Integrating knowledge tracing and item response theory: A tale of two frameworks

Traditionally, the assessment and learning science commu-nities rely on different paradigms to model student performance. The assessment community uses Item Response Theory which allows modeling different student abilities and problem difficulties, while the learning science community uses Knowledge Tracing, which captures skill acquisition. These two paradigms are complementary - IRT cannot be used to model student learning, while Knowledge Tracing assumes all students and problems are the same. Recently, two highly related models based on a principled synthesis of IRT and Knowledge Tracing were introduced. However, these two models were evaluated on different data sets, using different evaluation metrics and with different ways of splitting the data into training and testing sets. In this paper we reconcile the models' results by presenting a unified view of the two models, and by evaluating the models under a common evaluation metric. We find that both models are equivalent and only differ in their training procedure. Our results show that the combined IRT and Knowledge Tracing models offer the best of assessment and learning sciences - high prediction accuracy like the IRT model, and the ability to model student learning like Knowledge Tracing

Towards Interpretable Deep Learning Models for Knowledge Tracing

As an important technique for modeling the knowledge states of learners, the traditional knowledge tracing (KT) models have been widely used to support intelligent tutoring systems and MOOC platforms. Driven by the fast advancements of deep learning techniques, deep neural network has been recently adopted to design new KT models for achieving better prediction performance. However, the lack of interpretability of these models has painfully impeded their practical applications, as their outputs and working mechanisms suffer from the intransparent decision process and complex inner structures. We thus propose to adopt the post-hoc method to tackle the interpretability issue for deep learning based knowledge tracing (DLKT) models. Specifically, we focus on applying the layer-wise relevance propagation (LRP) method to interpret RNN-based DLKT model by backpropagating the relevance from the model's output layer to its input layer. The experiment results show the feasibility using the LRP method for interpreting the DLKT model's predictions, and partially validate the computed relevance scores from both question level and concept level. We believe it can be a solid step towards fully interpreting the DLKT models and promote their practical applications in the education domain

Beyond Standard Assumptions - Semiparametric Models, A Dyadic Item Response Theory Model, and Cluster-Endogenous Random Intercept Models

