The mind brain relationship as a mathematics problem solving strategies

the mind brain relationship as a mathematics problem solving strategies

In deciding which strategy to use when doing mental computing, is a complex mental procedure involving a frontoparietal network of brain regions .. later cannot be solved with the problem-solving patterns existing in their minds, . To solve such a mathematical task or problem, it is necessary to visually. Author content. Possible strategies for each problem in NPST encouraging to use non-routine problems in math teaching Specific facets of the brain and problem solving are to be .. This factor should be kept in mind. Keywords: problem solving, brain based learning, working memory, open problems teaching mathematical problem-solving with the brain in mind three basic notions (initial state, transformation (solution) steps, and goal state) is not exactly .. nobody could find he relationship between the number of parts, i.e. the main.

The second vital element of brain circuitry is structural plasticity, that is, dynamical changes in the synaptic connectivity not just during development but throughout adulthood. Abundant experimental evidence suggests that this form of plasticity is activity- and experience-dependent [ — ]. This is just one of many mechanisms underlying neural plasticity across spatial and temporal scales, from short- or long-term alterations in synaptic strengths to neurogenesis [ ], which are believed to support memory storage [].

Much as the brain is in constant flux and its functional connectivity continuously changes with every spike and synaptic discharge, so is the mind never exactly the same before and after instantiating each and every subjective representation.

Quantifying Declarative Mental States: But it is somehow even more peculiar that in spite of direct, detailed, continuous, and complete access to each and all conscious mental states we experience, we find it difficult to describe them comprehensively, let alone quantitatively. Indeed, it seems absurd that we can measure the concentration of Substance P in single neurons to the fifth significant digit; yet we can only measure the resulting sensation of pain semiqualitatively on a 7-point discrete scale.

In order to bring the study of conscious content into the realm of hard science, we need to devise a quantitative measurement system for subjective states [ ]. Language has often been considered a convenient proxy to access mental states, if not the most direct tool to describe them.

Want to Solve a Problem? Don’t Just Use Your Brain, but Your Body Too

The scientific characterization of the meaning of language, or semantic analysis, has a long history and remains one of the most active research areas in computational linguistics. Here we do not aim to review or even to provide a balanced commentary on the state of the art of semantic analysis techniques.

Instead, we introduce and explain a very specific, nonconventional approach to this problem that is particularly pertinent to the topic of this spotlight paper.

Most if not all of the best known computational methods of semantic analysis are based on variations of the common principle that the meaning of words relates to the contextual occurrence of their use in language [ ].

For example apples, oranges, and grapes tend to be used in similar contexts as reflected by their cooccurrence with similar words in the same sentence e. Thus, they share similar semantics they are all types of fruit. The notion that word meaning relates to the relative frequency of their cooccurrence is shared by many broadly adopted approaches, including Latent Semantic Indexing [ ], Latent Dirichlet Allocation [ ], Hyperspace Analogue to Language [ ], and many others [ ].

In practice, these techniques rely on the identification of statistical patterns of word usage in large-scale text corpora by computational parameter extraction. Although the details vary among types of computational semantic analysis, words or more generally, concepts are often allocated to a multidimensional abstract space such that the location of each concept reflects its meaning.

Alternatively, meanings can be identified with clusters of words in this space. For instance, all fruit words in the previous example would be located in the same region of the space.

the mind brain relationship as a mathematics problem solving strategies

By nature of its own principle, latent semantic analysis and its variations generate results that are highly context dependent. In other words, the semantics extracted from a cookbook are typically quite different from those detected in movie reviews or obituaries.

In fact, use of nonhomogeneous collections of corpora from different domains typically fails to yield meaningful semantics. Moreover, this general class of methodologies tends to produce a large number of highly specific dimensions. A rather complementary and historically precedent goal of lexical semantics has been to seek the fundamental or at least context independent dimensions of word meaning. In that work, subjects were asked to rate a large number of words in various hand-picked dimensions defined by two opposite extremes e.

Subsequent analysis identified three principal dimensions that were robust to cultural and geographical differences, namely, evaluation also known as valence: A limitation of these studies and other similar psychometric approaches [ ] is that they involve human subjects and arbitrary choices of starting terms.

the mind brain relationship as a mathematics problem solving strategies

Thus, they are not amenable to automated, high-throughput computational extraction. Word meaning has of course also been characterized for thousands of years in many languages and cultures through the creation of dictionaries. Researchers in computational linguistics are vigorously pursuing the topic of conceptual ontologies [ ].

the mind brain relationship as a mathematics problem solving strategies

Yet, it remains to be established if and how formal ontological theories could map semantic spaces such as those generated by latent semantic analyses.

