and emerging directions and goals for further implementation and research.

Learning environments

The research literature in statistics education is filled with success stories, which are of importance to the advancement of the field, but have not had a major impact on the way statistics is taught in all levels of education. We propose that one of the reasons for this is the lack of a systematic, comprehensive and integrated approach to design for educational change (e.g., [START_REF] Cuban | Why is it so hard to get good schools?[END_REF]. We suggest that what is needed is change in a combination of interrelated dimensions (content, pedagogy, space, time, tasks, tools, assessment, classroom culture etc.) that can bring about significant and sustainable reform in the teaching and learning of statistics by providing a coherent framework in which each dimension operates synergistically with others. [START_REF] Moore | New pedagogy and new content: The case of statistics[END_REF] similarly urged a reform of statistics instruction and curriculum based on strong synergies between content, pedagogy and technology. A learning environment perspective can provide such a framework. One of the major goals of statistics education is to educate critical, independent and statistically-literate learners who are able to study topics of their own interest and become involved in data-based decisions. A learning environment perspective can provide fertile affordances to support learners' growth and development in this direction as well as a guiding framework for teachers that can support their professional growth in statistics education.

Design dimensions of statistics learning environments that will be considered and discussed in this article are based on a number of principles arising from recent research. In particular we have drawn on research concerning the importance of prior knowledge and preference for depth over breadth (Bransford et al., 2000), the creation of failure-safe learning communities in which students can learn from their successes and mistakes [START_REF] Kapur | Designing for productive failure[END_REF], the nurture and articulation of learners' diverse expertise, encouragement of reflection and feedback [START_REF] Boud | Promoting reflection in learning: A model[END_REF], formative assessment [START_REF] Kingston | Formative assessment: A meta-analysis and a call for research[END_REF] and enculturation into the statistics discipline [START_REF] Edelson | Making authentic practices accessible to learning: Design challenges and strategies[END_REF].

The use of the metaphor of an environment emphasizes that classrooms are interacting social, cultural, physical, psychological and pedagogical systems rather than a collection of resources, tasks and activities or a list of separate factors that influence learning. Because of the complex nature of learning environments, successful design requires a working model of how components of the design that help frame forms of student participation and responsibility, are collectively constituted and orchestrated [START_REF] Lehrer | Designing to develop disciplinary knowledge: Modeling natural systems[END_REF]. To achieve this kind of balance and orchestration we argue that learning environments must be designed on the basis of learning theories, which can guide the design, help choose between the options and lead to better achievement of the pedagogical goals. In Ben-Zvi, Gravemeijer and Ainley (2018), we describe two theoretical frameworks that have been commonly used to guide the construction of learning environments: social constructivist theory as a background theory on teaching and learning and Realistic Mathematics Education (RME) as domain-specific instruction theory that is specific for mathematics education.

Design dimensions for statistics learning environment

In this section, we identify design dimensions that arise from theoretical and empirical sources. These dimensions are not meant to serve as a prescription for what teachers and designers should do, but rather to provide a wide spectrum of factors or starting points, that need to be considered, aligned and balanced in designing statistics learning environments. The goal of designing effective and positive statistics learning environments occurs in a wide continuum of settings. On this continuum, this article focuses on designed learning environments rather than informal and ambient ways of learning. The goal of these learning environments is for students to develop a deep and meaningful understanding of statistics and the ability to think and reason statistically. We discuss and expand on six dimensions of pedagogical design (Figure 1) proposed by [START_REF] Cobb | Principles of instructional design for supporting the development of students' statistical reasoning[END_REF], highlighting what we see as the important connections between them.

