Learner-centered schools, in which the responsibility for learning is returned to the students, provide enriched environments in which learners can explore and access information in whatever way best suits them at that moment. This often changes from one context to another, or one subject to another, so it’s ridiculous for teachers to think they have to or even CAN “diversify” learning from the top down. Yes, create an overarching framework and goals, but then turn the execution of the learning over to the learners and be there for them if/when they need help or other resources. The repeated use of the words “teach” and “manage” in this article demonstrates the belief that learning can’t take place without teaching–that children need adults to “manage” their learning. I suspect that would surprise any child under the age of 5! Can adults facilitate learning? Absolutely. But they do that by providing individuals with what they need/want (according to the individual) at that moment in time…not what the adult THINKS they need. They do that by encouraging the learners to ask and answer their own questions and to reflect on what works and what doesn’t.

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http://abcnews.go.com/WNT/PersonOfWeek/story?id=319677

http://articles.latimes.com/2005/jan/05/local/me-class5

Teaching Channel’s Video – Mathematical Argumentation: Which one doesn’t belong –demonstrations kindergarten and 1st grade teachers getting their students in the habit of explaining their thinking process at such a young age by implementing a different way to practice pattern recognition. Kids are asked to pick out the item that doesn’t belong when looking at a grouping of shapes or objects, as well as a reason as to why. Not only does this bolster logical reasoning skills, but it fosters a collaborative work environment where students become comfortable sharing their answers, supporting their claims, and listening to each other’s ideas. Instead of solving a math problem and arriving at a concrete answer, this proposes talking through problems in math classes. Thus, this approach stresses the open-endedness of questions. There is no wrong answer; every answer is correct as long as the student has evidence to explain their thinking process.

Daniel Willingham – a psychology professor at the University of Virginia – has called attention to two facets of mathematics that are not innate (and in turn need to be taught) in students: math facts and conceptual knowledge. While conceptual knowledge goes hand in hand with what this blog post is discussion – the development of explaining math thinking – math facts digress. Willingham stresses the importance of learning math facts to automaticity in early elementary education. While, yes, it is important to be able to explain one’s thinking process out loud, it is equally important to be able to recall math facts from long term memory instantaneously. Such automaticity frees up working memory, which can later be used in more complex problems. Thus, while I am a proponent of the mathematical argumentation learning approach, I deem it is necessary to couple it with the acquisition of automated arithmetic to encompass a multifaceted learning.

Moreover, it is interesting to look at teacher’s mathematical pedagogical knowledge. While we don’t have a recruitment problem of finding qualified teachers, we do see a deficit in our middle school teachers’ math pedagogy. This is a major finding of the 2012 Teacher Education and Development Study in Mathematics (TEDS-M). Pedagogy has a wide array of meanings – from specific teaching technique to the larger structure of the classroom. The finding that our middle school teachers lack the ability to fully be able to teach and explain the subject matter perhaps could be in part attributable to the core argument in this blog: our teachers never were asked to explain their math thinking in their own education. Implementing this learning process will not only aid students’ in their ability to make sense of things and articulate their thinking, but help bridge the gap or disconnect seen in US educator’s pedagogical knowledge.

While this learning approach clearly has vast applications in the realm of mathematics, I think that it could be pragmatic for developing better critical thinking skills, as well as metacognitive abilities. As Katrina Schwartz points out in this blog post, we can catalyze the development of critical thinking through teaching young learners what it really means to “explain your thinking.” A large issue with approaches to ingraining critical thinking skills in students rests on the idea that educators believe that development follows set stages, as proposed by Piaget: sensorimotor, preoperational, concrete operational, formal operational. Piaget argues that not until one reaches the last stage, around the age of 12, can one develop deductive logic, abstract thought, and hypothetical thinking. However, this blog post, along with further research, deviates from the theory of developmental stages. It’s not too early to start teaching mathematical logical processes. As seen in the video, children of the mere age of 5 and 6 were able to reason and explain why they chose which object does not belong. I believe that instilling these logical reasoning skills at a young age will bolster students’ critical thinking skills as subject matter gets more demanding, as it gets kids in the habit of reflecting upon their own (as well as their classmates’) ideas and thinking processes. For instance, when students reach higher levels of math, such as geometry or calculus, they will be able to more easily write out proofs to problems, essentially outlining their step by step thought process, than if these values were not instilled upon them in kindergarten and 1st grade.

As I mentioned above, being able to explain math thinking is closely related to metacognitive skills. Metacognition is a widely understood term, which directly translates to “thinking about thinking.” Essentially, metacognition is having awareness and understanding of one’s own thought process. Hence, the learning process that Schwartz is describing in this blog is just one way of applying metacognitive skills in mathematics at a very young age. In order to execute metacognition, Willingham proposes three different things students need to be able to do: plan, monitor, and evaluate their thought process. This is paralleled in the video when the teacher explains that her students – after practicing explaining their math thinking – have developed deeper levels of noticing and picking out more abstract similarities and differences between pattern sets. A common problem with metacognition is seen in the evaluation stage, where students frequently say, “I know it, I just can’t explain it.” If explaining thinking processes were stressed in early education, this problem could be avoided. This is backed up by the ideas of Christine McCormick, Carey Dimmitt, and Florence Sullivan in their scholarly article, Assessment of Metacognition: Metacognition, Learning, and Instruction. They argue that when utilized, metacognitive skills will facilitate deeper understanding and better learning, and provide evidence for this through analyzing students’ writing abilities – as brainstorming and writing can be viewed as a metacognitive process. Embedding metacognitive instruction in content matter – like the instruction described here in Schwartz’ blog – will help ensure connectivity and richer analysis in students.