Specifically, using a novel self-organization process, we constructed a semantic map of natural language that simultaneously represents synonymy and antonymy. Synonyms and antonyms are commonly listed in dictionaries for most terms.

For each dictionary and language, we initially allocated words at random locations in a finite, multidimensional spherical space. Then we started moving the position of every word following a simple rule: Thus, pairs of synonyms would tend to move closer to each other, and pairs of antonyms would move farther apart within the bounds of the multidimensional sphere. The semantic map is sufficiently robust to allow the automated extraction of synonyms and antonyms not originally present in the dictionaries used to construct the map, as well as to predict connotation from their coordinates.

Both the semantic content and the main geometric features of the map are consistent between dictionaries, among tested Western languages, and with previously established psychometric measures. Some of the mathematical formalism and speculative interpretations are elaborated in a second follow-up paper [ ]. Interestingly, the main emerging dimensions of this semantic map loosely correspond to the primary modulatory neurotransmitter systems in the mammalian brain [ ].

The previous paradigm can be expanded with appropriate adaptations to extract additional, independent dimensions of word meaning by considering other linguistic relations besides synonyms and antonyms. However, hypernyms and hyponyms are seldom listed in immediately machine-readable form in digital collections, the way synonyms and antonyms are. One exception is provided once again by WordNet, which explicitly provides is-a relationships among many of its terms.

Unlike synonyms and antonyms, which are symmetric relations if A is synonym of B, B is synonym of Ahypernyms and hyponyms are directional and mutually antisymmetric if A is hypernym of B, B is hyponym of A. We thus changed the form of the energy functional in the previously described optimization procedure [ ]. The resulting allocation of words in space yielded a ranking of all terms along a single dimension, that is, a simple scalar measure of their abstractness ontological generality.

The bottom 11 ex aequo of the list reads Edmontonia, Coelophysis, Deinocheirus, Struthiomimus, Deinonychus, dromaeosaur, Mononykus olecranus, oviraptorid, superslasher, Utahraptor, and Velociraptor. Moreover, because the measure is quantitative, it allows evaluation of relative comparisons. This opens the possibility to establish a probabilistic estimate of whether a word is more abstract than another. The metrics of context-independent word meaning along the principal dimensions described previously can be applied to characterize declarative mental states.

the mind brain relationship as a mathematics problem solving strategies

The most straightforward application is to quantify the content of verbal examples along the main dimensions of the map. This can help in relating semantic content to neural signals. It should be noted that the semantic map described here represents a complementary, rather than alternative, tool to more established latent semantic analyses. While maps produced by the latter are corpus and context dependent, this space adds general dimensions that are applicable to all corpora and context.

We indeed found that the first dimension good-bad was an excellent quantitative predictor not only of the movie critique score but also together with the second dimension of its genre high valence and arousal: A Radical View of Reality, Information, Consciousness, and Remaining Challenges Semantic mapping provides a possible approach to quantifying mental states that can be expressed declaratively.

In this framework, mental states and their relationships can be themselves represented as graphs of nodes and edges, respectively. If one believes that at least some mental states reflect properties of outer reality, it is possible to conceive reality itself as occurring in a giant graph in which any possible observable is a node, and edges correspond to probabilities that two observables would cooccur.

We call this conceptual construct the Universal Reality Graph. In this view, reality would unfold in time as a sequence of events constituting patterns of activation of subsets of nodes and all edges among them within the University Reality Graph. Any agent capable of observation will witness a subset of these activation patterns, that is, a sequence of partial events, each consisting of a collection of active nodes and edges.

Most importantly, the possibility to conceive reality as a graph offers interesting vistas on the solution of the mind-brain problem. If agents form graph-like minds to represent and therefore predict their experience of graph-like reality, it stands to reason that the fittest physical substrates selected by evolution to encode these representations be themselves graph like, namely, brain networks. The relationship between minds and brains could then be resolved as a mapping between their respective graphs and their embeddings.

In this framework, the fundamental operation to grow a mind is pairwise association between observables [ ], that is, establishment of edges between nodes in the mental graph based on corresponding experiences in the reality graph.

the mind brain relationship as a mathematics problem solving strategies

An interesting aspect of mental representation is that we only learn a small fraction of associations from those observed in reality. In particular, our ability to learn is gated by previously acquired background information. We have recently proposed that this constraint may be a consequence of the spatial relationship among the tree-like shaped neuronal axons and dendrites that underlie brain connectivity []. Specifically, in order for new synapses to be formed, the axon of the presynaptic neuron must be sufficiently close to a dendrite of the postsynaptic neuron, arguably because of preexisting connectivity with other neurons encoding for related knowledge.

