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• Published: 01 December 2021

Advancing mathematics by guiding human intuition with AI

• Alex Davies   ORCID: orcid.org/0000-0003-4917-5234 1 ,
• Petar Veličković 1 ,
• Lars Buesing 1 ,
• Sam Blackwell 1 ,
• Daniel Zheng 1 ,
• Nenad Tomašev   ORCID: orcid.org/0000-0003-1624-0220 1 ,
• Richard Tanburn 1 ,
• Peter Battaglia 1 ,
• Charles Blundell 1 ,
• András Juhász 2 ,
• Marc Lackenby 2 ,
• Geordie Williamson 3 ,
• Demis Hassabis   ORCID: orcid.org/0000-0003-2812-9917 1 &
• Pushmeet Kohli   ORCID: orcid.org/0000-0002-7466-7997 1  

Nature volume  600 ,  pages 70–74 ( 2021 ) Cite this article

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• Pure mathematics

The practice of mathematics involves discovering patterns and using these to formulate and prove conjectures, resulting in theorems. Since the 1960s, mathematicians have used computers to assist in the discovery of patterns and formulation of conjectures 1 , most famously in the Birch and Swinnerton-Dyer conjecture 2 , a Millennium Prize Problem 3 . Here we provide examples of new fundamental results in pure mathematics that have been discovered with the assistance of machine learning—demonstrating a method by which machine learning can aid mathematicians in discovering new conjectures and theorems. We propose a process of using machine learning to discover potential patterns and relations between mathematical objects, understanding them with attribution techniques and using these observations to guide intuition and propose conjectures. We outline this machine-learning-guided framework and demonstrate its successful application to current research questions in distinct areas of pure mathematics, in each case showing how it led to meaningful mathematical contributions on important open problems: a new connection between the algebraic and geometric structure of knots, and a candidate algorithm predicted by the combinatorial invariance conjecture for symmetric groups 4 . Our work may serve as a model for collaboration between the fields of mathematics and artificial intelligence (AI) that can achieve surprising results by leveraging the respective strengths of mathematicians and machine learning.

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One of the central drivers of mathematical progress is the discovery of patterns and formulation of useful conjectures: statements that are suspected to be true but have not been proven to hold in all cases. Mathematicians have always used data to help in this process—from the early hand-calculated prime tables used by Gauss and others that led to the prime number theorem 5 , to modern computer-generated data 1 , 5 in cases such as the Birch and Swinnerton-Dyer conjecture 2 . The introduction of computers to generate data and test conjectures afforded mathematicians a new understanding of problems that were previously inaccessible 6 , but while computational techniques have become consistently useful in other parts of the mathematical process 7 , 8 , artificial intelligence (AI) systems have not yet established a similar place. Prior systems for generating conjectures have either contributed genuinely useful research conjectures 9 via methods that do not easily generalize to other mathematical areas 10 , or have demonstrated novel, general methods for finding conjectures 11 that have not yet yielded mathematically valuable results.

AI, in particular the field of machine learning 12 , 13 , 14 , offers a collection of techniques that can effectively detect patterns in data and has increasingly demonstrated utility in scientific disciplines 15 . In mathematics, it has been shown that AI can be used as a valuable tool by finding counterexamples to existing conjectures 16 , accelerating calculations 17 , generating symbolic solutions 18 and detecting the existence of structure in mathematical objects 19 . In this work, we demonstrate that AI can also be used to assist in the discovery of theorems and conjectures at the forefront of mathematical research. This extends work using supervised learning to find patterns 20 , 21 , 22 , 23 , 24 by focusing on enabling mathematicians to understand the learned functions and derive useful mathematical insight. We propose a framework for augmenting the standard mathematician’s toolkit with powerful pattern recognition and interpretation methods from machine learning and demonstrate its value and generality by showing how it led us to two fundamental new discoveries, one in topology and another in representation theory. Our contribution shows how mature machine learning methodologies can be adapted and integrated into existing mathematical workflows to achieve novel results.

Guiding mathematical intuition with AI

A mathematician’s intuition plays an enormously important role in mathematical discovery—“It is only with a combination of both rigorous formalism and good intuition that one can tackle complex mathematical problems” 25 . The following framework, illustrated in Fig. 1 , describes a general method by which mathematicians can use tools from machine learning to guide their intuitions concerning complex mathematical objects, verifying their hypotheses about the existence of relationships and helping them understand those relationships. We propose that this is a natural and empirically productive way that these well-understood techniques in statistics and machine learning can be used as part of a mathematician’s work.

The process helps guide a mathematician’s intuition about a hypothesized function f , by training a machine learning model to estimate that function over a particular distribution of data P Z . The insights from the accuracy of the learned function \(\hat{f}\) and attribution techniques applied to it can aid in the understanding of the problem and the construction of a closed-form f ′. The process is iterative and interactive, rather than a single series of steps.

Concretely, it helps guide a mathematician’s intuition about the relationship between two mathematical objects X ( z ) and Y ( z ) associated with z by identifying a function \(\hat{f}\) such that \(\hat{f}\) ( X ( z ))  ≈  Y ( z ) and analysing it to allow the mathematician to understand properties of the relationship. As an illustrative example: let z be convex polyhedra, X ( z )  ∈   \({{\mathbb{Z}}}^{2}\times {{\mathbb{R}}}^{2}\) be the number of vertices and edges of z , as well as the volume and surface area, and Y ( z )  ∈   ℤ be the number of faces of z. Euler’s formula states that there is an exact relationship between X ( z ) and Y ( z ) in this case: X ( z ) · (−1, 1, 0, 0) + 2 =  Y ( z ). In this simple example, among many other ways, the relationship could be rediscovered by the traditional methods of data-driven conjecture generation 1 . However, for X ( z ) and Y ( z ) in higher-dimensional spaces, or of more complex types, such as graphs, and for more complicated, nonlinear \(\hat{f}\) , this approach is either less useful or entirely infeasible.

The framework helps guide the intuition of mathematicians in two ways: by verifying the hypothesized existence of structure/patterns in mathematical objects through the use of supervised machine learning; and by helping in the understanding of these patterns through the use of attribution techniques.

In the supervised learning stage, the mathematician proposes a hypothesis that there exists a relationship between X ( z ) and Y ( z ). By generating a dataset of X ( z ) and Y ( z ) pairs, we can use supervised learning to train a function \(\hat{f}\) that predicts Y ( z ), using only X ( z ) as input. The key contributions of machine learning in this regression process are the broad set of possible nonlinear functions that can be learned given a sufficient amount of data. If \(\hat{f}\) is more accurate than would be expected by chance, it indicates that there may be such a relationship to explore. If so, attribution techniques can help in the understanding of the learned function \(\hat{f}\) sufficiently for the mathematician to conjecture a candidate f ′. Attribution techniques can be used to understand which aspects of \(\hat{f}\) are relevant for predictions of Y ( z ). For example, many attribution techniques aim to quantify which component of X ( z ) the function \(\hat{f}\) is sensitive to. The attribution technique we use in our work, gradient saliency, does this by calculating the derivative of outputs of \(\hat{f}\) , with respect to the inputs. This allows a mathematician to identify and prioritize aspects of the problem that are most likely to be relevant for the relationship. This iterative process might need to be repeated several times before a viable conjecture is settled on. In this process, the mathematician can guide the choice of conjectures to those that not just fit the data but also seem interesting, plausibly true and, ideally, suggestive of a proof strategy.

Conceptually, this framework provides a ‘test bed for intuition’—quickly verifying whether an intuition about the relationship between two quantities may be worth pursuing and, if so, guidance as to how they may be related. We have used the above framework to help mathematicians to obtain impactful mathematical results in two cases—discovering and proving one of the first relationships between algebraic and geometric invariants in knot theory and conjecturing a resolution to the combinatorial invariance conjecture for symmetric groups 4 , a well-known conjecture in representation theory. In each area, we demonstrate how the framework has successfully helped guide the mathematician to achieve the result. In each of these cases, the necessary models can be trained within several hours on a machine with a single graphics processing unit.

Low-dimensional topology is an active and influential area of mathematics. Knots, which are simple closed curves in \({{\mathbb{R}}}^{3}\) , are one of the key objects that are studied, and some of the subject’s main goals are to classify them, to understand their properties and to establish connections with other fields. One of the principal ways that this is carried out is through invariants, which are algebraic, geometric or numerical quantities that are the same for any two equivalent knots. These invariants are derived in many different ways, but we focus on two of the main categories: hyperbolic invariants and algebraic invariants. These two types of invariants are derived from quite different mathematical disciplines, and so it is of considerable interest to establish connections between them. Some examples of these invariants for small knots are shown in Fig. 2 . A notable example of a conjectured connection is the volume conjecture 26 , which proposes that the hyperbolic volume of a knot (a geometric invariant) should be encoded within the asymptotic behaviour of its coloured Jones polynomials (which are algebraic invariants).

We hypothesized that there was a previously undiscovered relationship between the geometric and algebraic invariants.

Our hypothesis was that there exists an undiscovered relationship between the hyperbolic and algebraic invariants of a knot. A supervised learning model was able to detect the existence of a pattern between a large set of geometric invariants and the signature σ ( K ), which is known to encode important information about a knot K , but was not previously known to be related to the hyperbolic geometry. The most relevant features identified by the attribution technique, shown in Fig. 3a , were three invariants of the cusp geometry, with the relationship visualized partly in Fig. 3b . Training a second model with X ( z ) consisting of only these measurements achieved a very similar accuracy, suggesting that they are a sufficient set of features to capture almost all of the effect of the geometry on the signature. These three invariants were the real and imaginary parts of the meridional translation μ and the longitudinal translation λ . There is a nonlinear, multivariate relationship between these quantities and the signature. Having been guided to focus on these invariants, we discovered that this relationship is best understood by means of a new quantity, which is linearly related to the signature. We introduce the ‘natural slope’, defined to be slope( K ) = Re( λ/μ ), where Re denotes the real part. It has the following geometric interpretation. One can realize the meridian curve as a geodesic γ on the Euclidean torus. If one fires off a geodesic γ ⊥ from this orthogonally, it will eventually return and hit γ at some point. In doing so, it will have travelled along a longitude minus some multiple of the meridian. This multiple is the natural slope. It need not be an integer, because the endpoint of γ ⊥ might not be the same as its starting point. Our initial conjecture relating natural slope and signature was as follows.

a , Attribution values for each of the input X ( z ). The features with high values are those that the learned function is most sensitive to and are probably relevant for further exploration. The 95% confidence interval error bars are across 10 retrainings of the model. b , Example visualization of relevant features—the real part of the meridional translation against signature, coloured by the longitudinal translation.

Conjecture: There exist constants c 1 and c 2 such that, for every hyperbolic knot K ,

While this conjecture was supported by an analysis of several large datasets sampled from different distributions, we were able to construct counterexamples using braids of a specific form. Subsequently, we were able to establish a relationship between slope( K ), signature σ ( K ), volume vol( K ) and one of the next most salient geometric invariants, the injectivity radius inj( K ) (ref.  27 ).

Theorem: There exists a constant c such that, for any hyperbolic knot K ,

It turns out that the injectivity radius tends not to get very small, even for knots of large volume. Hence, the term inj( K ) −3 tends not to get very large in practice. However, it would clearly be desirable to have a theorem that avoided the dependence on inj( K ) −3 , and we give such a result that instead relies on short geodesics, another of the most salient features, in the Supplementary Information. Further details and a full proof of the above theorem are available in ref.  27 . Across the datasets we generated, we can place a lower bound of c  ≥ 0.23392, and it would be reasonable to conjecture that c is at most 0.3, which gives a tight relationship in the regions in which we have calculated.

The above theorem is one of the first results that connect the algebraic and geometric invariants of knots and has various interesting applications. It directly implies that the signature controls the non-hyperbolic Dehn surgeries on the knot and that the natural slope controls the genus of surfaces in \({{\mathbb{R}}}_{+}^{4}\) whose boundary is the knot. We expect that this newly discovered relationship between natural slope and signature will have many other applications in low-dimensional topology. It is surprising that a simple yet profound connection such as this has been overlooked in an area that has been extensively studied.

