Samy Skander Bahoura

UNIFORM RESULT FOR SOLUTIONS OF AN EQUATION WITH BOUNDARY SINGULARITY

Keywords: blow-up, boundary, singularity, a priori estimate, analytic domain, Lipschitz condition

We consider a variational problem with boundary singularity and Dirichlet condition. We give blow-up analysis for sequences of solutions of an equation with exponential nonlinearity. Also we derive a compactness criterion for this problem under some condition.

INTRODUCTION AND MAIN RESULTS

We set ∆ = ∂ 11 + ∂ 22 on open analytic domain Ω of R 2 .

We consider the following equation:

(P ) -∆u = |x| 2β V e u in Ω ⊂ R 2 , u = 0 in ∂Ω.
Here:

β ∈ (0, +∞), 0 ∈ ∂Ω, and, u ∈ W 1,1 0 (Ω), |x| 2β e u ∈ L 1 (Ω) and 0 ≤ V ≤ b.

We can see in [START_REF] Brezis | Semi-linear second-order elliptic equations in L1[END_REF] a nice formulation of this problem (P ) in the sens of the distributions. This Problem arises from geometrical and physical problems, see for example [START_REF] Aubin | Some Nonlinear Problems in Riemannian Geometry[END_REF][START_REF] Bandle | Isoperimetric Inequalities and Applications[END_REF][START_REF] Nagasaki | Asymptotic analysis for two-dimensional elliptic eigenvalue problems with exponentially dominated nonlinearities[END_REF][START_REF] Tarantello | Multiple condensate solutions for the Chern-Simons-Higgs theory[END_REF]. The above equation was studied by many authors, with or without the boundary condition, also for Riemannian surfaces, see , where one can find some existence and compactness results. In [START_REF] Brezis | Uniform estimates and Blow-up behavior for solutions of -∆u = V (x)e u in two dimension[END_REF] we have the following important Theorem (β = 0):

Theorem A(Brezis-Merle [START_REF] Brezis | Uniform estimates and Blow-up behavior for solutions of -∆u = V (x)e u in two dimension[END_REF]).For (u i ) i and (V i ) i two sequences of functions relative to (P ) with,

0 < a ≤ V i ≤ b < +∞ then for all compact set K of Ω it holds, sup K u i ≤ c, 1
with c depending on a, b, K and Ω.

One can find in [START_REF] Brezis | Uniform estimates and Blow-up behavior for solutions of -∆u = V (x)e u in two dimension[END_REF] an interior estimate if we assume a = 0, but we need an assumption on the integral of e u i , namely, we have (β = 0): Theorem B(Brezis-Merle [START_REF] Brezis | Uniform estimates and Blow-up behavior for solutions of -∆u = V (x)e u in two dimension[END_REF]).For (u i ) i and (V i ) i two sequences of functions relative to the problem (P ) with, 0 ≤ V i ≤ b < +∞ and Ω e u i dy ≤ C, then for all compact set K of Ω it holds; sup

K u i ≤ c,
with c depending on b, C, K and Ω.

We look to the uniform boundedness in all Ω of the solutions of the Problem (P ). When a = 0, the boundedness of Ω e u i is a necessary condition in the problem (P ) as showed in [START_REF] Brezis | Uniform estimates and Blow-up behavior for solutions of -∆u = V (x)e u in two dimension[END_REF] by the following counterexample (β = 0):

Theorem C(Brezis-Merle [START_REF] Brezis | Uniform estimates and Blow-up behavior for solutions of -∆u = V (x)e u in two dimension[END_REF]).There are two sequences (u i ) i and (V i ) i of the problem (P ) with,

0 ≤ V i ≤ b < +∞ and Ω e u i dy ≤ C, such that, sup Ω u i → +∞.
To obtain the two first previous results (Theorems A and B) Brezis and Merle used an inequality (Theorem 1 of [START_REF] Brezis | Uniform estimates and Blow-up behavior for solutions of -∆u = V (x)e u in two dimension[END_REF]) obtained by an approximation argument and used Fatou's lemma and applied the maximum principle in W 1,1 0 (Ω) which arises from Kato's inequality. Also this weak form of the maximum principle is used to prove the local uniform boundedness result by comparing a certain function and the Newtonian potential. We refer to [START_REF] Brezis | Nonlinear elliptic equations with measures revisited[END_REF] for a topic about the weak form of the maximum principle.

When β = 0, the above equation has many properties in the constant and the Lipschitzian cases:

Note that for the problem (P ) (β = 0), by using the Pohozaev identity, we can prove that Ω e u i is uniformly bounded when 0 < a ≤ V i ≤ b < +∞ and ||∇V i || L ∞ ≤ A and Ω starshaped, when a = 0 and ∇ log V i is uniformly bounded, we can bound uniformly Ω V i e u i . In [START_REF] Ma | Convergence for a Liouville equation[END_REF], Ma-Wei have proved that those results stay true for all open sets not necessarily starshaped in the case a > 0.

In [START_REF] Chen | A priori estimates for solutions to nonlinear elliptic equations[END_REF] (β = 0) Chen-Li have proved that if a = 0, ∇ log V i is uniformly bounded and u i is locally uniformly bounded in L 1 , then the functions are uniformly bounded near the boundary.

In [START_REF] Chen | A priori estimates for solutions to nonlinear elliptic equations[END_REF] (β = 0) Chen-Li have proved that if a = 0 and Ω e u i is uniformly bounded and ∇ log V i is uniformly bounded, then we have the compactness result directly. Ma-Wei in [START_REF] Ma | Convergence for a Liouville equation[END_REF], extend this result in the case where a > 0.

If we assume V more regular, we can have another type of estimates, a sup + inf type inequalities. It was proved by Shafrir see [START_REF] Shafrir | A sup+inf inequality for the equation -∆u = V e u[END_REF], that, if (u i ) i , (V i ) i are two sequences of functions solutions of the previous equation without assumption on the boundary and, 0 < a ≤ V i ≤ b < +∞, then we have the following interior estimate:

C a b sup K u i + inf Ω u i ≤ c = c(a, b, K, Ω).
One can see in [START_REF] Chen | A sharp sup+inf inequality for a nonlinear elliptic equation in R 2[END_REF] an explicit value of C a b = a b . In his proof, Shafrir has used a blow-up function, the Stokes formula and an isoperimetric inequality, see [START_REF] Bandle | Isoperimetric Inequalities and Applications[END_REF]. For Chen-Lin, they have used the blow-up analysis combined with some geometric type inequality for the integral curvature.

Now, if we suppose (V i ) i uniformly Lipschitzian with A the Lipschitz constant, then, C(a/b) = 1 and c = c(a, b, A, K, Ω), see Brezis-Li-Shafrir [START_REF] Brezis | A sup+inf inequality for some nonlinear elliptic equations involving exponential nonlinearities[END_REF]. This result was extended for Hölderian sequences (V i ) i by Chen-Lin, see [START_REF] Chen | A sharp sup+inf inequality for a nonlinear elliptic equation in R 2[END_REF]. Also, one can see in [START_REF] Yy | Harnack Type Inequality: the method of moving planes[END_REF], an extension of the Brezis-Li-Shafrir result to compact Riemannian surfaces without boundary. One can see in [START_REF] Yy | Blow-up analysis for solutions of -∆u = V e u in dimension two[END_REF] explicit form, (8πm, m ∈ N * exactly), for the numbers in front of the Dirac masses when the solutions blow-up. Here, the notion of isolated blow-up point is used.

In [START_REF] Chang | Scalar curvature equation on 2and 3-spheres[END_REF] we have some a priori estimates on the 2 and 3-spheres S 2 , S 3 .

Here we give the behavior of the blow-up points on the boundary and a proof of a Brezis-Merle's type Problem when β ≥ 0.

We have the following similar problem to Brezis and Merle problem:

Problem. Suppose that V i → V in C 0 ( Ω), with, 0 ≤ V i . Also, we consider a sequence of solutions (u i ) of (P ) relatively to (V i ) such that, Ω |x| 2β e u i dx ≤ C,
is it possible to have:

||u i || L ∞ ≤ C?
Here, we give a caracterization of the behavior of the blow-up points on the boundary and also a proof of the compactness theorem when V i are uniformly Lipschitzian. For the behavior of the blow-up points on the boundary, the following condition is enough,

0 ≤ V i ≤ b, The condition V i → V in C 0 ( Ω) is not necessary.
But for the proof of the compactness result (for the Brezis-Merle type problem) we assume that:

||∇V i || L ∞ ≤ A.
We have the following caracterization of the behavior of the blow-up points on the boundary. Theorem 1.1. Assume that max Ω u i → +∞, where (u i ) are solutions of the probleme (P ) with:

β ∈ (0, +∞), 0 ≤ V i ≤ b, and Ω |x| 2β e u i dx ≤ C, ∀ i,
then; after passing to a subsequence, there is a finction u, there is a number N ∈ N and N points x 1 , x 2 , . . . , x N ∈ ∂Ω, such that,

∂ ν u i → ∂ ν u + N j=1
α j δ x j , α j ≥ 4π weakly in the sens of measures.

and,

u i → u in C 1 loc ( Ω -{x 1 , . . . , x N }).
In the following theorem, we have a proof of a global a priori estimate which concern the problem (P ).

Theorem 1.2. Assume that (u i ) are solutions of (P ) relative to (V i ) with the following conditions:

β ∈ (0, +∞), 0 ∈ ∂Ω,
and,

0 ≤ V i ≤ b, ||∇V i || L ∞ ≤ A,
and

Ω |x| 2β e u i ≤ C, We have, ||u i || L ∞ ≤ c(b, β, A, C, Ω),

PROOF OF THE THEOREMS

Proof of theorem 1.1:

We have,

u i ∈ W 1,1 0 (Ω),
Since Ω |x| 2β e u i ≤ C, we have, by the Brezis-Merle result see [START_REF] Brezis | Uniform estimates and Blow-up behavior for solutions of -∆u = V (x)e u in two dimension[END_REF], e ku i ∈ L 1 (Ω) for k > 2 and the elliptic estimates imply that

u i ∈ W 2,k (Ω) ∩ C 1,ǫ ( Ω).
We denote by ∂ ν u i the inner normal derivative of u i . By the maximum principle, ∂ ν u i ≥ 0.