Focus on Developing Central Statistical Ideas Rather Than on Tools and Procedures

There are several key statistical ideas that students are expected to understand at a deep conceptual level (Garfield & Ben-Zvi, 2008). These ideas serve as overarching goals that direct teaching and motivate and guide students' learning. They include data, distribution, center, variability, comparing groups, sampling, modeling, inference and covariation. [START_REF] Garfield | Developing students' statistical reasoning: Connecting research and teaching practice[END_REF], in their Statistical Reasoning Learning Environment (SRLE), advocate a focus on key statistical ideas and the interrelations among them and suggest ways to present these ideas throughout a course, revisiting them in different contexts, illustrating their multiple representations and interrelationships, and helping students recognize how they form the supporting structure of statistical knowledge.

Use Well-Designed Tasks to Support the Development of Statistical Reasoning

An important part of a statistics learning environment is the use of carefully designed tasks, informed by research findings, that promote student learning through collaboration, interaction, discussion and addressing interesting problems [START_REF] Roseth | Collaboration in learning and teaching statistics[END_REF]. It may be argued that such tasks should be part of a well-considered instructional sequence, informed by the aim of developing central statistical ideas, which is underpinned by a local instruction theory. A local instruction theory typically consists of a theory about a potential learning process and theories about the means of supporting that learning process (Gravemeijer & Cobb, 2013). The former offers teachers background information, on the basis of which they may decide, on a daily basis, what learning goals to aim for. While the latter offer them information on how potential tasks, tools, ways of interacting and the classroom culture may support the intended learning process. This information will help teachers in choosing tasks and tools, anticipating the mental activities of the students, orchestrating classroom interaction and evaluating the implied hypothetical learning trajectories.

Use Real or Realistic and Motivating Data Sets

The design of pedagogic tasks in statistics must take account of the data that will be centrally involved. Data are at the heart of statistical work and data should be the focus for statistical learning as well [START_REF] Franklin | The Guidelines for Assessment and Instruction in Statistics Education (GAISE) project: Developing statistics education guidelines for pre K-12 and college courses[END_REF]. Throughout their experience of learning statistics, students need to consider methods of data collection and production and how these methods affect the quality of the data and the types of analyses that are appropriate. One approach can be to look for interesting data sets to motivate students to engage in activities, especially ones that ask them to make conjectures about a data set before analyzing it [START_REF] Ben-Zvi | Children's wonder how to wander between data and context[END_REF]. Another approach would be to start with a question and then discuss what data would be needed to answer it. However the provision of real or 'realistic' data is not always sufficient to engage students in tasks that develop statistical reasoning unless the task poses meaningful challenges and provides opportunities to use statistical ideas in realistic ways.

Consider two kinds of activities using real data which are relatively familiar within statistics education research. The first is exploratory data analysis based on a source of real data. Although data about students like themselves may have intrinsic interest, posing meaningful questions about the data can be challenging for school students [START_REF] Burgess | Investigating the nature of teacher knowledge needed and used in teaching statistics[END_REF]. Open-ended exploration of relationships in the data without a clear goal may not lead them to use statistical ideas in realistic ways. The second is a sampling task, such as repeatedly drawing small samples to estimate the proportion of sweets of a particular color within a bowl. Here the statistical idea of sampling is being used in a realistic way, to answer a specific question, but the task itself is not a meaningful challenge [START_REF] Ainley | Learning to reason from samples: commentary from the perspectives of task design and the emergence of "big data[END_REF]. If you really wanted to know the numbers of sweets of different colors it would be quicker and more reliable to empty the bowl and count them. What these tasks have in common is that, although based on real data, they do not emphasize opportunities for students to appreciate the utility of statistical ideas. As a result they may appear contrived and fail to engage and motivate students.

There is a further tension concerning the role and nature of data in statistics tasks. Students, particularly younger students, need to experience collecting, recording and cleaning their own data in order to develop their understandings of different forms of representation [START_REF] Neumann | Using real-life data when teaching statistics: Student perceptions of this strategy in an introductory statistics course[END_REF]. But data collection is time-consuming, often leaving relatively little time for analysis and discussion and the features of the resulting data sets cannot be predicted. Providing real world data sets, or devising data sets which are not authentic but embody the features that the teacher wants students to experience, will save time but students may find such data sets harder to understand and engage with (Arnold, 2004).