In most statistical analyses, quantitative education researchers often make simplifying assumptions regarding the manner in which their data was generated in order to answer some of these questions. These assumptions can help to reduce the complexity of the problem, and allow the researcher to describe their data using a simpler, and often times more interpretable, statistical model. However, making some of these assumptions when they are not true can lead to biased estimates and misleading answers. While the standard sets of assumptions associated with commonly-used statistical models are usually sufficient in a wide range of contexts, it will always be beneficial for education researchers to understand what they are, when they are reasonable, and how to modify them if necessary. This dissertation focuses on three of the most common models used in quantitative education research (viz. parametric models like Linear Models (LMs), Item Response Theory (IRT) models, and Random-Intercept Models (RIMs)), discusses the standard sets of assumptions that accompany these models, and then describes related models with less stringent sets of assumptions. In each of the following three chapters, we either explicitly unpack existing models that are useful but are currently still uncommon in the field of education research, or propose novel models and/or estimation strategies for these models. We begin in Chapter 1 with a common parametric model known as the Gaussian LM, and use it as a scaffold to better understand semiparametric models and their estimation. We begin by reviewing how the coefficients of the Gaussian LM are usually estimated using Maximum Likelihood (ML) or Least-Squares (LS). We then introduce the notion of an $m$-estimator as well as that of a Regular Asymptotically Linear estimator, and show how they relate to the ML estimator. In particular, we introduce the notion of influence functions/curves and discuss their geometry together with concepts such as Hilbert spaces and tangent spaces. We then demonstrate, concretely, how to derive the so-called efficient influence function under the Gaussian LM, and show that it is precisely the influence function of the ML and (Ordinary) LS estimators. This shows that the ML estimator (at least under the Gaussian LM) is efficient. Using the foundation built, we move on from the Gaussian LM by relaxing both the assumption that the residuals are normally distributed, as well as the assumption that they have a constant variance, and define this as the Heteroskedastic Linear Model. Unlike the Gaussian LM, this is a semiparametric model. Where possible, we make use of intuition and analogous results from the parametric setting to help describe the workflow for obtaining an efficient estimator for the coefficients of the Heteroskedastic Linear Model. In particular, we derive the nuisance tangent space for this semiparametric model, and use it to obtain the efficient influence function for our model. We then show how to use the efficient influence function to obtain an efficient estimator (which happens to be the Weighted LS estimator) from the (Ordinary) LS estimator via a one-step approach as well as an estimating equations approach. We then conclude by directing readers to more advanced material, including references on more modern approaches to estimating more general semiparametric models such as Targeted Maximum Likelihood Estimation. In Chapter 2, we focus on a class of measurement models known as Item Response Theory models which are useful for measuring latent traits of a subject based on the subject's response to items. We relax the condition that the responses are only a result of the individual's latent trait (and possibly an external rater), and propose a dyadic Item Response Theory (dIRT) model for measuring interactions of pairs of individuals when the responses to items represent the actions (or behaviors, perceptions, etc.) of each individual (actor) made within the context of a dyad formed with another individual (partner). Examples of its use in education include the assessment of collaborative problem solving among students, or the evaluation of intra-departmental dynamics among teachers. The dIRT model generalizes both Item Response Theory models for measurement and the Social Relations Model for dyadic data. Here, the responses of an actor when paired with a partner are modeled as a function of not only the actor's inclination to act and the partner's tendency to elicit that action, but also the unique relationship of the pair, represented by two directional, possibly correlated, interaction latent variables. We discuss generalizations such as accommodating triads or larger groups, but focus on demonstrating the key idea in the dyadic case. We show that estimation may be performed using Markov-chain Monte Carlo implemented in \texttt{Stan}, making it straightforward to extend the dIRT model in various ways. Specifically, we show how the basic dIRT model can be extended to accommodate latent regressions, random effects, distal outcomes. We perform a simulation study that demonstrates that our estimation approach performs well. In the absence of educational data of this form, we demonstrate the usefulness of our proposed approach using speed-dating data instead, and find new evidence of pairwise interactions between participants, describing a mutual attraction that is inadequately characterized by individual properties alone.Finally, in Chapter 3, we consider the often implicit assumption made when estimating the coefficients of structural Random Intercept Models (RIMs) that covariates at all levels do not co-vary with the random intercepts. A violation of this assumption (called cluster-level endogeneity) leads to inconsistent estimates when using standard estimation procedures. For two-level RIMs with such endogeneity, Hausman and Taylor (HT) devised a consistent multi-step instrumental variable estimator using only internal instruments. We, instead, approach this problem by explicitly modeling the endogeneity using a Structural Equation Model (SEM). In this chapter, we compare, through simulation, the HT and SEM estimators, and evaluate their asymptotic and finite sample properties. We show that the SEM approach is also flexible enough to deal with different exchangeability assumptions for the covariates (e.g., whether the correlations between pairs of all units in a cluster are the same) and investigate how these exchangeability assumptions affect finite sample properties of the HT estimator. For the simulations, we propose a new procedure for generating cluster- and unit-level covariates and random intercepts with a fully flexible covariance structure. We also compare our approach to another common approach known as Multilevel Matching using data from the High School and Beyond survey

I’ve (Urn)ed This: An Application and Criterion-based Evaluation of the Urnings Algorithm

