Van-Phu Ha
email: van-phu.ha@inria.fr

Olivier Sentieys
email: olivier.sentieys@inria.fr

Maximizing Computing Accuracy on Resource-Constrained Architectures

With the growing complexity of applications, designers need to fit more and more computing kernels into a limited energy or area budget. Therefore, improving the quality of results of applications in electronic devices with a constraint on its cost is becoming a critical problem. Word Length Optimization (WLO) is the process of determining bit-width for variables or operations represented using fixed-point arithmetic to trade-off between quality and cost. State-of-the-art approaches mainly solve WLO given a quality (accuracy) constraint. In this paper, we first show that existing WLO procedures are not adapted to solve the problem of optimizing accuracy given a cost constraint. It is then interesting and challenging to propose new methods to solve this problem. Then, we propose a Bayesian optimization based algorithm to maximize the quality of computations under a cost constraint (i.e., energy in this paper). Experimental results indicate that our approach outperforms conventional WLO approaches by improving the quality of the solutions by more than 170%.

I. INTRODUCTION

The rapid development in scientific and technological innovations during the last decade has driven the demand for more powerful chips with higher performance to handle complex tasks in various fields, such as artificial intelligence (AI), big data, and the Internet of Things (IoT). The requirement for integrating more applications and services, while ensuring at the same time high performance and quality of service (QoS) in a limited cost budget (energy and/or area), poses new challenges for energy-efficient computing. As an example, Deep Neural Networks (DNN) are used in multiple applications such as object detection, speech recognition, navigation, etc. Many of these applications operate on battery-powered intelligent embedded systems such as autonomous drones, self-driving cars, smart phones, and wearable devices. Thus, increasing the performance and/or QoS under a limited energy budget is a key competitive advantage among electronic device manufacturers.

Likewise, implementing applications on ASIC or FPGA devices also faces challenges of optimizing the performance and/or QoS within a constrained cost budget. In theses cases, besides energy, relevant cost constraints include either limited area inside a System on Chip (SoC) or limited resources like the number of LUTs, DSP blocks, memory size in the context of FPGA architectures. As an example, in the development life cycle of some SoC products, upgrading only some modules while keeping the remaining ones untouched helps shortening the time to market. Replaced modules usually have new features that enhance the performance of the system. This requires optimization in terms of maintaining high implementation performance and/or QoS within the available silicon area. This process takes more effort to optimize because the new modules are usually more sophisticated and complex than older ones, requiring more time and effort to create, test, and debug them to fit into the limited, unchanged area (or a given energy budget). Therefore, a design technique that optimizes for the performance and/or QoS within a constrained cost budget of the system would be of great concern, but is not really covered in the state of the art.

Approximate Computing (AC) emerged as an effective solution to adjust the trade-off between QoS and cost constraints, while satisfying design requirements. The use of fixed-point arithmetic in embedded systems, combined with a reduction of the precision, is one of the AC techniques for reducing cost and energy. This technique requires to determine the optimal fixed-point word-lengths (i.e., bit-widths) for representing all variables of the application that still fulfil some quality (i.e., accuracy of computations) requirement. This procedure is called Word-Length Optimization (WLO). Many methods are proposed to solve the WLO problem with QoS constraints quickly and efficiently [START_REF] Parashar | A polynomial time algorithm for solving the word-length optimization problem[END_REF]- [START_REF] Ha | Leveraging bayesian optimization to speed up automatic precision tuning[END_REF]. However, these methods traditionally explore the design space to minimize a cost function (e.g., area or energy) under a given constraint on the accuracy at the output of the system.

However, as mentioned previously, it might be of great interest to limit the area or energy cost budget of a computing kernel below a specified bound. In this case, the stringent constraint becomes the cost, and it is important to maximize accuracy (or the minimize accuracy degradation) as a quality metric under this cost constraint. In this paper, we propose new methods to solve this problem of maximizing QoS within a given cost budget. We first highlight the limitations of traditional methods to solve WLO under cost constraints. Then, we propose a method relying on Bayesian optimization to address the resource-constrained WLO problem. Experimental results show that our proposed algorithm outperforms the latest conventional approaches, improving solution quality by up to 300% compared to Uniform Word-Length Optimization (UWLO) and by up to 170% compared to a state-of-the-art classical WLO approach.

The remaining sections are organized as follows. In Section II, we provide background information and discuss related work, followed by an explanation of our motivations and problem statement in Section III. In Section IV, we describe our proposed Bayesian optimization-based method. In Section V, we compare the performance of our approach with traditional approaches on various benchmarks, before to conclude in Section VI.

II. RELATED WORK

In this section, the well-studied accuracy-constrained Word-Length Optimization (WLO) problem is presented, followed by the related state of the art. Then, we motivate the need for a new method to solve a cost-constrained WLO problem.

A number expressed using fixed-point arithmetic has both integer and fractional word-lengths (WLs), which are correspondingly represented by I and F bits, respectively. The dynamic range of the number represented with this format is covered by the integer WL, while its accuracy is controlled by the fractional WL. In this paper, we concentrate on the WLO for the fractional WL since it is the most time-consuming exploration in the design process. The dynamic range is usually evaluated through static analysis or simulation to determine the integer WL to guarantee no or low-probability overflow during computations.

Let the vector W = [W 0 , W 1 , . . . , W N -1 ] denote a wordlength configuration with N effective variables from the considered application to be explored for fixed-point conversion. The main objective of WLO is to determine a good-enough word-length configuration that minimizes a cost function under a quality (accuracy) constraint:

min C(W ) Subject to λ(W ) ≥ λ obj , (1)
where C and λ are the cost and accuracy functions depending on W , respectively. λ obj is the maximal degradation of the accuracy at a given application output that is acceptable for the required quality of service. Analytical models and simulationbased methodologies are both used to construct C and λ.

There are two approaches to solve this well-studied accuracyconstrained WLO problem: simulation-based and analytical approaches. The aim of analytical approaches is to approximate quality and cost functions of the WLO problem to be convex functions which can be solved quickly by some convex optimization algorithms, e.g., CVX [START_REF] Grant | cvx users' guide[END_REF]. Existing techniques are limited to modeling noise power metrics of Linear and Time-Invariant (LTI) systems (with some extensions) [START_REF] Parashar | A polynomial time algorithm for solving the word-length optimization problem[END_REF], [START_REF] Chan | Wordlength optimization of linear timeinvariant systems with multiple outputs using geometric programming[END_REF], [START_REF] Fiore | Efficient approximate wordlength optimization[END_REF]. Simulation-based approaches [START_REF] Lee | Accuracy-guaranteed bit-width optimization[END_REF]- [START_REF] Nguyen | Novel algorithms for word-length optimization[END_REF] use simulations and iterative search to address WLO. Uniform WordLength Optimization (UWLO) can quickly evaluates several solutions which are constructed by the same wordlength for each variable to obtain the best one. However, the quality of the solution obtained is much worse than the ones obtained by the nonuniform wordlength optimization methods [START_REF] Choi | Search-based wordlength optimization for vlsi/dsp synthesis[END_REF]. Some more advanced methods leverage noise budgeting techniques [START_REF] Novo | Cracking the complexity of fixed-point refinement in complex wireless systems[END_REF], [START_REF] Ha | Towards generic and scalable wordlength optimization[END_REF] to reduce the exploration time in solving WLO for large applications.

Bayesian Optimization (BO) is a machine-learning-based optimization technique [START_REF] Shahriari | Taking the human out of the loop: A review of bayesian optimization[END_REF] designed to optimize functions that often lack mathematical expressions and/or derivatives. BO typically consists of two essential components: i) a probabilistic surrogate model for modeling the unknown objective function based on previously observed samples and ii) an acquisition function that optimizes over the surrogate model to recommend further samples. The procedure of choosing surrogate models is a key distinction between BO approaches and those of others. While Gaussian Processes (GP) are often employed for moderate-size continuous-domain issues, tree-based models such as Random Forest and Tree-structured Parzen Estimator (TPE) are advantageous for large-size discrete-domain problems [START_REF] Eggensperger | Towards an empirical foundation for assessing bayesian optimization of hyperparameters[END_REF]. TPE identifies points that might have been drawn based on the assessment of a loss function at other points. BO was first used to speed up WLO in a hybrid method [START_REF] Ha | Leveraging bayesian optimization to speed up automatic precision tuning[END_REF].

III. RESOURCE-CONSTRAINED WLO

All the previously mentioned techniques solve the Word-Length Optimization problem without any constraint on the cost of the solution. The objective of the optimizer is to find a solution with minimum cost that guarantees the accuracy being higher than a given constraint. However, as motivated in the introduction, with the growing complexity of applications, it might be of great interest to limit the area or energy cost budget of a computing kernel under a restricted bound. In this case, the stringent constraint becomes the cost and it is important to find the maximal accuracy (or the minimum accuracy degradation) under this cost constraint.

The resource-constrained WLO problem can be stated as

max λ(W ) Subject to C(W ) ≤ C budget , (2)
where C and λ represent for the cost and accuracy functions of a WL configuration vector W . The objective of this problem is to find a WL configuration W o that maximizes the accuracy at the application output given C budget as a cost constraint. The WL components W i of W can be bounded as

m i ≤ W i ≤ n i m i , n i and W i ∈ Z + , (3)
to limit the exploration space.