By the Stokes formula, we obtain

∂Ω ∂ ν u i dσ ≤ C.
Thus, (using the weak convergence in the space of Radon measures), we have the existence of a nonegative Radon measure µ such that

∂Ω ∂ ν u i dσ ≤ C,
Without loss of generality, we can assume that ∂ ν u i > 0. Thus, (using the weak convergence in the space of Radon measures), we have the existence of a positive Radon measure µ such that,

∂Ω ∂ ν u i ϕdσ → µ(ϕ), ∀ ϕ ∈ C 0 (∂Ω).
We take an x 0 ∈ ∂Ω such that, µ(x 0 ) < 4π. Without loss of generality, we can assume that the following curve, B(x 0 , ǫ) ∩ ∂Ω := I ǫ is an interval.

We take a test function:

           η ǫ ≡ 1, on I ǫ , 0 < ǫ < δ/2, η ǫ ≡ 0, outside I 2ǫ , 0 ≤ η ǫ ≤ 1, ||∇η ǫ || L ∞ (I 2ǫ ) ≤ C 0 (Ω, x 0 ) ǫ .
We take a ηǫ such that,

-∆η ǫ = 0 in Ω ηǫ = η ǫ on ∂Ω.
Remark: We use the following steps in the construction of ηǫ :

We take a cutoff function η 0 in B(0, 2) or B(x 0 , 2):

1-We set η ǫ (x) = η 0 (|x -x 0 |/ǫ) in the case of the unit disk it is sufficient.
2-Or, in the general case: we use a chart (f, Ω) with f (0) = x 0 and we take µ ǫ (x) = η 0 (f (|x|/ǫ)) to have connected sets I ǫ and we take η ǫ (y

) = µ ǫ (f -1 (y)). Because f, f -1 are Lipschitz, |f (x) -x 0 | ≤ k 2 |x| ≤ 1 for |x| ≤ 1/k 2 and |f (x) -x 0 | ≥ k 1 |x| ≥ 2 for |x| ≥ 2/k 1 > 1/k 2 , the support of η is in I (2/k 1 )ǫ .            η ǫ ≡ 1, on f (I (1/k 2 )ǫ ), 0 < ǫ < δ/2, η ǫ ≡ 0, outside f (I (2/k 1 )ǫ ), 0 ≤ η ǫ ≤ 1, ||∇η ǫ || L ∞ (I (2/k 1 )ǫ ) ≤ C 0 (Ω, x 0 ) ǫ .
3-Also, we can take: µ ǫ (x) = η 0 (x/ǫ) and η ǫ (y) = µ ǫ (f -1 (y)), we extend it by 0 outside f (B 1 (0)). We have

f (B 1 (0)) = D 1 (x 0 ), f (B ǫ (0)) = D ǫ (x 0 ) and f (B + ǫ ) = D + ǫ (x 0 ) with f and f -1 smooth diffeomor- phism.            η ǫ ≡ 1, on a the connected set J ǫ = f (I ǫ ), 0 < ǫ < δ/2, η ǫ ≡ 0, outside J ′ ǫ = f (I 2ǫ ), 0 ≤ η ǫ ≤ 1, ||∇η ǫ || L ∞ (J ′ ǫ ) ≤ C 0 (Ω, x 0 ) ǫ .
And,

H 1 (J ′ ǫ ) ≤ C 1 H 1 (I 2ǫ ) = C 1 4ǫ, since f is Lipschitz.
Here H 1 is the Hausdorff measure.

We solve the Dirichlet Problem:

∆η ǫ = ∆η ǫ in Ω ⊂ R 2 , ηǫ = 0 in ∂Ω.
and finaly we set ηǫ = -η ǫ + η ǫ . Also, by the maximum principle and the elliptic estimates we have :

||∇η ǫ || L ∞ ≤ C(||η ǫ || L ∞ + ||∇η ǫ || L ∞ + ||∆η ǫ || L ∞ ) ≤ C 1 ǫ 2 ,
with C 1 depends on Ω.

We use the following estimate, see [START_REF] Boccardo | Nonlinear elliptic and parabolic equations involving measure data[END_REF][START_REF] Brezis | Semi-linear second-order elliptic equations in L1[END_REF][START_REF] Ding | The differential equation ∆u = 8π -8πhe u on a compact Riemann surface[END_REF][START_REF] Tarantello | Multiple condensate solutions for the Chern-Simons-Higgs theory[END_REF] ||∇u i || L q ≤ C q , ∀ i and 1 < q < 2.

We deduce from the last estimate that, (u i ) converge weakly in W 1,q 0 (Ω), almost everywhere to a function u ≥ 0 and Ω |x| 2β e u < +∞ (by Fatou lemma). Also, V i weakly converge to a nonnegative function V in L ∞ . The function u is in W 1,q 0 (Ω) solution of :

-∆u = |x| 2β V e u ∈ L 1 (Ω) in Ω u = 0 on ∂Ω,
As in the corollary 1 of Brezis-Merle result, see [START_REF] Brezis | Uniform estimates and Blow-up behavior for solutions of -∆u = V (x)e u in two dimension[END_REF], we have e ku ∈ L 1 (Ω), k > 1. By the elliptic estimates, we have u ∈ C 1 ( Ω).

We can write,

-∆((u i -u)η ǫ ) = |x| 2β (V i e u i -V e u )η ǫ + 2 < ∇(u i -u)|∇η ǫ > . (1)
We use the interior esimate of Brezis-Merle, see [START_REF] Brezis | Uniform estimates and Blow-up behavior for solutions of -∆u = V (x)e u in two dimension[END_REF],

Step 1: Estimate of the integral of the first term of the right hand side of (1).

We use the Green formula between ηǫ and u, we obtain,

Ω |x| 2β V e u ηǫ dx = ∂Ω ∂ ν uη ǫ ≤ 4ǫ||∂ ν u|| L ∞ = Cǫ (2)
We have,

-∆u i = |x| 2β V i e u i in Ω u i = 0 on ∂Ω,
We use the Green formula between u i and ηǫ to have:

Ω |x| 2β V i e u i ηǫ dx = ∂Ω ∂ ν u i η ǫ dσ → µ(η ǫ ) ≤ µ(I 2ǫ ) ≤ 4π -ǫ 0 , ǫ 0 > 0 (3)
From ( 2) and (3) we have for all ǫ > 0 there is i 0 = i 0 (ǫ) such that, for i ≥ i 0 ,

Ω |x| 2β |(V i e u i -V e u )η ǫ |dx ≤ 4π -ǫ 0 + Cǫ (4)
Step 2: Estimate of integral of the second term of the right hand side of (1).

Let Σ ǫ = {x ∈ Ω, d(x, ∂Ω) = ǫ 3 } and Ω ǫ 3 = {x ∈ Ω, d(x, ∂Ω) ≥ ǫ 3 }, ǫ > 0.
Then, for ǫ small enough, Σ ǫ is hypersurface.

The measure of Ω -Ω ǫ 3 is k 2 ǫ 3 ≤ µ L (Ω -Ω ǫ 3 ) ≤ k 1 ǫ 3 .
Remark: for the unit ball B(0, 1), our new manifold is B(0, 1 -ǫ 3 ).

(Proof of this fact; let's consider d(x, ∂Ω) = d(x, z 0 ), z 0 ∈ ∂Ω, this imply that (d(x, z 0 )) 2 ≤ (d(x, z)) 2 for all z ∈ ∂Ω which it is equivalent to (z -z 0 ) • (2x -z -z 0 ) ≤ 0 for all z ∈ ∂Ω, let's consider a chart around z 0 and γ(t) a curve in ∂Ω, we have;

(γ(t) -γ(t 0 ) • (2x -γ(t) -γ(t 0 )) ≤ 0 if
we divide by (t -t 0 ) (with the sign and tend t to t 0 ), we have γ ′ (t 0 ) • (x -γ(t 0 )) = 0, this imply that x = z 0 -sν 0 where ν 0 is the outward normal of ∂Ω at z 0 )) With this fact, we can say that S = {x, d(x, ∂Ω)

≤ ǫ} = {x = z 0 -sν z 0 , z 0 ∈ ∂Ω, -ǫ ≤ s ≤ ǫ}. It is sufficient to work on ∂Ω. Let's consider a charts (z, D = B(z, 4ǫ z ), γ z ) with z ∈ ∂Ω such that ∪ z B(z, ǫ z )
is cover of ∂Ω . One can extract a finite cover (B(z k , ǫ k )), k = 1, ..., m, by the area formula the measure of S ∩ B(z k , ǫ k ) is less than a kǫ (a ǫ-rectangle). For the reverse inequality, it is sufficient to consider one chart around one point of the boundary).

We write,

Ω | < ∇(u i -u)|∇η ǫ > |dx = Ω ǫ 3 | < ∇(u i -u)|∇η ǫ > |dx+ + Ω-Ω ǫ 3 < ∇(u i -u)|∇η ǫ > |dx.
(5)

Step 2.1:

Estimate of Ω-Ω ǫ 3 | < ∇(u i -u)|∇η ǫ > |dx.
First, we know from the elliptic estimates that

||∇η ǫ || L ∞ ≤ C 1 /ǫ 2 , C 1 depends on Ω.
We know that (|∇u i |) i is bounded in L q , 1 < q < 2, we can extract from this sequence a subsequence which converge weakly to h ∈ L q . But, we know that we have locally the uniform convergence to |∇u| (by Brezis-Merle theorem), then, h = |∇u| a.e. Let q ′ be the conjugate of q.

We have, ∀f ∈ L q ′ (Ω)

Ω |∇u i |f dx → Ω |∇u|f dx
If we take f = 1 Ω-Ω ǫ 3 , we have:

for ǫ > 0 ∃ i 1 = i 1 (ǫ) ∈ N, i ≥ i 1 , Ω-Ω ǫ 3 |∇u i | ≤ Ω-Ω ǫ 3 |∇u| + ǫ 3 . Then, for i ≥ i 1 (ǫ), Ω-Ω ǫ 3 |∇u i | ≤ mes(Ω -Ω ǫ 3 )||∇u|| L ∞ + ǫ 3 = ǫ 3 (k 1 ||∇u|| L ∞ + 1).
Thus, we obtain,

Ω-Ω ǫ 3 | < ∇(u i -u)|∇η ǫ > |dx ≤ ǫC 1 (2k 1 ||∇u|| L ∞ + 1) (6)
The constant C 1 does not depend on ǫ but on Ω.