There is increased interest in personalized learning and making e-learning environments more adaptable. Some e-learning systems may use an Item Response Theory (IRT)-based assessment system. An important distinction between assessment and learning contexts is that learner proficiency is expected to remain constant across an assessment, while it is expected to change over time in a learning context. Constant learner proficiency during an assessment enables conventional approaches to estimating person and item parameters using IRT. These IRT-based systems could be abandoned for alternative approaches to modeling learners and system learning content, but assessments may provide more functions than adapting learning material to students. Thus, there is the question, how can e-learning systems with IRT-based assessment components more dynamically adapt their learning content? Is there a solution that leverages IRT for adapting the learning content of the system? A promising solution is the Urnings algorithm. Like other candidate algorithms, it is computationally light, but this algorithm has mechanisms for preventing variance inflation and is suitable for e-learning contexts. It also provides a measure of uncertainty around estimates. It has been studied both through simulations and applications to e-learning systems. Results are promising; however, there has not been an application of the Urnings algorithm to an e-learning context where there are conventionally estimated person parameters to compare the algorithm estimates to. This study addresses this gap by applying the Urnings algorithm to a K–8 reading and mathematics learning platform. In data from this platform, we have person parameter estimates across academic years from an in-system diagnostic assessment. Results from this study will help industry researchers understand the feasibility of the Urnings algorithm for large e-learning systems with IRT-based assessment components

Psychometrics in Practice at RCEC

A broad range of topics is dealt with in this volume: from combining the psychometric generalizability and item response theories to the ideas for an integrated formative use of data-driven decision making, assessment for learning and diagnostic testing. A number of chapters pay attention to computerized (adaptive) and classification testing. Other chapters treat the quality of testing in a general sense, but for topics like maintaining standards or the testing of writing ability, the quality of testing is dealt with more specifically.\ud All authors are connected to RCEC as researchers. They present one of their current research topics and provide some insight into the focus of RCEC. The selection of the topics and the editing intends that the book should be of special interest to educational researchers, psychometricians and practitioners in educational assessment

IRT-Based Adaptive Hints to Scaffold Learning in Programming

Over the past few decades, many studies conducted in the field of learning science have described that scaffolding plays an important role in human learning. To scaffold a learner efficiently, a teacher should predict how much support a learner must have to complete tasks and then decide the optimal degree of assistance to support the learner\u27s development. Nevertheless, it is difficult to ascertain the optimal degree of assistance for learner development. For this study, it is assumed that optimal scaffolding is based on a probabilistic decision rule: Given a teacher\u27s assistance to facilitate the learner development, an optimal probability exists for a learner to solve a task. To ascertain that optimal probability, we developed a scaffolding system that provides adaptive hints to adjust the predictive probability of the learner\u27s successful performance to the previously determined certain value, using a probabilistic model, i.e., item response theory (IRT). Furthermore, using the scaffolding system, we compared learning performances by changing the predictive probability. Results show that scaffolding to achieve 0.5 learner success probability provides the best performance. Additionally, results demonstrate that a scaffolding system providing 0.5 probability decreases the number of hints (amount of support) automatically as a fading function according to the learner\u27s growth capability

New measurement paradigms

This collection of New Measurement Paradigms papers represents a snapshot of the variety of measurement methods in use at the time of writing across several projects funded by the National Science Foundation (US) through its REESE and DR K–12 programs. All of the projects are developing and testing intelligent learning environments that seek to carefully measure and promote student learning, and the purpose of this collection of papers is to describe and illustrate the use of several measurement methods employed to achieve this. The papers are deliberately short because they are designed to introduce the methods in use and not to be a textbook chapter on each method. The New Measurement Paradigms collection is designed to serve as a reference point for researchers who are working in projects that are creating e-learning environments in which there is a need to make judgments about students’ levels of knowledge and skills, or for those interested in this but who have not yet delved into these methods