These ideas are also consonant with the Information Integration Theory IIT of consciousness [ ], which is emerging as a leading candidate among the fundamental theories of mental content.

The underlying assumption of IIT is that consciousness is fundamentally a property of information processing. Specifically, according to the IIT, when a brain or in principle any other computing device is in a particular state, its amount of consciousness, called Phi, depends not only on the actual content represented in that state but also on the absence of all contents represented in the states that are not being but could be instantiated.

Thus silent neurons contribute to the conscious state as much as the active neurons, because consciousness depends as much on the content that could be represented by the network as on the content that is actually being represented. Therefore, consciousness is a product of the integrated activity in the network and is measured by information integration, a property that has been defined in graph structures [ ]. While the IIT profoundly links consciousness to information [ ], its cognitive underpinning is shared by other theories e.

A crucial and unique outcome of IIT, however, is that the definition of integrated information enables a geometric characterization of mental states or qualia [ ]. This can in principle provide a neurally based bottom-up correlate to the spaces that emerge from top-down semantic mapping of natural language. If the information processing product of neural network activity can be shown to correspond mathematically to a quantitative description of subjective mental content, the brain-mind problem would be effectively resolved.

Information is not only an essential element of consciousness, reality, and brain activity, but also of communication among conscious agents. Consider a dialogue between two individuals, in which one tells the other: I had such a day. More precisely, what does it mean to the speaking individual, and what does it mean to the listener? Assuming that the second individual had no idea of what time it was, whether the first person was rested or fatigued, and so forth, communication is indeed informative.

Mapping brain and mental states, however, opens another perspective on the meaning of communication. This simple example can be generalized to all of human communications. How much of the intended meaning is effectively transmitted between communicating conscious individuals on average? Even reminiscence and planning retrieval of autobiographic and prospective memories, resp. Such a type of communication between two instances of the same individual at different points in time can be expected to be much more effective than between different human beings, but even in those situations it will not be perfectly effective, as the mind is in constant flux.

In all cases, mental state quantification by semantic mapping and its corresponding neural correlates in brain activity spaces could dramatically enhance communication effectiveness, deeply altering human relationship. We have proposed that a satisfactory answer can ultimately come from mathematics, if the abstract spaces of brain activity and mental content can be quantitatively characterized and geometrically mapped onto each other.

  • Support of mathematical thinking through embodied cognition: Nondigital and digital approaches
  • International Scholarly Research Notices

We argued that semantic maps constitute a useful initial framework to establish a rigorous description of the mind and that network connectivity provides the most informative constraint on brain dynamics. However, defining the proper mathematical states to effectively bridge brain and mind still constitutes a formidable challenge.

State-of-the-art semantic maps only scratch the surface of the necessary quantification of the human mind. Next-generation voice recognition and optical character recognition software programs might soon enable real-time acquisition and analysis of the complete life-long natural language corpus experienced by an individual.

Exhaustive compendia of semantic relationships could be extracted from such a resource, enabling the creation of a comprehensive semantic map for that individual. Such a resource could then be used to systematically report subjective mental states.

While the main obstacle in quantifying mental content appears to be the required paradigm shift towards a science of first-person perspective, neuroscience faces mostly a technological hurdle in creating brain-wide neuron-level maps of network connectivity and activity.

Specifically, existing techniques can indeed map all of synaptic circuitries, but only in a very small volume a fraction of cubic millimeter of nervous tissue [ ], only in animal models, and not in vivo. Other techniques to analyze neuronal anatomy, only hinting at the potential connectivity [ 75], are possibly scalable to entire brains of live animals [], but again not human beings, let alone in normal behavioral conditions. The only noninvasive imaging techniques available to investigate the human brain e.

Grade 1 Math 3.12, Problem solving addition strategy, Draw a picture

One present-day partial solution is to use molecular homology to identify existing correspondences between neuron types in rodents and humans by comprehensive genetic mapping [ ] and single-neuron sequencing [ ]. The subsequent extension of rodent brain connectivity to human cognitive architecture would only be tentative, requiring extensive computational testing and refinement by multiscale simulation [ ].

An initial pilot project in this regard might tackle a suitable brain region and related computational functionssuch as the mammalian hippocampus [ ]. Assuming that, at least in principle, technological advancements enabled accumulation of sufficient datasets to adequately map the neuronal activity of the human brain, such a feat would likely involve massive automation.

High-throughput, machine-acquired, and large-scale data poses the outstanding matter of human interpretation [ — ]. This issue has recently promoted considerable growth in the field of neuroinformatics [ ], that is, the establishment of an information framework for neuroscience e. Recent initiatives have proposed a formalism to represent connectivity structure in neuronal network models [ ] and seeded web-based multimodal connectivity databases [ ].