Representation theory

Representation theory is the theory of linear symmetry. The building blocks of all representations are the irreducible ones, and understanding them is one of the most important goals of representation theory. Irreducible representations generalize the fundamental frequencies of Fourier analysis 28 . In several important examples, the structure of irreducible representations is governed by Kazhdan–Lusztig (KL) polynomials, which have deep connections to combinatorics, algebraic geometry and singularity theory. KL polynomials are polynomials attached to pairs of elements in symmetric groups (or more generally, pairs of elements in Coxeter groups). The combinatorial invariance conjecture is a fascinating open conjecture concerning KL polynomials that has stood for 40 years, with only partial progress 29 . It states that the KL polynomial of two elements in a symmetric group S N can be calculated from their unlabelled Bruhat interval 30 , a directed graph. One barrier to progress in understanding the relationship between these objects is that the Bruhat intervals for non-trivial KL polynomials (those that are not equal to 1) are very large graphs that are difficult to develop intuition about. Some examples of small Bruhat intervals and their KL polynomials are shown in Fig. 4 .

The combinatorial invariance conjecture states that the KL polynomial of a pair of permutations should be computable from their unlabelled Bruhat interval, but no such function was previously known.

We took the conjecture as our initial hypothesis, and found that a supervised learning model was able to predict the KL polynomial from the Bruhat interval with reasonably high accuracy. By experimenting on the way in which we input the Bruhat interval to the network, it became apparent that some choices of graphs and features were particularly conducive to accurate predictions. In particular, we found that a subgraph inspired by prior work 31 may be sufficient to calculate the KL polynomial, and this was supported by a much more accurate estimated function.

Further structural evidence was found by calculating salient subgraphs that attribution techniques determined were most relevant and analysing the edge distribution in these graphs compared to the original graphs. In Fig. 5a , we aggregate the relative frequency of the edges in the salient subgraphs by the reflection that they represent. It shows that extremal reflections (those of the form (0 , i ) or ( i ,  N  − 1) for S N ) appear more commonly in salient subgraphs than one would expect, at the expense of simple reflections (those of the form ( i ,  i  + 1)), which is confirmed over many retrainings of the model in Fig. 5b . This is notable because the edge labels are not given to the network and are not recoverable from the unlabelled Bruhat interval. From the definition of KL polynomials, it is intuitive that the distinction between simple and non-simple reflections is relevant for calculating it; however, it was not initially obvious why extremal reflections would be overrepresented in salient subgraphs. Considering this observation led us to the discovery that there is a natural decomposition of an interval into two parts—a hypercube induced by one set of extremal edges and a graph isomorphic to an interval in S N −1 .

a , An example heatmap of the percentage increase in reflections present in the salient subgraphs compared with the average across intervals in the dataset when predicting q 4 . b , The percentage of observed edges of each type in the salient subgraph for 10 retrainings of the model compared to 10 bootstrapped samples of the same size from the dataset. The error bars are 95% confidence intervals, and the significance level shown was determined using a two-sided two-sample t -test. * p < 0.05; **** p < 0.0001. c , Illustration for the interval 021435–240513  ∈   S 6 of the interesting substructures that were discovered through the iterative process of hypothesis, supervised learning and attribution. The subgraph inspired by previous work 31  is highlighted in red, the hypercube in green and the decomposition component isomporphic to an interval in S N − 1 in blue.

The importance of these two structures, illustrated in Fig. 5c , led to a proof that the KL polynomial can be computed directly from the hypercube and S N −1 components through a beautiful formula that is summarized in the  Supplementary Information . A further detailed treatment of the mathematical results is given in ref.  32 .

Theorem: Every Bruhat interval admits a canonical hypercube decomposition along its extremal reflections, from which the KL polynomial is directly computable.

Remarkably, further tests suggested that all hypercube decompositions correctly determine the KL polynomial. This has been computationally verified for all of the ∼ 3 × 10 6 intervals in the symmetric groups up to S 7 and more than 1.3 × 10 5 non-isomorphic intervals sampled from the symmetric groups S 8 and S 9 .

Conjecture: The KL polynomial of an unlabelled Bruhat interval can be calculated using the previous formula with any hypercube decomposition.

This conjectured solution, if proven true, would settle the combinatorial invariance conjecture for symmetric groups. This is a promising direction as not only is the conjecture empirically verified up to quite large examples, but it also has a particularly nice form that suggests potential avenues for attacking the conjecture. This case demonstrates how non-trivial insights about the behaviour of large mathematical objects can be obtained from trained models, such that new structure can be discovered.

In this work we have demonstrated a framework for mathematicians to use machine learning that has led to mathematical insight across two distinct disciplines: one of the first connections between the algebraic and geometric structure of knots and a proposed resolution to a long-standing open conjecture in representation theory. Rather than use machine learning to directly generate conjectures, we focus on helping guide the highly tuned intuition of expert mathematicians, yielding results that are both interesting and deep. It is clear that intuition plays an important role in elite performance in many human pursuits. For example, it is critical for top Go players and the success of AlphaGo (ref.  33 ) came in part from its ability to use machine learning to learn elements of play that humans perform intuitively. It is similarly seen as critical for top mathematicians—Ramanujan was dubbed the Prince of Intuition 34 and it has inspired reflections by famous mathematicians on its place in their field 35 , 36 . As mathematics is a very different, more cooperative endeavour than Go, the role of AI in assisting intuition is far more natural. Here we show that there is indeed fruitful space to assist mathematicians in this aspect of their work.

Our case studies demonstrate how a foundational connection in a well-studied and mathematically interesting area can go unnoticed, and how the framework allows mathematicians to better understand the behaviour of objects that are too large for them to otherwise observe patterns in. There are limitations to where this framework will be useful—it requires the ability to generate large datasets of the representations of objects and for the patterns to be detectable in examples that are calculable. Further, in some domains the functions of interest may be difficult to learn in this paradigm. However, we believe there are many areas that could benefit from our methodology. More broadly, it is our hope that this framework is an effective mechanism to allow for the introduction of machine learning into mathematicians’ work, and encourage further collaboration between the two fields.

Supervised learning

In the supervised learning stage, the mathematician proposes a hypothesis that there exists a relationship between X ( z ) and Y ( z ). In this work we assume that there is no known function mapping from X ( z ) to Y ( z ), which in turn implies that X is not invertible (otherwise there would exist a known function Y  °  X   −1 ). While there may still be value to this process when the function is known, we leave this for future work. To test the hypothesis that X and Y are related, we generate a dataset of X ( z ), Y ( z ) pairs, where z is sampled from a distribution P Z . The results of the subsequent stages will hold true only for the distribution P Z , and not the whole space Z . Initially, sensible choices for P Z would be, for example, uniformly over the first N items for Z with a notion of ordering, or uniformly at random where possible. In subsequent iterations, P Z may be chosen to understand the behaviour on different parts of the space Z (for example, regions of Z that may be more likely to provide counterexamples to a particular hypothesis). To first test whether a relation between X ( z ) and Y ( z ) can be found, we use supervised learning to train a function \(\hat{f}\) that approximately maps X ( z ) to Y ( z ). In this work we use neural networks as the supervised learning method, in part because they can be easily adapted to many different types of X and Y and knowledge of any inherent geometry (in terms of invariances and symmetries) of the input domain X can be incorporated into the architecture of the network 37 . We consider a relationship between X ( z ) and Y ( z ) to be found if the accuracy of the learned function \(\hat{f}\) is statistically above chance on further samples from P Z on which the model was not trained. The converse is not true; namely, if the model cannot predict the relationship better than chance, it may mean that a pattern exists, but is sufficiently complicated that it cannot be captured by the given model and training procedure. If it does indeed exist, this can give a mathematician confidence to pursue a particular line of enquiry in a problem that may otherwise be only speculative.

Attribution techniques

If a relationship is found, the attribution stage is to probe the learned function \(\hat{f}\) with attribution techniques to further understand the nature of the relationship. These techniques attempt to explain what features or structures are relevant to the predictions made by \(\hat{f}\) , which can be used to understand what parts of the problem are relevant to explore further. There are many attribution techniques in the body of literature on machine learning and statistics, including stepwise forward feature selection 38 , feature occlusion and attention weights 39 . In this work we use gradient-based techniques 40 , broadly similar to sensitivity analysis in classical statistics and sometimes referred to as saliency maps. These techniques attribute importance to the elements of X ( z ), by calculating how much \(\hat{f}\) changes in predictions of Y (z) given small changes in X ( z ). We believe these are a particularly useful class of attribution techniques as they are conceptually simple, flexible and easy to calculate with machine learning libraries that support automatic differentiation 41 , 42 , 43 . Information extracted via attribution techniques can then be useful to guide the next steps of mathematical reasoning, such as conjecturing closed-form candidates f ′, altering the sampling distribution P Z or generating new hypotheses about the object of interest z , as shown in Fig. 1 . This can then lead to an improved or corrected version of the conjectured relationship between these quantities.

Problem framing

Not all knots admit a hyperbolic geometry; however, most do, and all knots can be constructed from hyperbolic and torus knots using satellite operations 44 . In this work we focus only on hyperbolic knots. We characterize the hyperbolic structure of the knot complement by a number of easily computable invariants. These invariants do not fully define the hyperbolic structure, but they are representative of the most commonly interesting properties of the geometry. Our initial general hypothesis was that the hyperbolic invariants would be predictive of algebraic invariants. The specific hypothesis we investigated was that the geometry is predictive of the signature. The signature is an ideal candidate as it is a well-understood and common invariant, it is easy to calculate for large knots and it is an integer, which makes the prediction task particularly straightforward (compared to, for example, a polynomial).

Data generation

We generated a number of datasets from different distributions P Z on the set of knots using the SnapPy software package 45 , as follows.

All knots up to 16 crossings ( ∼ 1.7 × 10 6 knots), taken from the Regina census 46 .

Random knot diagrams of 80 crossings generated by SnapPy’s random_link function ( ∼ 10 6 knots). As random knot generation can potentially lead to duplicates, we calculate a large number of invariants for each knot diagram and remove any samples that have identical invariants to a previous sample, as they are likely to represent that same knot with very high probability.

Knots obtained as the closures of certain braids. Unlike the previous two datasets, the knots that were produced here are not, in any sense, generic. Instead, they were specifically constructed to disprove Conjecture 1. The braids that we used were 4-braids ( n  = 11,756), 5-braids ( n  = 13,217) and 6-braids ( n  = 10,897). In terms of the standard generators σ i for these braid groups, the braids were chosen to be \(({\sigma }_{{i}_{1}}^{{n}_{1}}{\sigma }_{{i}_{2}}^{{n}_{2}}...{\sigma }_{{i}_{k}}^{{n}_{k}}{)}^{N}\) . The integers i j were chosen uniformly at random for the appropriate braid group. The powers n j were chosen uniformly at random in the ranges [−10, −3] and [3, 10]. The final power N was chosen uniformly between 1 and 10. The quantity ∑ | n i | was restricted to be at most 15 for 5-braids and 6-braids and 12 for 4-braids, and the total number of crossings N ∑ | n i | was restricted to lie in the range between 10 and 60. The rationale for these restrictions was to ensure a rich set of examples that were small enough to avoid an excessive number of failures in the invariant computations.

For the above datasets, we computed a number of algebraic and geometric knot invariants. Different datasets involved computing different subsets of these, depending on their role in forming and examining the main conjecture. Each of the datasets contains a subset of the following list of invariants: signature, slope, volume, meridional translation, longitudinal translation, injectivity radius, positivity, Chern–Simons invariant, symmetry group, hyperbolic torsion, hyperbolic adjoint torsion, invariant trace field, normal boundary slopes and length spectrum including the linking numbers of the short geodesics.

The computation of the canonical triangulation of randomly generated knots fails in SnapPy in our data generation process in between 0.6% and 1.7% of the cases, across datasets. The computation of the injectivity radius fails between 2.8% of the time on smaller knots up to 7.8% of the time on datasets of knots with a higher number of crossings. On knots up to 16 crossings from the Regina dataset, the injectivity radius computation failed in 5.2% of the cases. Occasional failures can occur in most of the invariant computations, in which case the computations continue for the knot in question for the remaining invariants in the requested set. Additionally, as the computational complexity of some invariants is high, operations can time out if they take more than 5 min for an invariant. This is a flexible bound and ultimately a trade-off that we have used only for the invariants that were not critical for our analysis, to avoid biasing the results.

Data encoding

The following encoding scheme was used for converting the different types of features into real valued inputs for the network: reals directly encoded; complex numbers as two reals corresponding to the real and imaginary parts; categoricals as one-hot vectors.

All features are normalized by subtracting the mean and dividing by the variance. For simplicity, in Fig. 3a , the salience values of categoricals are aggregated by taking the maximum value of the saliencies of their encoded features.

Model and training procedure

The model architecture used for the experiments was a fully connected, feed-forward neural network, with hidden unit sizes [300, 300, 300] and sigmoid activations. The task was framed as a multi-class classification problem, with the distinct values of the signature as classes, cross-entropy loss as an optimizable loss function and test classification accuracy as a metric of performance. It is trained for a fixed number of steps using a standard optimizer (Adam). All settings were chosen as a priori reasonable values and did not need to be optimized.