Step 2.2:

Estimate of Ω ǫ 3 | < ∇(u i -u)|∇η ǫ > |dx.
We know that, Ω ǫ ⊂⊂ Ω, and ( because of Brezis-Merle's interior estimates)

u i → u in C 1 (Ω ǫ 3 ). We have, ||∇(u i -u)|| L ∞ (Ω ǫ 3 ) ≤ ǫ 3 , for i ≥ i 3 = i 3 (ǫ).
We write,

Ω ǫ 3 | < ∇(u i -u)|∇η ǫ > |dx ≤ ||∇(u i -u)|| L ∞ (Ω ǫ 3 ) ||∇η ǫ || L ∞ ≤ C 1 ǫ for i ≥ i 3 , For ǫ > 0, we have for i ∈ N, i ≥ max{i 1 , i 2 , i 3 }, Ω | < ∇(u i -u)|∇η ǫ > |dx ≤ ǫC 1 (2k 1 ||∇u|| L ∞ + 2) (7)
From ( 4) and (7), we have, for ǫ > 0, there is i

3 = i 3 (ǫ) ∈ N, i 3 = max{i 0 , i 1 , i 2 } such that, Ω |∆[(u i -u)η ǫ ]|dx ≤ 4π -ǫ 0 + ǫ2C 1 (2k 1 ||∇u|| L ∞ + 2 + C) (8)
We choose ǫ > 0 small enough to have a good estimate of (1).

Indeed, we have:

-∆[(u i -u)η ǫ ] = g i,ǫ in Ω, (u i -u)η ǫ = 0 on ∂Ω. with ||g i,ǫ || L 1 (Ω) ≤ 4π -ǫ 0 .
We can use Theorem 1 of [START_REF] Brezis | Uniform estimates and Blow-up behavior for solutions of -∆u = V (x)e u in two dimension[END_REF] to conclude that there is q > 1 such that:

Vǫ(x 0 ) e q|u i -u| dx ≤ Ω e q|u i -u|ηǫ dx ≤ C(ǫ, Ω).
where, V ǫ (x 0 ) is a neighberhood of x 0 in Ω.

Thus, for each x 0 ∈ ∂Ω -{x 1 , . . . , xm } there is ǫ x 0 > 0, q 1 > 1 such that:

B(x 0 ,ǫx 0 ) e q 1 u i dx ≤ C, ∀ i. Now, we consider a cutoff function η ∈ C ∞ (R 2 ) such that η ≡ 1 on B(x 0 , ǫ x 0 /2) and η ≡ 0 on R 2 -B(x 0 , 2ǫ x 0 /3).
We write

-∆(u i η) = |x| 2β V i e u i η -2∇u i • ∇η -u i ∆η.
By the elliptic estimates (see [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second order[END_REF]) (u i ) i is uniformly bounded in W 2,q 1 (V ǫ (x 0 )) and also, in C 1 (V ǫ (x 0 )). Finaly, we have, for some ǫ > 0 small enough,

||u i || C 1,θ [B(x 0 ,ǫ)] ≤ c 3 ∀ i.

Proof of theorem 1.2:

Without loss of generality, we can assume that 0 is a blow-up point. Since the boundary is an analytic curve γ(t), there is a neighborhood of 0 such that the curve γ can be extend to a holomorphic map such that γ ′ (0) = 0 (series) and by the inverse mapping one can assume that this map is univalent around 0. In the case when the boundary is a simple Jordan curve the domain is simply connected, see [START_REF] Stoker | Differential Geometry[END_REF]. In the case that the domains has a finite number of holes it is conformally equivalent to a disk with a finite number of disks removed, see [START_REF] Krantz | Geometric functions theory[END_REF]. Here we consider a general domain. Without loss of generality one can assume that γ(B + 1 ) ⊂ Ω and also γ(B - 1 ) ⊂ ( Ω) c and γ(-1, 1) ⊂ ∂Ω and γ is univalent. This means that (B 1 , γ) is a local chart around 0 for Ω and γ univalent. (This fact holds if we assume that we have an analytic domain, in the sense of Hofmann see [START_REF] Hofmann | Geometric and transformational properties of Lipschitz domains, Semmes-Kenig-Toro domains, and other classes of finite perimeter domains[END_REF], (below a graph of an analytic function), we have necessary the condition ∂ Ω = ∂Ω and the graph is analytic, in this case γ(t) = (t, ϕ(t)) with ϕ real analytic and an example of this fact is the unit disk around the point (0, 1) for example).

By this conformal transformation, we can assume that Ω = B + 1 , the half ball, and ∂ + B + 1 is the exterior part, a part which not contain 0 and on which u i converge in the C 1 norm to u. Let us consider B + ǫ , the half ball with radius ǫ > 0. Also, one can consider a C 1 domain (a rectangle between two half disks) and by charts its image is a C 1 domain). We know that:

u i ∈ C 2,ǫ ( Ω).
Thus we can use integrations by parts (Gauss-Green-Riemann-Stokes formula). The second Pohozaev identity applied around the blow-up 0 see for example [START_REF] Fusco | Functions of Bounded variations and Free discontinuity Problems[END_REF][START_REF] Ma | Convergence for a Liouville equation[END_REF][START_REF] Necas | Direct Methods in the Theory of Elliptic Equations[END_REF] gives :

2(1 + β) B + ǫ |x| 2β V i e u i dx + B + ǫ < x|∇V i > |x| 2β V i e u i dx + ∂B + ǫ < ν|x > |x| 2β V i e u i dσ = = ∂ + B + ǫ g(∇u i )dσ, (9) with,
g(∇u i ) =< ν|∇u i >< x|∇u i > -< x|ν > |∇u i | 2 2 .
also,

2(1 + β) B + ǫ |x| 2β V e u dx + B + ǫ < x|∇V > |x| 2β V e u dx + ∂B + ǫ < ν|x > |x| 2β V e u dσ = = ∂ + B + ǫ g(∇u)dσ, (10)
Thus,

2(1 + β) B + ǫ |x| 2β V i e u i dx -2(1 + β) B + ǫ |x| 2β V e u dx+ + B + ǫ < x|∇V i > |x| 2β V i e u i dx - B + ǫ < x|∇V > |x| 2β V e u dx + o(1) = = ∂ + B + ǫ g(∇u i ) -g(∇u)dσ = o(1),
First, we tend i to infinity after ǫ to 0, we obtain:

lim ǫ→0 lim i→+∞ 2(1 + β) B + ǫ |x| 2β V i e u i dx = 0, (11) however γ
(B + ǫ ) |x| 2β V i e u i dx = ∂γ(B + ǫ ) ∂ ν u i dσ = α 1 + O(ǫ) + o(1) > 0. ( 12
)
which is a contradiction.

Here we used a theorem of Hofmann see [START_REF] Hofmann | Geometric and transformational properties of Lipschitz domains, Semmes-Kenig-Toro domains, and other classes of finite perimeter domains[END_REF], which gives the fact that γ(B + ǫ ) is a Lipschitz domain. Also, we can see that γ((-ǫ, ǫ)) and γ(∂ + B + ǫ ) are submanifolds.

We start with a Lipschitz domain B + ǫ because it is convex and by the univalent and conformal map γ the image of this domain γ(B + ǫ ) is a Lipschitz domain and thus we can apply the integration by part and here we know the explicit formula of the unit outward normal it is the usual unit outward normal (normal to the tangent space of the boundary which we know explicitly because we have two submanifolds).

In the case of the disk D = Ω, it is sufficient to consider B(0, ǫ) ∩ D which is a Lipschitz domain because it is convex (and not necessarily γ(B + ǫ )).

There is a version of the integration by part which is the Green-Riemann formula in dimension 2 on a domain Ω. This formula holds if we assume that there is a finite number of points y 1 , ..., y m such that ∂Ω -(y 1 , ..., y m ) is a C 1 manifold and for C 1 tests functions, see [START_REF] Fusco | Functions of Bounded variations and Free discontinuity Problems[END_REF], for the Gauss-Green-Riemann-Stokes formula, for C 1 domains with singular points (here a finite number of singular points).

Remark 1: Note that a monograph of Droniou contain a proof of all fact about Sobolev spaces (with Strong Lipschitz property) with only weak Lipschitz property (Lipschitz-Charts), we start with Strong Lipschitz property and by γ we have weak Lipschtz property.

Remarks about the conformal map :

1-It sufficient to prove that γ 1 ((-ǫ, ǫ)) = ∂Ω∩ γ1 (B ǫ ) = ∂Ω∩ γ1 (B ǫ )∩{|abscissa| < ǫ}, for ǫ > 0 small enough. Where γ1 is the holomorphic extension of γ 1 (t) = t + iϕ(t). For this, we argue by contradiction, we have for z ǫ ∈ B ǫ , γ1 (z ǫ ) = (t ǫ , ϕ(t ǫ )) for |t ǫ | ≥ ǫ. Because γ1 is injective on B 1 and γ1 = γ 1 = t+iϕ(t) on the real axis, we have necessirely |t ǫ | ≥ 1. But, by continuity |γ 1 (z ǫ )| → 0 because z ǫ → 0. And, we use the fact that |γ 1 (z ǫ )| = |(t ǫ , ϕ(t ǫ ))| ≥ |t ǫ | ≥ 1, to have a contradiction.) (This means that for a small radius when the graph go out from the ball, it never retruns to the ball). (This fact imply that, when we have a curve which cut ∂Ω in γ1 (B ǫ ) then the point have an abscissa such that |abscissa| < ǫ. This fact (by a contradiction with the fact ∂ Ω = ∂Ω and consider paths), imply that the image of the upper part of the ball is one side of the curve and the image of the lower part is in the other side of the curve.

2-Also, we can consider directly the coordinate T and change the function u

(z) → u(T ) by z = T /a 1 + x 0 . Set: ψ(λ 1 , λ 2 ) → M ∈ Ω such that, --→ x 0 M = λ 1 i ′ 1 + λ 2 j ′ 1 where (i ′ 1 , j ′ 1 ) is a basis such that i ′ 1 = e -iθ i 1 , j ′ 1 = e -iθ j 1 . And, ϕ(x 1 , x 2 ) → M such that, --→ OM = x 1 i 1 + x 2 i 2 the canonical basis (i 1 , j 1 )
. Then, we have two charts ϕ and ψ and the complex affix T M = λ 1 + iλ 2 and z M = x 1 + ix 2 are such that (transition map):

T M /e iθ + x 0 = z M = ϕ -1 oψ(λ 1 , λ 2 ),
We have:

∂ λ 1 = cos θ∂ x 1 + sin θ∂ x 2 , ∂ λ 2 = -sin θ∂ x 1 + cos θ∂ x 2 ,
Thus, the metric in the chart ψ or coordinates (λ 1 , λ 2 ) is : g λ ij = δ ij and the Laplacian in the two charts, ψ and ϕ are the usual Laplacian ∂ λ 1 λ 1 + ∂ λ 2 λ 2 .