A. Room for Accuracy Improvement Given a Cost Budget

We use a subset of the solution space of a 5-stage 33-tap Finite Impulse Response (FIR) filter as an example to motivate our work. The filter has 17 different variables represented in fixed-point arithmetic. Fig 1 shows 10 5 solutions obtained from a random search. Each solution evaluates the FIR filter output accuracy (PSNR) and the associated cost (energy in nJ) for a random WL configuration. The integer WL is set to 12 bits for all variables to avoid overflow and the fractional WL of each variable is varied in the range of 3 to 20 bits, which can be considered large enough to reduce the chance of missing good solutions. The energy is evaluated based on a library of arithmetic operators characterized for various WL in a 28nm FDSOI technology (see Sec.V-A for more details). Given an energy level, there is a large set of solutions consuming the same amount of energy but resulting in a significantly different quality at the application output. For instance, with 0.0075 nJ considered as a maximum energy budget, the quality of satisfied solutions can vary from -20dB to 60dB. Hence, to maximize the performance or quality of the system given a cost budget, it is worth to design an optimization method that can obtain the highest quality solution satisfying the cost constraint from possible solutions. PSNR (dB)

B. Resource-Constrained WLO as a New Problem

The classical accuracy-constrained WLO focuses on minimizing cost as much as possible while satisfying QoS at the system output. Meanwhile, the objective of the resourceconstrained WLO is to exploit the available resource (energy and/or area) of the system to enhance its overall performance. The objective of these two problems are totally different, leading to the different exploration targets. For the design purpose that enhances quality on a system with specific resources as much as possible, the search objective of the original problem is no longer relevant. Hence, it is important to design new methods to solve the resource-constrained WLO problem. We illustrate the objective target of the classical WLO for FIR filter in the context of a system with an energy budget. We use TPE as a search method for the classical problem with a PSNR quality target of 25 dB. The use of TPE follows the setup in [START_REF] Ha | Leveraging bayesian optimization to speed up automatic precision tuning[END_REF].

In Fig. 2 Fig. 3: Different exploration trajectories of Tabu search method corresponding to different starting points for the resourceconstrained WLO problem on the FIR application. The energy budget is 0.0075 nJ. Each color corresponds to a search procedure with a given starting point. Fixed-point variables of each procedure are chosen with uniform word-lengths (UWLs) at the starting point. The number in the legend indicates the wordlength given for the UWL. For each procedure, the starting point is further from the cost budget line than the ending point. Some procedures overlapped during the exploration process.

the classical problem are compared to those discovered by Random Search (RS) in the context of a system with an energy budget as 0.0075 nJ. Clearly, TPE focuses on exploring lowcost solutions around the accuracy constraint, i.e. around the vertical line 20 dB, missing possible solutions with higher accuracy and still satisfying the energy budget, i.e., solutions found by RS range in 30 to 40 dB of PSNR. Similar to TPE, recent simulation-based techniques mentioned in [START_REF] Ha | Leveraging bayesian optimization to speed up automatic precision tuning[END_REF], [START_REF] Nguyen | Novel algorithms for word-length optimization[END_REF] such as Min+1 and Tabu Search also rely on a search model to restrict the exploration space, followed by a fine tuning around the quality constraint to find a better solution.

C. Limitations of Classical WLO Methods

Numerous methods to address the WLO problem have been proposed in the literature. In [START_REF] Ha | Leveraging bayesian optimization to speed up automatic precision tuning[END_REF], [START_REF] Cantin | A comparison of automatic word length optimization procedures[END_REF], the performance and execution time of several state-of-the-art techniques, including Min+1, Max-1, Heuristic approach, Tabu Search, and GRASP, have been compared. The comparison shows that variants of bi-directional searches that combine steepest-descent and mildest-ascent procedures, such as heuristic approach [START_REF] Sung | Simulation-based word-length optimization method for fixed-point digital signal processing systems[END_REF] and Tabu search [START_REF] Nguyen | Novel algorithms for word-length optimization[END_REF], outperform the mono-directional searches. However, due to the greedy nature of those algorithms, the final solution is not guaranteed to be a global minima and is dependent on the starting point. Additionally, most of these searches are based on local iterative searches that travel across small distances in the discrete domain. As a result, if the starting point is distant from a local minimum, convergence will be slow [START_REF] Ha | Leveraging bayesian optimization to speed up automatic precision tuning[END_REF].

Fig. 3 illustrates the limitations of the classical methods. We choose Tabu search as a typical method to address the resourceconstrained WLO. The FIR filter with a maximum energy supply as 0.0075 nJ is still used for the illustration. The search procedure is initialized at different points. Each initial point is constituted from fixed-point variables with UWL. The integer word-length of variables is fixed as 12 bits. The fractional wordlength is incremented every two bits from 2 bits to 20 bits, thus the UWL varies from 14 bits to 32 bits for each starting point. The quality of obtained solutions with Tabu search and different starting points significantly differs and strongly depends on the initialization. At starting points with UWL equal to 30 or 32, the obtained solutions have a bad PSNR of around -30 dB. At the remaining initial points, Tabu search produced better solutions than RS from 10 to 50 dB of PSNR. However, the solution quality is varying dramatically. Besides, procedures that start farther away from the cost constraint tend to converge more slowly than those that start closer to it. The trends are consistent with other benchmarks evaluated in Section V.

IV. SOLVING RESOURCE-CONSTRAINED WLO

In this section, we introduce a BO-based method to address the resource-constraint WLO problem. The loss function is constructed first, followed by the description of our BO-based approach.

A. Loss Function

The resource-constrained WLO problem statement of Eq. 2 is changed as follows using the Lagrange multipliers approach to turn a constrained problem into an unconstrained one:

f (W ) = -λ(W ) + α(C(W ) -C budget ), (4)
with a positive and big enough α. By minimizing the loss function, TPE tends to sample more frequently solutions that are of high quality and satisfy the cost constraint. The factor α is important for exploration. If α is very small, TPE will ignore the restricted cost condition and concentrate on exploring solutions with quality as high as possible. If α is very large, the solution of TPE would be highly dependent on the restricted cost condition. That is, TPE focuses on finding solutions that cost less than C budget while neglecting solutions of high quality.

Based on many WLO experiments, we found that the cost and quality functions tend to be proportional to W . A high quality solution usually comes at a high cost. Thus, the best solutions to satisfy the cost constraint with high quality are likely to be those around the cost budget C budget . Therefore, for faster convergence, we force the loss function to cover only a narrow range [c l , c h ] around C budget . Solutions with costs outside this range of interest are penalized with the positive infinite loss value. The loss function is then defined as

f (W ) = -λ(W ) + α(C(W ) -C budget ) if C(W ) ∈ [c l , c h ], +∞ otherwise. (5)

B. TPE Algorithm

For the resource-constrained WLO problem of Eq. 2, a WL configuration formed by the choice of each variable, i.e., W i ∈ [m i , n i ] represents a solution. All possible combination of different values of W i creates a solution space of this WLO problem. With the leverage of Bayesian Optimization relying on TPE [START_REF] Bergstra | Algorithms for hyper-parameter optimization[END_REF] to solve the WLO problem, each W i is considered as a hyper-parameter to be tuned. Each W i is initially mapped to a prior of quantized uniform distribution in which the value is sampled uniformly in [m i , n i ] and then rounded to the nearest integer value.

Algorithm 1 describes the TPE algorithm for the resourceconstrained WLO problem. The algorithm is first initialized by uniform word-length (UWL) configurations and corresponding loss values evaluated by the loss function. These configurations are then served as first samples in an observation history H. TPE works as an iterative approach. At each iteration, a WL configuration W i is evaluated by the loss function f (W i ). Then, obtained samples {W i , f (W i )} stored in the observation history H are divided into two groups. The first group contains good samples where the loss f i is less than a threshold γ * , whereas the second group consists of the remaining, considered as bad samples. TPE uses these two sample groups to model two likelihood probability density functions l(W ) = p(W |f (W ) < γ * ) and g(W ) = p(W |f (W ) ≥ γ * ), respectively. Then, the TPE algorithm decides which hyperparameter to try in the next iteration by maximizing the ratio

l(W ) g(W ) = p(W |f (W ) < γ * ) p(W |f (W ) ≥ γ * ) . Algorithm 1 TPE algorithm 1: H ← Uniform Word-Length Initialization 2: for i ∈ [1, . . . , T ] do 3: W * = argmax l(W ) g(W )
4:

Evaluate f (W * )

5:

H ← H ∪ (W * , f (W * )) 6:
Update l(W ) and g(W ) given H 7: end for Our initialization procedure constructed from UWL configurations samples the solution space at some design points ranging from low-accuracy-and-low-cost solutions to highaccuracy-and-high-cost solutions. These design points are then used to construct density functions at the very first iteration of the algorithm. Under the evaluation of the loss function, TPE can start exploring from the points that yield low loss values. This initialization procedure provides better results and faster convergence than an initialization constructed from random configurations.

V. EVALUATION

A. Experiment setup

We choose Uniform Word-Length Optimization (UWLO) as our baseline. Our goal is to evaluate how much our approach can perform better than UWLO, a fast and simple method considering all variables having the same WL. In our approach, we employed Adaptive TPE (ATPE), an improved extension of TPE that was implemented on Hyperopt [START_REF] Bergstra | Algorithms for hyper-parameter optimization[END_REF] to update the hyper-parameters of TPE in real time. The cost and quality functions, λ(W ) and C(W ), in the loss function are normalized within a range. First, UWLO is performed to obtain the quality and the cost of uniform-wordlength solutions. λ(W ) is normalized between upper and lower bounds. The lower bound is the solution which satisfies the cost constraint and has the highest quality. The upper bound is the highest quality value obtained from those solutions without the cost constraint. The normalization range of C(W ) is [c l , c h ], where c l and c h are separated from the central value, C budget , by a distance of 10% of the difference between the minimum and the maximum cost values obtained from the UWL solutions. We conduct experiments on our benchmarks with five values of α, i.e., α = {0.1, 0.2, 0.3, 0.4, 0.5}, to evaluate its impact to the approach. The number of iterations for each experiment is chosen as 5000 which can cover mostly the possibility of improving the solution quality as much as possible. Note that more complex applications would need more iterations to increase the possibility of obtaining better solutions. To further demonstrate the limitations of classical methods, we conduct several experiments with Tabu search [START_REF] Nguyen | Novel algorithms for word-length optimization[END_REF] on our benchmarks. In each benchmark, Tabu Search is initialized at different uniform wordlengths.

All experiments were conducted on a Linux-powered Intel Xeon E5640 processor at 2.67GHz with 4GB of RAM. Energy is the basis for our cost model. The energy model counts the operations carried out by each operator and determines the overall energy cost using the empirically determined energy consumption of an operation from several synthesis for various WLs. An operator is characterized by the WLs of the operands, the WL of the result, and the arithmetic operation performed. A 28nm FDSOI technology is used for characterization along with Synopsys Design Compiler and Prime Time.