Bridging Mathematics with Word Problems

The aim of this thesis was to explore several important aspects of word problems: the nature of word problems used in school mathematics textbooks and the difficulty level of different types of word problems. The specific goals were to investigate students’ performance when solving various types of word problems and to determine whether students’ word-problem skills and their beliefs about word problem-solving can be improved by enriching word problems used in mathematics teaching. To achieve the goals, this thesis reports on five original studies, as follows. Study I showed a comparison between the characteristics of word problems presented in Thai and Finnish school mathematics textbooks. The analyses included 1,565 word problems from a series of second- to fourth-grade Thai and Finnish mathematics textbooks. The overall results show that the nature of word problems used in Finnish textbooks vary from Thai textbooks in many ways. Finnish textbooks contain more multistep word problems, while in Thai textbooks, one-step word problems appear more frequently. Thai textbooks have a smaller percentage of repetitive sections (ones that include only the same type of problems) than Finnish textbooks. In both countries, the percentage of word problems requiring the use of realistic considerations is extremely low, less than five percent of the total. Studies II and III presented the impacts of a Word Problem Enrichment (WPE) programme, developed to encourage teachers to use innovative self-created word problems to improve student mathematical modelling and problem-solving skills. Participants comprised 10 classroom teachers and their 170 students from fourth and sixth grades, from elementary schools in southwest Finland. In Study II, the intervention effectiveness on student problem-solving performance was investigated. The results suggested that enriching word problems used in mathematics teaching is a promising method for improving student problem-solving skills when solving non-routine and application word problems. However, it is not known if WPE has an effect on student beliefs about word problem-solving, and how the programme works for students with different initial motivation in learning mathematics. Study III examined the effectiveness of WPE on student beliefs about word problem-solving by using latent profile analysis (LPA) and structural equation modelling (SEM) to analyse relationships among the different cognitive, motivation, and belief factors. Results indicated that the impacts of WPE are various depending upon the initial motivation level of students. The effects of WPE on student beliefs appeared only in students with a low initial motivation level, while its impacts on student problem-solving performance were found only in students with a high initial motivation level. Studies IV and V were conducted to examine hypotheses regarding (1) the dimensionality of students’ performance on word problems and (2) difficulty level of three types of word problems: routine, non-routine and application word problems by utilizing item response theory (IRT) modelling. The data used in Study IV was collectedas part of the Word Problem project (Studies II and III). Participants comprised 170 fourth- and sixth-grade students. Students’ problem-solving performance was assessed with a word problem-solving test, including five word problems: one routine, three non-routine, and one application. The results of Study IV show that students’ performance on word problems can be seen as a unidimensional construct that denies the original assumption. The results of the IRT model indicate that the theoretically demanding application word problem has a higher difficulty level than non-routine and routine word problems. Nevertheless, the results are obscure if this application word problem (used in Study IV) is harder because of its demand for realistic considerations or other possibly relevant factors (e.g. decimal numbers included, division, more problem-solving