We write:

∆u(M ) = ∆ λ (uoψ(λ 1 , λ 2 ))
And then we apply the conformal map γ1 which send the affix T M , M in a neighborhood of x 0 ∈ ∂Ω to B ǫ with the fact that send T M , M ∈ ∂Ω to the real axis (-ǫ, ǫ) and the other parts of Ω and Ωc .

We have: ψ : (λ 1 , λ 2 ) → M and γ1 : (λ 1 , λ 2 ) → (µ 1 , µ 2 ). Then the map considerded is: ψoγ -1

1 . It is a chart. Indeed the first chart is: ψog -1 where g is g : (λ 1 , λ 2 ) → [λ 1 , λ 2 + ϕ 0 (λ 1 )] with ϕ 0 the "graph map" as in the defiinition of ∂Ω.

Then the transition map is:

goγ -1 1
is smooth from a neighborhood of 0 to a neighborhood of 0 and send the upper part of a ball (lower part) to an upper part of a domain (lower part). And then ψoγ -1

1 is a chart with the property that γ1 is conformal.

Remark that: ψ and ϕ are charts for Ω and ψ is almost chart for ∂Ω. But ψoγ -1 1 and ψog -1 are the "real" charts of the boundary.

3-We can remark that a definition of C k , k ≥ 1 domain, is equivalent to a definition of a submanifold with the condition ∂ Ω = ∂Ω or Ω = Ω.

Remark 2: about a variational formulation.

we consider a solutions in the sense of distribution. By the same argument (in the proof of the maximum principle obtained by Kato's inequality W 1,1 0 is sufficient), see [START_REF] Brezis | Nonlinear elliptic equations with measures revisited[END_REF], we prove that the solutions are in the sense C 2 0 of Agmon, see [START_REF] Aubin | Some Nonlinear Problems in Riemannian Geometry[END_REF]. Also, we have corollary 1 of [START_REF] Brezis | Uniform estimates and Blow-up behavior for solutions of -∆u = V (x)e u in two dimension[END_REF]. We use Agmon's regularity theorem. We return to the usual variational formulation in W 1,2 0 and thus we have the estimate of [START_REF] Brezis | Semi-linear second-order elliptic equations in L1[END_REF] or by Stampacchia duality theorem in W 1,2 0 .


Ryan S J D Baker
email: ryan.baker@nottingham.ac.uk

Albert T Corbett
email: corbett@cmu.edu

Kenneth R Koedinger
email: koedinger@cmu.edu

Shelley Evenson
email: evenson@andrew.cmu.edu

Ido Roll
email: iroll@cmu.edu

Angela Z Wagner
email: awagner@cmu.edu

Meghan Naim
email: naimm@nhsd.k12.pa.us

Jay Raspat
email: raspatj@nhsd.k12.pa.us

Daniel J Baker
email: bakerd1@umdnj.edu

Joseph E Beck
email: joseph.beck@gmail.com

Adapting to When Students Game an Intelligent Tutoring System

It has been found in recent years that many students who use intelligent tutoring systems game the system, attempting to succeed in the educational environment by exploiting properties of the system rather than by learning the material and trying to use that knowledge to answer correctly. In this paper, we introduce a system which gives a gaming student supplementary exercises focused on exactly the material the student bypassed by gaming, and which also expresses negative emotion to gaming students through an animated agent. Students using this system engage in less gaming, and students who receive many supplemental exercises have considerably better learning than is associated with gaming in the control condition or prior studies.

Introduction

In recent years, increasing attention has been paid to the subject of how students choose to use intelligent tutoring systems. Recent models have suggested that students adopt a variety of strategies for using intelligent tutoring systems and other interactive learning environments, with different strategies potentially leading to different learning outcomes [START_REF] Aleven | Toward tutoring help seeking: Applying cognitive modeling to meta-cognitive skills[END_REF][START_REF] Arroyo | Inferring learning and attitudes from a Bayesian Network of log file data[END_REF][START_REF] Baker | Off-Task Behavior in the Cognitive Tutor Classroom: When Students "Game The System[END_REF][START_REF] Stevens | Modeling the Development of Problem-Solving Skills in Chemistry with a Web-Based Tutor[END_REF]. One strategy in particular, gaming the system, has been found to be associated with poorer learning gains in intelligent tutoring systems [START_REF] Baker | Do Performance Goals Lead Students to Game the System[END_REF][START_REF] Baker | Off-Task Behavior in the Cognitive Tutor Classroom: When Students "Game The System[END_REF]. We define gaming the system as attempting to succeed in an educational environment by exploiting properties of the system rather than by learning the material and trying to use that knowledge to answer correctly. Gaming has been observed in a variety of types of learning environments, from educational games [START_REF] Klawe | Designing Game-based Interactive Multimedia Mathematics Learning Activities[END_REF] to online newsgroups [START_REF] Cheng | Adaptive Reward Mechanism for Sustainable Online Learning Community[END_REF], and has been repeatedly documented in intelligent tutoring systems [START_REF] Aleven | Helping Students to Become Better Help Seekers: Towards Supporting Metacognition in a Cognitive Tutor[END_REF][START_REF] Baker | Off-Task Behavior in the Cognitive Tutor Classroom: When Students "Game The System[END_REF][START_REF] Beck | Engagement tracing: using response times to model student disengagement[END_REF][START_REF] Mostow | A La Recherche du Temps Perdu, or As Time Goes By: Where does the time go in a Reading Tutor that listens[END_REF][START_REF] Murray | Effects of Dissuading Unnecessary Help Requests While Providing Proactive Help[END_REF]. Within the specific intelligent tutoring system that we will discuss in this paper, gaming behavior consists of systematic guessing and rapid-fire hint requests [START_REF] Baker | Detecting Student Misuse of Intelligent Tutoring Systems[END_REF].

Baker and his colleagues [START_REF] Baker | Detecting Student Misuse of Intelligent Tutoring Systems[END_REF] have determined that gaming can be divided in some systems into two distinct behaviors -"harmful" gaming, which typically occurs on the problem steps the student knows least well, and is associated with poor learning outcomes, and "non-harmful" gaming, which typically occurs on problem steps the student already knows, and is not associated with poor learning outcomes.

In this paper, we present a tutor component that responds to harmful gaming, in order to improve gaming students' learning. This tutor incorporates an animated agent, Scooter the Tutor, who observes students as they interact with the tutor, looks increasingly unhappy when students game and gives a student supplementary exercises on the exact steps of the problem-solving process that the student gamed.

Design

Two previous attempts to address gaming in intelligent tutoring systems took a "preventative" approach to addressing gaming, attempting to directly prevent known gaming behaviors [START_REF] Aleven | Helping Students to Become Better Help Seekers: Towards Supporting Metacognition in a Cognitive Tutor[END_REF][START_REF] Beck | Engagement tracing: using response times to model student disengagement[END_REF]. Researchers at Carnegie Mellon and Carnegie Learning introduced a two-second delay between each level of a multi-level hint, to prevent a student from clicking through hints at high speed, and gave mandatory hints ("proactive help") when a student commits more than three errors on a single step, preventing systematic guessing [START_REF] Aleven | Helping Students to Become Better Help Seekers: Towards Supporting Metacognition in a Cognitive Tutor[END_REF]. Researchers at the University of Massachusetts redesigned their system to not give help until a student had spent a minimum amount of time on the current problem [START_REF] Beck | Engagement tracing: using response times to model student disengagement[END_REF].

In [START_REF] Baker | Off-Task Behavior in the Cognitive Tutor Classroom: When Students "Game The System[END_REF], we hypothesized that students using a system re-designed to directly prevent gaming would attempt to discover new ways to game. Shortly after, [START_REF] Murray | Effects of Dissuading Unnecessary Help Requests While Providing Proactive Help[END_REF] found that students using a tutor with two-second help delays developed new strategies for gaming -for example, rapidly repeating the same error several times in a row in order to elicit delay-free proactive help. An additional concern with direct prevention is that students game features which are used in more positive ways by the majority of students who do not game.

Our design approach, by contrast, attempted to meet two conditions: First, the design must improve the learning of students who currently game. Second, the design must change the tutor minimally for students who do not game.

In accordance with these design goals, we developed a new component for the students' intelligent tutoring software -an animated agent named "Scooter the Tutor", developed using graphics from the Microsoft Office Assistant [START_REF]Microsoft Office 97[END_REF] but modifying those graphics to enable a wider range of emotions. Scooter was designed to both reduce the incentive to game, and to help students learn the material that they were avoiding by gaming, while affecting non-gaming students as minimally as possible.

When the student is not gaming, Scooter looks happy and occasionally gives the student positive messages (see the top-left of Figure 1). Scooter's behavior changes when the student is detected to be gaming harmfully (using an updated version of the gaming detector presented in [START_REF] Baker | Detecting Student Misuse of Intelligent Tutoring Systems[END_REF][START_REF] Baker | Detecting When Students Game The System, Across Tutor Subjects and Classroom Cohorts[END_REF]). If the detector assesses that the student has been gaming harmfully, but the student has not yet obtained the answer, Scooter displays increasing levels of displeasure (culminating in the expression shown on the bottomleft of Figure 1), to signal to the student that he or she should now stop gaming, and try to get the answer in a more appropriate fashion.

If the student obtains a correct answer through gaming, Scooter gives the student a set of supplementary exercises designed to give the student another chance to cover the material that the student bypassed by gaming this step. The supplementary exercises have three levels, each multiple-choice -the student is only given one chance to answer each level. In each of the first two levels of an exercise, the student is asked to answer a question that either requires understanding one of the concepts required to answer the step the student gamed through, or a question which is about what role the step they gamed through plays in the overall problem-solving process. If the student gets both the first and second levels wrong, he or she is given a third level, which is still relevant to the step the student gamed through, but which is very easy, in order to prevent indefinite floundering.

If the student gets any level right on the first try, Scooter lets the student return to the regular tutor exercise; if the student gets all three levels (including the very easy third level) wrong, Scooter assumes that the student was trying to game him, asks the student to attempt to get his exercises correct on the first try, and marks the problem step involved to receive supplementary exercises in future problems. If the student tries to game a supplementary exercise, Scooter displays anger.

Our goal, in designing Scooter, was to benefit students in three fashions. First, by representing how much each student had been gaming, Scooter both serves as acontinual reminder that the student should not game, and lets teachers know which students were gaming recently. Second, Scooter was intended to invoke social norms in students by expressing negative emotion when students game. Scooter's display of anger is a natural social behavior in this context; if a student systematically guessed every number from 1 to 38 when working with a human tutor, it seems reasonable to expect that the human tutor would become impatient or upset. Therefore, we hypothesized that when Scooter becomes angry, he will invoke social norms, leading the student to game the system less. Third, by giving students supplemental exercises targeted to the material the student was gaming through, Scooter gives students a second chance and another way to learn material he or she may otherwise miss entirely. Additionally, supplemental exercises may change the incentive to gamewhereas gaming might previously have been seen as a way to avoid work, it now leads to extra work. Thus, we predicted that Scooter would both reduce gaming and improve gaming students' learning, either by reducing their gaming or giving them a second chance to learn the material they miss by gaming.

