Teaching Mathematics through Guided Discovery
As with every academic subject, there are a variety of strategies for teaching mathematics to school-aged students. Some strategies seem to be better than others, especially when tackling certain topics. There is the direct instruction approach where students are given the exact tools and formulas they need to solve a problem, sometimes without a clear explanation as to why. The student is told to do certain steps in a certain order and in turn expects to do them as such at all times. This leaves little room for solving varying types of problems. It can also lead to misconceptions and students may not gain the full understanding that their teachers want them to have. So how can mathematics teachers get their students to better understand the concepts that are being taught?
A somewhat underused strategy for teaching mathematics is that of guided discovery. With this strategy, the student arrives at an understanding of a new mathematical concept on his or her own. An activity is given in which "students sequentially uncover layers of mathematical information one step at a time and learn new mathematics" (Gerver & Sgroi, 2003). This way, instead of simply being told the procedure for solving a problem, the student can develop the steps mainly on his own with only a little guidance from the teacher.
The ability for children to discover is innate. From birth children discover all sorts of different things about the world around them. It has even been said that "babies are as good at discovery as the smartest adult" (Gopnik, 2005). Discovering is the natural way that children learn. By interacting with the world around them, they ar...
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... a sense of accomplishment, something they cannot get through direct instruction alone. This sense of accomplishment will raise their mathematical self-esteem. This can, in turn, help students appreciate and enjoy mathematics even more. Few would argue against the idea that any teaching strategy that gets students to believe in themselves and enjoy the subject is a good one.
Works Cited
1. Begley, Sharon. The Best Ways to Make Schoolchildren Learn? We Just Don't Know. Wall
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2. Gerver, Robert K. and Richard J. Sgroi. Creating and Using Guided-Discovery Lessons.
Mathematics Teacher. Vol. 96, No. 1. January 2003. pg. 6.
3. Gopnik, Alison. How We Learn. The New York Times. New York, NJ: Sunday January 16,
2005. Section 4A; Column 1; Education Life Supplement; pg. 26.
I remember how mathematics was incredibly difficult for me and because of this I can relate to the struggles students have with math. For a teacher to be successful they need to create relevance for the students. I understand how to relate the various topics of mathematics to topics of the world, which for most students is difficult to do, For example, I remember at the CREC School I was observing at, there was a student of Bosnian decent who was having trouble understanding how to read a map of the United States. So I showed her a map of Bosnia with the same map key, and we discerned what everything meant (where the capital was, where the ocean was, major port cities were, etc…). She caught on quickly as she already had an understanding of Bosnia and it quickly transferred over to the map of the thirteen colonies. This skill is easily transferrable to mathematics by using relevant, real-world examples of concepts learned by
Reys, R., Lindquist, M. Lambdin, D., Smith, N., and Suydam, M. (2001). Helping Children Learn Mathematics. New York: John Wiley & Sons, Inc.
While children can remember, for short periods of time, information taught through books and lectures, deep understanding and the ability to apply learning to new situations requires conceptual understanding that is grounded in direct experience with concrete objects. The teacher has a critical role in helping students connect their manipulative experiences, through a selection of representations, to essential abstract mathematics. Together, outstanding teachers and regular experiences with hands-on learning can bestow students with powerful learning in
All children learn differently and teachers, especially those who teach mathematics, have to accommodate all children’s different capacities for learning information. When teaching mathematics, a teacher has to be able to use various methods of presenting the information in order to help the students understand the concepts they are being taught. Most teachers in the past have taught mathematics through procedural lessons. Procedural lessons consist of having the students work with a concept over and over again until it is memorized. For example, children could be given homework assignments with the equation three times five over and over again until that equation is memorized.
Now days we can see that young children are very inquisitive about finding the reason behind every occurrence. They are self motivated to learn about the “Hows” and “Whys” of the world. It can be said that the children are almost like scientist as they collect evidences by scrutinizing and experiencing the world. Children are generally involved in the process making hypotheses; they are also engaged in evaluating the statistical data and releasing prior beliefs when they are presented by other stronger evidences. All this they are doing even when they are searching for their toys, arranging blocks in any random manner or playing with toys with their friends. Children also show amazing psychological intuition by watching the actions of other people and can also determine underlying enthusiasm, desires and preferences (Kushnir and Wellman, 2010).