We use three applications as the benchmarks to evaluate our approach: FIR, IIR and NLM. The FIR filter, also used in previous sections, is implemented with a 5-stage cascaded structure of 33 taps each. The IIR filter has a 5-stage cascaded structure of 2 nd order filters. The number of effective variables to optimize is 17 for FIR and 33 for IIR. Non-local means (NLM) is a denoising approach in image processing. To denoise the target pixel, the algorithm calculates the mean of all pixels in the picture, weighted by their similarity to the target pixel. The implementation of NLM uses 19 effective variables. We explore only the fractional word-lengths in the resourceconstrained WLO problem. The integer word-length of each signal in the applications is determined beforehand so that the dynamic range of each signal is covered. The integer word-length of FIR, IIR and NLM are set to 12, 12 and 6, respectively. The word-lengths of the variables used in FIR and IIR range from 13 to 32 bits, whereas NLM is explored from 8 to 32 bits.

We use Peak Signal to Noise Ration (PSNR) for filters and Structural Similarity Index Measure (SSIM) for NLM as quality metrics. We choose three cost budget objectives for every application as shown in Table I, thus resulting in nine different benchmarks. The targets are selected proportionally to the operation counts of each application, i.e., the number of additions and multiplications. The cost budgets of NLM are higher than the remaining applications due to its higher computation complexity.

B. Performance Evaluation

Figure 4 presents a comparison between our approach and UWLO serving as a baseline. In overall, our approach outperforms UWLO by obtaining solutions with better quality for a given cost budget. The results show that solutions obtained by our approach with different values of α are always better than those obtained by UWLO. Table II summarizes the quality improvement of our solutions compared to those obtained by UWLO. In several cases, including FIR (Target 1, 2 and 3), IIR (Target 1 and 2), and NLM (Target 1), our approach significantly improves the quality of the solutions. This indicates that a solution can be obtained quickly by UWLO, but which is far from the optimal; this also confirms the conclusion in [START_REF] Choi | Search-based wordlength optimization for vlsi/dsp synthesis[END_REF]. Fig. 4: Performance comparison between our approach and UWLO in terms of quality of results for different benchmarks. The results are normalized with the highest quality value indicated in parentheses for each benchmark. 3.32% For some benchmarks, the quality of solutions obtained by explorations with small α, such as α = {0.1, 0.2}, is worse than for higher α values. As explained in Section IV-A, the cost constraint will be not respected with small α. This means Fig. 5: Comparison of the normalized quality of solutions obtained by Tabu Search and our approach for different benchmarks. The solutions obtained by Tabu search are the best found solutions under the target cost constraint, but for different starting points. Our solutions are obtained with different α. The normalization range are mentioned in parentheses, where the numbers above and below represent for upper and lower bounds, respectively. Letter "T" is the abbreviation of "Target". that the exploration strategy will focus more on improving the solution quality without being strictly constrained by the cost budget, which causes an excessive search in the infeasible region. As a result, this will reduce the possibility of improving the quality of solutions in the feasible region. Figure 5 illustrates a comparison of the quality of solutions found by our proposed method and Tabu Search using various benchmarks. The quality of results found using Tabu Search varies a lots among benchmarks, the solution strongly depending on its starting point. In the meanwhile, the quality of solutions obtained via our method is much more stable. Figure 5 also illustrates that the average quality obtained by our method is always superior than that of Tabu search. Table III reports the average improvements of the quality of solutions provided by our method over Tabu search. The table shows that the solutions can be improved from 20% to 173% using our optimized BO-based algorithm to solve the resourceconstrained WLO problem. Especially, for IIR with target 1 and target 2, our method outperforms Tabu by up to 158% and 173%, respectively.

VI. CONCLUSION

In this paper, we present our Bayesian-Optimization-based approach to maximize the quality of digital applications im-plemented under a resource-constrained budget. We first show the importance of the resource-constrained word-length optimization (WLO) problem in systems which have limited silicon area and energy supply. Then, we highlight the limitations of classical approaches for WLO, such as Tabu search, which are not well adapted to solve the resource-constrained WLO problem. We thus propose a method based on BO and the TPE algorithm with an adaptive loss function. The results from our experiments show that our approach significantly outperforms UWLO and Tabu search, a state-of-the-art method. Our method is the first efficient approach addressing the resourceconstrained WLO problem and also the starting point for further research that will address other aspects on the scalability and optimality of this problem, as well as other cost constraints, such as limited area or maximum number of resources.

Fig. 1 :Fig. 2 :

 12 Fig. 1: 10 5 random solutions as a representative subset of all possible solutions, for the FIR filter benchmark. The red line represents an energy budget of 0.0075 nJ

  , the explored solutions discovered by TPE for

TABLE I :

 I The benchmarks for the evaluation

	Application Target 1 (nJ) Target 2 (nJ) Target 3 (nJ)
	FIR	0.0065	0.0070	0.0075
	IIR	0.0010	0.0012	0.0014
	NLM	250	300	400

TABLE II :

 II Quality improvement of our solutions in percentage compared to UWLO solutions. Given value for each benchmark is the average of those obtained by different values of α.

	Application Target 1 Target 2 Target 3
	FIR	321.19% 200.48% 90.68%
	IIR	52.39%	50.67% 16.91%
	NLM	160.50%	25.63%

TABLE III :

 III Average improvement of the solutions provided by our approach over those obtained by Tabu Search

	Benchmark	Improvement
	FIR -Target 1	85.93%
	FIR -Target 2	93.40%
	FIR -Target 3	59.40%
	IIR -Target 1	158.93%
	IIR -Target 2	173.46%
	IIR -Target 3	35.94%
	NLM -Target 1	21.79%
	NLM -Target 2	20.61%
	NLM -Target 3	55.52%


Utilization Deficiencies and Transfer of Strategies in Preschoolers

Jérôme Clerc, Patricia H. Miller

The transfer of new cognitive skills learned in one context to another context has been one of the thorniest problems facing both developmental and educational psychologists (e.g., [START_REF] Cox | The rediscovery of the active learner in adaptive contexts: A developmental-historical analysis of transfer of training[END_REF][START_REF] Salomon | Rocky roads to transfer: Rethinking mechanisms of a neglected phenomenon[END_REF]. For example, why does a child trained to use a new strategy on one task have such difficulty transferring it to another, seemingly similar, task? In educational settings, the issue is a practical one---teaching children to use strategies that will improve their learning across various academic tasks, materials, and settings. For developmental psychology, the theoretical issue has been: What are the developmental processes involved in moving from first acquiring a strategy to its later transferable, general, cross-task efficient application? We take a new perspective on the issue of transfer by examining it in terms of strategy utilization deficiencies (e.g., [START_REF] Miller | Strategy utilization deficiencies in children: When, where, and why[END_REF]), in which a child produces a relevant strategy but it provides little or no benefit for cognitive performance (e.g., recall).

Over 40 years ago, Flavell (e.g., [START_REF] Keeny | Spontaneous and induced verbal rehearsal in a recall task[END_REF] remarked on the problem of transfer in children when he prompted nonstrategic children to produce a good memory strategy, and they were able to do so, but they often did not transfer the strategy to a later session in which the strategy was not prompted. Insuring the transfer of a prompted or instructed strategy continues to be a challenge today [START_REF] Klahr | Finding one's place in transfer space[END_REF]. The issue remains an important one, for rarely are two tasks calling for the same strategy exactly the same in children's daily lives; at least modest transfer nearly always is required.

One theoretical concept from strategy research that might shed light on transfer is a utilization deficiency (UD). Numerous researchers (e.g., [START_REF] Coyle | The development of strategic memory: A modified microgenetic assessment of utilization deficiencies[END_REF][START_REF] Cox | The rediscovery of the active learner in adaptive contexts: A developmental-historical analysis of transfer of training[END_REF][START_REF] Miller | Strategy utilization deficiencies in children: When, where, and why[END_REF][START_REF] Schwenck | Factors influencing the incidence of utilization deficiencies and other patterns of recall/strategy-use relations in a strategic memory task[END_REF] have found that memory strategy production sometimes does not result in the expected enhancement of recall. A UD can occur both when the initial strategy production is spontaneous [START_REF] Miller | Strategy utilization deficiencies in children: When, where, and why[END_REF] and when it requires an adult's prompt (Bjorklund, Miller, Coyle Slawinsky, 1997). [START_REF] Miller | Strategy utilization deficiencies in children: When, where, and why[END_REF] argued that UDs frequently can be attributed to a greater cognitive load than the child can support. This quantity-of-resource explanation particularly applies to young children, because of their limited processing capacity in working memory. That is, once a strategy becomes stronger and more automatic so that it requires less cognitive load, then resources are released for the successful application of the strategy in the service of recall. As evidence of the link between cognitive resources and UD, children with greater memory spans are less likely to show a UD [START_REF] Woody-Dorning | Children's individual differences in capacity: Effects on strategy production and utilization[END_REF]. Moreover, when the effortfulness of producing a strategy is reduced by having an adult execute the strategy, UDs decline (DeMarie-Dreblow & Miller, 1988;[START_REF] Miller | The role of strategy effortfulness in strategy effectiveness[END_REF].

During the last two decades, considerable research has focused on the specific conditions in which a UD does and does not occur (e.g., [START_REF] Bjorklund | Utilization deficiencies in the development of memory strategies[END_REF][START_REF] Miller | Strategy utilization deficiencies in children: When, where, and why[END_REF][START_REF] Schneider | The development of young children's memory strategies: Evidence from the Würzburg Longitudinal Memory Study[END_REF][START_REF] Schwenck | Factors influencing the incidence of utilization deficiencies and other patterns of recall/strategy-use relations in a strategic memory task[END_REF]. We propose that transfer may be one such condition causing a UD, due to the challenge of utilizing a strategy when the task is changing.