Study Methods

We studied Scooter's effectiveness in the context of a year-long Cognitive Tutor curriculum for middle school mathematics, within 5 classes at 2 schools in the Pittsburgh suburbs. The study was conducted in the spring semester, after students had already used the Cognitive Tutor for several months. Initially, the study was designed such that every student used both a version of the tutor with Scooter (experimental condition), and a version of the tutor without Scooter (control condition). Each student was randomly assigned to use one of two lessons (a lesson on percents, and a lesson on scatterplots) with Scooter, and the other lesson without Scooter. All students completed the control condition of the study first, and the experimental condition second. However, due to a scheduling error, the experimental condition of the study took place in the same week as subject material on percents was being taught in class. To avoid bias in favor of the experimental condition, we will therefore limit our discussion to data from the scatterplot lesson. 51 students participated in the experimental condition for the scatterplot lesson (12 were absent for either the pre-test or post-test, and thus their data will not be included in analyses relevant to learning gains); 51 students participated in the control condition for the scatterplot lesson (17 were absent for either the pre-test or post-test).

Before using the tutor, all students first viewed conceptual instruction, delivered via a PowerPoint presentation with voiceover and simple animations [cf. 4]. In the experimental condition, a brief description of Scooter was incorporated into the instruction. Then students completed a pre-test, used the tutor lesson for 80 minutes across multiple class periods, and completed a post-test. Test items were counterbalanced across the pre-test and post-test, and were identical to items used in past studies using this tutor lesson [START_REF] Baker | Detecting Student Misuse of Intelligent Tutoring Systems[END_REF]. Log files were used to distill measures of Scooter's interactions with each student, including the frequency with which Scooter got angry, and the frequency with which Scooter gave a student supplementary exercises. In addition, observational data was collected to determine each student's frequency of gaming, using the quantitative observational method as in [START_REF] Baker | Off-Task Behavior in the Cognitive Tutor Classroom: When Students "Game The System[END_REF], in order to analyze Scooter's effects on gaming frequency. Another potential measure, the gaming detector [START_REF] Baker | Detecting Student Misuse of Intelligent Tutoring Systems[END_REF], was not used because of risk of bias in using the same metric both to drive interventions and as a measure of the intervention's effectiveness.

Results

Scooter was associated with a sizeable, though only marginally significant, reduction in the frequency of observed gaming. 33% of students were seen gaming in the control condition, while 18% of students were seen gaming in the experimental condition, a marginally significant difference, χ 2 (1,N=102)= 3.30, p=0.07. However, although fewer students gamed, those students who did game did not appear to game less. The average gamer in the control condition gamed 17% of the time, while the average gamer in the experimental condition gamed 14% of the time, which was not a significant difference, t(23)=0.74, p=0.47.

Despite the apparent reduction in gaming, however, there was not an overall improvement in learning. Overall, students in the control condition averaged a 22 point pre-post gain (44%->66%), while students in the experimental condition averaged a 25 point pre-post gain (37%->62%), which was not a significant difference, t(70)=0.34, p=0.73. However, analyzing overall learning may not be the most appropriate way to test the intervention's effect on learning. Gamers are a fairly small subset of the overall population, both in this study and past studies [cf. [START_REF] Baker | Detecting When Students Game The System, Across Tutor Subjects and Classroom Cohorts[END_REF][START_REF] Baker | Off-Task Behavior in the Cognitive Tutor Classroom: When Students "Game The System[END_REF]. Therefore, differences in gamers' learning may be swamped by normal variation in the rest of the population. Additionally, since students engaged in different degrees of gaming, and the detector was accurate but not perfect [cf.4], not all students who in engaged in harmful gaming received the same number of interventions from Scooter. Thus, in the following sections, we will look at the students who got a considerable amount of each type of intervention from Scooter, to see if and how the students' behavior and learning was affected by Scooter. We will analyze the two types of interventions separately, since the two types of interventions were given in different situations and may have had different effects.

Supplementary Exercises

Overall, Scooter gave a fairly small number of exercises: no student received a set of exercises from Scooter on more than 3.2% of problem steps (12 sets), and the median student received a set of exercises on only 1.1% of problem steps (3 sets). However, % of students ever seen gaming Scooter's exercises were assigned to exactly the problem steps students gamed on (according to the detector), and were significantly correlated to the frequency of observed gaming, r=0.43, F(1,38)=8.24, p<0.01, so the exercises might have had more effect on learning than their low frequency might otherwise indicate. One possible model for how learning could relate to the number of supplementary exercises received is a linear relationship -the more supplementary exercises a student receives, the more they learn. However, students who never receive supplementary exercises don't receive supplementary exercises precisely because they don't engage in harmful gaming, and not engaging in harmful gaming is generally associated with better learning [cf. 4]. Therefore, if supplementary exercises positively affect learning, it may be more reasonable to expect students who receive either many or very few supplementary exercises to show good learning, with the students in the middle showing poorer learning.

In fact, this is exactly the relationship we find, as shown in Figure 3. The third of students that received the most supplementary exercises had significantly better learning than the other two thirds, t(37)=2.25, p=0.03; the overall difference between all three groups was also significant, F(2,36)=3.10, p=0.06. This occurred because the students who received the most supplementary exercises started out behind the rest of the class (common among students who frequently game [cf. 7]), but caught up by the post-test (see Figure 4 Left). There was a statistically significant interaction between pre-test and post-test scores, and how many supplementary exercises the student received (top third versus other two thirds), F(1,37) = 5.07, p=0.03, for a repeated measures ANOVA. Note that there was not a ceiling in the mid-60s, nor a post-test floor effect: students in each group had perfect post-test scores, or low post-test scores.

In considering the evidence that students who received many supplemental exercises caught up to the rest of the class, it is worth remembering that students receive supplemental exercises because they are detected to be engaging in a large amount of harmful gaming. In both the control condition (see Figure 4 Right), and in prior studies involving the same tutor lesson [START_REF] Baker | Detecting Student Misuse of Intelligent Tutoring Systems[END_REF][START_REF] Baker | Do Performance Goals Lead Students to Game the System[END_REF], frequent harmful gaming is associated with starting out lower than the rest of the class, and falling further behind by the post-test, rather than catching up. As shown in Table 1, students in the control condition and past studies who did not use Scooter and engaged in more than the median amount of harmful gaming (among harmful gamers) averaged a 22 point Interestingly, although Scooter's exercises appear to be associated with improved learning, Scooter's exercises were not directly associated with the decrease in gaming reported in the previous section. If receiving an exercise from Scooter led a student to reduce his/her gaming, we would expect the students who received more exercises to reduce their gaming over time. There is no evidence of such a decrease. Figure 5 (left) shows the frequency in gaming over the 3 days of the study among the students who received many exercises (top third) in the experimental condition, compared to the other students. Among the students who received more exercises, neither the apparent increase in gaming from day 1 to day 2, nor the apparent decrease in gaming from day 2 to day 3, was statistically significant, χ 2 (1,N=155)= 0.31, p=0.58, χ 2 (1,N=105)= 0.17, p=0.68. Overall, the students who received more exercises gamed significantly more often than the students who received fewer exercises, χ 2 (1,N=388)= 24.33, p<0.001.

Expressions of Anger

Overall, Scooter became angry considerably more often than he gave supplementary exercises. The median student saw an angry Scooter 13% of the time, and the student who saw an angry Scooter the most often saw an angry Scooter 38% of the time.

There did not appear to be an association between viewing an angry Scooter more often, and better learning. Students who received more expressions of anger did not have a significantly larger average learning gain than other students, whether we compared the top quartile to the other students, t(37)=0.48, p=0.63, effect size = 0.20σ , the top third, t(37)=0.16, p=0.87, or the top half, t(37)=0.15, p=0.88.

Additionally, there was no evidence of a relationship between Scooter's frequency of expressions of anger, and a reduction in gaming over time (as shown in Figure 5, Fig. 5. Observed Gaming Over Time, in the Experimental Condition right). Among the students who saw an angry Scooter the most often (top quartile), there was not a significant change either from day 1 to day 2, or day 2 to day 3, χ 2 (1,N=79)= 0.04, p=0.84, χ 2 (1,N=50)= 0.83, p=0.36.

Conclusions

In this paper, we present a re-designed tutor that responds to when students game the system, incorporating an animated agent, Scooter the Tutor. Students who received a large number of supplementary exercises from Scooter had high learning gains, and caught up to the rest of the class. This result is quite different from the pattern observed in the control condition and past studies [START_REF] Baker | Detecting Student Misuse of Intelligent Tutoring Systems[END_REF][START_REF] Baker | Do Performance Goals Lead Students to Game the System[END_REF], where students who game harmfully start out with lower pre-test scores, and fall further behind the rest of the class by the post-test.

Since students tend to game harmfully on the steps they know least well [START_REF] Baker | Detecting Student Misuse of Intelligent Tutoring Systems[END_REF], the supplementary exercises may have been effective in large part because they offered additional learning support (and, perhaps, different learning support) for each student on the exact steps which that student found most difficult. Hence, we may be able to use a student's choice to game as an opportunity to learn more about where the student is having difficulty.

Incorporating Scooter into the tutor also led to about half as many students choosing to game. It is not entirely clear what aspect of the modified tutor led to the reduction in gaming. Neither students who saw an angry Scooter more often, nor students who received more supplementary exercises, reduced their gaming over time. One possibility is that simply knowing Scooter was present, and that he would make it impossible to hide gaming, led some students to game less. Thus, although Scooter's actions may not have directly affected the students who saw an angry Scooter, Scooter's presence may have motivated some students to avoid gaming during the entire lesson.

Overall, these results suggest that there is value to detecting and responding to differences in how students choose to use intelligent tutoring systems. By responding to gaming, we can develop tutors that help lower-performing students catch up to the rest of the class, and come closer to the goal of developing educational systems that help all students achieve.

Fig. 1 .

 1 Fig. 1. Scooter the Tutor -looking happy when the student has not been gaming harmfully (top-left), giving a supplementary exercise to a gaming student (right), and looking angry when the student is believed to have been gaming heavily, or attempted to game Scooter during a supplementary exercise (bottom-left).