Wu, Y. (2008). Experimental Study on Effect of Different Mathematical Teaching Methodologies on Students’ Performance. Journal of Mathematics Studies. Vol 1(1) 164-171.
Jean Piaget (1896 – 1980), a Swiss psychologist, portrayed the child as a ‘lone scientist’, creating their own sense of the world. Their knowledge of relationships among ideas, objects and events is constructed by the active processes of internal assimilation, accommodation and equilibration. (Hughes, 2001). He also believed that we must understand the child’s understandings of the world, and this should guide the teaching practises and evaluation. The fundamental basis of learning was discovery. To understand is reconstruct by discovery, and such conditions must be compiled...
Automaticity of math facts is beneficial to all mathematics learning. Fortunately, there are ways to help students learn basic facts without skill and drill. Explicit strategy instruction is more effective than encouraging strict rote memorization (Woodward, 2006). Yet, many educators are unsure of how to help students master facts. Too many educators still have misconceptions of how students learn facts and how they commit them to long-term memory (Baroody, 1985).
[2]Hanna, Gila (2000), “Proof, Explanation and Exploration: An Overview,” Educational Studies in Mathematics, V44, pp. 5-23
By the age of three a child's brain is three quarters of its adult size. From infancy to the age of two development is very rapid (Santrock, 1996). For this reason it is essential for the child to be able to explore their world around them. By exploring children will increase their knowledge and understanding of the world.
Using literacy strategies in the mathematics classroom leads to successful students. “The National Council of Teachers of Mathematics (NCTM, 1989) define mathematical literacy as an “individual's ability to explore, to conjecture, and to reason logically, as well as to use a variety of mathematical methods effectively to solve problems." Exploring, making conjectures, and being able to reason logically, all stem from the early roots of literacy. Authors Matthews and Rainer (2001) discusses how teachers have questioned the system of incorporating literacy with mathematics in the last couple of years. It started from the need to develop a specific framework, which combines both literacy and mathematics together. Research was conducted through
“Infancy and childhood are the most critical periods of life in a person’s development; the body is growing rapidly, and motor, speech, and cognitive skills are evolving” (“What Does a Pediatrician Do?”, 2013, par 1). During that stage children learn very fast and they acquire a lot of information and uses them very frequently. Berk stated in her book Exploring Lifespan Development “ Infants and toddler’s mental representation are impressive but in early childhood , representational capacity blossom” (2015, pg,175). Children are little explorers they learn by watching and doing. According to Piaget’s Cognitive development theory “children actively construct knowledge as they manipulate and explore their world” (2015,pg,17). As children explore their brain get the ability to acquire new information and
Sherley, B., Clark, M. & Higgins, J. (2008) School readiness: what do teachers expect of children in mathematics on school entry?, in Goos, M., Brown, R. & Makar, K. (eds.) Mathematics education research: navigating: proceedings of the 31st annual conference of the Mathematics Education Research Group of Australia, Brisbane, Qld: MERGA INC., pp.461-465.
Mathematics teachers teach their students a wide range of content strands – geometry, algebra, statistics, and trigonometry – while also teaching their students mathematical skills – logical thinking, formal process, numerical reasoning, and problem solving. In teaching my students, I need to aspire to Skemp’s (1976) description of a “relational understanding” of mathematics (p. 4). Skemp describes two types of understanding: relational understanding and instrumental understanding. In an instrumental understanding, students know how to follow steps and sequential procedures without a true understanding of the mathematical reasons for the processe...
One of the widest used methods is learning through discovery. Discovery is finding out information using hands on experiments. The children can discover what happens in science and why. They answer the problems for themselves. They use their schema, prior knowledge of science, to search for information. The cycle of scientific discovery is first a question or series of questions are raised. Second, through a discussion a problem is identified and narrowed so that the kids can solve the problem. Third, with the assistance of the teacher, the children propose a way of looking at the problem and then collect the...