First, to assess whether there may be a relationship between the geometry and algebra of a knot, we trained a feed-forward neural network to predict the signature from measurements of the geometry on a dataset of randomly sampled knots. The model was able to achieve an accuracy of 78% on a held-out test set, with no errors larger than ±2. This is substantially higher than chance (a baseline accuracy of 25%), which gave us strong confidence that a relationship may exist.

To understand how this prediction is being made by the network, we used gradient-based attribution to determine which measurements of the geometry are most relevant to the signature. We do this using a simple sensitivity measure r i  that averages the gradient of the loss L with respect to a given input feature x i  over all of the examples x  in a dataset \({\mathscr{X}}\) :

This quantity for each input feature is shown in Fig. 3a , where we can determine that the relevant measurements of the geometry appear to be what is known as the cusp shape: the meridional translation, which we will denote μ , and the longitudinal translation, which we will denote λ . This was confirmed by training a new model to predict the signature from only these three measurements, which was able to achieve the same level of performance as the original model.

To confirm that the slope is a sufficient aspect of the geometry to focus on, we trained a model to predict the signature from the slope alone. Visual inspection of the slope and signature in Extended Data Fig. 1a, b shows a clear linear trend, and training a linear model on this data results in a test accuracy of 78%, which is equivalent to the predictive power of the original model. This implies that the slope linearly captures all of the information about the signature that the original model had extracted from the geometry.

The confidence intervals on the feature saliencies were calculated by retraining the model 10 times with a different train/test split and a different random seed initializing both the network weights and training procedure.

For our main dataset we consider the symmetric groups up to S 9 . The first symmetric group that contains a non-trivial Bruhat interval whose KL polynomial is not simply 1 is S 5 , and the largest interval in S 9 contains 9! ≈ 3.6 × 10 5 nodes, which starts to pose computational issues when used as inputs to networks. The number of intervals in a symmetric group S N is O ( N ! 2 ), which results in many billions of intervals in S 9 . The distribution of coefficients of the KL polynomials uniformly across intervals is very imbalanced, as higher coefficients are especially rare and associated with unknown complex structure. To adjust for this and simplify the learning problem, we take advantage of equivalence classes of Bruhat intervals that eliminate many redundant small polynomials 47 . This has the added benefit of reducing the number of intervals per symmetric group (for example, to ~2.9 million intervals in S 9 ). We further reduce the dataset by including a single interval for each distinct KL polynomial for all graphs with the same number of nodes, resulting in 24,322 non-isomorphic graphs for S 9 . We split the intervals randomly into train/test partitions at 80%/20%.

The Bruhat interval of a pair of permutations is a partially ordered set of the elements of the group, and it can be represented as a directed acyclic graph where each node is labelled by a permutation, and each edge is labelled by a reflection. We add two features at each node representing the in-degree and out-degree of that node.

For modelling the Bruhat intervals, we used a particular GraphNet architecture called a message-passing neural network (MPNN) 48 . The design of the model architecture (in terms of activation functions and directionality) was motivated by the algorithms for computing KL polynomials from labelled Bruhat intervals. While labelled Bruhat intervals contain privileged information, these algorithms hinted at the kind of computation that may be useful for computing KL polynomial coefficients. Accordingly, we designed our MPNN to algorithmically align to this computation 49 . The model is bi-directional, with a hidden layer width of 128, four propagation steps and skip connections. We treat the prediction of each coefficient of the KL polynomial as a separate classification problem.

First, to gain confidence that the conjecture is correct, we trained a model to predict coefficients of the KL polynomial from the unlabelled Bruhat interval. We were able to do so across the different coefficients with reasonable accuracy (Extended Data Table 1 ) giving some evidence that a general function may exist, as a four-step MPNN is a relatively simple function class. We trained a GraphNet model on the basis of a newly hypothesized representation and could achieve significantly better performance, lending evidence that it is a sufficient and helpful representation to understand the KL polynomial.

To understand how the predictions were being made by the learned function \(\hat{f}\) , we used gradient-based attribution to define a salient subgraph S G for each example interval G , induced by a subset of nodes in that interval, where L is the loss and x v is the feature for vertex v :

We then aggregated the edges by their edge type (each is a reflection) and compared the frequency of their occurrence to the overall dataset. The effect on extremal edges was present in the salient subgraphs for predictions of the higher-order terms ( q 3 , q 4 ), which are the more complicated and less well-understood terms.

The threshold C k for salient nodes was chosen a priori as the 99th percentile of attribution values across the dataset, although the results are present for different values of C k in the range [95,  99.5]. In Fig. 5a , we visualize a measure of edge attribution for a particular snapshot of a trained model for expository purposes. This view will change across time and random seeds, but we can confirm that the pattern remains by looking at aggregate statistics over many runs of training the model, as in Fig. 5b . In this diagram, the two-sample two-sided t -test statistics are as follows—simple edges: t  = 25.7, P  = 4.0 × 10 −10 ; extremal edges: t  = −13.8, P  = 1.1 × 10 −7 ; other edges: t  = −3.2, P  = 0.01. These significance results are robust to different settings of the hyper-parameters of the model.

Code availability

Interactive notebooks to regenerate the results for both knot theory and representation theory have been made available for download at https://github.com/deepmind .

Data availability

The generated datasets used in the experiments have been made available for download at https://github.com/deepmind .

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Acknowledgements

We thank J. Ellenberg, S. Mohamed, O. Vinyals, A. Gaunt, A. Fawzi and D. Saxton for advice and comments on early drafts; J. Vonk for contemporary supporting work; X. Glorot and M. Overlan for insight and assistance; and A. Pierce, N. Lambert, G. Holland, R. Ahamed and C. Meyer for assistance coordinating the research. This research was funded by DeepMind.

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A.D., D.H. and P.K. conceived of the project. A.D., A.J. and M.L. discovered the knot theory results, with D.Z. and N.T. running additional experiments. A.D., P.V. and G.W. discovered the representation theory results, with P.V. designing the model, L.B. running additional experiments, and C.B. providing advice and ideas. S.B. and R.T. provided additional support, experiments and infrastructure. A.D., D.H. and P.K. directed and managed the project. A.D. and P.V. wrote the paper with help and feedback from P.B., C.B., M.L., A.J., G.W., P.K. and D.H.

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Extended data fig. 1 empirical relationship between slope and signature..

a Signature vs slope for random dataset. b Signature vs slope for Regina dataset.

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Understanding and promoting students’ mathematical thinking: a review of research published in ESM

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• Volume 103 , pages 7–25, ( 2020 )

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In this paper, we offer a comparative review of research on understanding and promoting students’ mathematical thinking. The sources for the review are papers that were published in Educational Studies in Mathematics (ESM) during two windows of time: 1994–1998 and 2014–2018. Selection of these two time periods enables us to comment on the “state of the art” in research as well as identify changes over the past 25 years. The review is guided by an analysis of conceptualizations of “mathematical thinking” proposed in the research literature, selected curriculum documents, and international assessment programs such as the OECD’s Programme for International Student Assessment (PISA). The review not only documents salient features of research studies, such as the country of origin of the authors, educational level of the participants, research aims, theoretical perspectives, and methodological approaches, but also identifies the contribution to knowledge made by this body of work as well as future research directions and opportunities.

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This paper extends the review of research on mathematical thinking presented by the first author at the 2018 Regional Conference of the International Group for the Psychology of Mathematics Education (PME). The conference theme was Understanding and promoting mathematical thinking .

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Goos, M., Kaya, S. Understanding and promoting students’ mathematical thinking: a review of research published in ESM . Educ Stud Math 103 , 7–25 (2020). https://doi.org/10.1007/s10649-019-09921-7

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STAR: One thing we know from psychology about the learning process is that the act of reaching into your brain, grabbing some knowledge, pulling it out, chewing on it, talking about it, and putting it back helps you learn. Psychologists call this elaborative encoding. The more times you can do that process — putting knowledge in, getting it out, elaborating on it, putting it back in — the more you will have learned, remembered, and understood the material. We’re trying to get math teachers to help students engage in that process of elaborative encoding.

GAZETTE: How did you learn math yourself?

STAR: Learning math should involve some sense-making. It’s necessary that we listen to what our teacher tells us about the math and try to make sense of it in our minds. Math learning is not about pouring the words directly from the teacher’s mouth into the students’ ears and brains. That’s not the way it works. I think that’s how I learned math. But that’s not how I hope students learn math and that’s not how I hope teachers think about the teaching of math. Teachers should teach math in a way that encourages students to engage in sense-making and not merely to memorize or internalize exactly what the teacher says or does.

GAZETTE: Tell us about the teaching method described in the research.

STAR: One of the strategies that some teachers may use when teaching math is to show students how to solve problems and expect that the student is going to end up using the same method that the teacher showed. But there are many ways to solve math problems; there’s never just one way.

The strategy we developed asks that teachers compare two ways for solving a problem, side by side, and that they follow an instructional routine to lead a discussion to help students understand the difference between the two methods. That discussion is really the heart of this routine because it is fundamentally about sharing reasoning: Teachers ask students to explain why a strategy works, and students must dig into their heads and try to say what they understand. And listening to other people’s reasoning reinforces the process of learning.

GAZETTE: Why is this strategy an improvement over just learning a single method?

STAR: We think that learning multiple strategies for solving problems deepens students’ understanding of the content. There is a direct benefit to learning through comparing multiple methods, but there are also other types of benefits to students’ motivation. In this process, students come to see math a little differently — not just as a set of problems, each of which has exactly one way to solve it that you must memorize, but rather, as a terrain where there are always decisions to be made and multiple strategies that one might need to justify or debate. Because that is what math is.

For teachers, this can also be empowering because they are interested in increasing their students’ understanding, and we’ve given them a set of tools that can help them do that and potentially make the class more interesting as well. It’s important to note, too, that this approach is not something that we invented. In this case, what we’re asking teachers to do is something that they do a little bit of already. Every high school math teacher, for certain topics, is teaching students multiple strategies. It’s built into the curriculum. All that we’re saying is, first, you should do it more because it’s a good thing, and second, when you do it, this is a certain way that we found to be especially effective, both in terms of the visual materials and the pedagogy. It’s not a big stretch for most teachers. Conversations around ways to teach math for the past 30 or 40 years, and perhaps longer, have been emphasizing the use of multiple strategies.

GAZETTE: What are the potential challenges for math teachers to put this in practice?

STAR: If we want teachers to introduce students to multiple ways to solve problems, we must recognize that that is a lot of information for students and teachers. There is a concern that there could be information overload, and that’s very legitimate. Also, a well-intentioned teacher might take our strategy too far. A teacher might say something like, “Well, if comparing two strategies is good, then why don’t I compare three or four or five?” Not that that’s impossible to do well. But the visual materials you would have to design to help students manage that information overload are quite challenging. We don’t recommend that.

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Research shows the best ways to learn math.

Students learn math best when they approach the subject as something they enjoy. Speed pressure, timed testing and blind memorization pose high hurdles in the pursuit of math, according to Jo Boaler, professor of mathematics education  at Stanford Graduate School of Education and lead author on a new working paper called "Fluency Without Fear."

"There is a common and damaging misconception in mathematics – the idea that strong math students are fast math students," said Boaler, also cofounder of YouCubed at Stanford, which aims to inspire and empower math educators by making accessible in the most practical way the latest research on math learning.

Fortunately, said Boaler , the new national curriculum standards known as the Common Core Standards for K-12 schools de-emphasize the rote memorization of math facts. Maths facts are fundamental assumptions about math, such as the times tables (2 x 2 = 4), for example. Still, the expectation of rote memorization continues in classrooms and households across the United States.

While research shows that knowledge of math facts is important, Boaler said the best way for students to know math facts is by using them regularly and developing understanding of numerical relations. Memorization, speed and test pressure can be damaging, she added.

Number sense is critical

On the other hand, people with "number sense" are those who can use numbers flexibly, she said. For example, when asked to solve the problem of 7 x 8, someone with number sense may have memorized 56, but they would also be able to use a strategy such as working out 10 x 7 and subtracting two 7s (70-14).

"They would not have to rely on a distant memory," Boaler wrote in the paper.

In fact, in one research project the investigators found that the high-achieving students actually used number sense, rather than rote memory, and the low-achieving students did not.

The conclusion was that the low achievers are often low achievers not because they know less but because they don't use numbers flexibly.

"They have been set on the wrong path, often from an early age, of trying to memorize methods instead of interacting with numbers flexibly," she wrote. Number sense is the foundation for all higher-level mathematics, she noted.

