In the article Procedural knowledge vs conceptual knowledge in mathematics education I expand on James E Schwartz’s piece based around procedural and conceptual knowledge. To paraphrase Schwartz, “mathematical procedures, known to mathematicians as algorithms (are) procedures (which) enable (us) to find answers to problems according to set rules.” The rules based approach:I argue that mathematical education has historically focused on imparting procedural knowledge, through a ‘rulesbased approach’. I use the term ‘rulesbased approach to refer to a traditional approach which looks something like the following: “Here’s a rule … The problem with a teaching and learning process that focuses on mathematical procedures is not that it teaches students mathematical procedures – the problem is that such learning comes at the expense of mathematical understanding. Mathematical understanding is the realm of conceptual knowledge. In the Procedural knowledge vs conceptual knowledge in mathematics education article I propose that in order for students to acquire conceptual knowledge, the rulesbased (procedural) approach needs to be replaced by a conceptual approach. When we use an effective conceptual approach we are teaching math conceptually. When we teach math conceptually the following pedagogical dynamics apply:
In the online course ‘Engagement: Winning over your mathematics class' the above teaching and learning aspects of the conceptual approach to teaching mathematics are explored in depth and numerous, specific examples of what ’teaching math conceptually’ looks like are presented. Shifting from a traditional approach to teaching mathematics to a conceptual approach is a challenging transition for teachers to make. Adding to the challenge of making such a transition are the misconceptions commonly possessed by teachers who have not yet made the shift. Four common misconceptionsBelow are four common misconceptions held by teachers about a conceptual approach to teaching mathematics Misconception 1:
Conceptuallybased teaching is synonymous with handson activities, and handson activities are extremely difficult to manage with my students. Few would disagree that handson activities are difficult to manage in many secondary classrooms. In an ideal world, all students would explore handson materials within a wellstructured activity, in a highlyengaged, questioning, selfdirected manner. Clearly, many secondary students have become removed from that ideal world! Therefore, setting students free amongst abundant handson materials is likely to be ineffective in bringing about conceptual understanding. Grant Wiggins excellently explores this point in his Experiential Learning article by expanding on the statement “Just because it’s handson doesn’t mean it’s mindson”. However, the belief that conceptuallybased teaching is synonymous with handson activities is, in my view, a misconception. A quality, conceptual approach can incorporate some use of handson materials as part of wellscaffolded, teacherled activities containing multiple leading questions. Student interaction can be governed by the teacher in accordance with the capacity for the particular group of students for learning. Misconception 2: Conceptuallybased teaching is too time consuming to be realistically implemented. This misconception usually stems from a couple of sources. Firstly, it can be linked to misconception 1, i.e. that a conceptual approach is one where lots of materials are distributed to students who are expected to spend considerable time investigating mathematically with the materials in order to gain the required insights. Commonly, teachers with misconception 2 have experienced ‘losing’ significant time because such activities have provided very little insight for students due to the activities failing dismally through student disengagement, poor implementation or both. Such an experience causes teachers to retreat to the comfort of a traditional approach. Secondly, teachers can conclude that conceptuallybased teaching is more time consuming than (traditional) procedural teaching because their efforts – even when not involving handson materials – simply take longer than the approach they are used to. The reason this is a misconception, however, is that there are conceptuallybased approaches which have proven to be more efficient than traditional approaches. But as I’ve already mentioned, an effective, conceptuallybased approach requires a very different pedagogy to a traditional, procedural, let’skeep‘emalltogether approach. I experienced this timesaving effect over many years. So, too, have many of the teachers who have undertaken the 15hour course ‘Engagement: Winning over your mathematics class' where this approach is comprehensively explored. Misconception 3: Conceptuallybased teaching is too difficult to manage. Is a successful conceptuallybased approach too difficult for a teacher to manage? You bet it is – if the teacher is not equipped with the necessary pedagogies. However there are highly successful, highly skilled primary teachers (and high school teachers) all over the world who are able to successfully engage their students and allow for individual progression, who have multiple activities occurring simultaneously and whose students are learning MUCH more than set routines for answering text questions. Ask these teachers if what they are doing is too difficult for them to manage! Of course it isn’t. To these teachers a conceptuallybased approach is an obvious one to use. The real issue is that the required skills are very different to (what I refer to as) a traditional approach and therefore need to be acquired. Misconception 4 I’ll lose my role as teacher, and therefore my enjoyment that comes with that role. This is absolutely a misconception and stems from the idea that teaching conceptually and becoming more of a facilitator means the teacher has less opportunity to ‘work the group’; to pour out charisma into the classroom. Arguably, a studentcentred, conceptual approach offers more avenues for a teacher’s charisma to shine. However, those avenues will look somewhat different to the approach the teacher is familiar with. In addition, The Conceptual Approach offers more chances to connect with students oneonone and in small groups because, at any given time, the remaining students are better occupied and do not require the teacher’s direct input. Summary From the outset the challenge of teaching math conceptually appears significant, inefficient and counterintuitive. However, with measured guidance and copious, ‘feet on the ground’ examples of teaching math conceptually, the challenge, although requiring significant time, can turn into reality with relative ease. This is the primary purpose of the Learn Implement Share course ‘Engagement: Winning over your mathematics class'.
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AuthorRichard Andrew likes to write about stuff that matters in education. That boils down to 'anything that helps teachers to betterengage their students'. His view? "Not much else really matters  engaged students learn. Disengaged students do not." ArchivesCategories
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