steps required). Moreover, the sample size of Study IV could be considered relatively small for this kind of complicated IRT model. Therefore, Study V uses a larger sample size and a bigger set of word problems with more variety in application and non-routine word problems. The data used in Study V was collected as part of the Quest for Meaning project. Participants comprised 891 fourth-grade students (446 boys and 445 girls) from different elementary schools situated in cities, small towns, and rural communities in southern Finland. On the same lines as Study IV, the results of Study V indicated that students’ performance on word problems can be seen as a unidimensional construct. Concerning item difficulty level, the results of the IRT model do not show a clear distinction among word-problem types and reject the hypothesis that application word problems have a higher difficulty level than non-routine word problems. Some non-routine word problems appear to be more difficult than the application word problem, even though other characteristics of these two types of word problems were very similar (e.g., they required the same type of operation and the same number of problem-solving steps). The results of the five studies reveal that even though the mathematics textbooks were highly regarded in Thailand and Finland, most given word problems frequently include a simple goal without demanding any realistic considerations. These results strongly suggest that more innovative application word problems are definitely needed in classroom mathematics. In our study, we developed the WPE to encourage teachers to develop their own meaningful non-routine and applications word problems, and to use these self-created word problems to improve mathematical modelling and students’ word problem-solving performance. The results show that WPE is a promising approach to improve not only student problem-solving skills but also student beliefs about word problem-solving. The impacts of WPE are different depending upon students’ initial motivation level. The impacts of WPE on student beliefs were found only in students with a low initial motivation level, while its impacts on student problem-solving performance were found only in students with a high initial motivation level. These results suggest that in classroom practice, it is important that teachers provide enough support for students to be more confident and feel less overwhelmed when facing non-routine and application word problems. Teachers should be aware of differences of word-problem types and utilise this information in planning how to scaffold students’ word problem-solving by giving word problems based on their difficulty level.Väitöskirjatyö kohdistuu matematiikan sanallisten tehtävien tärkeisiin ominaisuuksiin: koulumatematiikassa hyödynnettävien sanallisten tehtävien luonteen sekä erityyppisten sanallisten tehtävien vaikeustason tarkasteluun. Keskeisinä tavoitteina oli tarkastella oppilaiden suoriutumista heidän ratkaistessaan erityyppisiä sanallisia tehtäviä ja selvittää, voidaanko oppilaiden sanallisten tehtävien ratkaisutaitoja ja heidän uskomuksiaan sanallisten tehtävien ratkaisuun liittyen parantaa rikastamalla matematiikan opetuksessa käytettäviä sanallisia tehtäviä. Näiden tavoitteiden saavuttamiseksi tässä väitöstutkimuksessa toteutettiin viisi osatutkimusta. Osatutkimuksessa I vertailtiin suomalaisissa ja thaimaalaisissa matematiikan oppikirjoissa käytettävien sanallisten tehtävien ominaisuuksia. Tutkimuksessa analysoitiin 1565 sanallista tehtävää suomalaisista ja thaimaalaisista eri oppikirjasarjojen toisen–neljännen luokan matematiikan oppikirjoista. Tulokset osoittivat, että suomalaisissa oppikirjoissa esiintyvät sanalliset tehtävät eroavat monin tavoin Thaimaassa käytössä olevien oppikirjojen tehtävistä. Suomalaisissa oppikirjoissa on enemmän useita välivaiheita sisältäviä sanallisia tehtäviä, kun taas thaimaalaisissa oppikirjoissa esiintyy enemmän yksivaiheisia sanallisia tehtäviä. Thaimaalaisissa oppikirjoissa on prosentuaalisesti