Role of the brain

Boaler said that some students will be slower when memorizing, but still possess exceptional mathematics potential.

"Math facts are a very small part of mathematics, but unfortunately students who don't memorize math facts well often come to believe that they can never be successful with math and turn away from the subject," she said.

Prior research found that students who memorized more easily were not higher achieving – in fact, they did not have what the researchers described as more "math ability" or higher IQ scores. Using an MRI scanner, the only brain differences the researchers found were in a brain region called the hippocampus, which is the area in the brain responsible for memorizing facts – the working memory section.

But according to Boaler, when students are stressed – such as when they are solving math questions under time pressure – the working memory becomes blocked and the students cannot as easily recall the math facts they had previously studied. This particularly occurs among higher achieving students and female students, she said.

Some estimates suggest that at least a third of students experience extreme stress or "math anxiety" when they take a timed test, no matter their level of achievement. "When we put students through this anxiety-provoking experience, we lose students from mathematics," she said.

Math treated differently

Boaler contrasts the common approach to teaching math with that of teaching English. In English, a student reads and understands novels or poetry, without needing to memorize the meanings of words through testing. They learn words by using them in many different situations – talking, reading and writing.

"No English student would say or think that learning about English is about the fast memorization and fast recall of words," she added.

Strategies, activities

In the paper, coauthored by Cathy Williams, cofounder of YouCubed, and Amanda Confer, a Stanford graduate student in education, the scholars provide activities for teachers and parents that help students learn math facts at the same time as developing number sense. These include number talks, addition and multiplication activities, and math cards.

Importantly, Boaler said, these activities include a focus on the visual representation of number facts. When students connect visual and symbolic representations of numbers, they are using different pathways in the brain, which deepens their learning, as shown by recent brain research.

"Math fluency" is often misinterpreted, with an over-emphasis on speed and memorization, she said. "I work with a lot of mathematicians, and one thing I notice about them is that they are not particularly fast with numbers; in fact some of them are rather slow. This is not a bad thing; they are slow because they think deeply and carefully about mathematics."

She quotes the famous French mathematician, Laurent Schwartz. He wrote in his autobiography that he often felt stupid in school, as he was one of the slowest math thinkers in class.

Math anxiety and fear play a big role in students dropping out of mathematics, said Boaler.

"When we emphasize memorization and testing in the name of fluency we are harming children, we are risking the future of our ever-quantitative society and we are threatening the discipline of mathematics," she said. "We have the research knowledge we need to change this and to enable all children to be powerful mathematics learners. Now is the time to use it."

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The Year in Math

December 22, 2023

Video : In 2023, mathematicians improved bounds on Ramsey numbers, a central measure of order in graphs; found a new aperiodic monotile; and discovered a new upper bound to the size of sets without 3-term arithmetic progressions.

Christopher Webb Young, Kristina Armitage and Merrill Sherman/ Quanta Magazine

Introduction

Mathematical truths are often born of the conflict between order and disorder. Mathematicians discover patterns, and, to better understand the mysterious forces at play, they look for countervailing impulses that disrupt those patterns.

That tension came up repeatedly in our coverage this past year. We covered breakthroughs in graph theory, combinatorics, number theory and geometry — areas where patterns arise in unexpected ways, sometimes because of connections between seemingly distinct mathematical structures, and sometimes because of hidden intrinsic mechanisms uncovered by mathematicians in new proofs.

In a riveting interview with our senior writer Jordana Cepelewicz, Andrew Granville discussed how calculation and experimentation can, in sometimes forgotten ways, help mathematicians search for hidden patterns. He also spoke about changes in what it takes to convince other mathematicians that a result is true, and why he believes that examining the social nature of mathematics is essential to understanding what a proof is.

This was one of several conversations we published this past year about the nature of mathematical truth. Eugenia Cheng spoke with Joy of Why podcast host Steven Strogatz about category theory , a sort of “mathematics of mathematics” that can scare off other mathematicians with its level of abstraction. And Justin Moore spoke with Strogatz about the limits of the axioms — basic, obvious truths — of set theory and why there will always be important, unanswerable mathematical questions.

Though the bulk of our coverage fell squarely in the abstract realm, Minhyong Kim spoke with Kevin Hartnett about Mathematics for Humanity, an organization he founded to support mathematicians who want to use math to solve social challenges. And Mike Orcutt reported on how mathematics is used to ascertain the fairness of legislative district maps and to draw more equitable ones.

DVDP for Quanta Magazine

A Big Year in Graph Theory

If there is one area of math that was particularly fruitful in 2023, it’s graph theory. One of the biggest mathematical discoveries of the past year was the proof of a new, tighter upper bound to Ramsey numbers . These numbers measure the size that graphs must reach before inevitably containing objects called cliques. The discovery, announced in March, was the first advance of its type since 1935. It pertained to so-called symmetric Ramsey numbers. This was followed in June by a new result on the more general asymmetric case.

Both of these papers concerned what happens as graphs grow infinitely large. But Quanta also pondered the middle distance , looking at what mathematicians can prove about graphs that are too large to analyze using brute force, but smaller than the infinite, asymptotic limit.

We chronicled new results on how networks of connected oscillators come into synchrony and how graph theory connects to quantum field theory . We reported a new discovery about the possibilities of subdividing mathematical objects called vector spaces in a particular way into subsets called designs . And Patrick Honner, our Quantized Academy columnist, wrote about the way that local properties of graphs govern their global structure.

Quanta also published articles on two long-standing coloring problems. One explored the proof of the famous four-color theorem , which shows how four colors are enough to color any map on the plane so that no two adjacent regions have the same color. The other covered a new result on a less well-known but equally intriguing question, which asks how much of a plane can be colored in a way that ensures that no two points that are exactly one unit apart have the same color.

Samuel Velasco/ Quanta Magazine

Making Combinatorics Conjectures Count

Graph theory can be thought of as a branch of combinatorics — the mathematical study of counting. Counting what can happen with collections of nodes and edges is, in some sense, a special case of counting combinations more generally.

The year ended with a landmark proof by four prominent mathematicians of a longstanding conjecture that relates combinatorics to the algebraic structure of sets.

Back in February, two computer scientists, Zander Kelley and Raghu Meka, stunned mathematicians with news of an out-of-left-field breakthrough on an old combinatorics question: How many integers can you throw into a bucket while making sure that no three of them form an evenly spaced progression (like 3, 8 and 13 or 101, 201 and 301)? Kelley and Meka smashed a long-standing upper bound on the number of integers smaller than some cap N that could be put in the bucket without creating such a pattern.

The previous month, Kevin Hartnett reported on a paper from November 2022 by another outsider — a researcher at Google named Justin Gilmer who had left mathematics years before, but had never stopped thinking about a combinatorial problem called the union-closed conjecture. This conjecture concerns families of sets like {1}, {1, 2}, {2, 3, 4}, {1, 2, 3, 4}. This family is “union-closed” because if you combine any two sets in the family, the combination is also in the family. The conjecture says that if a family is union-closed, it must have at least one number that appears in at least half the sets. Gilmer used an argument drawn from information theory that relied on randomly choosing two sets from a union-closed family that met certain characteristics to prove a result that is an important step towards the full conjecture. His argument is yet another example of how randomness can be used as a tool to infer the existence of structure.

By contrast, an April article by Kevin Hartnett described an instance where intricate but simple structures surprisingly turn out to be possible. Bernardo Subercaseaux and Marijn Heule showed that it’s possible to fill an infinite grid with numbers in such a way that the distance between two occurrences of the same number must be greater than the number itself — using only the numbers between 1 and 15.

And longtime Quanta contributor Erica Klarreich wrote about the surprising prevalence of so-called intransitive dice . These are, for example, sets of three dice A, B and C in which A is likely to beat (roll a higher number than) B, B is likely to beat C and C is likely to beat A. A new paper showed that if you know only that die A beats die B and B beats C, that gives no information about whether A or C is likely to prevail in a head-to-head matchup.

Courtesy of Samuel Jinglian Li

New Connections in Number Theory

Perhaps more than in any other area of mathematics, number theorists can prove simple-sounding theorems using incredibly complicated technical constructions. This year, Quanta took readers on a tour of some of those constructions. We published an in-depth visual explainer of modular forms, which have been described as the “fifth fundamental operation” of math, along with addition, subtraction, multiplication and division. And we took readers on a historical tour of quadratic reciprocity , one of number theory’s most powerful tools. The modular-forms explainer was inspired by an article about so-called noncongruence modular forms — a less well-studied type of function that nevertheless has major implications for physics.

Max Levy, who wrote the quadratic-reciprocity explainer, got interested in the subject while reporting about a surprising summer discovery about patterns that circles can make. Levy recounted how two students working on a summer research project helped disprove a long-standing conjecture about how circles can be harmoniously nested, called the local-to-global conjecture. It was one of many developments this year that showcased the increasing utility of computational tools in mathematics. The students and their co-authors first found evidence that the conjecture was false by poring over computer-generated plots they’d created in an effort to see it at work.

Modular forms are closely related to elliptic curves — smooth functions of two variables where one variable is squared and the other cubed. (The functions also satisfy some particular mathematical constraints.) The relationship between the two was central to Andrew Wiles’ 1994 proof of Fermat’s Last Theorem. Hartnett wrote about advances in researchers’ understanding of that relationship for elliptic curves that are defined with variables drawn from imaginary quadratic fields — numbers of the form a + b $latex \sqrt{-5}$ where a and b are both rational numbers, or fractions.

He also wrote about a long-awaited magnum opus — a 451-page manuscript by the Fields medalist Akshay Venkatesh, together with Yiannis Sakellaridis and David Ben-Zvi, which elaborates further connections between objects related to modular forms and L -functions, an important type of infinite sum with a deep relationship to prime numbers.

Number theorists pay particular attention to prime numbers and the subtle and beautiful ways they’re distributed among the other integers. Intriguingly, if you consider them going out to infinity, it has long been known that the primes leave equal numbers of remainders when divided by some number — for instance, if you divide all the prime numbers by 5, you’ll get equal numbers of the remainders 1, 2, 3 and 4. But mathematicians keep striving to prove results about how quickly primes even out. In October, we reported on a new generation of mathematicians proving theorems about the ways in which primes are distributed.

We also introduced — and reintroduced — a fun mathematical game called Hyperjumps that explores the tension between structure and randomness by challenging players to create simple sequences of numbers using basic arithmetic.

Aperiodic Monotile Found After Long Search

It was also an exciting year in geometry. The most attention-getting result of the year was the discovery of a new kind of tile that covers the plane in a pattern that never repeats. A two-tile combination that does this has been known since the 1970s, but the single tile, discovered by a hobbyist named David Smith and announced in March, was a sensation. Fans used the simple design as a cookie cutter and sewed it into quilts. We followed our news coverage with a column explaining some of the underlying math and another giving a brief history of tiling .

Speaking of needles, it was also a year of progress on the Kakeya conjecture, which asks how small a volume of space an idealized needle can occupy while spinning in all directions. A new proof of a special case of the conjecture (called the “sticky” Kakeya conjecture) gives strong evidence that the more general conjecture is true.

The conjecture turns out to have implications not only for geometry, but also for harmonic analysis and the study of partial differential equations. A follow-up explainer examines those implications. And a Quantized Academy column takes readers through the conjecture’s underlying logic.

In other geometry news, a long-standing idea about maps between spheres of different dimensionality, called the telescope conjecture, was shown to be false . Particular types of contact structures (patterns of planes that satisfy certain mathematical properties) that had long been thought to be impossible turned out to exist .

We interviewed Emmy Murphy , a geometer who studies such contact structures. Murphy describes contact geometry (and its sibling, symplectic geometry) as existing in the middle of a spectrum of rigidity and flexibility. In rigid geometry, much depends, she said, on precise measurements, while flexible geometry tends to resemble algebra. But in between, she said, is where “visual thinking is more useful.”

In January, the mathematician Assaf Naor and the computer scientist Oded Regev proved the existence of so-called spherical cubes. These are objects whose surface area grows slowly — as does the surface area of spheres in higher dimensions — but which can completely fill space the way cubes can.

One of the most prominent geometers of the 20th century, Eugenio Calabi, died at age 100 on September 25. Jerry Kazdan, one of his longtime colleagues, said that Calabi would “ask interesting questions that no one else was thinking about.” Our obituary of Calabi explores those questions, focusing particularly on his best-known discovery, Calabi-Yau manifolds, which later became central to string theory in physics.

Harol Bustos for Quanta Magazine

It’s an Unstable World After All

Speaking of physics, we also published several new results about the mathematics of black holes, a favorite subject of contributing writer Steve Nadis. He wrote about a new paper that found an infinite number of different black hole shapes in higher dimensions, and another paper that clarifies the mathematics of the boundaries of black holes .