To explain this possibility, it first should be noted that researchers have focused on possible internal causes of UD such as limited working memory capacity, poor metacognitive or declarative knowledge, and lower intelligence (e.g., Miller & Seier,1994;[START_REF] Schneider | The development of young children's memory strategies: First findings from the Würzburg Longitudinal Memory Study[END_REF]. This focus on internal causes ignores important transactions between individuals and their contexts (external causes), including the physical aspects of the task. Only an integration of both internal (resources, knowledge) and external (environmental variations) determinants of strategy use can produce a satisfactory understanding of both UD and transfer. A child, as an "in-context" strategy user and rememberer, may or may not manifest a UD, depending on task features and especially tasks' familiarity: The more familiar the task is, the easier and the less resource-demanding it likely will be for children to apply their memory strategies to it. A transfer task, because of its new physical features, may be an external influence on performance. Thus, whether the available resources are sufficient for avoiding a UD depends not only on internal factors (e.g., working memory) but also on task demands [START_REF] Logan | Toward an instance theory of automatization[END_REF].

Prior resource accounts have had a limited focus--on freeing resources for encoding items to be remembered and thus reducing UDs--rather than the high resource-demanding entire strategy cycle that includes not only producing the strategy and self-monitoring, but also adapting the strategy to the specific task demands. The latter is particularly important when a child attempts to transfer a strategy to a new context and is faced with a similar but slightly different task. During transfer, children try to adapt their strategy to a changed task, in order to benefit both from what they already understand of the task and from what they currently are discovering as new unexpected features.

Several studies of memory organizational strategies (grouping, clustering) are relevant to the issue of transfer in that they have introduced systematic variations from one phase to another during the same session or from one session to another. In particular, Bjorklund and Schneider and their colleagues [START_REF] Bjorklund | Developmental differences in the acquisition and maintenance of an organizational strategy: Evidence for the utilization deficiency hypothesis[END_REF][START_REF] Bjorklund | Training and extension of a memory strategy: Evidence for utilization deficiencies in the acquisition of an organizational strategy in high-and low-IQ children[END_REF][START_REF] Coyle | The development of strategic memory: A modified microgenetic assessment of utilization deficiencies[END_REF][START_REF] Schwenck | Factors influencing the incidence of utilization deficiencies and other patterns of recall/strategy-use relations in a strategic memory task[END_REF][START_REF] Schwenck | Developmental and individual differences in young children's use and maintenance of a selective memory strategy[END_REF] introduced new words and categories at these transitions on free recall tasks. UDs sometimes emerged. For example, children might remain strategic over the transitions, but recall declined, thus demonstrating a UD [START_REF] Bjorklund | Training and extension of a memory strategy: Evidence for utilization deficiencies in the acquisition of an organizational strategy in high-and low-IQ children[END_REF] However, simply varying the items introduced very little task change; the phases or sessions were virtually identical in the layout of the task, instructions, etc. Thus, a stronger test of transfer, with physical changes in the transfer task that make it less clear that it is the same task, is needed. Our study asks: How do broader physical task changes force children to deal with a changing task and what is the cost to the child's resources and, consequently, strategy transfer?

In these more challenging transfer situations, looking at UDs as a partial cause of the difficulty of transfer of recall may provide a new avenue for understanding and facilitating transfer. Even if it appears that a memory strategy has been transferred successfully, the operation of a UD may be reflected in lower recall on the transfer task. Also, by shifting the focus of UD from internal factors to their interaction with external factors involved during transfer, we hope to advance theorizing concerning UDs. UDs may be especially frequent during attempted transfer because of the cognitive resources required to analyze the new task with respect to strategy choice. Thus, UDs may occur not only when producing a strategy in a single task but also when attempting to transfer an effective strategy to related tasks. In this sense, this UD would be a new type of UD.

The choice of transfer tasks is critical and must be based on theoretical considerations in order to advance our understanding of processes underlying transfer. In contemporary studies of cognitive transfer, the original task and the transfer task can be compared along two different but complementary dimensions: structural similarities and surface similarities (e. g., [START_REF] Gamo | Transfer of strategy use by semantic recoding in arithmetic problem solving[END_REF][START_REF] Gick | The cognitive basis of knowledge transfer[END_REF]Holyoak & Koh,1987;[START_REF] Sternberg | Mechanisms of transfer[END_REF]. Structural elements of a task are major ones, which define the causal chain one has to understand in order to solve the task. Surface elements are rather minor ones that can be modified from one task to another without changing any causal elements.

Because children often fail to transfer even to a structurally similar task if there are surface changes (for a review, see [START_REF] Pressley | Cognitive strategies[END_REF], our study focuses on this sort of transfer. Even after a child grasps the structural similarity of the two tasks, what does the child do about the changed surface features? A child must decide whether the transfer task calls for the strategy used in the first task and then figure out how to adapt the strategy to the specific features of the transfer task. Will the child ignore surface dissimilarities between tasks? This may reduce cognitive load because it reduces the amount of information that has to be dealt with, but at the risk of not recognizing an important element for successful adaptation of the strategy. Or, will the child take the high-effort route of carefully examining all surface dissimilarities in order to achieve an optimal adaptation of the strategy? That is, during transfer the child's perspective may be: "I know what to do (produce the strategy), but how do I do it (what about surface elements?)?" Uncertainty concerning how to take surface dissimilarities into account always is present in a case of strategy transfer, and children must manage this uncertainty-a resource-consuming process that may contribute to a UD.

In our first experiment, we hypothesized that the introduction of transfer tasks would cause a UD. That is, the strategy would transfer but strategy effectiveness would not. A selective attention strategy task developed by [START_REF] Miller | Children's attention allocation, understanding of attention, and performance on the incidental learning task[END_REF] served as the common structure between tasks. That is, the goal of all tasks was to remember only one category of items and thus to attend selectively to only those items, and not to items from another category, during study time. Indeed, our work is confined to memory strategies and does not address other kinds of cognitive strategies. Each of two transfer tasks varied systematically from the main task in two out of three specific surface elements: a two-dimensional versus three-dimensional layout, a single unified array versus that array arranged in 12 pieces, and vertical (upright) versus horizontal (flat on table or floor) orientation of the materials. These visual differences were associated with minor motor differences in the selective strategy, specifically, opening a little door (3D) versus turning over a card (2D), moving the hand a small distance (one array) versus a further distance (12-piece array), and moving the hand higher and lower (vertical array) versus towards and away from the self (horizontal). This systematic variation permits us to determine the features that affect transfer or UDs, if the two transfer tasks differed in transfer or UDs. It also permits some generalization of our findings.

One theoretically important comparison in our study is children who initially were strategic ("spontaneous strategic") versus initially were nonstrategic and became strategic only after being prompted to produce the strategy ("prompted strategic"). One question was whether the latter could easily be prompted to adopt and to utilize the selective attention strategy. It typically is difficult to prompt or teach 3-to-5-year-olds standard memory strategies used by older children (e.g., [START_REF] Schwenck | Developmental and individual differences in young children's use and maintenance of a selective memory strategy[END_REF]. Because preschoolers' memory strategy production usually is low, it has been difficult to test them for UDs, and thus few studies (Woody-Ramsey & Miller, 1988) have examined UDs in preschool children.

Another question about spontaneous strategic and prompted strategic children was whether they would differ in the degree of strategy transfer and/or UD during transfer. On the one hand, spontaneous strategy use has been considered effortful [START_REF] Miller | Age differences in the capacity demands of a strategy among spontaneously strategic children[END_REF], and thus likely to cause a UD, especially when the strategy is recently acquired. Accessing possible strategies in one's repertoire, selecting one strategy, and executing it appropriately are all resource demanding and thus might lead to UDs during transfer. On the other hand, spontaneous strategy production could be advantageous for recall, because these children would not need to devote resources to learning the strategy but prompted strategic children would. Note that both groups of children would appear equally strategic on the surface (both highly strategic) on both main and transfer tasks. However, underlying differences might surface during transfer when children are faced with the uncertainty of whether and how to transfer, with one group showing more UDs (good strategy production but less recall) than the other.

In order to provide a fine-grained look at the above issues, we added trial-by-trial, including microgenetic, analyses to the overall group analyses. Such analyses provide a more detailed picture of change as it is occurring [START_REF] Siegler | The microgenetic method: A direct means for studying cognitive development[END_REF].

1. Experiment 1 1.1. Method 1.1.1. Participants
The 113 children included 38 4-year-olds (46 to 49 months, 18 boys and 20 girls), 38 4.5-year-olds (53 to 55 months, 19 boys and 19 girls), and 37 5-year-olds (60 to 62 months, 19 boys and 18 girls), with parental permission. The children were French and predominantly middle class. They lived in a suburban area surrounding the city of Lille in France. Children were tested individually in a quiet room at their school.

Apparatus and Materials

Pretest, prompt, and main task phases. For the first three phases we used an apparatus similar to that of [START_REF] Miller | Children's attention allocation, understanding of attention, and performance on the incidental learning task[END_REF]. It consisted of a wooden box which was 65 cm long, 25 cm high and 11 cm deep. On the face of the box were twelve doors, each of which could be opened by a knob. Each door (9.5 cm square) had a drawing (7 cm square) of a house (designating a household item behind the door) or a cage (designating an animal). Once children opened a door, they saw a black and white drawing, either of an animal (e.g. duck, horse, butterfly) for "cage" doors or of a household object (e.g., table, chair, broom) for "house" doors. The drawings of animals and household objects in all tasks came from a study by Cycowicz, Friedman, Rothstein, and Snodgrass (1997) in which norms of familiarity and visual complexity were computed based on answers from 5-and 6-year-olds. This database was adapted for French studies by Alario andFerrand (1999). From this, we selected items with rather high levels of familiarity and rather low visual complexity. These drawings were placed 9 cm behind the door so that the entire wooden box presentation was 3D. The front of the box presented two spatially corresponding rows of six doors each. In each row, no more two items from the same category (house or cage) could be placed next to each other, and in each column the two items could not be from the same category. The back of the wooden box, where the 12 line drawings were inserted, could be opened in order to change the locations of drawings from trial to trial. Children had to remember where the six relevant line drawings (house items or animals, depending on their condition) were located in the box. An identical second set of these six drawings was used to test recall. Because the box was upright, with all doors facing the child, the presentation was vertical rather than flat on the table (horizontal) as in one transfer task.

