In the first article, “Understanding Children’s Reasoning,” Bright tells us that although observing is a great way to see what children are doing, the best way to understand what children are actually thinking is to simply ask them questions about their answers and how they got them. This is important because performance is basically a symptom of mathematical understanding, so it is more beneficial to figure out the child’s understanding than to base judgments on mere performance. When asking children for their explanations, there are various types of questions that teachers can ask in order to receive varied responses and thought-out explanations, rather than “I don’t know” or “I guessed.” Questions should be specific to the problems being solved, or else the children may begin to see the questions as a routine instead of understanding that the teacher is genuinely interested in their reasoning. Once the teacher has a grasp on the children’s understanding of a certain concept, it is imperative that he or she does something with this knowledge. There are a variety of methods that are available to use in order to assist children individually, as well as a whole class, but that does not mean that it is a simple task to decide which strategy will best fit each student’s learning abilities. Some of these strategies include adjusting numbers for individual instruction, presenting problems based on information about what types of thinking are demonstrated, small group instruction, suggesting alternative strategies in order to expose the students to a wider range of techniques, and monitoring the effectiveness of instruction.
In the second article, “Teaching Strategies: Teaching Arithmetic through Problem Solving,” I really liked that they discussed the fact that it’s most important for the student to understand the concept and the process and that the method is not the most important part of mathematics. Children learn in many different ways and use a variety of methods depending on their conceptual level, which is perfectly fine as long as they understand the concept and can solve the problem in a way that makes sense to them. Also, they pointed out that teachers should allow students to explain their solutions because it helps them verbalize their thought processes and it shows that there are multiple ways to solve one problem. This article also stresses that working in partners is beneficial to the students because it allows them to work things out on their own instead of having the teacher show or tell them what to do and they are able to help their classmates if someone is having a more difficult time on a certain problem.
In the last article, “Kindergarten is More Than Counting,” I learned that fluency with small numbers is crucial to children’s learning in mathematics. Fluency means being able to put together and take apart numbers without even thinking about it. Manipulatives are very helpful in teaching children numbers and number operations, but it is important that they do not become dependant on them. Children must develop number fluency and a mental imagery of numbers, and this is usually accomplished pretty easily through the use of quick images. Quick images promote flexible thinking and encourage sharing of ideas. Being flexible while working with numbers will provide a foundation for students in relation to mental mathematics with larger numbers and in developing computation strategies. The students should be encouraged to move past counting one-by-one and move towards using mental imagery (mental math) because it will be very beneficial to them when solving problems once they are fluent in the basic numbers.
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