In April, we described how mathematicians are teaming up with physicists to understand new kinds of symmetries in quantum field theories.

Kathryn Mann and Thomas Barthelmé, along with Steven Frankel, published a series of papers characterizing dynamical systems called Anosov flows that balance chaos and stability. At any given point, the flows converge in one direction and diverge in another.

And in what might be the most unsettling math article of the year, we related news of a series of three papers by Marcel Guàrdia, Jacques Fejoz and Andrew Clarke showing that planetary orbits in a model solar system will always be unstable. The good news is that their model is quite unlike our solar system, although Clarke thinks similar instabilities may exist here as well.

But if they do, they’re not going to send any of the planets out of their orbits anytime soon, so you can look forward to another year of math coverage from Quanta in 2024.

Correction:   January 12, 2024 An earlier version of this story said that Justin Gilmer had proved the union-closed conjecture. That is incorrect; he proved a result that is an important step towards that conjecture, but not the full result. The text has been updated.

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Mathematicians Prove Hawking Wrong About the Most Extreme Black Holes

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all the numbers are changing, but what doesn't change is the relationship between x and y: y is always one more than twice x. That is, y=2x+1. Finding what doesn't change "tames" the situation. So, you have tamed this problem! Yay. And if you want a fancy mathematical name for things that don’t vary, we call these things "invariants." The number of messed-up recruits is invariant, even though they are all wiggling back and forth, trying to figure out which way is right!

3) Encourage generalizations

So, of course, the next question that comes to my mind is how to generalize what you’ve already discovered: there are 15 ways that 2 mistakes can be arranged in a line of 6 recruits. What about a different number of mistakes? Or a different number of recruits? Is there some way to predict? Or, alternatively, is there some way to predict how these 15 ways of making mistakes will play out as the recruits try to settle themselves down? Which direction interests you?

4) Inquire about reasoning and rigor

The students were looking at the number of ways the recruits could line up with 2 out of n faced the wrong way: Anyway, I had a question of my own. It looks like the number of possibilities increases pretty fast, as the number of recruits increases. For example, I counted 15 possibilities in your last set (the line of six). What I wonder is this: when the numbers get that large, how you can possibly know that you've found all the possibilities? (For example, I noticed that >>>><< is missing.) The question "How do I know I've counted 'em all?" is actually quite a big deal in mathematics, as mathematicians are often called upon to find ways of counting things that nobody has ever listed (exactly like the example you are working on).

The students responded by finding a pattern for generating the lineups in a meaningful order: The way that we can prove that we have all the possibilities is that we can just add the number of places that the second wrong person could be in. For example, if 2 are wrong in a line of 6, then the first one doesn’t move and you count the space in which the second one can move in. So for the line of six, it would be 5+4+3+2+1=15. That is the way to make sure that we have all the ways. Thanks so much for giving challenges. We enjoyed thinking!

5) Work towards proof

a) The group wrote the following: When we found out that 6 recruits had 15 different starting arrangements, we needed more information. We needed to figure out how many starting positions are there for a different number of recruits.

By drawing out the arrangements for 5 recruits and 7 recruits we found out that the number of starting arrangements for the recruit number before plus that recruit number before it would equal the number of starting arrangements for that number of recruits.

We also found out that if you divide the starting arrangements by the number of recruits there is a pattern.

To which the mentor replied: Wow! I don't think (in all the years I've been hanging around mathematics) I've ever seen anyone describe this particular pattern before! Really nice! If you already knew me, you'd be able to predict what I'm about to ask, but you don't, so I have to ask it: "But why?" That is, why is this pattern (the 6, 10, 15, 21, 28…) the pattern that you find for this circumstance (two recruits wrong in lines of lengths, 4, 5, 6, 7, 8…)? Answering that—explaining why you should get those numbers and why the pattern must continue for longer lines—is doing the kind of thing that mathematics is really about.

b) Responding to students studying a circular variation of raw recruits that never settled down: This is a really interesting conclusion! How can you show that it will always continue forever and that it doesn’t matter what the original arrangement was? Have you got a reason or did you try all the cases or…? I look forward to hearing more from you.

6) Distinguish between examples and reasons

a) You have very thoroughly dealt with finding the answer to the problem you posed—it really does seem, as you put it, "safe to say" how many there will be. Is there a way that you can show that that pattern must continue? I guess I’d look for some reason why adding the new recruit adds exactly the number of additional cases that you predict. If you could say how the addition of one new recruit depends on how long the line already is, you’d have a complete proof. Want to give that a try?

b) A student, working on Amida Kuji and having provided an example, wrote the following as part of a proof: In like manner, to be given each relationship of objects in an arrangement, you can generate the arrangement itself, for no two different arrangements can have the same object relationships. The mentor response points out the gap and offers ways to structure the process of extrapolating from the specific to the general: This statement is the same as your conjecture, but this is not a proof. You repeat your claim and suggest that the example serves as a model for a proof. If that is so, it is up to you to make the connections explicit. How might you prove that a set of ordered pairs, one per pair of objects forces a unique arrangement for the entire list? Try thinking about a given object (e.g., C) and what each of its ordered pairs tells us? Try to generalize from your example. What must be true for the set of ordered pairs? Are all sets of n C2 ordered pairs legal? How many sets of n C2 ordered pairs are there? Do they all lead to a particular arrangement? Your answers to these questions should help you work toward a proof of your conjecture.

9) Encourage extensions

What you’ve done—finding the pattern, but far more important, finding the explanation (and stating it so clearly)—is really great! (Perhaps I should say "finding and stating explanations like this is real mathematics"!) Yet it almost sounded as if you put it down at the very end, when you concluded "making our project mostly an interesting coincidence." This is a truly nice piece of work!

The question, now, is "What next?" You really have completely solved the problem you set out to solve: found the answer, and proved that you’re right!

I began looking back at the examples you gave, and noticed patterns in them that I had never seen before. At first, I started coloring parts red, because they just "stuck out" as noticeable and I wanted to see them better. Then, it occurred to me that I was coloring the recruits that were back-to-back, and that maybe I should be paying attention to the ones who were facing each other, as they were "where the action was," so I started coloring them pink. (In one case, I recopied your example to do the pinks.) To be honest, I’m not sure what I’m looking for, but there was such a clear pattern of the "action spot" moving around that I thought it might tell me something new. Anything come to your minds?

10) Build a Mathematical Community

I just went back to another paper and then came back to yours to look again. There's another pattern in the table. Add the recruits and the corresponding starting arrangements (for example, add 6 and 15) and you get the next number of starting arrangements. I don't know whether this, or your 1.5, 2, 2.5, 3, 3.5… pattern will help you find out why 6, 10, 15… make sense as answers, but they might. Maybe you can work with [your classmates] who made the other observation to try to develop a complete understanding of the problem.

11) Highlight Connections

Your rule—the (n-1)+(n-2)+(n-3)+… +3+2+1 part—is interesting all by itself, as it counts the number of dots in a triangle of dots. See how?

12) Wrap Up

This is really a very nice and complete piece of work: you've stated a problem, found a solution, and given a proof (complete explanation of why that solution must be correct). To wrap it up and give it the polish of a good piece of mathematical research, I'd suggest two things.

The first thing is to extend the idea to account for all but two mistakes and the (slightly trivial) one mistake and all but one mistake. (If you felt like looking at 3 and all but 3, that'd be nice, too, but it's more work—though not a ton—and the ones that I suggested are really not more work.)

The second thing I'd suggest is to write it all up in a way that would be understandable by someone who did not know the problem or your class: clear statement of the problem, the solution, what you did to get the solution, and the proof.

I look forward to seeing your masterpiece!

Advice for Keeping a Formal Mathematics Research Logbook

As part of your mathematics research experience, you will keep a mathematics research logbook. In this logbook, keep a record of everything you do and everything you read that relates to this work. Write down questions that you have as you are reading or working on the project. Experiment. Make conjectures. Try to prove your conjectures. Your journal will become a record of your entire mathematics research experience. Don’t worry if your writing is not always perfect. Often journal pages look rough, with notes to yourself, false starts, and partial solutions. However, be sure that you can read your own notes later and try to organize your writing in ways that will facilitate your thinking. Your logbook will serve as a record of where you are in your work at any moment and will be an invaluable tool when you write reports about your research.

Ideally, your mathematics research logbook should have pre-numbered pages. You can often find numbered graph paper science logs at office supply stores. If you can not find a notebook that has the pages already numbered, then the first thing you should do is go through the entire book putting numbers on each page using pen.

• Date each entry.

• Work in pen.

• Don’t erase or white out mistakes. Instead, draw a single line through what you would like ignored. There are many reasons for using this approach:

– Your notebook will look a lot nicer if it doesn’t have scribbled messes in it.

– You can still see what you wrote at a later date if you decide that it wasn’t a mistake after all.

– It is sometimes useful to be able to go back and see where you ran into difficulties.

– You’ll be able to go back and see if you already tried something so you won’t spend time trying that same approach again if it didn’t work.

• When you do research using existing sources, be sure to list the bibliographic information at the start of each section of notes you take. It is a lot easier to write down the citation while it is in front of you than it is to try to find it at a later date.

• Never tear a page out of your notebook. The idea is to keep a record of everything you have done. One reason for pre-numbering the pages is to show that nothing has been removed.

• If you find an interesting article or picture that you would like to include in your notebook, you can staple or tape it onto a page.

Advice for Keeping a Loose-Leaf Mathematics Research Logbook

Get yourself a good loose-leaf binder, some lined paper for notes, some graph paper for graphs and some blank paper for pictures and diagrams. Be sure to keep everything that is related to your project in your binder.

– Your notebook will look a lot nicer if it does not have scribbled messes in it.

• Be sure to keep everything related to your project. The idea is to keep a record of everything you have done.

• If you find an interesting article or picture that you would like to include in your notebook, punch holes in it and insert it in an appropriate section in your binder.

Making Mathematics Home | Mathematics Projects | Students | Teachers | Mentors | Parents | Hard Math Café |

• Open access
• Published: 11 March 2019

Enhancing achievement and interest in mathematics learning through Math-Island

• Charles Y. C. Yeh   ORCID: orcid.org/0000-0003-4581-6575 1 ,
• Hercy N. H. Cheng 2 ,
• Zhi-Hong Chen 3 ,
• Calvin C. Y. Liao 4 &
• Tak-Wai Chan 5  

Research and Practice in Technology Enhanced Learning volume  14 , Article number:  5 ( 2019 ) Cite this article

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Conventional teacher-led instruction remains dominant in most elementary mathematics classrooms in Taiwan. Under such instruction, the teacher can rarely take care of all students. Many students may then continue to fall behind the standard of mathematics achievement and lose their interest in mathematics; they eventually give up on learning mathematics. In fact, students in Taiwan generally have lower interest in learning mathematics compared to many other regions/countries. Thus, how to enhance students’ mathematics achievement and interest are two major problems, especially for those low-achieving students. This paper describes how we designed a game-based learning environment, called Math-Island , by incorporating the mechanisms of a construction management game into the knowledge map of the elementary mathematics curriculum. We also report an experiment conducted with 215 elementary students for 2 years, from grade 2 to grade 3. In this experiment, in addition to teacher-led instruction in the classroom, students were directed to learn with Math-Island by using their own tablets at school and at home. As a result of this experiment, we found that there is an increase in students’ mathematics achievement, especially in the calculation and word problems. Moreover, the achievements of low-achieving students in the experimental school outperformed the low-achieving students in the control school (a control group in another school) in word problems. Moreover, both the low-achieving students and the high-achieving students in the experimental school maintained a rather high level of interest in mathematics and in the system.

Introduction

Mathematics has been regarded as a fundamental subject because arithmetic and logical reasoning are the basis of science and technology. For this reason, educational authorities emphasize students’ proficiency in computational skills and problem-solving. Recently, the results of the Program for International Student Assessment (PISA) and the Trends in Mathematics and Science Study (TIMSS) in 2015 (OECD 2016 ; Mullis et al. 2016 ) revealed a challenge for Taiwan. Although Taiwanese students had higher average performance in mathematics literacy compared to students in other countries, there was still a significant percentage of low-achieving students in Taiwan. Additionally, most Taiwanese students show low levels of interest and confidence in learning mathematics (Lee 2012 ).