This main task thus can be labelled a 3D-one piece-vertical (upright) task. The two transfer tasks that followed--2D-one piece-horizontal (flat on table) and 3D-12 pieceshorizontal--constituted variations of this main one. They were designed to be moderately different in layout from the main task (each with two surface dissimilarities with the main task), while still having the same basic logic of selecting items from one category and not selecting items from a second category present (structure similarity). Items were comparable on both criteria of familiarity (high) and visual complexity (low) in the three tasks (F<1, ns).

Moreover, all 36 items, when pronounced, had one or two syllables, with three exceptions (one item in each task was three syllables long).

Cardboard transfer task: 2D-one piece-horizontal (flat on table).

A 56 cm X 20 cm piece of white cardboard was laid out horizontally (flat) on the table. On the cardboard, twelve square outlines held twelve cards, with each card separated from its neighbour by 2 cm. Each card was a 7 cm square and had on its visible side a categorical sign (house or cage, the same as in the main task) and on its hidden back the drawing of either an animal or a household object, depending on the category of the card. All the animals and household objects were different from those used in the main task (see Appendix). Cards were displayed in two spatially corresponding rows as in the main task, except that they were horizontal.

Placement had the same constraints concerning the locations of categories as in the main task.

Carpet transfer task: 3D-12 pieces-horizontal. We used a children's colorful carpet, 131 cm long and 90.5 cm wide, designed for playing with little cars. It contained several drawn items that usually surround automobiles, such as roads, factories, houses, and parking places. The carpet was set on the floor and we placed on it twelve 10 cm X 10 cm X 10 cm wooden 3D boxes. On the top of each box was a door, and 9 cm behind the opened door one could see a 7 cm square line drawing of either an animal or a household object. All items were different from those used in the other tasks (see Appendix). Again each box had a categorical sign (house or cage, the same as in the main task) indicating a household object or an animal inside. The boxes were positioned in two rows of six boxes, as in the two other tasks, but the distance was 10 cm between two boxes on the same row and 30 cm between one row and the row above it. Consequently, boxes were spread over the entire carpet. Thus, this was a 3D-12 pieces-horizontal task. Again, this transfer task differed from the main one only in surface features, particularly 12 pieces-horizontal versus one piece-vertical, since the same selective attention strategy was appropriate.

Procedure

The study included five phases-each a selective memory task in which children were to remember the exact location of the six items belonging to the relevant category when a total of 12 items was present. All children first were given a pretest (2 trials), which assessed whether they were strategic or nonstrategic. Nonstrategic children then, in the second phase, were prompted to use the selective strategy-selecting and looking at only the relevant items, the items to be remembered (2 trials), while children who were strategic simply had two more trials as in the pretest. Next, after a half-day (i.e., afternoon of the same day), came the main task (6 trials). These first three phases used the first task described above. On the next day were two transfer tasks (6 trials in each), separated by a half-day (i.e., morning and afternoon).

Pretest strategy assessment (2 trials). Spontaneously strategic versus nonstrategic children were identified on two trials in a pretest at the start of the first session. At the beginning of the task, participants were allowed a few minutes to freely manipulate the box and the recall set of drawings to ensure that they would not open all the doors during the experiment simply out of curiosity. Then they were told that each door would only reveal one type of item, depending on its category (e.g., animals are under cage doors). Four questions ensured that the children understood this systematic categorical correspondence between a door sign and the hidden drawing behind it: "Behind a cage door, can you find a table ?   A duck? Behind a house door, can you find a butterfly? A chair?" The experimenter then told participants that they could open any doors they wanted in any order, but one at a time (each door had to be closed before opening the next one) during thirty seconds, and that immediately afterward they would have to recall the exact location of all animal items (vs. household items, depending on the trial). After 30 seconds of free door-opening, the children were given the six drawings of the relevant category one by one. For each item, participants had to touch or point to the door hiding the identical drawing. Their recall score for each trial was the total number of relevant item locations recalled. The relevant category for the two trials was the same--either animals or household objects, with half of the children given each category. Opening doors for 30 s and trying to recall the location of the six drawings constituted a trial. Between the two trials the back of the box was opened, allowing the experimenter to change the drawings' locations, but the same drawings, signs on the doors, and systematic categorical correspondence of doors and drawings were kept.

Strategy prompt phase (2 trials).

Children who failed to use the selective attention strategy on the two assessment trials described above (i.e., who opened more than one irrelevant door on both trials) were considered not spontaneously strategic and thus were prompted to produce the selective strategy. The experimenter prompted the strategy by telling the children which doors to open (i.e., only the ones for the particular category to be recalled), explaining why they should open only those doors (opening others is unnecessary and resource-consuming), and demonstrating the selective strategy (i.e., correctly opening only the relevant doors) twice. Then the two selective memory trials were given (with no demonstrations or explanations), to ensure that these originally-nonstrategic children understood and correctly executed the strategy, which all these children were able to do.

Children had the same category (animal or household object) as relevant as in the pretest.

The spontaneously-strategic children (who had opened only relevant doors in the two trials of the pretest, allowing for one mistake per trial), did not receive the strategy prompting.

Instead, after the two trials of the pretest they simply performed the two additional selective memory trials given to the prompted strategic children (i.e., with no demonstrations or explanations). This equalized the amount of experience of the groups; both had a total of four trials before beginning the main task.

Main task (6 trials): 3D-one piece-vertical (upright).

The main task used the same procedure, apparatus, and type of materials as in the strategy assessment phase. The same 12 drawings were used on all trials of this main task; only their locations varied from trial to trial (randomly). Each participant was given 6 trials, three with the animal category and then three with the household object category (or vice versa for half the children). In between each set of three trials was a brief discussion with the child intended to avoid proactive interference and perseveration. The main task lasted roughly 20 minutes per child.

Cardboard transfer task (6 trials): 2D-one piece-horizontal (flat on table).

The procedure was the same as in the main task: after a brief period of letting children freely manipulate the materials, they could turn over whichever cards they wanted one at a time during a 30-second period and had to recall the locations of items from the relevant category.

Again, there were three trials per category. Thus, the differences between this transfer task and the main task lie in the task's presentation format (2D-horizontal vs 3D-vertical) but not in the logical strategic response to be applied to the material. That is, the tasks had only surface dissimilarities, not structural ones.

Carpet transfer task (6 trials): 3D-12 pieces-horizontal. The same procedure as for the other tasks was used (i.e., free manipulation then a study period followed by a recall test with cards with line drawings, 6 trials). Half of the children were given the cardboard then carpet transfer tasks, whereas the other half had the reverse order.

A pilot test was conducted on six children to ensure that the tasks were ageappropriate. Two 4-year-olds, two 4.5-year-olds and two 5-year-olds (a boy and a girl at each age) were presented with all the materials. They could label the drawings, and understood the correspondence between categorical signs and their corresponding objects, and task instructions, indicating that both new transfer tasks were feasible for such young participants.

Scoring and identification of spontaneous strategic and prompted strategic children

Following Miller's procedure, in all phases of the study a strategic trial (use of a selective strategy) was defined as a trial in which the child opened only relevant doors, allowing for one error (a maximum of one non-relevant door opened). Children who met this criterion on both pretest trials were designated as spontaneous strategic. The remaining children, who were nonstrategic in the pretest but, after prompting, used the strategy on the next two trials, were designated as prompted strategic.

Results

We first examine strategies and recall separately and then look at whether there are discrepancies between the two, which would be evidence for a UD. Finally we will present microgenetic trial-by-trial analyses. Two preliminary analyses of variance (ANOVA) performed on type of transfer task (2D-one piece-horizontal vs 3D-12 pieces-horizontal ) showed no significant difference, p>.10, for either strategic score or recall, and did not interact with age. Thus, type of transfer task was collapsed in subsequent analyses so that we could examine whether performance declined from the first transfer task to the second one.

Strategic score

We conducted a 3 (task: main, transfer 1, transfer 2) X 3 (age) X 2 (strategy group: spontaneous strategic, n=49, versus prompted strategic, n=64) ANOVA on the number of trials (6 per task for a total of 18) on which the selective strategy was used. There was a significant effect of strategy group, F(1,107)=9.63, p0.01,  2 =0.08, and a significant Age X Strategy Group interaction, F(2,107)=3.31, p0.05,  2 =0.06. As Figure 1 indicates, children identified as spontaneous strategic on the pretest more frequently used the selective attention strategy over the three tasks than did children who became strategic only after the prompts, but this is due to the youngest group-the 4-year-olds (M=6.0 [maximum possible=6], SD=0 and M=4.7, SD=2.2 for spontaneous strategic versus prompted strategic, respectively), as shown by the planned comparison for these groups, F(1,107)=12.5, p0.01,  2 =0.11.

Children of all ages were almost perfectly strategic on the three tasks if they had been spontaneously strategic on the pretest. The same was true for the two older groups, even if they had been nonstrategic on the pretest; that is, once prompted they continued to be strategic. The youngest prompted strategic group, however, had some trouble maintaining the prompted strategy. Still, they were rather strategic even on the transfer tasks, with a mean of 4.2 strategic trials out of 6 trials, which is significantly better than chance, t(26)=10.1, p<0.01.

In fact, all six subgroups (3 ages X spontaneous strategic vs. prompted strategic) had strategic scores greater than chance on the transfer trials, all p's<0.01. For these t tests we computed a mean strategic (selectivity) score to compare with chance (.50) for each subgroup as follows:

First, for each transfer trial for each child, the number of relevant doors opened was divided by the total number of doors opened to obtain a selectivity score. Then, we computed the child's mean selectivity score across the 12 transfer trials, and then the mean selectivity score for each of the 6 groups, which was compared to .50 (chance). Thus, all groups were generally strategic, even on the transfer tasks. Consistent with this, there was no effect of task (i.e., strategy production did not decline between the main task and the transfer tasks), p0.10.

That is, the children successful transferred the strategy from the main task to both transfer tasks. Finally, age was nonsignificant, p0.10.

