phenomenon · claim-type · cross-domain-transfer · mechanism · claim-relation · replication-status · limitation — grounded in the full text| phenomenon | conceptual and procedural knowledge in mathematics teaching— “The terms conceptual knowledge and procedural knowledge are often used by teachers and never more so than when discussing how teachers teach, and children learn mathematics.” | |
| claim-type | critique— “For example, the false positioning of procedural and conceptual knowledge as being two extremes of a continuum, rather than them having the capacity to occupy different places on that continuum depending on the situation, does little to acknowledge their complexity.” | |
| method — evidence | — “A thorough synthesis of extant and seminal literature will provide advice to teachers and teacher educators on how a deeper insight into conceptual and procedural knowledge could improve the quality of mathematics teaching.” | |
| mechanism | once procedures have been memorised and practised without understanding, students lose the motivation to look for their meaning, which is why a procedure-first sequence blocks the conceptual half of the iteration— “who asserted that once students have memorised and practised procedures (including written algorithms, that they do not necessarily understand) they have less motivation to comprehend their meaning or the reasoning behind them.” | mechanisms[0] |
| mechanism | acquiring procedural knowledge reduces the load on working memory, and the freed capacity can then be spent on building conceptual knowledge— “There is also evidence that gaining procedural knowledge can lessen the cognitive demands on working memory. This lessening of demand can then free the working memory to focus on conceptual knowledge development” | mechanisms[1] |
| mechanism | a powerful earlier understanding (e.g. reading the equals sign as 'give me an answer') obtrudes into later learning and obfuscates the teaching that follows— “Cognitive interference, considered by Pesek and Kirshner as being the more important interference, is when previous understandings of something are so poweful they obtrude into subsequent learning.” | mechanisms[2] |
| contradicts | the belief, held by the experienced teachers the author surveyed, that conceptual and procedural knowledge are mutually exclusive and that conceptual is always the more necessary— “the teachers indicated that they carried the belief that conceptual and procedural knowledge were mutually exclusive, and that conceptual knowledge was, without exception, more appropriate or necessary than procedural knowledge” | claim_relations[0] |
| sense | concept - a mental representation embodying the essential features of an object, situation or idea, whose function is to let phenomena be classified as belonging together or not— “According to Westwood (2008) a concept can be defined as “a mental representation that embodies all the essential features of an object, a situation, or an idea. Concepts enable us to classify phenomena as belonging, or not belonging, together in certain categories” | senses[0] |
| sense | procedural knowledge - characterised by some researchers as the capacity to follow steps in sequence to reach a mathematical goal— “Procedural knowledge is characterised by some researchers (Canobi, 2009; Miller & Hudson, 2007; Rittle-Johnson & Schneider, 2015) as the capacity to follow steps in sequence to solve mathematical problems or reach a mathematical goal.” | senses[1] |
| sense | the conceptual/procedural relation - bi-directional but NOT always symmetrical, with conceptual knowledge at times the stronger support— “although the relationship between conceptual and procedural knowledge is bi-directional, it is not always symmetrical, and that, at times, conceptual knowledge is stronger and more consistent in supporting procedural knowledge, than the reverse.” | senses[2] |
| sense | conceptual/procedural - a single distinction carrying eight different name pairs across eight traditions (Skemp's instrumental/relational, Tulving's episodic/semantic, Anderson's procedural/declarative...), which the paper asserts are the same division— “In the past, the terms have been couched differently (Table 1), but regardless of the labels, the divisions regarding the types of knowledge are consistent” | senses[3] |
| sense | deep - in Star's matrix a property that BOTH knowledge types can have, not a property of conceptual knowledge alone— “The matrix indicates that for each knowledge type (conceptual and procedural) there is the possibility of developing a superficial knowledge or a deep knowledge” | senses[4] |
| sense | adaptive expertise - both knowledge types deep AND applicable to unfamiliar situations; routine expertise is the same knowledge without the transfer— “Routine expertise is where there is a superficial conceptual and/or procedural knowledge, which is able to be applied to familiar situations, but not unfamiliar ones, or to new tasks. Adaptive expertise is where both conceptual and procedural knowledge is deep, and where that knowledge can be aplied creatively, flexibly and appropriately to… | senses[5] |
| note | the belief, held by the experienced teachers the author surveyed, that conceptual and procedural knowledge are mutually exclusive and that conceptual is always the more necessary — The paper's target is a belief held by practitioners rather than a published claim, and it is quoted from the author's own observation of them. | claim_relations[0] |
| note | conceptual knowledge —renamed-across-fields→ Relational Knowledge — An in-text renaming attributed to a named source: 'notably characterised by Skemp, 1978, as Relational…'. | concept_edges[0] |
| weight | conceptual knowledge —renamed-across-fields→ Relational Knowledge — 0.9 Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concept_edges[0] |
| evidence status note | conceptual knowledge —renamed-across-fields→ Declarative — PAGE FURNITURE IS SPLICED INTO THE 'evidence' QUOTE - the journal's page footer and a page number, between the sentence and Table 1: 'Vol 46, 2, February 2021 60 Australian Journal of Teacher…'. The words on either side of it were never continuous on the page; read the quote with it deleted. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concept_edges[1] |
| note | conceptual knowledge —renamed-across-fields→ Declarative — TABLE 1 IS THIS PROJECT'S THESIS WITH A TABLE. Eight research traditions across 1976-1984 - Skemp, Piaget, Gelman & Gallistel, Resnick, Tulving, Anderson, Van Lehn, Baroody - each name the SAME two categories differently, and the paper says so outright: 'regardless of the labels, the divisions regarding the types…'. The claim of sameness is the paper's, not the tagger's. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concept_edges[1] |
| weight | conceptual knowledge —renamed-across-fields→ Declarative — 0.8 Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concept_edges[1] |
| evidence status note | procedural knowledge —renamed-across-fields→ Instrumental — PAGE FURNITURE IS SPLICED INTO THE 'evidence' QUOTE - the journal's page footer and a page number, between the sentence and Table 1: 'Vol 46, 2, February 2021 60 Australian Journal of Teacher…'. The words on either side of it were never continuous on the page; read the quote with it deleted. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concept_edges[2] |