vähemmän toistavia osioita (sisältävät ainoastaan tietyn tyyppisiä tehtäviä) kuin suomalaisissa oppikirjoissa. Molempien vertailtavien maiden oppikirjoissa sellaisten tehtävien osuus, joiden ratkaiseminen vaatii todellisten arkielämän näkökohtien huomioimista, on todella vähäinen, vain noin viisi prosenttia kaikista sanallisista tehtävistä. Osatutkimukset II ja III esittelivät niin sanotun Sanallisten Tehtävien Rikastaminen (STR) –ohjelman vaikutuksia, joka kehitettiin tarkoituksena rohkaista opettajia hyödyntämään opetuksessaan innovatiivisia, itse kehittelemiään sanallisia ongelmia parantamaan oppilaiden matemaattisen mallintamisen ja ongelmanratkaisun taitoja. Tutkittavina oli 10 luokanopettajaa ja heidän 170 oppilastaan neljänneltä ja kuudennelta luokalta varsinaissuomalaisista kouluista. Osatutkimuksessa II selvitettiin intervention vaikuttavuutta suhteessa oppilaiden ongelmanratkaisutaitoihin. Tulokset osoittivat, että matematiikan opetuksessa sanallisten tehtävien rikastaminen on lupaava menetelmä oppilaiden ongelmanratkaisutaitojen parantamiseksi, kun ratkaistaan ei-rutiininomaisia ja soveltamista vaativia sanallisia ongelmia. Tässä osatutkimuksessa jäi kuitenkin vielä epäselväksi, onko STR:llä vaikutusta oppilaiden uskomuksiin sanallisten ongelmanratkaisutehtävien ratkaisua kohtaan ja kuinka ohjelma vaikuttaa erilaisen motivaation matematiikan opiskelua kohtaan omaavien oppilaiden oppimiseen. Osatutkimuksessa III selvitettiin STR-ohjelman vaikuttavuutta oppilaiden uskomuksiin sanallisiin ongelmanratkaisutehtäviin liittyen hyödyntäen latenttia profiilianalyysia (LPA) ja rakenneyhtälömallinnusta (structural equation modelling, SEM), joiden avulla analysoitiin erilaisten kognitiivisten, motivationaalisten ja uskomuksiin liittyvien tekijöiden välisiä suhteita. Tulokset indikoivat, että STR-ohjelman vaikutukset ovat erilaisia riippuen oppilaiden motivaatiotasosta matematiikan opiskelua kohtaan. STR:n vaikutukset uskomuksiin näkyivät ainoastaan niiden oppilaiden kohdalla, joilla oli alhainen motivaatio, kun taas ohjelmalla oli vaikutuksia ongelmanratkaisutaitojen tasoon vain sellaisten oppilaiden osalta, joiden motivaatio oli korkea. Osatutkimuksissa IV ja V selvitettiin (1) sijoittuvatko oppilaiden suoritukset sanallisissa tehtävissä yhdelle vaikeusdimensiolle vai onko sanallisten tehtävien vaikeudessa eri dimensioita ja (2) kolmen tyyppisten sanallisten tehtävien (rutiininomaisetn, ei-rutiininomaiset ja soveltamista vaativat tehtävät) vaikeustasoa hyödyntämällä modernia testiosioiden mallinnusmenetelmääa (item response theory modelling, IRT). Tutkimuksen IV aineisto kerättiin osana sanallisten tehtävien interventioprojektia (vrt. Osatutkimukset II ja III). Tutkittavina oli 170 neljännen ja kuudennen luokan oppilasta. Oppilaiden suoriutumista sanallisista tehtävistä arvioitiin ongelmanratkaisutestillä, joka piti sisällään viisi sanallista tehtävää: yhden rutiininomaisen tehtävän, kolme ei-rutiininomaista tehtävää ja yhden soveltamista vaativan tehtävän. Osatutkimuksen IV tulokset osoittavat, että oppilaiden suoriutuminen sanallisista tehtävistä voidaan odotusten vastaisesti nähdä yksiulotteisena rakenteena. IRT-mallin tulokset antavat viitteitä, että teoreettisesti vaativampi soveltamista vaativa sanallinen tehtävä on vaikeustasoltaan haastavampi kuin ei-rutiininomaiset ja rutiininomaiset tehtävät. Tulosten avulla ei kuitenkaan voitu vielä selittää, johtuiko soveltamista vaativan tehtävän (vrt. Osatutkimus IV) vaikeus siitä, että sen ratkaiseminen edellytti realististen näkökohtien huomioimista vai mahdollisesti jotkin muut relevantit tekijät (esim. desimaalilukujen tai jakolaskujen sisältyminen, monivaiheisempi ongelmanratkaisuprosessi). Tämän lisäksi otoskoko Osatutkimuksessa IV oli suhteellisen pieni monimutkaisen testiosioiden mallinnusmenetelmän hyödyntämiseen. Tästä syystä Osatutkimuksessa V hyödynnettiin suurempaa otoskokoa ja laajempaa sanallisten tehtävien joukkoa, joka sisälsi monipuolisempia rutiininomaisia ja ei-rutiininomaisia tehtäviä. Osatutkimuksen V aineistona oli