A parallel informatics effort is required to enable storage, manipulation, and analysis of machine- and human-readable empirical data on cognitive functions, behaviors, and introspection []. The neuroinformatics of language might provide a useful bridge between neural and cognitive frameworks [ ]. The proposed vision offers a path towards a mathematical solution of the relationship between brain and mind that is consistent with contemporary philosophical positions [ ].

Tremendous advancements in physics, chemistry, and biology provided an increasingly unified understanding of the material world. Bridging neuroscience with a quantitative description of inner subjective life may provide a fundamental closure to human scientific inquiry. Acknowledgments The author is grateful to Professor Aurelio Ascoli the author's father for his useful feedback on this paper, with particular attention to historical details. He also thanks Professor James L.

Olds for his encouragement and suggestion to write this material for publication after its presentation at the Krasnow Institute retreat. View at Google Scholar M. View at Google Scholar A. View at Google Scholar T. From an embodied cognition perspective, we will first discuss previous research in the context of manipulatives, hand gestures, and whole-body movement because those areas have shown promise of learning effectiveness in mathematics education and have been popular topics of research.

Based on this background, we present technologically enhanced examples of embodied mathematical activities. Finally, we end with recommendations for future research and development directions related to embodied cognition, technology, and mathematics learning. Review Embodied cognition defined Embodied cognition is a decades-long branch of research that encompasses a diverse set of theories that are based on the idea that human cognition is rooted in the bidirectional perceptual and physical interactions of the body with the world Gibson, ; Wilson, Ways of thinking, such as representations of knowledge and methods of organizing and expressing information, are influenced by the perceptual and motor systems including body shape and movement, neural systems engaged in action planning, and systems involved in sensation and perception Glenberg, Embodied cognition implicates a perception-action cycle in which behavior consists of a succession of adaptive motor reactions to changes in external e.

The perception-action cycle that underlies embodied cognition also holds true for visual and symbolic representations of actions. For example, mental imagery is used to understand the positions of three-dimensional objects after rotations e. As we further detail in our article, gestures are also an outgrowth of simulated action and perception. While the hand movements are clearly actions, they do not have a direct effect on the world and are instead representational.

Support of mathematical thinking through embodied cognition: Nondigital and digital approaches

Perceptual symbols are believed to be multimodal traces of neural activity that contain at least some of the motor information present during actual sensorimotor experience Barsalou, Memories from movement can prepare learners for future action and can be retrieved to solve related tasks in different situations that no longer engage physical movement, but rather, a mental transformation of those motor processes. For example, instead of trying to imagine how an object would appear when rotated, learners can reduce this burden of tracking information by allowing their hands to do it and seeing what happens.

Freeing up those mental resources can allow them to think more deeply about spatial relations and have a better understanding of that concept before transitioning to spatial reasoning more abstractly. This natural desire to situate cognition with real contexts is reflected in the mind-body connections of mathematical concepts such as embodied numerosity Moeller et al.

Integrating the body into the learning experience can, therefore, improve mathematical understanding by providing a connection between concrete referents and abstract concepts. Embodied cognition and mathematics understanding There is ample evidence that different aspects of mathematics are embodied.

It is quite common to observe that especially young children use their fingers in combination with mathematical tasks. Such an observation is not surprising given the fact that fingers are typically readily available and cover the number range within which children are usually introduced to counting and arithmetic.

This usage of fingers is a manifestation of embodied cognition and in the case of counting has been labeled embodied numerosity Moeller et al. The link between numerical representation and finger representation has been established in different studies and experiments. For example, one line of research has been investigating the relationship between finger gnosis and mathematical abilities.

The participant then has to identify which finger or fingers were touched. The use of fingers is also an impressive example to illustrate how physical body features influence how individuals process numbers.

The Mind-Brain Relationship as a Mathematical Problem

Somewhat related, it has been shown that finger representations get internalized especially in the first years of schooling. These errors originate in situations where sums are larger than In these situations, a two-digit result must somehow be represented with only 10 fingers.

In order to accomplish this, a full hand must be reused and a failure to keep track of reused full hands results in a specific error, called a split-5 error. Such strong sub-base 5 effects have not been found in adults which seems to indicate that the relationship of finger and number representation is stronger in early elementary school and then diminishes from that point on Domahs et al. The results of this study indicate that finger use initially provides a natural scaffolding structure for calculation but then the benefits of using fingers fade, likely because fingers do not effectively deal with the complexity of later mathematics.

What can be learned from this literature is how interlinked at least parts of our bodies and thought processes are. But it is also clear that this relationship is not a simple one and varies across the lifespan see also Newman,