The existence of a significant percentage of low-achieving students is probably due to teacher-led instruction, which still dominates mathematics classrooms in most Asian countries. It should be noted that students in every classroom possess different abilities and hence demonstrate different achievements. Unfortunately, in teacher-led instruction, all the students are required to learn from the teacher in the same way at the same pace (Hwang et al. 2012 ). Low-achieving students, without sufficient time, are forced to receive knowledge passively. Barr and Tagg ( 1995 ) pointed out that it is urgent for low-achieving students to have more opportunities to learn mathematics at their own pace. Researchers suggested one-to-one technology (Chan et al. 2006 ) through which every student is equipped with a device to learn in school or at home seamlessly. Furthermore, they can receive immediate feedback from Math-Island, which supports their individualized learning actively and productively. Thus, this may provide more opportunities for helping low-achieving students improve their achievement.

The low-interest problem for almost all students in Taiwan is usually accompanied by low motivation (Krapp 1999 ). Furthermore, students with continuously low performance in mathematics may eventually lose their interest and refuse to learn further (Schraw et al. 2001 ). This is a severe problem. To motivate students to learn, researchers design educational games to provide enjoyable and engaging learning experiences (Kiili and Ketamo 2007 ). Some of these researchers found that game-based learning may facilitate students’ learning in terms of motivation and learning effects (Liu and Chu 2010 ), spatial abilities and attention (Barlett et al. 2009 ), situated learning, and problem-solving (Li and Tsai 2013 ). Given these positive results, we hope that our educational game can enhance and sustain the student’s interest in learning mathematics.

In fact, many researchers who endeavored to develop educational games for learning mathematics have shown that their games could facilitate mathematics performance, enjoyment, and self-efficacy (Ku et al. 2014 ; McLaren et al. 2017 ). Although some of the studies were conducted for as many as 4 months (e.g., Hanus and Fox 2015 ), one may still criticize them for the possibility that the students’ interest could be a novelty effect—meaning their interest will decrease as the feeling of novelty diminishes over time (Koivisto and Hamari 2014 ). Due to the limitations of either experimental time or sample sizes, most studies could not effectively exclude the novelty effect of games, unless they were conducted in a natural setting for a long time.

In this study, we collaborated with an experimental elementary school for more than 2 years. The mathematics teachers in the school adopted our online educational game, Math-Island . The students used their own tablet PCs to learn mathematics from the game in class or at home at their own pace. In particular, low-achieving students might have a chance to catch up with the other students and start to feel interested in learning mathematics. Most importantly, because the online educational game was a part of the mathematics curriculum, the students could treat the game as their ordinary learning materials like textbooks. In this paper, we reported a 2-year study, in which 215 second graders in the school adopted the Math-Island game in their daily routine. More specifically, the purpose of this paper was to investigate the effect of the game on students’ mathematics achievement. Additionally, we were also concerned about how well the low-achieving students learned, whether they were interested in mathematics and the game, and how their interest in mathematics compared with that of high-achieving students. In such a long-term study with a large sample size, it was expected that the novelty effect would be considerably reduced, allowing us to evaluate the effect of the educational game on students’ achievement and interest.

The paper is organized as follows. In the “ Related works ” section, we review related studies on computer-supported mathematics learning and educational games. In the “ Design ” section, the game mechanism and the system design are presented. In the “ Method ” section, we describe the research method and the procedures of this study. In the “ Results ” section, the research results about students’ achievement and interest are presented. In the “ Discussion on some features of this study ” section, we discuss the long-term study, knowledge map design, and the two game mechanisms. Finally, the summary of the current situation and potential future work is described in the “ Conclusion and future work ” section.

Related works

Computer-supported mathematics learning.

The mathematics curriculum in elementary schools basically includes conceptual understanding, procedural fluency, and strategic competence in terms of mathematical proficiency (see Kilpatrick et al. 2001 ). First, conceptual understanding refers to students’ comprehension of mathematical concepts and the relationships between concepts. Researchers have designed various computer-based scaffolds and feedback to build students’ concepts and clarify potential misconceptions. For example, for guiding students’ discovery of the patterns of concepts, Yang et al. ( 2012 ) adopted an inductive discovery learning approach to design online learning materials in which students were provided with similar examples with a critical attribute of the concept varied. McLaren et al. ( 2017 ) provided students with prompts to correct their common misconceptions about decimals. They conducted a study with the game adopted as a replacement for seven lessons of regular mathematics classes. Their results showed that the educational game could facilitate better learning performance and enjoyment than a conventional instructional approach.

Second, procedural fluency refers to the skill in carrying out calculations correctly and efficiently. For improving procedural fluency, students need to have knowledge of calculation rules (e.g., place values) and practice the procedure without mistakes. Researchers developed various digital games to overcome the boredom of practice. For example, Chen et al. ( 2012a , 2012b ) designed a Cross Number Puzzle game for practicing arithmetic expressions. In the game, students could individually or collaboratively solve a puzzle, which involved extensive calculation. Their study showed that the low-ability students in the collaborative condition made the most improvement in calculation skills. Ku et al. ( 2014 ) developed mini-games to train students’ mental calculation ability. They showed that the mini-games could not only improve students’ calculation performance but also increase their confidence in mathematics.

Third, strategic competence refers to mathematical problem-solving ability, in particular, word problem-solving in elementary education. Some researchers developed multilevel computer-based scaffolds to help students translate word problems to equations step by step (e.g., González-Calero et al. 2014 ), while other researchers noticed the problem of over-scaffolding. Specifically, students could be too scaffolded and have little space to develop their abilities. To avoid this situation, many researchers proposed allowing students to seek help during word problem-solving (Chase and Abrahamson 2015 ; Roll et al. 2014 ). For example, Cheng et al. ( 2015 ) designed a Scaffolding Seeking system to encourage elementary students to solve word problems by themselves by expressing their thinking first, instead of receiving and potentially abusing scaffolds.

Digital educational games for mathematics learning

Because mathematics is an abstract subject, elementary students easily lose interest in it, especially low-achieving students. Some researchers tailored educational games for learning a specific set of mathematical knowledge (e.g., the Decimal Points game; McLaren et al. 2017 ), so that students could be motivated to learn mathematics. However, if our purpose was to support a complete mathematics curriculum for elementary schools, it seemed impractical to design various educational games for all kinds of knowledge. A feasible approach is to adopt a gamified content structure to reorganize all learning materials. For example, inspired by the design of most role-playing games, Chen et al. ( 2012a , 2012b ) proposed a three-tiered framework of game-based learning—a game world, quests, and learning materials—for supporting elementary students’ enjoyment and goal setting in mathematics learning. Furthermore, while a game world may facilitate students’ exploration and participation, quests are the containers of learning materials with specific goals and rewards. In the game world, students receive quests from nonplayer virtual characters, who may enhance social commitments. To complete the quests, students have to make efforts to undertake learning materials. Today, quests have been widely adopted in the design of educational games (e.g., Azevedo et al. 2012 ; Hwang et al. 2015 ).

However, in educational games with quests, students still play the role of receivers rather than active learners. To facilitate elementary students’ initiative, Lao et al. ( 2017 ) designed digital learning contracts, which required students to set weekly learning goals at the beginning of a week and checked whether they achieved the goals at the end of the week. More specifically, when setting weekly goals, students had to decide on the quantity of learning materials that they wanted to undertake in the coming week. Furthermore, they also had to decide the average correctness of the tests that followed the learning materials. To help them set reasonable and feasible goals, the system provided statistics from the past 4 weeks. As a result, the students may reflect on how well they learned and then make appropriate decisions. After setting goals, students are provided with a series of learning materials for attempting to accomplish those goals. At the end of the week, they may reflect on whether they achieved their learning goals in the contracts. In a sense, learning contracts may not only strengthen the sense of commitment but also empower students to take more control of their learning.

In textbooks or classrooms, learning is usually predefined as a specific sequence, which students must follow to learn. Nevertheless, the structure of knowledge is not linear, but a network. If we could reorganize these learning materials according to the structure of knowledge, students could explore knowledge and discover the relationships among different pieces of knowledge when learning (Davenport and Prusak 2000 ). Knowledge mapping has the advantage of providing students concrete content through explicit knowledge graphics (Ebener et al. 2006 ). Previous studies have shown that the incorporation of knowledge structures into educational games could effectively enhance students’ achievement without affecting their motivation and self-efficacy (Chu et al. 2015 ). For this reason, this study attempted to visualize the structure of knowledge in an educational game. In other words, a knowledge map was visualized and gamified so that students could make decisions to construct their own knowledge map in games.

To enhance students’ mathematics achievement and interests, we designed the Math-Island online game by incorporating a gamified knowledge map of the elementary mathematics curriculum. More specifically, we adopt the mechanisms of a construction management game , in which every student owns a virtual island (a city) and plays the role of the mayor. The goal of the game is to build their cities on the islands by learning mathematics.

System architecture

The Math-Island game is a Web application, supporting cross-device interactions among students, teachers, and the mathematics content structure. The system architecture of the Math-Island is shown in Fig.  1 . The pedagogical knowledge and learning materials are stored in the module of digital learning content, organized by a mathematical knowledge map. The students’ portfolios about interactions and works are stored in the portfolio database and the status database. When a student chooses a goal concept in the knowledge map, the corresponding digital learning content is arranged and delivered to his/her browser. Besides, when the student is learning in the Math-Island, the feedback module provides immediate feedback (e.g., hints or scaffolded solutions) for guidance and grants rewards for encouragement. The learning results can also be shared with other classmates by the interaction module. In addition to students, their teachers can also access the databases for the students’ learning information. Furthermore, the information consists of the students’ status (e.g., learning performance or virtual achievement in the game) and processes (e.g., their personal learning logs). In the Math-Island, it is expected that students can manage their learning and monitor the learning results by the construction management mechanism. In the meantime, teachers can also trace students’ learning logs, diagnose their weaknesses from portfolio analysis, and assign students with specific tasks to improve their mathematics learning.

The system architecture of Math-Island

• Knowledge map

To increase students’ mathematics achievement, the Math-Island game targets the complete mathematics curriculum of elementary schools in Taiwan, which mainly contains the four domains: numerical operation , quantity and measure , geometry , and statistics and probability (Ministry of Education of R.O.C. 2003 ). Furthermore, every domain consists of several subdomains with corresponding concepts. For instance, the domain of numerical operation contains four subdomains: numbers, addition, and subtraction for the first and second graders. In the subdomain of subtraction, there are a series of concepts, including the meaning of subtraction, one-digit subtraction, and two-digit subtraction. These concepts should be learned consecutively. In the Math-Island system, the curriculum is restructured as a knowledge map, so that they may preview the whole structure of knowledge, recall what they have learned, and realize what they will learn.

More specifically, the Math-Island system uses the representational metaphor of an “island,” where a virtual city is located and represents the knowledge map. Furthermore, the island comprises areas, roads, and buildings, which are the embodiments of domains, subdomains, and concepts in the curriculum, respectively. As shown in Fig.  2 , for example, in an area of numeral operation in Math-Island, there are many roads, such as an addition road and a subtraction road. On the addition road, the first building should be the meaning of addition, followed by the buildings of one-digit addition and then two-digit addition. Students can choose these buildings to learn mathematical concepts. In each building, the system provides a series of learning tasks for learning the specific concept. Currently, Math-Island provides elementary students with more than 1300 learning tasks from the first grade to the sixth grade, with more than 25,000 questions in the tasks.

The knowledge map

In Math-Island, a learning task is an interactive page turner, including video clips and interactive exercises for conceptual understanding, calculation, and word problem-solving. In each task, the learning procedure mainly consists of three steps: watching demonstrations, practicing examples, and getting rewards. First, students learn a mathematical concept by watching videos, in which a human tutor demonstrates examples, explains the rationale, and provides instructions. Second, students follow the instructions to answer a series of questions related to the examples in the videos. When answering questions, students are provided with immediate feedback. Furthermore, if students input wrong answers, the system provides multilevel hints so that they could figure out solutions by themselves. Finally, after completing learning tasks, students receive virtual money according to their accuracy rates in the tasks. The virtual money is used to purchase unique buildings to develop their islands in the game.

Game mechanisms

In the Math-Island game, there are two game mechanisms: construction and sightseeing (as shown in Fig.  3 ). The former is designed to help students manage their learning process, whereas the latter is designed to facilitate social interaction, which may further motivate students to better develop their cities. By doing so, the Math-Island can be regarded as one’s learning portfolio, which is a complete record that purposely collects information about one’s learning processes and outcomes (Arter and Spandel 2005 ). Furthermore, learning portfolios are a valuable research tool for gaining an understanding about personal accomplishments (Birgin and Baki 2007 ), because learning portfolios can display one’s learning process, attitude, and growth after learning (Lin and Tsai 2001 ). The appearance of the island reflects what students have learned and have not learned from the knowledge map. When students observe their learning status in an interesting way, they may be concerned about their learning status with the enhanced awareness of their learning portfolios. By keeping all activity processes, students can reflect on their efforts, growth, and achievements. In a sense, with the game mechanisms, the knowledge map can be regarded as a manipulatable open learner model, which not only represents students’ learning status but also invites students to improve it (Vélez et al. 2009 ).