Integrate the Use of Technological Tools That Allow Students to Explore and Analyze Data

The design of tasks and the ways in which students may access and explore data are significantly influenced by the range of technological tools available to support the development of students' understanding and reasoning, such as computers, graphing calculators, statistical software and web applets [START_REF] Biehler | Interrelated learning and working environments for supporting the use of computer tools in introductory classes[END_REF]. Students no longer have to spend time performing tedious calculations, or drawing graphs and can focus instead on the more important task of learning how to choose appropriate analytic methods and how to interpret results. Technological tools are used not only to generate statistics, graph or analyze data, but also to help students visualize concepts and develop an understanding of abstract ideas through simulations. For examples of innovative tools and ways to use them to help develop students' reasoning, see [START_REF] Biehler | Technology for enhancing statistical reasoning at the school level[END_REF].

Establish a Classroom Culture that Fosters Statistical Arguments

The design of tasks, technological and assessment tools has to take into account the expected forms of classroom discourse. In statistics learning environments, the use of activities and technology allows for a form of classroom discourse in which student learn to question each other and respond to such questions, as well as explaining their answers and arguments. [START_REF] Cobb | Principles of instructional design for supporting the development of students' statistical reasoning[END_REF] describe the effective classroom discourse in which statistical arguments explain why the organization of data gives rise to insights about the phenomenon under investigation and students engage in sustained exchanges that focus on significant statistical ideas.

It can be challenging to create a statistics learning environment with classroom discourse that enables students to engage in discussions in which significant statistical issues emerge and where arguments are presented and their meaning is openly negotiated. Creating a classroom climate where students feel safe expressing their views, even if they are tentative is another challenging task and is related to classroom culture, in which the teacher and students have to develop the corresponding classroom social norms and socio-mathematical (or socio-statistical) norms [START_REF] Yackel | Sociomathematical norms, argumentation, and autonomy in mathematics[END_REF]. These norms encompass the obligation for the students to explain and justify their solutions, to try to understand the explanations and reasoning of the other students, to ask for clarification when needed and eventually to challenge the ways of thinking with which they do not agree. The teacher is not expected to give explanations, but to pose tasks and ask questions that may foster students' thinking. Socio-statistical norms would be tailored to what it means to do statistics, for example what a statistical problem is, what a statistical argument is and so forth.

The shift in the classroom culture is related to a potential shift in the role of the students, from problem solvers to statisticians who analyze and represent data to make them easily accessible for decision makers. When adopting the role of a data analyst, or data detective [START_REF] Pfannkuch | An exploration of students' statistical thinking with given data[END_REF], students can start reflecting on the adequacy and clarity of condensed descriptions and representations of data, which may foster the reinvention of more sophisticated representations and concepts.

Use Assessment to Monitor the Development of Students' Statistical Learning and to Evaluate Instructional Plans

Assessment should be aligned to well-designed tasks that focus on central statistical ideas in a discourse-rich classroom. Much of the value of changes in the other design dimensions will be lost if assessment practices are not aligned in this way, since the attention of students and teachers will be shaped by the requirements of assessment. In recent years, many alternative forms of assessment have been used in statistics classes. In addition to quizzes, homework and exams, many teachers use statistical projects as a form of assessment [START_REF] Macgillivray | Teaching statistical thinking through investigative projects[END_REF]. Other forms of alternative assessment are also used to assess students' statistical literacy (e.g., critique a graph in a newspaper) and reasoning (e.g., write a meaningful short essay), or to provide feedback to the teacher (e.g., minute papers).

Assessments need to be aligned with learning goals, focusing on understanding key ideas and not just on skills, procedures and computed answers. This can be done with formative assessments used during a course (e.g., quizzes, small projects, or observing and listening to students in class) as well as with summative evaluations (course grades). Useful and timely feedback is essential for assessments to lead to learning. Types of assessment may be more or less practical in different types of courses. However, it is possible, even in large classes, to implement good assessment practices (Garfield & Ben-Zvi, 2008, pp. 65-89).