Fig. 2 .

 2 Fig. 2. The frequency of gaming (observed) in each condition.

Fig. 3 .Fig. 4 .

 34 Fig. 3. The Learning Gains Associated With Receiving Different Levels of Supplemental Exercises From Scooter

Table 1 .

 1 Learning gains for students who received large numbers of supplementary exercises from Scooter, and for students who did not use Scooter and engaged in more than the median amount of harmful gaming, among harmful gamers. All students used the same lesson on Scatterplots

	Group

Acknowledgements

We would like to thank Steve Ritter, Tom McGinnis, Sue Cameron, Russ Hall, Dina Crimone, Pat Battaglia, Shaaron Ainsworth, Cecily Heiner, Peter Scupelli, and Cristen Torrey for helpful suggestions and assistance. This work was funded by NSF grant REC-043779 to "IERI: Learning-Oriented Dialogs in Cognitive Tutors: Toward a Scalable Solution to Performance Orientation".


Ton De Jong

Utwente Nl)

LEARNING COMPLEX DOMAINS AND COMPLEX TASKS, THE PROMISE OF SIMULATION BASED TRAINING

Computer simulations enable the realization of didactical concepts such as inquiry learning, collaborative learning, and situated learning. These didactical approaches are not novel in the sense that they have just emerged, but simulation learning environments help to realize these approaches in new type of contexts. The design of such simulation learning environments is a multifaceted endeavour. First, the processes and difficulties involved in inquiry, collaboration, and knowledge application have to be identified so that measures that actually support learners in these learning processes can be created and incorporated in the application. Second personal, curricular, and organizational requirements and constraints for the multimedia supported didactical innovations should be recognized. This article presents examples from projects (SIMQUEST and Co-Lab 1 ) in which simulation based learning environments in engineering and science domains have been developed and evaluated.

INTRODUCTION

New types of (on-line) learning environments are becoming available for use in the actual classroom rapidly. Trends that nowadays dominate the field of learning and instruction are constructivism, situationism, and collaborative learning. More specifically, we can say that the new view on learning entails that learners are encouraged to construct their own knowledge (instead of copying it from an authority be it a book or a teacher), in realistic situations (instead of merely decontextualised, formal situations such as the classroom), together with others (instead of on their own). These new trends have not emerged just by themselves; they are based on changing epistemological views. First, knowledge is not seen anymore as something that is assessed in relation to an external objectivistic "truth", but as individually flavoured and thus potentially different between people. Second, these individual knowledge states are exchanged between professionals that seek for mutual understanding and agreement. In this respect knowledge has a strong social character. Third, we have started to value knowledge that is applicable in realistic situations, and thus is not restricted to abstract knowledge.

1 SIMQUEST and Co-Lab were and are partially supported by European Community under the Telematics and Information Society Technology (IST) RTD programmes, under contracts SERVIVE (ET 1020), and Co-Lab (IST-2000-25035). The author is solely responsible for the content of this article. It does not represent the opinion of the European Community or SURF, and the European Community or SURF is not responsible for any use that might be made of data appearing therein. For project details see the project websites: www.simquest.nl, and www.co-lab.nl. My thanks go to all the persons who have or are working on one of the projects and whose ideas are represented in this paper.

SIMQUEST

SIMQUEST was created to serve teachers and learners involved in discovery learning. SIMQUEST is an authoring system dedicated to simulations for discovery learning. It has the following two goals:

• To provide learners with supportive environments for discovery learning, in the form of cognitive tools directed at scaffolding the processes of discovery leaning; • To provide authors with a flexible tool for creating simulation-based discovery learning environments, containing both technical and conceptual support for the authoring process. The central focus of this presentation is the learning aspect, for the authoring aspect see van Joolingen and de Jong (2003) and de Jong et al., (in press). The primary learning goal for learners who work with a SIMQUEST learning environment is to construct knowledge of the domain under inquiry. This does not necessarily imply that the learner must know the model underlying the simulation in all detail. The goal is to understand the principles of the domain that account for the observed behaviour and/or the effects of actions performed within the domain (we have called this "intuitive knowledge", see [START_REF] Swaak | Discovery simulations and the assessment of intuitive knowledge[END_REF]. To reach this goal, students have to overcome the many problems they have with discovery learning (see for an overview de Jong & van Joolingen, 1998). In SIMQUEST applications, simulations are embedded in instructional support which aims at supporting learners in the discovery process. Currently, SIMQUEST provides four types of instructional support for learners: • Model progression. A learning environment created with SIMQUEST may contain a number of different simulation models, ordered along dimensions such as difficulty, 'order' (qualitative vs. quantitative), or perspective on the domain; • Assignments. Assignments provide the learner with short-term goals, like finding a specified relation, predicting the behaviour of the simulation or achieving a specified simulation state.

In co-operation with model progression, assignments decompose the overall learning goal of a simulation into a number of subgoals. Learners can answer assignments and then receive feedback on their answer; • Explanations. In the SIMQUEST authoring system, the author can define textual, graphical, and multimedia explanations. These explanations can be used to provide learners with extra information on variables, relations, or events in the simulation; • Monitoring. The monitoring tool helps learners monitor, compare, and replay the experiments they have been doing, and that, in relation to answers to investigation assignments, can provide feedback on the relation between the experiments and answers chosen. Figure 1 gives an example of a part of a learning environment created with SIMQUEST (see [START_REF] Van Der Meij | Learning with Multiple Representations. Supporting students' translation between representations in a simulation-based learning environment[END_REF]. In this learning environment (called "Moment") learners explore the behaviour of moment in two situations: moment on a bolt caused by a force on an open-end spanner (displayed in Figure 1), and moment on a hoisting crane caused by a load. This figure does not include instructional support. Instructional support (e.g., assignments) and the simulation are fully integrated in SIMQUEST. The simulation can be set in a certain state by the assignment and can be controlled from the assignment by the learner. Also, values from the assignment are used for generating adequate feedback.

CO-LAB

In today's working environment, employees have to collect, make sense of and use more and more information to keep up with developments in their field. To make the most of this information they need to acquire new knowledge and skills and develop better ways to collaborate with fellow workers based at different locations. The Co-Lab project is designed to develop a learning environment that will give users remote access to a virtual workspace for collaborative inquiry-based learning using experimentation and modelling. Initially, Co-Lab develops demonstration software for the fields of water management and climate control in greenhouses. The objective of the Co-Lab project is to design, develop, and evaluate a new system for collaborative, inquiry-based learning. Basically, Co-Lab follows an inquiry based approach with a number of (not necessarily sequential) phases: o rientation, hypothesis, experiment, data interpretation, conclusion, and evaluation (see [START_REF] Njoo | Exploratory learning with a computer simulation for control theory: Learning processes and instructional support[END_REF][START_REF] De | Co-Lab Specifications. Part 1 -Theoretical background[END_REF]. Compared to simulation-based learning environments Co-Lab has a number of specific characteristics:

• In Co-Lab a series of instructional support measures is built in, as it is in the SIMQUEST learning environments; • In Co-Lab expressing the conclusions of the discovery process is done is a specific way, namely by "modelling". Learners in the end create a runnable model of the domain. To create this model, learners are provided with modelling tools in the phases orientation, hypothesis, and conclusion; • In Co-Lab, for discovery, learners do not only have a simulation environment available (as is the case in SIMQUEST) but also local and remote laboratories and databases. This means that learners can work with "real" data;

• In Co-Lab the learning process is a collaborative endeavour. Over and in all phases of the inquiry cycle three learners work together to reach the ultimate goal (a runnable model).

Inquiry learning forms an excellent basis for collaboration since at a number of points in the learning process specific decisions need to be taken (e.g., which hypothesis to test, which variable values to change etc.). To facilitate this collaboration they have access to collaborative workspaces (based on whiteboards) and dedicated communication facilities. The design of Co-Lab is based on a "building metaphor". Learners enter a building for a specific topic and may move between floors (different levels of the same topic) and at each floor move between a laboratory (to do experiments), a theory room (for the modelling aspect), and a meeting room (where overall planning and discussion takes place). Initially, in Co-Lab we will develop demonstration software for the fields of water management and climate control in greenhouses. For water management we have access to large external databases of water flow in actual rivers, for climate control in greenhouses an external laboratory will be built. Figure 2 shows an overview of a mock-up of one level in a Co-Lab building. In parallel with the technical development of the Co-Lab environment, a comprehensive support system (similar to the one created for SIMQUEST) will be developed to help learners in their experimentation, collaboration, and modelling activities. In addition, Co-Lab will be designed to be integrated with the curriculum.

CONCLUSIONS

This paper started with an overview of trends in learning and instruction: constructivism, collaborative learning, and situated learning. These trends can be recognized in the examples that were presented: SIMQUEST and Co-Lab.

In SIMQUEST the first focus was on constructivism in the form of discovery learning. The core of SIMQUEST learning environments is always a simulation in which learners change values of input variables and observe values of output variables to discover the underlying model. By creating a structure around the simulation as we do in SIMQUEST, part of this constructivism is taken away. The design challenge is to give students just enough support to enable them to engage in discovery with giving away as little as possible (both from a content and from a discovery process perspective). Most of the SIMQUEST learning environments that we created have a realistic context, as for example could be seen in the learning environment displayed in Figure 1.

Other examples of realistic simulations we have created are "growth of tomatoes in greenhouses" and a "sewage plant". Both these simulation learning environments are very encompassing environments with submodules on more fundamental science issues. This principle of a family of simulations under one specific theme has also been used in the Co-Lab learning environment. At the start SIMQUEST learning environments were designed for individual learning. Recently we have added facilities for collaborative learning. Now, the prior knowledge of learners can be indicated with their views on propositions in the domain (true, not true, to be tested, etc.) and after having completed their views on a list of propositions, the lists of two students can be combined and students are offered chat facilities to discuss their individual views. When they collaboratively decide to test a specific proposition the SIMQUEST collaboration tools can present the students with an appropriate assignment (see Gijlers & de Jong, in preparation).