[Insert Figure 1 about here]

Recall score

A similar 3 (task) X 3 (age) X 2 (strategic spontaneity) ANOVA on recall showed significant effects of strategic spontaneity, F(1,107)=4.3, p0.05,  2 =0.04, and task, F(2,214)=18.5, p0.01,  2 =0.15; see Figure 2. Spontaneously strategic children recalled better (M=2.58 per task, SD=0.12) than prompted strategic ones (M=2.28, SD=0.10). Recall was better on the main task (M=2.69, SD=1) than on the first transfer task (M=2.19, SD=0.96), F(1,107)=24.9, p0.01,  2 =0.19) and the second transfer task (M=2.25, SD=0.95), F(1,107)=26.5, p0.01,  2 =0.2. A correction for multiple testing was conducted: Bonferroni tests confirmed significant differences between the main and first transfer task, t(214)=5.87, p0.01, and main and second transfer task, t(214)=5.16, p0.01.The two transfer tasks did not differ (F1, ns). Age showed a marginally significant effect, F(2,107)=2.69, p=0.07,  2 =0.05.

All interaction effects were nonsignificant, p0.10. Thus, in contrast to continued strategic behavior during transfer, recall declined during transfer, which suggests a utilization deficiency. Overall, recall was lower for prompted strategic children.

[Insert Figure 2 about here] One alternative interpretation needs to be addressed, however. Because the pretest and prompted trials provided four trials very similar to the trials in the main task (same big wooden box and items), perhaps these extra four trials facilitated recall on this task. In this case, better recall in the main task than in both transfer tasks would reflect inflated recall on the main task, due to a training effect from these four trials, rather than decreased recall due to transfer to a different task. To rule out this possibility, we conducted another 3 (task) X 3 (age) X 2 (strategic spontaneity) ANOVA, with recall scores unchanged for tasks 2 and 3.

However, the recall scores for the main task differed from those used in the previous analysis since they now were based on the first six trials the children received-the two trials of the pretest, the two trials of the prompted trials (or the two extra pretest trials for the spontaneously strategic children), and only the two first trials of the main task. In this way these first six trials with the main task could be compared with the six trials of each transfer task. The results were very similar to the initial ANOVA: strategic spontaneity, F(1,107)=5.77, p<.05,  2 =0.05, age, F(2,107)=4.25, p<.05,  2 =0.07, and task, F(2,214)=7.25, p<.01,  2 =0.06, had an effect. Spontaneous strategic children recalled more items (M=2.58 per task, SD=1.02) than prompted strategic ones (M=2.16, SD=0.88). The only significant effect of age was between 4 (M=2.03, SD=0.92) and 5-year-olds (M=2.62, SD=0.99), F(1,71)=7.8, p<.01,  2 =0.10). Concerning tasks (the main factor of interest in the present analysis), recall was higher for task one (the "new main task") than for the first transfer task, (M=2.57, SD=0.99 and M=2.19, SD=0.96 respectively), F(1,107)=10.94, p<.01,  2 =0.09), and second transfer task (M=2.25, SD=0.95, F(1,107)=7. 4, p<.01,  2 =0.07).

Bonferroni tests confirmed significant differences between the main and first transfer task, t(214)=3.93, p0.01, and the main and second transfer task, t(214)=3.3, p0.01. The two transfer tasks did not differ (F<1, ns). Thus, again the results suggest a UD.

To more clearly establish the presence of UDs, we conducted another 3 (task) X 3 (age) X 2 (strategic spontaneity) ANOVA, but only on the 101 children who were in fact strategic in the main task (i.e., on at least 5 out of 6 trials). This eliminates those prompted strategic children who were only moderately strategic on the main task and thus permits a clearer test of transfer and UD by assessing recall only in children who in fact had a solid strategy to try to transfer. The resulting number of spontaneous strategic and prompted strategic children was 11 and 21, respectively, for 4-year-olds, 18 and 17 for 4.5-year-olds, and 20 and 14 for 5-year-olds. Task remained highly significant, F(2,190)=14.490, p.01,  2 =0.13. Planned comparisons showed that recall differed significantly between Tasks1 (M=2.75, SD=1.04) and 2 (M=2.32, SD=0.98), F(1,95)=16.26, p<.01,  2 =0.15), and between Tasks 1 and 3 (M=2.30, SD=0.99), F(1,95)=26.18, p<.01,  2 =0.22). Bonferroni tests confirmed significant differences between the main and first transfer task, t(190)=4.71, p0.01, and main and second transfer task, t(190)=4.93, p0.01. All other effects were nonsignificant. Thus, the UD (high strategic performance but lower recall) during transfer held up even with this more stringent test. The lack of significance of strategic spontaneity in this ANOVA, compared to the first one, suggests that if children who first are not strategic actually succeed, after the prompts, in producing the strategy consistently on the main task, their recall over the three tasks is not different from that of spontaneously strategic children.

That is, it did not matter whether children's high strategic score was pre-existing or prompted--both groups showed a decline in recall during transfer.

Individual analysis of transfer-UD patterns

Following recommendations by Schneider [START_REF] Schlagmüller | The development of organizational strategies in children: Evidence from a microgenetic longitudinal study[END_REF][START_REF] Schneider | The development of young children's memory strategies: Evidence from the Würzburg Longitudinal Memory Study[END_REF]), we followed these group analyses with an individual one, to determine how many children displayed a transfer-UD pattern. Children were considered utilization deficient only if they demonstrated either similar strategic scores on the main and first transfer tasks as well as a greater recall score in the main than in the first transfer task, or similar strategic scores on the main and second transfer tasks as well as a greater recall score in the main than in the second transfer task, or both. Following [START_REF] Bjorklund | Developmental differences in the acquisition and maintenance of an organizational strategy: Evidence for the utilization deficiency hypothesis[END_REF], we used the Newman-Keuls test. We used the MSerror revealed by three (one per age group) 3 (task) X 2 (strategic spontaneity) ANOVAs over the three tasks to determine critical values for change in strategic and recall scores: age 4 (strategy : 0.71, recall : 0.39), 4.5 (strategy : 0.58, recall : 0.42) and 5-year-olds (strategy : 0.33, recall : 0.47). For each child, the differences in recall scores and in strategic scores between the main and the first transfer task were compared to these critical values. All children showing a decline in the recall score that was greater than the critical value, and an absence of such a difference in strategic scores, were considered utilization deficient. The same procedure was used between the main and the second transfer task. Thus, this is a rather strict criterion for a UD. Overall, 56 children met these criteria, and after exclusion of 7 who actually were not strategic in the main task (i.e., fewer than 5 strategic trials), 49 were classified as UD -43% of the total sample. Specifically, 10 4-yearolds (4 spontaneous strategic and 6 prompted strategic), 22 4.5-year-olds (12 spontaneous strategic and 10 prompted strategic), and 17 5-year-olds (9 spontaneous strategic and 8 prompted strategic) were considered UD with this procedure.

Microgenetic analysis of individual strategic performance

An examination of individual participants' strategic performance over trials supported and clarified the group analyses. We compared, at each age, spontaneous strategic and prompted strategic children (as assessed on the pretest). Each task was considered either as a strategic session (if the child was strategic on at least 5 of the 6 trials) or a non-strategic session (fewer than five strategic trials). The numbers of prompted strategic and spontaneous strategic children were 27 and 11 for 4-year-olds, 20 and 18 for 4.5-year-olds, and 17 and 20 for 5-year-olds. Among prompted strategic 4-year-olds, 44% had at least one non-strategic session over the three sessions (sign test, p0.10, bilateral), whereas none of the spontaneous strategic children of this age had such a session (sign test, p0.01, bilateral). Thus, consistent with the group analyses, some of the prompted strategic children in the youngest age group sometimes had trouble maintaining their selective strategy. Few of the older children had this problem. Among the 4.5-year-olds, only 20% of the prompted strategic children (sign test, p0.05, bilateral) and 6% of the spontaneous strategic children (sign test, p0.01, bilateral) had at least one non-strategic session. Similarly, among the 5-year-olds, only 18% of the prompted strategic children (sign test, p0.05, bilateral) and 10% of the spontaneous strategic children (sign test, p0.01, bilateral) had at least one non-strategic session.

Figure 3 shows representative examples of strategic and non-strategic performance over trials in four 4-year-old prompted strategic children--two girls (Sophie and Chloé) and two boys (Bruno and Hervé) 1 . Bruno, the most strategic, began the main task by opening each relevant door once and no irrelevant doors, and continued with that through all three tasks except for possibly exploring a bit by opening a couple of irrelevant doors for the first two trials of the last transfer task. Chloé also was generally strategic, but showed more variability over trials than did Bruno in the total number of doors she opened. Sophie moved from being nonstrategic to being strategic but then, interestingly, returned to being nonstrategic for five trials before settling on the selective strategy for the rest of the trials. Hervé was strategic on the main task but then had some trouble maintaining the strategy on the transfer tasks, returning to selectivity only occasionally on those tasks. Both strategic children were rather consistent in the total number of doors opened: all but two trials with 6 opened doors-Bruno ; 8 trials with 6 opened doors and 4 with 7 -Chloé. Sophie and Hervé, with their nonstrategic patterns, showed more intra-child variability in total number of doors opened: 4 trials with 6 opened doors, 3 with 7, 5 with 5-Sophie ; no trials with 6 opened doors, 2 with 8, 4 with 9, 6 with 10-Hervé. Also of interest is the temporary drop in doors opened (relevant doors and total doors) on the first trial of each transfer task (i.e., trials 7 and 13) for all except Bruno.