Discussion: Contemporary issues and emerging directions

The goal of this article has been to draw attention to the need to think about learning environments and their design in statistics education as a way of considering how sustainable change in the learning and teaching of statistics can be supported. We have provided six dimensions of statistics learning environments. Designing for educational change to support the development of students' statistical reasoning is a challenging task. Using a lever to make a one-dimensional change (e.g., formulate new tasks, use of a new pedagogical strategy) may make a difference that is not necessarily a sustainable change in students' understanding of statistical ideas. This article has argued for a holistic and integrated approach that advocates a learning environment where students are engaged in making and testing conjectures using data, discussing and explaining statistical reasoning, focusing on the important big ideas of statistics, using innovative tools in creative ways to assist their learning and being assessed in appropriate ways.

A key factor in this discussion is that these six dimensions, which are inter-related (see Figure 1), must be aligned and balanced. Issues of alignment are important for accelerating statistics learning both within and outside of schools. The meaning of these design principles being part of integrative whole is that using one of them separately is not enough to make deep and sustainable change in students' learning. The learning environment approach helps to interlink them. For example, the design of motivating tasks is linked to real data collection; these data can be used to build students' statistical understanding taking advantages of the innovative affordances of technological tools; productive classroom discourse is supported by the design of open-ended tasks that support argumentation and by appropriate responses by the teacher; assessment methods need to align with the design of tasks; a provision of a new tool must consider the potential interactions with content and pedagogy [START_REF] Moore | New pedagogy and new content: The case of statistics[END_REF].

Thus we argue that pedagogical and research efforts for change must consider the interactions among these dimensions. There are however other important dimensions of learning environments that were not included in this article. One example is the emotional aspects of engagement and identity to motivate all students to participate and reflect on their experiences [START_REF] Heyd-Metzuyanim | The co-construction of learning difficulties in mathematicsteacher-student interactions and their role in the development of a disabled mathematical identity[END_REF]. In addition, the broader community (school level policy makers, local and national authorities, etc.) plays a significant role in the constitution of the learning environment. For example, tensions may arise between required traditional examinations and alternative assessment methods employed in a learning environment or between national curricula and an emergent and dynamic learning trajectory in the learning environment.

Further research is crucially needed to provide more well-researched holistic examples in different contexts and age levels. Systematic studies are also needed about the effectiveness of statistics learning environments, learning environment design issues, the role of alignment between the various dimensions of statistics learning environments, new possibilities for teaching and learning in innovative designs; and opportunities in cutting edge areas, such as, model-based reasoning, visual representation to teach complex abstract concepts, learning in virtual worlds, net-based collaborative teams and communities and big data.

The difficulty of demonstrating the effectiveness of learning environments, raises profound methodological issues in researching them. The learning environment approach acknowledges a complex system or ecology in which a traditional methodology is not sustainable. Instead, a design research approach [START_REF] Cobb | Design Experiments in Educational Research[END_REF] is needed, where iterative design of the learning environment sensitizes the research team to the key mechanisms for learning within the design. We recommend that more attention be given to methodological aspects of researching the design of learning environments.

First steps in moving towards the learning environment perspective in the statistics education community are for researchers to consider the implications of this approach in their studies and for professional development to support teachers to consider how current curricula and materials align in the context of social, cultural, physical, psychological and pedagogical components of a learning environment. Careful and steady change over a period of time, rather than a push for radical change, may lead to a successful implementation of a learning environment in the statistics education world, both among researchers and teachers.

Figure 1 :

 1 Figure 1: A web of interrelated dimensions of a learning environment

This article is based on a chapter by Ben-Zvi, Gravemeijer and[START_REF] Ben-Zvi | Design of statistics learning environments[END_REF].