Lastly, I think it is fair to say that this learning approach for explaining math thinking is just the start; educators need to transfer this method into other domains and continually ask more abstract and open-ended questions, which in turn demand deeper analysis and aid the generation of independent thoughts. Moreover, implementing this method of instruction can help bridge the gap to development of critical thinking and metacognitive skills, which in turn will bolster the students’ future educational endeavors.

]]>Having students explain how they got to an answer or argue why a shape/concept does not belong to a category directly contradicts the false belief that if a student understands algorithms, then they will pick up on the conceptual basis behind math. Teaching students conceptual meaning behind basic math will not hinder or confuse their subsequent performance. In the attached video, it is clear that even young elementary students are more than capable of articulating their reasoning behind a problem to themselves and their peers, and appear to be more engaged in the process of learning math as opposed to route memorization.

Moving forward, I am curious to see more schools and educators adopt a more conceptually based curriculum and teaching strategies in regards to mathematics. Since this type of inquisitive mindset is so crucial to have as a foundation, a push for early interventions or exposure to mathematic argumentation before even entering kindergarten is inevitable. If a student is already accustomed to that deeper level of abstract thinking in regards to equations, building up to higher mathematics will become a much more natural and less overwhelming process—leading to more competent students.

]]>The words “equal,” “average,” and “individual” in this excerpt are being misused to make a tendentious argument which has no reason for being without a set of systematic distortions.

(1) “Equal access” *never* meant “as ensuring that everyone has access to the same experiences.” The equality applies to the *access* not to the system: if there is equal access to an education, everyone still gets a unique educational experience.

(2) “But now we know there is no such thing as an average person….” I don’t believe with know this now. “Average” refers to a statistical norm. Individuals approach this norm to varying degrees. Does the author really believe that a man in North American who is 5’10” tall is *not* closer to average height? By saying “average person,” the author evades the entanglements of specificity.

(3) “[W]e can require that educational assessments be built to measure individual learning and development rather than simply ranking students against one another.” Educational assessments *do* measure individual learning and development: by comparing scores year after year. There are other ways to do this besides testing instruments that are valid and reliable, but that does not mean valid and reliable tests fail to measure anything. This is like saying “my growth can’t be measured by a yardstick, because a yardstick only gives one number!”

‘Everyone is unique! Education must change entirely to accommodate this keen insight!’ Were this argument true, our educational systems would have created only automata. And yet, everywhere you go, people are different. It seems that the standardized and standardizing educational system that so badly needs decrying is actually terribly ineffective.

The author has to mis-state the problem so that the remedy on offer seems desirable.

A better solution would come from a less distorted estimation of the problem.

]]>Tutrang Nguyen et al 2015 illustrates a strong correlation between early math ability and later achievement. The article suggests that elementary curricula should focus on precursory math skills in order to provide a foundation for greater conceptual understanding. Pattern recognition exercises like those conducted in both videos of the blog are one of many ways to teach these precursory skills; the more important aspect of this blog is that the teacher provides valuable aid by prompting the students to explain their reasoning. According to Cecil Mercer and Susan Miller 1992, struggling students require scaffolding in both concrete and conceptual areas of math; the teachers in this blog employ scaffolding to force the student to think more deeply.

An important aspect of developing a strong conceptual understanding of math is an emphasis on depth, not just breadth of teaching. According to Gilbert Valverde and William Schmidt 1998, many math implements such as unnecessarily large textbooks prevent a deep conceptual understanding because of the emphasis on too many broad topics. The teachers in these videos do a good job of exploring each scenario with depth by allowing multiple students to explain their evaluations and explanations. These multiple explanations allow other students to expand beyond their initial understandings and therefore further their conceptual knowledge of these mathematical scenarios.

Although the teachers in these videos do prompt students for explanation, the students are not expected to explain beyond an initial observation. The scaffolding provided to these early elementary students provides the initial reasoning behind their evaluation of the scenario, but their conceptual understandings could be strengthened if the teachers prompted for further elaboration. In addition, further elaboration would allow fellow students to listen and incorporate the deeper analysis into their own understanding of the problem.

The techniques employed by teachers in these classrooms provide young students with the valuable opportunity of deepening their conceptual understanding of math concepts. The fact that students must explain the reasoning behind their evaluations forces them to turn their thoughts into coherent sentences, a valuable means of furthering conceptual knowledge.

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Animation Notes ]]>

I think some of the other compromise projections like Natural Earth and Winkel Tripel would make better standard maps. Form there kids should be getting into other issues of map distortions using any and all of the already mentioned projections.

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