aiemmassa Merkitystä etsimässä –projektisa koottu laaja aineisto. Tutkittavina oli 891 neljännen luokan oppilasta (446 poikaa ja 445 tyttöä) suurehkoissa kaupungeissa, pikkukaupungeissa ja maaseudulla sijaitsevista alakouluista eripuolilta eteläistä Suomea. Linjassa Osatutkimuksen IV tulosten kanssa, myös Osatutkimuksen V tulokset antoivat viitteitä, että oppilaiden suoriutuminen sanallisissa tehtävissä voidaan selittää yksiulotteisella rakenteella. IRT-mallin tulokset eivät osoita selkeää eroa sanallisten tehtävien eri vaikeustasotyyppien välillä ja hylkäävät hypoteesin siitä, että soveltamista vaativien sanallisten tehtävien vaikeustaso olisi korkeampi kuin ei-rutiininomaisten tehtävien. Jotkut ei-rutiininomaiset sanalliset tehtävät näyttivät olevan vaikeampia kuin soveltamista vaativat tehtävät, vaikka muut ominaisuudet näiden kahden erityyppisten sanallisten tehtävien välillä olivat hyvin samankaltaiset (esim. vaativat samanlaisia laskutoimintoja ja yhtä monta välivaihetta). Viiden osatutkimuksen tulokset paljastavat, että vaikka matematiikan oppikirjoja pidetään yleisesti korkeatasoisina Thaimaassa ja Suomessa, suurin osa niissä olevista sanallisista tehtävistä sisältävät yksinkertaisen tavoitteen ilman että ne edellyttäisivät todellisen elämän tilanteiden huomioon ottamista Nämä tulokset osoittavat selkeästi, että kouluissa tarvitaan innovatiivisempia, soveltamista vaativia matematiikan sanallisia tehtäviä. Tutkimuksissamme kehitimme STR-ohjelman rohkaisemaan opettajia kehittämään itse omia ei-rutiininomaisia ja soveltamista vaativia tehtäviä ja hyödyntämään näitä itse kehitettyjä sanallisia tehtäviä parantaakseen matemaattista mallintamista ja oppilaiden sanallisissa ongelmanratkaisutehtävissä suoriutumista. Tulokset osoittavat, että STR tarjoaa lupaavan lähestymistavan parantaa oppilaiden ongelmanratkaisutaitojen lisäksi myös oppilaiden uskomuksia sanallisten tehtävien ratkaisemiseen liittyen. STR:n vaikutukset olivat erilaisia riippuen oppilaiden motivaatiotasosta. STR vaikutti vain sellaisten oppilaiden uskomuksiin, joilla oli alhainen motivaatio, kun taas ohjelman vaikutukset ongelmanratkaisutehtävissä suoriutumiseen oli nähtävissä ainoastaan niiden oppilaiden keskuudessa, joilla oli korkea motivaatio. Näiden tulosten mukaisesti on tärkeää, että opettajat tarjoavat riittävästi tukea oppilaille, jotta oppilaiden itsevarmuus parantuisi ja he tuntisivat itsensä vähemmän lannistuneiksi kohdatessaan ei-rutiininomaisia ja soveltamista vaativia sanallisia tehtäviä. Opettajien tulisi olla tietoisia erityyppisistä sanallisista tehtävistä ja hyödyntää tätä tietoa suunnitellessaan, kuinka tukea oppilaiden sanallisten tehtävien ongelmanratkaisua tarjoamalla vaikeustasoltaan erilaisia sanallisia tehtäviä.Siirretty Doriast

Separating cognitive and content domains in mathematical competence

The present study investigates the empirical separability of mathematical (a) content domains, (b) cognitive domains, and (c) content-specific cognitive domains. There were 122 items representing two content domains (linear equations vs. theorem of Pythagoras) combined with two cognitive domains (modeling competence vs. technical competence) administered in a study with 1,570 German ninth graders. A unidimensional item response theory model, two two-dimensional multidimensional item response theory (MIRT) models (dimensions: content domains and cognitive domains, respectively), and a four-dimensional MIRT model (dimensions: content-specific cognitive domains) were compared with regard to model fit and latent correlations. Results indicate that the two content and the two cognitive domains can each be empirically separated. Content domains are better separable than cognitive domains. A differentiation of content-specific cognitive domains shows the best fit to the empirical data. Differential gender effects mostly confirm that the separated dimensions have different psychological meaning. Potential explanations, practical implications, and possible directions for future research are discussed. (DIPF/Orig.