[Insert Figure 3 here]

Trial-by-trial analysis of subgroups

To provide a fine-grained analysis of trial-by-trial changes, we examined strategies and recall for the 6 cells (3 ages X spontaneously strategic vs prompted strategic) over the 18 trials (see Figure 4). We paid special attention to the temporal evolution of strategic trials within a single task and during the transition from one task to another. Several patterns emerged that go beyond those provided by the above analyses. First, strategy performance was impressively high even on transfer tasks, but recall generally was moderate, with mean recall never reaching 60% (3.6 items) on any trial. In particular, the spontaneous strategic 4year-olds were almost perfectly strategic on the main task but recalled only approximately 2.5 items. This pattern of high strategy production and moderate recall overall, and less recall in the youngest children (even though strategic) is consistent with a UD. That is, fully producing a good strategy does not ensure complete strategy effectiveness, and strategy effectiveness increases with age. Second, the strategy and recall curves over the 18 trials show asynchrony much of the time. That is, the curves are not parallel. For example, an increase in strategic behavior from one trial to the next is not consistently accompanied by an increase in recall. This is considered evidence for a UD. Third, most of the 6 subgroups showed a slight decline in strategic behavior or recall or both on the first trial of the transfer tasks (trials 7 and 13), but then recovered. Thus, there was a slight disruption to performance at first during transfer, consistent with uncertainty and increased cognitive load (see Discussion). In summary, overall the trial-by-trial changes show the sorts of inconsistency in the relations between strategies and recall predicted by a UD perspective. Although any one piece of evidence for UD in this trial-by-trial analysis is not flawless, the set of evidence, together with the earlier analyses, provides strong support for UDs, especially during transfer.

[Insert Figure 4 about here]

Discussion

In general, children transferred the selective strategy to tasks that were superficially different but conceptually the same (i.e., selective recall requiring a selective study strategy).

However, recall declined during the transfer tasks, even when considering only children who were highly strategic on the main task. In other words, with ceiling or near ceiling, and equivalent levels of, strategy scores in the three tasks, recall was significantly lower in the transfer tasks. The children thus showed a UD during transfer when the task changed.

Importantly, these UDs occurred on the transfer tasks regardless of whether children initially, before the main task, were spontaneously strategic or became strategic only after prompts. Specifically, recall scores fell during both transfer tasks for all groups, and no interaction appeared between task and spontaneity. In short, being spontaneously strategic did not help children maintain their recall during transfer at the same level as in the main task. This outcome indicates the pervasiveness of UDs across children during transfer. Moreover, by showing UDs in both spontaneous strategic and prompted strategic children, the present study bridges the two major reviews of UDs-one reporting UDs in spontaneously strategic children [START_REF] Miller | Strategy utilization deficiencies in children: When, where, and why[END_REF] and one reporting UDs in prompted strategic children [START_REF] Bjorklund | Instructing children to use memory strategies: Evidence of utilization deficiencies in memory training studies[END_REF].

This study is the first to clearly show transfer UDs with a memory strategy in preschool children. The few previous transfer studies examining UDs either studied older children, could not clearly show UDs in preschool children because these children could not become strategic enough even with training, examined non-memory tasks, only changed particular items rather than the physical format on transfer tasks, or changed particular items on the training tasks which introduced change even during training and not just during transfer [START_REF] Bjorklund | Utilization deficiencies in the development of memory strategies[END_REF][START_REF] Bjorklund | Training and extension of a memory strategy: Evidence for utilization deficiencies in the acquisition of an organizational strategy in high-and low-IQ children[END_REF]Blöte, Resing, Mazer, & Van Noort,1999;[START_REF] Lange | Memory-strategy learning and maintenance in preschool children[END_REF][START_REF] Pierce | Instructing young children in the use of memorizing strategies: Effects at study and recall[END_REF][START_REF] Ringel | On knowing how well one is remembering: The persistence of strategy use during transfer[END_REF][START_REF] Schwenck | Developmental and individual differences in young children's use and maintenance of a selective memory strategy[END_REF].

The lack of difference in strategies or recall on the two types of transfer tasks (2D-one piece-horizontal vs 3D-12 pieces-horizontal) shows that children demonstrated UDs on two different transfer tasks that differed systematically from each other and from the initial task. This shows some generality to the transfer-UD phenomenon. The fact that the external changes posed by a transfer task generally affected recall but not strategic behavior shows the importance of looking at external factors influencing whether a UD occurs.

Two other results of secondary interest are notable. One is that even though spontaneous strategic and prompted strategic children were similar with respect to UDs, spontaneous strategic children did show some advantages in other ways. Spontaneously strategic 4-year-olds (but not the older children) were better able to maintain their strategic behavior on the transfer tasks than were the spontaneously non-strategic 4-year-olds. In addition, at all ages spontaneously strategic children had greater recall on all three tasks (i.e., there was no interaction of task and spontaneity, see Figure 2).

The other notable secondary result, important to the strategy and transfer literatures, is that all children participating in the experiment, from every age and spontaneous strategic status, did quickly learn the selective strategy and also continued to use it most of the time in all three tasks, thus exhibiting strategy transfer. Transfer to tasks with different visual layouts is particularly impressive in children this young, especially given the half-day delay between each of the three tasks.

One possible objection to our conclusion that a UD occurred during transfer is one that can be raised about transfer experiments in general: Our transfer tasks may inadvertently have been more difficult than the main task, so that what appears to be a problem of recall during task transfer may instead be a problem of task difficulty. That is, in the present study the observed UD on the transfer tasks might simply be showing that tasks 2 and 3 are more difficult recall tasks per se, regardless of whether they are transfer tasks. This is somewhat unlikely, given that they did not pose greater difficulty for strategy production. Still, to rule out this objection, a second experiment addressed this issue by counterbalancing the order of all three tasks such that a given task (e.g., Miller's) might be given first, second, or third, and thus sometimes was a main task and sometimes a transfer task.

Experiment 2

Method

Participants were 17 spontaneously strategic 4-year-olds (M age: 48 months, range: from 46 to 50 months) who did not participate in the first experiment. Because strategy spontaneity did not interact with the effect of task in the first experiment, we included only spontaneously strategic children. Moreover, this group provides a better test of UD.

The procedures and test materials were the same as in the first experiment, except that the three tasks were presented in a counterbalanced order. That is, the Miller task was not always the first task. For convenience, we will use Task A to refer to the original Miller task (3D-one piece-vertical), Task B for one transfer task (2D-one piece-horizontal) and Task C for the other transfer task (3D-12 pieces-horizontal). Six children received the tasks in the order A-B-C, six in B-C-A and five in C-A-B. If the same pattern as in Experiment 1 occurs, then the results from that experiment cannot be attributable simply to the nature of the three tasks.

Results

One-way ANOVAS examined the effect of task order (1 st vs. 2 nd vs. 3 rd task) on strategic and recall scores, to look at transfer. Due to the small sample, we first ensured that parametric tests were appropriate. For strategies, all three distributions were normally distributed (Kolmogorov-Smirnov test, ns) and the variance-covariance matrix respected the sphericity condition (Mauchly's test, ns). Strategic scores in the three tasks (1 st : M=4.35, SD=1.97; 2 nd : M=4.06, SD=2.19; 3 rd : M=4.47, SD=2.15) did not differ, F(2,32)=0.39, ns (though a power analysis showed that 1-ß=0.11, a low value). Nonsignificant Kolmogorov-Smirnov and Mauchly tests also permitted parametric tests on recall. The ANOVA yielded an effect of task order, F(2,32)=3.31, p<.05,  2 =0.17. Planned comparisons showed that this was due to a significant difference between the first-presented and third-presented tasks, F(1,16)=5.99, p<.05,  2 =0.27, recall being better in the first (M=1.96, SD=0.85) than in the third task (M=1.54, SD=0.69); recall in the second task was M=1.64, SD=0.83. A Bonferroni test confirmed a significant difference between the first and third transfer task, t(32)=2.45, p0.05. This overall pattern suggests a UD, as in Experiment 1. Thus, the particular materials used in the main task versus each transfer task did not matter, and the pattern of results of Experiment 1 was replicated. Another way to examine task difficulty is to perform an ANOVA on overall (across 18 trials) recall scores with Task (A, B, C) as a repeated factor. If any task is more difficult than the others it should result in lower recall overall. No effect of task appeared, F(2,32)=.20, ns. Recall scores were very similar for tasks A (M=1.71, SD=1.43), B (M=1.66, SD=1.23) and C (M=1.8, SD=1.3). This is additional evidence that different task difficulty due to the materials did not account for the results of Experiment 1.

As in Experiment 1, we analyzed the individual strategy-recall patterns of the 17 children, to detect how many displayed a transfer-UD pattern. Children were considered UD only if they had similar strategic scores on the first and third task and greater recall in the first task than in the third. Again we computed the Newman-Keuls test, using the MSerror revealed by the ANOVA over the three tasks to determine critical values for both strategic (1.04) and recall (0.42) scores. For each child, the differences in recall scores and in strategic scores between the first and the third task were compared to these critical values: All children showing a decline in recall that was greater than the critical value, and an absence of such a difference in strategic scores, were considered UD. Seven of the 17 children (41%) were considered UD using this procedure, which is consistent with the percentage of such children that was observed in Experiment 1. This classification is particularly appropriate in this second experiment, given that 6 of these 7 children were strategic on the main task.

As in Experiment 1, to more clearly assess UDs on the transfer task, we conducted an ANOVA on only the children (N=11) who met the criteria for being spontaneously strategic on the main task. A Friedman ANOVA performed on recall revealed a significant effect of task (3), p<.05. Recall was higher on the first task, the main task (M=2.32, SD=0.68), than on the second task, the first transfer task (M=1.7, SD=0.91, Cohen's d=0.78), or on the third task, the second transfer task (M=1.7, SD=0.72, Cohen's d=0.93). Wilcoxon tests were significant at p=.05 and p.01, respectively. Thus, again, a UD was indicated.

Discussion

Results of this second study rule out the possibility that the decreased recall on transfer tasks in the first study could have been caused by different levels of inherent task difficulty rather than difficulty due to whether the task was a main task or a transfer task. Since the three tasks were presented in different orders, any change in recall score from one task to another can be interpreted as an effect of transfer to a somewhat different task. This study also confirms and generalizes the UD results of the first one, with a different group of children, by showing that children could maintain their strategies during transfer, but failed to recall as many items in the third-presented task (a transfer task) as in the first one (the main task).

Still, one objection to our conclusion that Experiments 1 and 2 demonstrate a transfer UD might be that strategic children, especially the spontaneous strategic ones, find the task rather boring after a few trials and simply lose interest and invest less effort over the 18 trials. This could show the same pattern of strategy transfer but reduced recall. That is, perhaps the reduced recall is not really due to the tasks being transfer tasks. This explanation seems unlikely because in Experiment 1 strategic performance remained high and there was not a gradual and steady decline in recall over the 18 trials that might suggest fatigue or boredom. Also, any fatigue or boredom should have reduced strategy production as well. Finally, since a half day separated the main task and the first transfer task, as well as the two transfer tasks, fatigue seems unlikely. However, to completely rule out this possibility we conducted a third experiment in which children received the same task over 18 trials, to see if recall decreased.