Two game mechanisms for Math-Island

First, the construction mechanism allows students to plan and manage their cities by constructing and upgrading buildings. To do so, they have to decide which buildings they want to construct or upgrade. Then, they are required to complete corresponding learning tasks in the building to determine which levels of buildings they can construct. As shown in Fig.  4 , the levels of buildings depend on the completeness of a certain concept, compared with the thresholds. For example, when students complete one third of the learning tasks, the first level of a building is constructed. Later, when they complete two thirds of the tasks, the building is upgraded to the second level. After completing all the tasks in a building, they also complete the final level and are allowed to construct the next building on the road. Conversely, if students failed the lowest level of the threshold, they might need to watch the video and/or do the learning tasks again. By doing so, students can make their plans to construct the buildings at their own pace. When students manage their cities, they actually attempt to improve their learning status. In other words, the construction mechanism offers an alternative way to guide students to regulate their learning efforts.

Screenshots of construction and sightseeing mechanisms in Math-Island

Second, the sightseeing mechanism provides students with a social stage to show other students how well their Math-Islands have been built. This mechanism is implemented as a public space, where other students play the role of tourists who visit Math-Island. In other words, this sightseeing mechanism harnesses social interaction to improve individual learning. As shown in Fig.  4 , because students can construct different areas or roads, their islands may have different appearances. When students visit a well-developed Math-Island, they might have a positive impression, which may facilitate their self-reflection. Accordingly, they may be willing to expend more effort to improve their island. On the other hand, the student who owns the island may also be encouraged to develop their island better. Furthermore, when students see that they have a completely constructed building on a road, they may perceive that they are good at these concepts. Conversely, if their buildings are small, the students may realize their weaknesses or difficulties in these concepts. Accordingly, they may be willing to make more effort for improvement. On the other hand, the student who owns the island may also be encouraged to develop their island better. In a word, the visualization may play the role of stimulators, so that students may be motivated to improve their learning status.

This paper reported a 2-year study in which the Math-Island system was adopted in an elementary school. The study addressed the following two research questions: (1) Did the Math-Island system facilitate students’ mathematics achievement in terms of conceptual understanding, calculating, and word problem-solving? In particular, how was the mathematics achievement of the low-achieving students? (2) What was students’ levels of interest in mathematics and the system, particularly that of low-achieving students?

Participants

The study, conducted from June 2013 to June 2015, included 215 second graders (98 females and 117 males), whose average age was 8 years old, in an elementary school located in a suburban region of a northern city in Taiwan. The school had collaborated with our research team for more than 2 years and was thus chosen as an experimental school for this study. In this school, approximately one third of the students came from families with a low or middle level of socioeconomic status. It was expected that the lessons learned from this study could be applicable to other schools with similar student populations in the future. The parents were supportive of this program and willing to provide personal tablets for their children (Liao et al. 2017 ). By doing so, the students in the experimental school were able to use their tablets to access the Math-Island system as a learning tool at both school and home. To compare the students’ mathematics achievement with a baseline, this study also included 125 second graders (63 females and 62 males) from another school with similar socioeconomic backgrounds in the same region of the city as a control school. The students in the control school received only conventional mathematics instruction without using the Math-Island system during the 2-year period.

Before the first semester, a 3-week training workshop was conducted to familiarize the students with the basic operation of tablets and the Math-Island system. By doing so, it was ensured that all participants had similar prerequisite skills. The procedure of this study was illustrated in Table  1 . At the beginning of the first semester, a mathematics achievement assessment was conducted as a pretest in both the experimental and the control school to examine the students’ initial mathematics ability as second graders. From June 2013 to June 2015, while the students in the control school learned mathematics in a conventional way, the students in the experimental school learned mathematics not only in mathematics classes but also through the Math-Island system. Although the teachers in the experimental school mainly adopted lectures in mathematics classes, they used the Math-Island system as learning materials at school and for homework. At the same time, they allowed the students to explore the knowledge map at their own pace. During the 2 years, every student completed 286.78 learning tasks on average, and each task took them 8.86 min. Given that there were 344 tasks for the second and third graders, the students could finish 83.37% of tasks according to the standard progress. The data also showed that the average correctness rate of the students was 85.75%. At the end of the second year, another mathematics achievement assessment was administered as a posttest in both schools to evaluate students’ mathematics ability as third graders. Additionally, an interest questionnaire was employed in the experimental school to collect the students’ perceptions of mathematics and the Math-Island system. To understand the teachers’ opinions of how they feel about the students using the system, interviews with the teachers in the experimental school were also conducted.

Data collection

Mathematics achievement assessment.

To evaluate the students’ mathematics ability, this study adopted a standardized achievement assessment of mathematics ability (Lin et al. 2009 ), which was developed from a random sample of elementary students from different counties in Taiwan to serve as a norm with appropriate reliability (the internal consistency was 0.85, and the test-retest reliability was 0.86) and validity (the correlation by domain experts in content validity was 0.92, and the concurrent validity was 0.75). As a pretest, the assessment of the second graders consisted of 50 items, including conceptual understanding (23 items), calculating (18 items), and word problem-solving (9 items). As a posttest, the assessment of the third graders consisted of 60 items, including conceptual understanding (18 items), calculating (27 items), and word problem-solving (15 items). The scores of the test ranged from 0 to 50 points. Because some students were absent during the test, this study obtained 209 valid tests from the experimental school and 125 tests from the control school.

Interest questionnaire

The interest questionnaire comprised two parts: students’ interest in mathematics and the Math-Island system. Regarding the first part, this study adopted items from a mathematics questionnaire of PISA and TIMSS 2012 (OECD 2013 ; Mullis et al. 2012 ), the reliability of which was sound. This part included three dimensions: attitude (14 items, Cronbach’s alpha = .83), initiative (17 items, Cronbach’s alpha = .82), and confidence (14 items Cronbach’s alpha = .72). Furthermore, the dimension of attitude was used to assess the tendency of students’ view on mathematics. For example, a sample item of attitudes was “I am interested in learning mathematics.” The dimension of initiatives was used to assess how students were willing to learn mathematics actively. A sample item of initiatives was “I keep studying until I understand mathematics materials.” The dimension of confidences was used to assess students’ perceived mathematics abilities. A sample item was “I am confident about calculating whole numbers such as 3 + 5 × 4.” These items were translated to Chinese for this study. Regarding the second part, this study adopted self-made items to assess students’ motivations for using the Math-Island system. This part included two dimensions: attraction (8 items) and satisfaction (5 items). The dimension of attraction was used to assess how well the system could attract students’ attention. A sample item was “I feel Math-island is very appealing to me.” The dimension of satisfaction was used to assess how the students felt after using the system. A sample item was “I felt that upgrading the buildings in my Math-Island brought me much happiness.” These items were assessed according to a 4-point Likert scale, ranging from “strongly disagreed (1),” “disagreed (2),” “agreed (3),” and “strongly agreed (4)” in this questionnaire. Due to the absences of several students on the day the questionnaire was administered, there were only 207 valid questionnaires in this study.

Teacher interview

This study also included teachers’ perspectives on how the students used the Math-Island system to learn mathematics in the experimental school. This part of the study adopted semistructured interviews of eight teachers, which comprised the following three main questions: (a) Do you have any notable stories about students using the Math-Island system? (b) Regarding Math-Island, what are your teaching experiences that can be shared with other teachers? (c) Do you have any suggestions for the Math-Island system? The interview was recorded and transcribed verbatim. The transcripts were coded and categorized according to the five dimensions of the questionnaire (i.e., the attitude, initiative, and confidence about mathematics, as well as the attraction and satisfaction with the system) as additional evidence of the students’ interest in the experimental school.

Data analysis

For the first research question, this study conducted a multivariate analysis of variance (MANOVA) with the schools as a between-subject variable and the students’ scores (conceptual understanding, calculating, and word problem-solving) in the pre/posttests as dependent variables. Moreover, this study also conducted a MANOVA to compare the low-achieving students from both schools. In addition, the tests were also carried out to compare achievements with the norm (Lin et al. 2009 ). For the second research question, several z tests were used to examine how the interests of the low-achieving students were distributed compared with the whole sample. Teachers’ interviews were also adopted to support the results of the questionnaire.

Mathematics achievement

To examine the homogeneity of the students in both schools in the first year, the MANOVA of the pretest was conducted. The results, as shown in Table  2 , indicated that there were no significant differences in their initial mathematics achievements in terms of conceptual understanding, calculating, and word problem-solving (Wilks’ λ  = 0.982, F (3330) = 2.034, p  > 0.05). In other words, the students of both schools had similar mathematics abilities at the time of the first mathematics achievement assessment and could be fairly compared.

At the end of the fourth grade, the students of both schools received the posttest, the results of which were examined by a MANOVA. As shown in Table  3 , the effect of the posttest on students’ mathematics achievement was significant (Wilks’ λ  = 0.946, p  < 0.05). The results suggested that the students who used Math-Island for 2 years had better mathematics abilities than those who did not. The analysis further revealed that the univariate effects on calculating and word problem-solving were significant, but the effect on conceptual understanding was insignificant. The results indicated that the students in the experimental school outperformed their counterparts in terms of the procedure and application of arithmetic. The reason may be that the system provided students with more opportunities to do calculation exercises and word problems, and the students were more willing to do these exercises in a game-based environment. Furthermore, they were engaged in solving various exercises with the support of immediate feedback until they passed the requirements of every building in their Math-Island. However, the students learned mathematical concepts mainly by watching videos in the system, which provided only demonstrations like lectures in conventional classrooms. For this reason, the effect of the system on conceptual understanding was similar to that of teachers’ conventional instruction.

Furthermore, to examine the differences between the low-achieving students in both schools, another MANOVA was also conducted on the pretest and the posttest. The pretest results indicated that there were no significant differences in their initial mathematics achievement in terms of conceptual understanding, calculating, and word problem-solving (Wilks’ λ  = 0.943, F (3110) = 2.210, p  > 0.05).

The MANOVA analysis of the posttest is shown in Table  4 . The results showed that the effect of the system on the mathematics achievement of low-achieving students was significant (Wilks’ λ  = 0.934, p  < 0.05). The analysis further revealed that only the univariate effect on word problem-solving was significant. The results suggested that the low-achieving students who used Math-Island for 2 years had better word problem-solving ability than those students in the control school, but the effect on conceptual understanding and procedural fluency was insignificant. The results indicated that the Math-Island system could effectively enhance low-achieving students’ ability to solve word problems.

Because the mathematics achievement assessment was a standardized achievement assessment (Lin et al. 2009 ), the research team did a further analysis of the assessments by comparing the results with the norm. In the pretest, the average score of the control school was the percentile rank of a score (PR) 55, but their average score surprisingly decreased to PR 34 in the posttest. The results confirmed the fact that conventional mathematics teaching in Taiwan might result in an M-shape distribution, suggesting that low-achieving students required additional learning resources. Conversely, the average score of the experimental school was PR 48 in the pretest, and their score slightly decreased to PR 44 in the posttest. Overall, both PR values were decreasing, because the mathematics curriculum became more and more difficult from the second grade to the fourth grade. However, it should be noted that the experimental school has been less affected, resulting in a significant difference compared with the control school (see Table  5 ). Notably, the average score of word problem-solving in the posttest of the experimental school was PR 64, which was significantly higher than the nationwide norm ( z  = 20.8, p  < .05). The results were consistent with the univariate effect of the MANOVA on word problem-solving, suggesting that the Math-Island system could help students learn to complete word problems better. This may be because the learning tasks in Math-Island provided students with adequate explanations for various types of word problems and provided feedback for exercises.

To examine whether the low-achieving students had low levels of interest in mathematics and the Math-Island system, the study adopted z tests on the data of the interest questionnaire. Table  5 shows the descriptive statistics and the results of the z tests. Regarding the interest in mathematics, the analysis showed that the interest of the low-achieving students was similar to that of the whole sample in terms of attitude, initiative, and confidence. The results were different from previous studies asserting that low-achieving students tended to have lower levels of interest in mathematics (Al-Zoubi and Younes 2015 ). The reason was perhaps that the low-achieving students were comparably motivated to learn mathematics in the Math-Island system. As a result, a teacher ( #T-301 ) said, “some students would like to go to Math-Island after school, and a handful of students could even complete up to forty tasks (in a day),” implying that the students had a positive attitude and initiative related to learning mathematics.