| note | procedural knowledge —renamed-across-fields→ Instrumental — TABLE 1 IS THIS PROJECT'S THESIS WITH A TABLE. Eight research traditions across 1976-1984 - Skemp, Piaget, Gelman & Gallistel, Resnick, Tulving, Anderson, Van Lehn, Baroody - each name the SAME two categories differently, and the paper says so outright: 'regardless of the labels, the divisions regarding the types…'. The claim of sameness is the paper's, not the tagger's. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concept_edges[2] |
| weight | procedural knowledge —renamed-across-fields→ Instrumental — 0.8 Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concept_edges[2] |
| note | CRA —same-as→ CPA — 'also referred to as' - one instructional sequence, two acronyms live in the same literature. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concept_edges[3] |
| weight | CRA —same-as→ CPA — 0.9 Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concept_edges[3] |
| differs in | adaptive expertise —contrasted-with→ routine expertise — whether the knowledge transfers to unfamiliar situations - routine expertise applies to familiar ones only Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concept_edges[5] |
| weight | adaptive expertise —contrasted-with→ routine expertise — 0.8 Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concept_edges[5] |
| for which use | limitation — the conceptual/procedural split as a sound partition: the paper reports Kieran declaring the dichotomy fundamentally unsound because procedures are conceptual during elaboration Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | limitations[2] |
| for which use | limitation — a placement of a task or a learner at one end of the conceptual/procedural continuum: the paper calls that positioning false, since either can occupy different places depending on the situation Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | limitations[3] |
| replication | untested | |
| no type for this | conceptual knowledge → procedural knowledge — THE DISTINCTION ITSELF IS DECLARED UNSOUND, and no type can say that. Kieran (2013) 'declared the dichotomy between them to be fundamentally unsound'. 'distinct-from' asserts the two ARE distinct; there is no way to record a cited author denying that the boundary can be drawn - the same gap quine1951 sits in. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | unhoused_relations[0] |
| no type for this | procedures → concepts — THE CATEGORIES INTERPENETRATE RATHER THAN RELATE: 'during any period of elaboration, procedures are conceptual in nature'. Not 'is-a', not 'part-of', not 'same-as' - a claim that membership is temporary and phase-dependent, which no edge type carries. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | unhoused_relations[1] |
| limitation | The paper's claim that misunderstanding of the two knowledge forms is widespread among teachers rests on anecdote rather than on collected data. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | limitations[0] |
| limitation | The paper concedes that the concept-first iterative approach it endorses has little empirical evidence behind it. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | limitations[1] |
| limitation | The dichotomy the paper's title turns on is reported as having been declared fundamentally unsound, on the ground that procedures are conceptual in nature. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | limitations[2] |
| limitation | Treating the two knowledge forms as two ends of a continuum is called false; the paper holds that either can occupy different positions depending on the situation. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | limitations[3] |
| limitation | Two cells of Star's own matrix have no worked exemplars - the paper reports that deep-procedural and shallow-conceptual knowledge are hard even to illustrate. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | limitations[4] |
finding · quantity · variable · definition · key-term — grounded in the full text| finding | Experienced primary and lower-secondary teachers believed that conceptual and procedural knowledge are mutually exclusive and that conceptual knowledge is always the more appropriate thing to teach. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.findings[0] |
| finding | Procedural and conceptual knowledge both matter and each strengthens the other. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.findings[1] |
| finding | Either procedural or conceptual knowledge can be shallow or deep. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.findings[2] |
| finding | The synthesis endorses teaching conceptual knowledge before procedural knowledge in most cases. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.findings[3] |
| finding | Students are likelier to develop conceptual knowledge if they begin with it and then move to procedural knowledge. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.findings[4] |
| finding | The empirical support for concept-first is that students taught conceptually and then procedurally outperform students taught procedurally and then conceptually. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.findings[5] |
| finding | Sequencing arithmetic problems to expose conceptual relationships was efficacious; in random order the effect was not seen. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.findings[6] |
| finding | The conceptual-procedural relationship is bi-directional but not always symmetrical: at times conceptual knowledge supports procedural knowledge more strongly than the reverse. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.findings[7] |
| finding | The paper's answer to its own title is that teachers need conceptual AND procedural knowledge, with the timing and placement left to the teacher's professional judgement. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.findings[8] |
| finding | Kieran (2013) declared the conceptual-procedural dichotomy fundamentally unsound. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.findings[9] |
| definition | concept Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.definitions[0] |
| definition | conceptual knowledge Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.definitions[1] |
| definition | procedure Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.definitions[2] |
| definition | procedural knowledge Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.definitions[3] |
| definition | routine expertise Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.definitions[4] |
| definition | adaptive expertise Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.definitions[5] |
| definition | cognitive interference Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.definitions[6] |
| definition | attitudinal interference Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.definitions[7] |
| definition | iterative view Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.definitions[8] |
| concept | conceptual knowledge Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[0] |
| concept | procedural knowledge Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[1] |
| concept | concept Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[2] |
| concept | Relational Knowledge Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[3] |
| concept | routine expertise Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[4] |
| concept | adaptive expertise Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[5] |
| concept | CRA Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[6] |
| concept | CPA Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[7] |
| concept | Declarative Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[8] |
| concept | Instrumental Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[9] |
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