Experiment 3

Method

Participants were 9 spontaneously strategic 4-year-olds (M age=48.5 months, range= 45-51 months, 5 girls and 4 boys) and 21 spontaneously strategic 5-year-olds (M age=60.5 months, range=57-63 months, 11 girls and 10 boys), who were not in the earlier experiments.

They received the main task only (but for 18 trials instead of 6), with the same experimental design as previously: 4 pretest trials and then three sessions of six trials each with a half-a-day break between each of the four sessions. Overall, half of the trials required the children to memorize household objects and half animals. Order of presentation was counterbalanced as previously.

Results and Discussion

One-way ANOVAS examined the effect of task repetition (1 st vs. 2 nd vs. 3 rd presentation) on strategic and recall scores. Strategic scores did not differ (F<1, ns) with M=5. 90,SD=.31;M=5.93,SD=.25;and M=5.93,SD=.25, for 1 st , 2 nd and 3 rd presentation respectively. Most relevant is that the effect of task repetition was also nonsignificant for recall scores (F<1, ns) with M=2.66, SD=.60;M=2.63,SD=.76;and M=2.48,SD=.67, for 1 st , 2 nd and 3 rd presentation, respectively. Even the younger children, who might be least likely to maintain their effort during a boring task, showed no sign of decreased recall: M=2.74 (SD=0.84), 2.59 (SD=0.90), and 2.70 (SD=0.74). Also, they maintained their strategic behavior (M=5. 77,SD=0.44;M=5.77,SD=0.44;and M=5.88,SD=0.33 for the three tasks, respectively). Thus, Experiment 3 eliminates the fatigue explanation and suggests that the pattern of results of Experiments 1 and 2 do in fact reflect a UD triggered by strategy transfer.

General Discussion

In both Experiment 1 and 2, children continued to be strategic during transfer, but recall declined, which is evidence for a utilization deficiency. Trial-by-trial analyses of patterns over trials provided converging evidence. Thus, we ask: Why would superficial changes in tasks, like the ones used here, have caused a UD? In other words, why is the minor adaptation of a known strategy, to a similar task, so difficult that its benefit for recall is compromised? This is a critical question for transfer of strategy effectiveness (benefit for recall) in school settings because rarely is a given problem exactly the same at different times.

Several explanations can be considered. Given that the children could transfer the strategy, we can eliminate the explanation that children show poorer recall during transfer due to lack of understanding of the common structural similarity and a lack of active matching between tasks [START_REF] Brown | Young children's mental models determine analogical transfer across problems with a common goal structure[END_REF][START_REF] Gentner | The roles of similarity in transfer: separating retrievability from inferential soundness[END_REF]. They figured out rather quickly that the selective attention strategy would still be relevant.

The most interesting possible explanation is that the reduced benefit to recall on transfer tasks is due to the change of task per se. When faced with new tasks, children have to devote some of their cognitive resources to self-monitor the uncertainty posed by a new task.

Uncertainty inevitably occurs during transfer, because children have to decide both whether to use the selective attention strategy and, if so, how to adapt it to the transfer tasks. Selfmonitoring of uncertainty is one aspect of more general self-regulation skills. Several authors have argued that poor self-regulation skills [START_REF] Miller | Strategy utilization deficiencies in children: When, where, and why[END_REF] or the cognitive load of self-regulation when it is used [START_REF] Bjorklund | Developmental differences in the acquisition and maintenance of an organizational strategy: Evidence for the utilization deficiency hypothesis[END_REF], could cause a UD. Furthermore, other researchers recently have shown that cognitive strategies' production and monitoring always imply good self-regulation skills [START_REF] Glaser | Improving fourth grade students' composition skills: Effects of strategy instruction and self-regulation procedures[END_REF][START_REF] Montague | Self-regulation strategies to improve mathematical problem solving for students with learning disabilities[END_REF][START_REF] Zito | Self-regulated strategy development: Relationship with the social-cognitive perspective and the development of selfregulation[END_REF]. In short, although the children still were able to keep producing the strategy during transfer, the high cost of engaging in self-regulation when dealing with the new surface features of the task may have interfered with their fully applying the strategy to recall.

Future research could test the cognitive load/self-regulation explanation for UD during transfer in several ways. One way would be to examine correlations between several selfregulation behaviors and UDs. Another approach would be to see whether reducing the uncertainty and thus the self-regulation requirement, perhaps by providing a prompt to transfer the strategy, results in greater recall than in children who transfer without this prompt.

What specific mechanisms of self-regulation might be required for dealing with the uncertainty of the transfer task? Some plausible candidates are suggested by the highly active area of research on the executive function aspect of self-regulation (see Best, Miller, &[START_REF] Best | Executive functions after age 5: Changes and correlates[END_REF]Best &Miller, 2010, for reviews). Executive function refers to cognitive skills involved in adaptive, goal directed behaviors, particularly in new situations. From this perspective, a transfer task would be a new situation to which a strategy, a goal-directed behavior, could be applied. Executive function is thought to consist of several related but separable components--inhibiting a tendency to produce a prepotent response, updating working memory, and shifting from one cognitive set to another (e.g., [START_REF] Miyake | The unity and diversity of executive functions and their contributions to complex 'frontal lobe' tasks: A latent variable analysis[END_REF].

Children likely call on these skills when trying to transfer a strategy to another context, in both laboratory and school settings. When faced with a transfer task, children must inhibit their tendency to jump into the task and instead analyze the nature of the task and consider whether they have a useful strategy in their repertoire. They also might have to inhibit irrelevant strategies suggested by surface differences in the transfer task. For example, in the play context suggested by our car-playing carpet in one transfer task, they have to inhibit the tendency to play with the road-village layout. Regarding updating working memory, they must hold in mind, and compare, the characteristics of the present task and the earlier one.

Finally, they must shift from the specific motor patterns used by the selective strategy in the main task to slightly different ones required by the transfer task (e.g., manipulating 2D rather than 3D materials, opening doors far apart rather than close together). Such an analysis based on components of executive function provides a new perspective to research on transfer.

The present study advances our understanding of UDs in several ways. First, it revealed and conceptualized a different kind of UD than those already identified in the literature-a transfer UD that occurs when shifting to physically different versions of the same task. Thus, a UD can emerge either a) during a single task or b) on a transfer task. That is, during a single task children can move from a UD to no UD, as in previous UD studies, as the strategy becomes more automatic and less resource-demanding. A UD during transfer also may result from high resource demands, but mainly from the uncertainty as to whether to transfer the strategy, which calls on self-regulation skills and strategy modification.

Second, demonstrating UDs during transfer to similar but not identical tasks is compelling evidence that UDs would occur in natural settings, which typically present this kind of transfer situation. For example, even when children have learned a memory or math strategy at school they may not benefit from it when encountering a new problem that looks slightly different. Third, the definition of a UD typically is that a strategy provides "… either no benefit for performance, such as recall, or leads to little benefit for strategic younger children in comparison to strategic older children" [START_REF] Miller | How best to utilize a deficiency[END_REF](Miller, , p. 1013)). The present study extends this definition by showing that being highly strategic also can be associated with an actual decline in recall, in the case of transfer. Fourth, our use of a strategy that preschoolers easily could grasp after a prompt and most continued to use at a high level created a condition in which a UD could be demonstrated even in preschool children.

Fifth, by examining trial-by-trial changes in both strategy production and recall, we were able to provide a fine-grained analysis of transfer at each age, in both spontaneous and prompted strategic children. This analysis revealed discrepancies between change in strategies and recall from one trial to the next, individual differences in strategic behavior throughout the three tasks, individual-based patterns of UD, and a temporary decline in performance on trial one of each transfer task, when demands for self-regulation are highest.

The present study also has implications for work on transfer, which is important, since decades of research have not produced a unified view of strategy transfer and how it is affected by developmental level. Our results suggest that transfer research could benefit from examining different aspects of performance, such as strategy production and recall, separately. On the surface it may appear that transfer has occurred (e.g., continued use of a trained strategy in the present study) but a closer look (e.g., at recall) may indicate otherwise. Some aspects may transfer while others do not. Moreover, by drawing on a trial-by-trial analysis, transfer studies could track more specifically where a failure to transfer occurs.

The study has several limitations. One is that there may be individual differences in IQ, verbal competence, and working memory linked to some of the variables studied (e.g., being spontaneously strategic or not, showing a UD or not), which should be examined in future research. Another limitation is that although the two transfer tasks chosen for this study did not differ in their impact on UD, transfer tasks with more extreme physical (external) differences from the main task might effect a different outcome.

In conclusion, the present study advances work on both UDs and strategy transfer. For UD work, it contributes to the current issue of when, where, and how UDs occur. For transfer work, it directs attention to transfer of strategy effectiveness as well as the strategy per se.

For both areas of research, the results argue for increased attention to cognitive control, including executive function, as well as the fruitfulness of examining trial-by-trial changes.

Figure captions

Fig. 1 .

 1 Fig. 1. Mean number of strategic trials in spontaneously strategic and prompted strategic

Fig. 2 .

 2 Fig. 2. Mean recall scores in each task for spontaneously strategic and prompted strategic

Fig. 3 .

 3 Fig. 3. Examples of strategic (3a) and non strategic (3b) sessions in 4 year-olds (original

Fig. 4 .

 4 Fig. 4. Microgenetic evolution of recall and strategy scores over all three tasks.

Acknowledgements

We thank Céline Leboucq, Caroline Chochoy, Cécile Grulois and Juline Fiancette for collecting the data.

Appendix

Names of visual items (French translation in bracket) used in the three tasks. In the original database (Cycowicz et al., 1997), familiarity ranged from 1 (less familiar) to 5 (most familiar), and visual complexity ranged from 1 (low complexity) to 5 (high complexity).

Item