Another teacher ( T-312 ) also indicated “some students who were frustrated with math could regain confidence when receiving the feedback for correct answers in the basic tasks. Thanks to this, they would not feel high-pressure when moving on to current lessons.” In a sense, the immediate feedback provided the low-achieving students with sufficient support and may encourage them to persistently learn mathematics. Furthermore, by learning individually after class, they could effectively prepare themselves for future learning. The results suggested that the system could serve as a scaffolding on conventional instruction for low-achieving students. The students could benefit from such a blended learning environment and, thus, build confidence in mathematics by learning at their own paces.

The low-achieving students as a whole were also attracted to the system and felt satisfaction from it. Teacher ( #T-307 ) said that, “There was a hyperactive and mischievous student in my class. However, when he was alone, he would go on to Math-Island, concentrating on the tasks quietly. He gradually came to enjoy learning mathematics. It seemed that Math-Island was more attractive to them than a lecture by a teacher. I believed that students could be encouraged, thus improve their ability and learn happily.” Another teacher ( #T-304 ) further pointed out that, “For students, they did not only feel like they were learning mathematics because of the game-based user interface. Conversely, they enjoyed the contentment when completing a task, as if they were going aboard to join a competition.” In teachers’ opinions, such a game-based learning environment did not disturb their instruction. Instead, the system could help the teachers attract students’ attention and motivate them to learn mathematics actively because of its appealing game and joyful learning tasks. Furthermore, continuously overcoming the tasks might bring students a sense of achievement and satisfaction.

Discussion on some features of this study

In addition to the enhancement of achievement and interest, we noticed that there are some features in this study and our design worth some discussion.

The advantages of building a long-term study

Owing to the limitations of deployment time and sample sizes, it is hard for most researchers to conduct a longitudinal study. Fortunately, we had a chance to maintain a long-term collaboration with an experimental school for more than 2 years. From this experiment, we notice that there are two advantages to conducting a long-term study.

Obtaining substantial evidence from the game-based learning environment

The research environment was a natural setting, which could not be entirely controlled and manipulated like most experiments in laboratories. However, this study could provide long-term evidence to investigate how students learned in a game-based learning environment with their tablets. It should be noted that we did not aim to replace teachers in classrooms with the Math-Island game. Instead, we attempted to establish an ordinary learning scenario, in which the teachers and students regarded the game as one of the learning resources. For example, teachers may help low-achieving students to improve their understanding of a specific concept in the Math-Island system. When students are learning mathematics in the Math-Island game, teachers may take the game as a formative assessment and locate students’ difficulties in mathematics.

Supporting teachers’ instructions and facilitating students’ learning

The long-term study not only proved the effectiveness of Math-Island but also offered researchers an opportunity to determine teachers’ roles in such a computer-supported learning environment. For example, teachers may encounter difficulties in dealing with the progress of both high- and low-achieving students. How do they take care of all students with different abilities at the same time? Future teachers may require more teaching strategies in such a self-directed learning environment. Digital technology has an advantage in helping teachers manage students’ learning portfolios. For example, the system can keep track of all the learning activities. Furthermore, the system should provide teachers with monitoring functions so that they know the average status of their class’s and individuals’ learning progress. Even so, it is still a challenge for researchers to develop a well-designed visualization tool to support teachers’ understanding of students’ learning conditions and their choice of appropriate teaching strategies.

Incorporating a gamified knowledge map of the elementary mathematics curriculum

Providing choices of learning paths.

Math-Island uses a representational metaphor of an “island,” where a virtual city is located and represents the knowledge map. Furthermore, the island comprises areas, roads, and buildings, which are the embodiments of domains, subdomains, and concepts in the curriculum, respectively. Because the gamified knowledge map provides students with multiple virtual roads to learn in the system, every student may take different routes. For instance, some students may be more interested in geometry, while others may be confident in exploring the rules of arithmetic. In this study, we noticed that the low-achieving students needed more time to work on basic tasks, while high-achieving students easily passed those tasks and moved on to the next ones. As a result, some of the high-achieving students had already started to learn the materials for the next grade level. This was possibly because high-achieving students were able to respond well to challenging assignments (Singh 2011 ). Therefore, we should provide high-achieving students with more complex tasks to maintain their interest. For example, Math-Island should provide some authentic mathematical problems as advanced exercises.

Visualizing the learning portfolio

In this study, we demonstrated a long-term example of incorporating a gamified knowledge map in an elementary mathematical curriculum. In the Math-Island game, the curriculum is visualized as a knowledge map instead of a linear sequence, as in textbooks. By doing so, students are enabled to explore relationships in the mathematics curriculum represented by the knowledge map; that is, the structure of the different roads on Math-Island. Furthermore, before learning, students may preview what will be learned, and after learning, students may also reflect on how well they learned. Unlike traditional lectures or textbooks, in which students could only follow a predefined order to learn knowledge without thinking why they have to learn it, the knowledge map allows students to understand the structure of knowledge and plan how to achieve advanced knowledge. Although the order of knowledge still remains the same, students take primary control of their learning. In a sense, the knowledge map may liberate elementary students from passive learning.

Adopting the mechanisms of a construction management game

This 2-year study showed that the adaptation of two game mechanisms, construction and sightseeing, into the elementary mathematical curriculum could effectively improve students’ learning achievement. The reason may be that students likely developed interests in using Math-Island to learn mathematics actively, regardless of whether they are high- and low-achieving students.

Gaining a sense of achievement and ownership through the construction mechanism

Regardless of the construction mechanism, Math-Island allows students to plan and manage their cities by constructing and upgrading buildings. Math-Island took the advantages of construction management games to facilitate elementary students’ active participation in their mathematical learning. Furthermore, students may manage their knowledge by planning and constructing of buildings on their virtual islands. Like most construction management games, students set goals and make decisions so that they may accumulate their assets. These assets are not only external rewards but also visible achievements, which may bring a sense of ownership and confidence. In other words, the system gamified the process of self-directed learning.

Demonstrating learning result to peers through the sightseeing mechanism

As for the sightseeing mechanism, in conventional instruction, elementary students usually lack the self-control to learn knowledge actively (Duckworth et al. 2014 ) or require a social stage to show other students, resulting in low achievement and motivation. On the other hand, although previous researchers have already proposed various self-regulated learning strategies (such as Taub et al. 2014 ), it is still hard for children to keep adopting specific learning strategies for a long time. For these reasons, this study uses the sightseeing mechanism to engage elementary students in a social stage to show other students how well their Math-Islands have been built. For example, in Math-Island, although the students think that they construct buildings in their islands, they plan the development of their knowledge maps. After learning, they may also reflect on their progress by observing the appearance of the buildings.

In brief, owing to the construction mechanism, the students are allowed to choose a place and build their unique islands by learning concepts. During the process, students have to do the learning task, get feedback, and get rewards, which are the three major functions of the construction functions. In the sightseeing mechanism, students’ unique islands (learning result) can be shared and visited by other classmates. The student’s Math-Island thus serves as a stage for showing off their learning results. The two mechanisms offer an incentive model connected to the game mechanism’s forming a positive cycle: the more the students learn, the more unique islands they can build, with more visitors.

Conclusion and future work

This study reported the results of a 2-year experiment with the Math-Island system, in which a knowledge map with extensive mathematics content was provided to support the complete elementary mathematics curriculum. Each road in Math-Island represents a mathematical topic, such as addition. There are many buildings on each road, with each building representing a unit of the mathematics curriculum. Students may learn about the concept and practice it in each building while being provided with feedback by the system. In addition, the construction management online game mechanism is designed to enhance and sustain students’ interest in learning mathematics. The aim of this study was not only to examine whether the Math-Island system could improve students’ achievements but also to investigate how much the low-achieving students would be interested in learning mathematics after using the system for 2 years.

As for enhancing achievement, the result indicated that the Math-Island system could effectively improve the students’ ability to calculate expressions and solve word problems. In particular, the low-achieving students outperformed those of the norm in terms of word problem-solving. For enhancing interest, we found that both the low-achieving and the high-achieving students in the experimental school, when using the Math-Island system, maintained a rather high level of interest in learning mathematics and using the system. The results of this study indicated some possibility that elementary students could be able to learn mathematics in a self-directed learning fashion (Nilson 2014 ; Chen et al. 2012a , b ) under the Math-Island environment. This possibility is worthy of future exploration. For example, by analyzing student data, we can investigate how to support students in conducting self-directed learning. Additionally, because we have already collected a considerable amount of student data, we are currently employing machine learning techniques to improve feedback to the students. Finally, to provide students appropriate challenges, the diversity, quantity, and difficulty of content may need to be increased in the Math-Island system.

Abbreviations

Program for International Student Assessment

The percentile rank of a score

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The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors would like to thank the Ministry of Science and Technology of the Republic of China, Taiwan, for financial support (MOST 106-2511-S-008-003-MY3), and Research Center for Science and Technology forLearning, National Central University, Taiwan.

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CYCY contributed to the study design, data acquisition and analysis, mainly drafted the manuscript and execution project. HNHC was involved in data acquisition, revision of the manuscript and data analysis.ZHC was contributed to the study idea and drafted the manuscript. CCYL of this research was involved in data acquisition and revision of the manuscript. TWC was project manager and revision of the manuscript. All authors read and approved the final manuscript.

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Charles Y.C. Yeh is currently an PhD student in Graduate Institute of Network Learning Technology at National Central University. The research interests include one-to-one learning environments and game-based learning.

Hercy N. H. Cheng is currently an associate professor and researcher in National Engineering Research Center for E-Learning at Central China Normal University, China. His research interests include one-to-one learning environments and game-based learning.

Zhi-Hong Chen is an associate professor in Graduate Institute of Information and Computer Education at National Taiwan Normal University. His research interests focus on learning technology and interactive stories, technology intensive language learning and game-based learning.

Calvin C. Y. Liao is currently an Assistant Professor and Dean’s Special Assistant in College of Nursing at National Taipei University of Nursing and Health Sciences in Taiwan. His research focuses on computer-based language learning for primary schools. His current research interests include a game-based learning environment and smart technology for caregiving & wellbeing.

Tak-Wai Chan is Chair Professor of the Graduate Institute of Network Learning Technology at National Central University in Taiwan. He has worked on various areas of digital technology supported learning, including artificial intelligence in education, computer supported collaborative learning, digital classrooms, online learning communities, mobile and ubiquitous learning, digital game based learning, and, most recently, technology supported mathematics and language arts learning.

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Yeh, C.Y.C., Cheng, H.N.H., Chen, ZH. et al. Enhancing achievement and interest in mathematics learning through Math-Island. RPTEL 14 , 5 (2019). https://doi.org/10.1186/s41039-019-0100-9

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• Olga Dg. Unay
• Published in Asian Journal of Education… 20 August 2024
• Mathematics, Education

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RESEARCH ASSISTANT I MATH (STUDENT TEMP)

Job summary.

The Department of Mathematics & Statistics at the University of Michigan-Dearborn invites applications for undergraduate research in applied mathematics during Fall 2024 on the theme Inverse Problems in Signal and Image Processing. The successful applicant for this position will help in designing, developing and testing mathematical theory and computational algorithms for a research project in data-driven signal/image processing under the supervision of Drs. Aditya Viswanathan and Yulia Hristova. The project will incorporate elements of applied linear and matrix algebra, numerical analysis, and Fourier analysis, among others. The prerequisite for this project is a course in linear algebra.

Responsibilities*

The key responsibilities of this position are:

• Developing mathematically rigorous theory supporting the research objectives of the project.
• Developing, implementing, and validating computational algorithms using a software package such as MATLAB.
• Documenting research findings through means of project reports and presentations.
• Communicating with investigators and other project participants regularly to report project progress and discuss potential issues/problems.

Required Qualifications*

• Current UM-Dearborn undergraduate student.
• Successful completion of a Linear Algebra course (e.g. Math 227).
• Ability to work effectively in a remote setting (access to a computer with internet connectivity required).
• Excellent organizational, analytic, and communication skills.

Desired Qualifications*

• Familiarity with the MATLAB programming environment,
• Familiarity using the LaTeX typesetting system,
• Ability to prepare and edit mathematically rigorous reports, presentations and other
• Ability to work independently
• Ability to prioritize, pay close attention to detail, follow through with tasks, and work as
• part of a team of professionals
• Excellent verbal and written communication skills

Background Screening

The University of Michigan conducts background checks on all job candidates upon acceptance of a contingent offer and may use a third party administrator to conduct background checks.  Background checks are performed in compliance with the Fair Credit Reporting Act.

U-M EEO/AA Statement

The University of Michigan is an equal opportunity/affirmative action employer.