In Co-Lab the situatedness of the learning environment is amplified by making data sources such as real (remote and local) laboratories available to students. The collaborative aspect that is inherent in Co-Lab also introduces a realistic aspect. Collaboration is included to foster learning but it is also an important aspect of real scientific discovery (see for example [START_REF] Dunbar | What scientific thinking reveals about the nature of cognition[END_REF]. Taking this into account, our learning goals have also shifted from specific domain related learning goals to include other professional goals such as "collaboration", "communication", and "working methods". In Co-Lab the constructivistic aspect is found in the discovery learning that is the primary mode of learning in Co-Lab but, in addition to what happens in SIMQUEST, also in the modelling that students are supposed to do and which gives them the opportunity to express their thoughts in a formal way. Developing learning environments that follow innovative pedagogical concepts is one thing, having them introduced in actual instruction and training is yet another accomplishment. For SIMQUEST we have now found cooperation with professional publishers and have produced simulation environments that form a unit with other course material (most particular a book). This helps teachers in the way that they now can choose for one integrated course and do not need to use the SIMQUEST authoring facilities to adapt a simulation to their own situation. Although, intended originally for use by teachers directly, we now have found that teachers do not have the time, skills, and interest to design or even adapt a computer based learning environment. Having the software fit with the curriculum is also a major concern in the Co-Lab project. One issue that is particularly important here is that working in the Co-Lab environment almost inevitably means that a) this concerns a multi-hours involvement of learners and b) there is an integration of different science domains (physics, chemistry, biology etc.). The demonstrators to be developed in Co-Lab (in the areas of water management and climate control in greenhouses) are situated and appealing topics, but both require an integration of domains and an extensive time investment. Currently, this does not always line up with the curriculum structure.

Related to these issues is the very important place of the teacher or trainer (or training department). For SIMQUEST we have found that teachers have little experience in discovery learning, and especially the teachers we have been working with (who came from middle vocational training) had fear to give their students the freedom and responsibilities that these open learning environments require. What we found was that teachers sometimes restricted the freedom of discovery learning again and changed the learning environment in a very structured, set-by-step, learning experience (see De Jong et al., 1998). We try to overcome this by including suggestions and information in an advice tool for authors in the SIMQUEST authoring environment (see Pieters, Limbach, & de Jong, in press). Also, a specific training course for teachers has been developed. A third point that hampers the introduction of innovative learning environments is the need for new ways of assessment. Introducing new ways of learning necessarily means that new goals are reached and thus new assessment methods should be used. For SIMQUEST we have developed a new type of test, the intuitive knowledge test [START_REF] Swaak | M easuring intuitive knowledge in science: the development of the what-if test[END_REF]. For collaborative learning environments assessment methods that takes the learning process into account need to be developed. Many innovative approaches to learning and instruction developed in projects fail to make it to the actual schools or companies. To have a lasting place in the actual curriculum, a strong relation between the software and the rest of the curriculum (in content, timing, and approach) and/or the conditions in the working environment is a necessary condition. In addition teachers should have the adequate skills and the necessary commitment. Finally, there should be a readiness (and the formal authority) to use new ways of assessment.

Figure 1 .

 1 Figure 1. Example of a SIMQUEST simulation

Figure 2 .

 2 Figure 2. Overall impression of the Co-Lab environment. Upper part: Co-Lab city with different buildings; middle section: one floor in a building; bottom part: specific tools.


Ola Berge
email: ola.berge@intermedia.uio.no

Richard Edvin Borge
email: richared@ifi.uio.no

Annita Fjuk
email: annita.fjuk@intermedia.uio.no

Jens Kaasbøll

Terje Samuelsen
email: terjesam@ifi.uio.no

Learning Object-Oriented Programming

Keywords: K.3.2 [Computer and Information Science Education] Learning, teaching, phenomenographic, pedagogy

Loud discussions concerning various ways of teaching object-orientation have taken place without much empirical evidence for any position. This paper reports qualitative observations of learning of object-oriented programming in an introductory course. The students were found to cope reasonably well with the object-oriented concepts, and they had learnt procedural programming first. However, they modeled the real world domain to be represented in the program by imagination and through coding. Their problems may be attributed to the high complexity generated by having to relate to four -six areas of attention. Three ways of improving teaching are suggested, making the areas of attention and the ways to relate them more explicit for the students, forcing modeling by means of a tool, and reducing complexity by means of programming environments that visualize objects and their behavior.

INTRODUCTION

Teaching programming and modeling by means of object-oriented methods has become a common mode of introductory computer science training during the last ten years. Research indicates that learning object-orientation is more difficult than coming to grips with other paradigms [START_REF] Vessey | Requirements Specification: Learning Object, Process, and Data Methodologies[END_REF], while a more recent study of programming learning showed no significant difference between procedural and OO learners [START_REF] Widenbeck | Novice comprehension of small programs written in the procedural and object-oriented styles[END_REF]. Repeated discussions on e-mail lists reveal many frustrated teachers (SIGCSE.MEMBERS), and high drop out and failure rates in introductory programming at universities world wide also add to the impression of a subject difficult to teach and learn. Previous studies of novices learning programming have investigated students learning procedural [START_REF] Spohrer | Novice Mistakes: Are the Folk Wisdoms Correct?[END_REF] and logic programming [START_REF] Booth | Learning to program. A phenomenographic perspective[END_REF]. They conclude that after an introductory period, most students have learnt the syntax and semantics of the programming language, but they struggle with composing their programs. This would imply that courses which go beyond the simplest programs will constitute a much larger challenge to the students, regardless of the programming paradigm and the sequence of instruction. When learning object-oriented programming, the language has more concepts than previously used languages, which could imply that learning this paradigm becomes more difficult. On the other hand, the

BACKGROUND

The students who are studied attend an introductory course in object-oriented programming with Java. The sequence of instruction is variables, control structures, methods, arrays, file operations, graphical interfaces, classes, objects and pointers, object-sets, inheritance, sub-and superclasses, abstract classes and exception handling. The course is scheduled to 50% workload during a semester, with four hours of lecture, two hours of classroom problem solving and two hours of closed labs per week. The students are given five mandatory exercises during the semester. Only a minority of the students has any prior programming experience.

METHOD

Since little is known concerning learning of programming, devising categories that lend themselves to quantitative studies are difficult, so a qualitative approach was selected. In order to elicit how the students understand programming, their conversations and programming were observed. Observation has long been used for studying learning of a diverse set of subjects, logic programming included [START_REF] Marton | Learning and awareness[END_REF]. The method is called phenomenographic, since it aims at describing people's experience of the world. The students were observed for eight hours in the lectures. A request for permission to observe the students in the classrooms and the labs was sent to the instructors, Three out of 11 instructors were positive, and the students in these groups were observed for a total of 20 hours, 18 of which were done in the lab. During the observations, the observers also asked the students some questions, mostly concerning what they experienced as most difficult. Notes were taken during the observations. Audio tape recording was not considered giving sufficient added value, since much of the conversation was concerned with code on the computer screen, and understanding the recording without the programs being discussed would be difficult. Recording the program code with video was considered impractical due to the small fonts and flickering screen on the computer terminals. On some occasions, the observers also guided the students, taking over the role of the instructor. This happened when there were more students requesting assistance than the instructors could handle. Acting as an instructor provided opportunities for getting the students to talk about their programs, but the interaction implied that fewer comments from the students were noted down.

AREAS OF ATTENTION

The tasks in the education studied concerned developing applications for domains that the students should be somewhat familiar with, eg, spectators in a movie theatre. When developing object-oriented applications, the students have to consider several domains. First, the students have to consider the real world domain that the application is supposed to serve, eg, the movies, the theatre with its rows and seats and the audience. Second, the students have to consider the objects that the program will generate during run-time, and these objects of the classes "customer," "movie" and "seat" have to represent the actual customers, movies and seats. People can imagine that these objects exist in the computer during program execution, but they do not lend themselves to experience. Third, the interaction with the running program catches the students' attention when running a program that is void of syntax errors, and they can experience the windows on the screen the input they type. Fourth, the code with its class descriptions has to be written and discussed, and this area tends to be in the focus of the students' attention during their labs. The classes constitute the prescriptors of the objects, the descriptors of the real world concepts, and the organizing units for the user interface.

Keeping these four areas in mind when programming may be trivial for the experienced programmer, who may consciously choose to disregard interface and objects for periods, and only concentrate on constructing class descriptions that represent real world phenomena. But some of the students observed did not have any clear conceptions of objects during program execution, so they stored every change of the program on file. Another student struggled for two hours with obtaining consistency between the data entered at the user interface and what her program stored on file (JK 03.04.03) 1 . She could specify how the file should look like, but she was incapable of sketching the objects generated during program execution and the pointers, which kept the objects coherent. The five areas that the students should consider are illustrated in Figure 1.

Figure 1. Areas that the programmer should keep attention to when constructing an object-oriented program

Three of these areas are visible for the programmer, the code, the file, and the user input. The student has to recall the real world domain from her memory, which is probably trivial when the student is familiar with it. She can also refer to the problem specification. The program execution with its objects is something that the student has to imagine, and the teachers advice the students to draw sketches like those depicted in the upper right corner of Figure 1 When the novices have to grapple with five areas and the 10 relations that we can consider between them, finding that most students do not master these relations should not come as a surprise.

The student who struggled with getting the correct data on file was talking about her program in the following way: "the program reads from the keyboard and writes on the file," indicating that she regarded the world as "input → processing in program → output," see Figure 2, and not as a set of interlinked objects that represent an area in the world. We would say that this student had a procedural and not an object-oriented conception when programming. (The instructor draws object structures) I: Such a one which controls whether we have this one already. S: And that is the HashMap? I: Yes If they were talking about the real world cinema, they would hardly speak of a show as something that contains a theatre, but rather about a theatre that hosts a show. Assuming that all students are familiar with going to the movies, they probably use this experience as a frame of reference when discussing the objects, even if their conversations do not mention the real world domain. Their framework of understanding can therefore be illustrated as in Figure 3.

Modelling by imagination

Modelling by coding

Design by copying

RELATING THE AREAS

The movies task studied was the students' first assignment of size more than a page of code and a somewhat complex domain, so they had to learn two new skills. The first is modelling the reality by means of object-oriented concepts, and we will refer to this activity as "modelling." The other activity that they have to learn is to structure their object-oriented program by means of classes and methods for file handling, user interface, etc., and we call this process "program design."

The concepts that the students had available for modelling were objects, pointers, arrays, and HashMaps. These concepts were clearly implementation oriented, and no analysis concepts like associations and cardinalities, were known to the students. The observation that the students made design models rather than models of the real world domain, is therefore not surprising. Moreover, only occasionally did the students draw models on paper, and even if they did, these models disappeared under other papers, or the students were not able to translate them into code. Their conversations about arrays and pointers were not anchored in these drawings, but rather in the visible code on the computer screens. Referring to Figure 3, they imagined the two upper areas rather than visualizing them, so we can call their modelling activity "modelling by imagination." A typical situation arouse when the students modelled by imagination was that the student asked the instructor for assistance. She replied by drawing what she perceived as the student's model, and commented upon it, e.g. (TS 27.03.03): I: Isn't it easier to put the seat into the theatre class instead of 100 seat objects? Those students who were unable to code based on the drawn model could only work in the upward direction from Code to Program execution in Figure 3, so we will call this activity "modelling by coding." The opposite direction, which the more advanced students could handle, would be "coding the model."

The students arrived at their program design by copying a program made by the lecturer and modifying it, we call it "design by copying." [START_REF] Booth | Learning to program. A phenomenographic perspective[END_REF],p.207] observed that students use the same strategy when writing their first program. Some students also reported uncertainty with the imported structure (TS 27.03.03):

S: We are sure about what to do, but we don't know where. This statement indicates that they have a certain mastery of the details without a complete understanding of the big picture.

STRUCTURING THE PROGRAM

Being novices, the students needed lots of assistance in structuring their program, e.g. ( Again, the student is modelling by imagination, and the instructor provides visualisation, forcing the student to rethink his design. The students were not familiar with the power of pointers, objects, arrays and HashMaps, e.g., the instructor comments upon a student's array for pointing to seat objects in the theatre (JK 20.03.03):

I: But it is three-dimensional! Even though the students struggled with putting the pieces together, they all tried to use the set-concepts when storing many objects of a kind and never when creating only one object. Also, no students were reported mixing up Class with Array or HashMap. When designing by copying, the students were told to use a Database class and a Control class, but these classes were frequently misunderstood (RB 24.03.03). Since the students had no intuition concerning them, they merely followed the lecturers' examples, hence there is ample room for misunderstanding.

OBJECT-ORIENTED CONCEPTS

Personal experience shows that the students' speed of learning in programming courses differ vastly. One example of the slow learners was a student who should program a class that took care of customer information, and he made his program generate a new customer-object for every piece of information for each customer (RB 31.03.03). Most of the students observed had a more sophisticated understanding of objects. They knew that many objects could be generated for a class, that each customer should have one object. Further, they knew how to declare private variables and public methods in the classes. Some students also demonstrated a good understanding of the difference between object and class (static) variables. Their task given was to make two Rectangle objects, the first with height equal 2, width 3 and colour red; the second 4, 6 and yellow. After having made the two rectangles, the program shall display this information for each rectangle. The code is: Yes Pointers were also problematic in some instances. The students knew that they should create a pointer and assign it to the new object when created, but pointer assignments in general was more difficult (TS 27.03.03): I: Why do you say (while pointing at the code, which was like): Seat S = new Seat(); Seat T = new Seat(); S = T; The student was not aware that the first Seat object would be lost. Questions concerning most of the concepts in Java were also raised by the students. However there were few syntax related ones, and none of the observers overheard any questions concerning the basics of variables, if and while. While teachers have reported that parameters are on the top of the list of difficult to learn programming concepts, parameters were not mentioned more frequently than other concepts by the students observed.

DISCUSSION

Previous studies of learning procedural and logic programming [START_REF] Spohrer | Novice Mistakes: Are the Folk Wisdoms Correct?[END_REF], [START_REF] Booth | Learning to program. A phenomenographic perspective[END_REF] have pointed to that most students learn syntax and semantics within reasonable time, and thereafter they struggle with learning how to design their programs. This study shows a similar pattern when learning object-oriented programming. However the students had learnt the basics of procedure oriented programming first, so a certain correspondence with the previous findings should be expected. A study of students who learn the objectoriented concepts first is needed in order to find out whether the pattern also applies in the pure objectoriented case. Widenbeck, and Ramalingam [START_REF] Widenbeck | Novice comprehension of small programs written in the procedural and object-oriented styles[END_REF] distinguish between what they call program model and domain model in their study of novice OO programmers. The program model covers the elementary operations and the control flow, while the domain model concerns to the problem situation referred to in the program, specified as data flow and function information. In addition, they have studied students' understanding of program state. Their elementary operations seem to correspond to what is regarded as code here, while control flow, data flow and state are aspects of the program execution. Their functions are abstractions of input/output, referring to the screen display and typing in the areas presented here. There is no specific mentioning of issues of the real world domain in [START_REF] Widenbeck | Novice comprehension of small programs written in the procedural and object-oriented styles[END_REF], which may be due to the fact that the categories used in [START_REF] Widenbeck | Novice comprehension of small programs written in the procedural and object-oriented styles[END_REF] were based on previous studies of procedural programming. A basic OO perspective of the real world domain described in code and represented in the program execution constitute a perspective that does not easily apply to the procedural approach. The procedural bias of the study is shown in its view of how programs work: "The transformation of data from an input state to a desired output state is the fundamental purpose of a program…" [START_REF] Widenbeck | Novice comprehension of small programs written in the procedural and object-oriented styles[END_REF] Classes and objects pose two challenges in learning program design. First, the students have to learn how to make a model of reality that their objects are going to represent, and second, they have to learn how to design the other components of the program, like user interface and file handling, and how these components should be linked to the reality model. The students studied were working on their first assignment where both kinds of structuring were necessary. Concerning the modelling process, they worked with mapping implementation concepts like HashMap to the program execution that they imagined, hence we called this process "designing by imagination." Few signs of explicit checking against reality were performed. A possible way of strengthening the learning of modelling would be to introduce modelling of reality explicitly by means of object diagrams that illustrate cardinalities. An additional modelling step may, however, also be counterproductive to student learning since it entails more to be learnt. A study of professionals doing ER modelling showed that they did not consider the ER diagrams as a conceptual model of reality but as database design [START_REF] Hitchman | An interpretive study of how practitioners use entity-relationship modelling in a ternary relationship situation[END_REF]. If professional modellers do not separate between analysis and design models, there is no point in introducing this notion in introductory courses. However, the professionals consistently checked their model with the reality, while the students seemed to have their attention mostly at the code and the program execution. When structuring the rest of their programs, the students copied solutions presented by the teachers or found in the course material. Copying and modifying is a well known first step in learning programming, and it seems to work in this setting also. Therefore, the program design that the teachers preach should be tuned to the students' tasks at the novice level, being general enough to a large set of tasks, without bringing in unnecessary sophistication. The concrete model-view-control pattern that was applied in the course seemed to fit the students' tasks. The students demonstrated a seemingly hazardous programming development by ignoring reality checking and by coding without visual models. While the instructors repeatedly assisted the students in drawing models, the students did not take up this habit, something that might have happened because they were unable to base their coding on the models. This inability may stem from the way of teaching, or novices may in any case find it easier to code based on an example and their conception of the domain of the program rather than on models. Explicit teaching of the areas of attention and the ways programmers think when making inferences from one area to the other, see , might strengthen the students' abilities to use models more actively. The empirical study made by Widenbeck and Ramalingam [START_REF] Widenbeck | Novice comprehension of small programs written in the procedural and object-oriented styles[END_REF] shows that the worst performing OO novices make more errors on operation and control flow, while the procedural learners fail more on data flow and function. More advanced beginners do not exhibit these differences. They conclude that learners need both domain and program knowledge in order to boost their comprehension. Although they use basically procedural categories of analysis, pointing to that students need to understand relations between several ways of regarding a program fits well with our observations of OO learners. An alternative approach is to reduce the number of areas of attention from the four in figure 4, and possibly six when bringing in user interface and files like in Figure 1, to only two, by means of special programming environments. Karel J Robot [START_REF] Bergin | Robot: A Gentle Introduction to the Art of Object-Oriented Programming in Java[END_REF], Robocode and Alice [START_REF] Cooper | Teaching Objects-first In Introductory Computer Science[END_REF] are programming environments where the objects generated in the programs become visible, such that the students do not have to imagine objects that are supposed to exist inside the computer. Also, the students do not have to consider relations between phenomena outside the computer and the objects inside, thereby reducing the number of areas of attention to two only, see Figure 5. The students studied here learnt procedural programming first, which may can provide some indication as to whether starting with the procedural paradigm has had any effect of the students' understanding of the object-oriented paradigm. One student regarded the operation of her program as an input-transformation-output sequence, see Figure 2. This sequence corresponds to the procedural view and contradicts the object-oriented paradigm. No other student seemed to have such a conception, however, so teaching the procedural way first did not seem to have much impact in this respect. Another interpretation of the students' poor performance of modelling is that modelling is not encouraged in procedural programming, so that they have learnt coding first. Instead of modelling being helpful for programming, it becomes an additional burden that has to be learnt together with the object-oriented concepts. There are also programming environments that force modelling, and at the same time generate code skeletons from the models drawn [START_REF] Bluej | The Interactive Java Environment[END_REF]. Such environments may also be useful for students who have started out with procedural programming, since they may turn "modelling by imagination" into modelling and "Modelling by coding into "Coding the model."

Figure 2 .

 2 Figure 2. One student's conception of her program.

Figure 3 .

 3 Figure 3. The students' areas of attention and directions of inference. The Program execution is dotted due to the students' habits of only imagining this area.

Figure 4 .

 4 Figure 4. Areas of attention and ways of programming

Figure 5 .

 5 Figure 5. Areas of attention in special programming environments.

Code Visible Class Food{ String name; Int price; void beOrdered() } Class Guest { Program execution

	The real world domain To be represented in the	Cannot be experienced
	program execution
		Screen display and typing
		Direct experience

Guest no: 14 Food: Shrimps Drink: Wine

	Guest	Food
	No 14	Name Shrimp
	Total 130	Price 100
	Order	beOrdered
	dishes
	Permanent data file

Visible Shrimps 73 Salmon 34

  Yes, static means that the colour is the same for all the objects of this class. Subclasses were introduced in the lectures during the time of observations, but most of the students had not started to use them in the programs. Questions during the lecture indicated a good understanding among some of the students (OB,AF,RB 31.03.03) S: Do you need a constructor in the sub-class? I: Well, yes, we do so here, due to the parameter However, other students had not grasped the idea (TS 04.04.03):

	colour = c;
	}
	...
	}
	The students were asked what would be displayed, and they said (RB 31.03.03):
	S1: 3, 4 and yellow. And the other 4, 6 and yellow
	I: Any other suggestions?
	S2: Red and then yellow
	I: Any other?
	S3: Yellow and yellow.
	I: Why
	S: The colour is static
	I:
	class Rectangle
	{
	private double heigth;
	private double width;
	private static String colour;
	Rectangle(double h, double w, String c)
	{
	heigth = h;
	width = w;

S: Do all variables have to be declared twice in sub-classes? Students also seemed to have grasped the generality of the class concept, e.g. (JK 20.03.03) S1: Arrays are actually objects, see S2: So what applies there applies here as well S1:

Observer initials and date of observation.

Code

Visible