According
to the Ministry of Education in Trinidad and Tobago, their curriculum guide
states that;
Numeracy refers to the ability and competence to apply
mathematical concepts (ideas) and skills (processes) to effectively engage in
and manage diverse situations in real life. It facilitates the development of
higher-order thinking skills that equip students with a solid foundation for
the problem-solving challenges of the future.
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They
also state that “fraction is used to describe equal parts of a whole or equal
parts of a collection of objects. It is a number (e.g. ¼ is a number on a
number line).
After
teaching fractions they want to ensure that students would be able to explain
the ways fractions are used outside of school.
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Our Group Determined,
Students can be introduced to
fractions informally by the use of discrete objects,
such as pencils, pies, chairs, etc.
If the whole is a collection of 4 pencils, then one
pencil is 1/4 of the whole. If the whole
is 5 chairs, then one chair is 1/5 of the whole, etc.
This is an appropriate way to
introduce students to the so-called unit fractions,1/2, 1/3 ,1/4 , . . .
The pros and cons of using discrete
objects to model fractions are clear. It is simple, but it limits students to
thinking only about \how many" but not “how much". Thus if the whole
is 4 pencils, we can introduce the fractions 2/4 and 3/4 by counting the number
of pencils, but it would be unnatural to use this method to introduce the whole.
As many of us know, learning and teaching fractions and the various fraction operations can be difficult for many teachers as well as children. It's not the concept of a fraction that is difficult - it is the various operations: addition, subtraction, multiplication, division, comparing, simplifying, etc. of fractions. Other factors which contribute to the students’ weakness in fractions is that fractions cannot be counted and there are infinite numbers of fractions between any two fractions, as reported by Robert Siegler (2010). Students tend to memorize formulas or algorithms instead of understanding them. Students also have a difficulty in incorporating concept into practice, example is that students do not know why addition and subtraction require a common denominator. Although being exposed to the computing of fractions from primary school, students in secondary school still make significant error in the addition and subtraction of fractions.
ReplyDeleteOur group believe that the teaching of fractions procedurally was the major reason why we had difficult time learning fractions at the primary school level. We use procedures without fully understanding the concepts and this is what made it very difficult when moving on to a more complex fractions. For example, 7/11 = ?/121. We would cross-multiply, which provides the correct solution of 77 (7 x 121 / 11). Or we will use the equivalent fractions approach, which also provides the correct solution (121/11 = 11; 11 x 7 = 77). Which approach is more efficient? Well, which is easier: multiplying 7 x 121 or 7 x 11?
One reason we don't always choose the most efficient method is that we don't really choose any method. We just apply an algorithm without thinking. We perform procedures accurately but not always conceptually, flexibly or efficiently. In other words, we lacked the conceptual knowledge. In the above problem, for example, the equivalent fractions approach is only more efficient than cross-multiplying if students have memorized multiples of 11 however, we were not taught to look at problems conceptually we would just apply what was learnt not really understanding know and why it was done.
We are not saying the teaching procedurally is totally wrong but after completing Math 1 and currently perusing Math 2 we saw that teaching conceptually would have been more beneficial to our understanding fractions and we believe that it would also benefit children doing fractions now if they were taught conceptually before procedurally.
References:
Skemp, RR (1964).A three-part theory for learning mathematics. In FW Land, New Approaches to Mathematics Teaching. Macmillan & Co. Ltd: London.
Thompson, PW (1994). Concrete materials and teaching for mathematical understanding. In Arithmetic Teacher 41 (9) NCTM.
Van de Walle, JA (2004). Elementary and middle school mathematics – teaching developmentally. Pearson: New York.
You have made such a valid point Leah...such 'procedural' knowledge spreads across the curriculum. Too often are students taught the 'how' instead of the 'why'. Hence the saying, 'he have lots of book sense and zero common sense!'
ReplyDeleteIn the new draft Primary School Curriculum, the formal teaching of Fractions is introduced in Standard Two. However, students have informal knowledge of Fractions from very early stages in their lives.
ReplyDeleteFor example, it is not uncommon for a young child to ask for "half" of something. This fraction vocabulary would have been taught to them implicitly. Children would have also constructed within their minds how much one half should represent. This may be the case because young children are often given portions of a quantity of something in order to make their share child sized.
While a child may not understand that fractions really refer to a type of mathematical relationship among quantities, they are able to deduce how it is represented in the real world.
Teachers must try to understand their students' perspectives of fractions before they can add to that knowledge. It cannot be overstated how much students need a teacher to bridge the gap between their constructed knowledge and their instructed knowledge of Fractions.
Reference
Making Sense of Fractions, Ratios and Proportions by Bonnie Litwiller and George Bright.
Group members: Rishmattie Maharaj, Christine Jianath, Angalie Maraj, Maya Dass
ReplyDeleteyes I agree with you because as teachers we all need to scaffold the child. Here the teacher modifies the amount of support according to the needs of the child by modelling the behaviour, for example possible methods
of approaching a problem. The teacher breaks down the task and makes the task manageable for the individual child, thus supporting the development of the child’s own problem-solving skills.
Young children are egocentric, and it is through social interaction that they can begin to appreciate the points of view of other people. Sequences of instruction involve discussion, hands-on experience and practical exploration. As adults we expect objects to behave in a stable and predictable manner. Children have to learn to recognize these attributes. They need to handle and use a variety of objects in order to form their own rules and structures for dealing with the world. This is of particular importance in mathematics.
Group Member: C. Ramdhanee, R. Simbhoo, C. Bisram, E. Sucre, K. Khan
ReplyDeleteAlthough according to the Curriculum, fractions are being introduced quiet late the students must have some sort of knowledge in their early stages for things such as sharing would come into play. Especially when building social interaction and they have to share with their friends. If someone says well i want a piece of the chocolate they would be inturn sharing either 1/2 or 1/4 without initially knowing that is the amount. Therefore they must have knowledge about fractions and its uses.
According to Cassandra's Group example, there will be some pros and cons in introducing them to fractions through the use of discrete materials such as pencils and chairs.At the same time is is very good because will be able to differentiate 1 chair out for 4 chairs and say it is 1 out of four, in turn our teaching of 1/4 comes from that. From here on we can switch the resources to something like a pizza which will interest them as we go along. Show them that if they have a small pizza of 4 pieces and John ate 1 slice he ate 1/4.
Yes there will be the cons of introducing wholes this way but we as teachers must be able to crucially think about the various fraction vocabulary we would want the children to know at the end of the lesson and teach them efficiently and according to the Curriculum.
Therefore according to MOE Curriculum guide we engage students in the application of mathematics through the use of everyday real life situations. As stated above with the fractions, yes they would know to share but they would be doing it unknowingly that there is some specifics to what they are doing. Therefore we as teachers must take the guide to help the students understand and link their experiences in someway to what they are doing in school.
Learning the various fractions operations can be difficult for many children. It is not the concept of a fraction that is difficult it is the various operations of addition, subtraction, multiplication, division, comparing, simplifying etc of fractions and how they are taught proves difficult for children. There are many ways to teach fractions and the success or failure of the student to understand fractions will depend on the teaching method used. It has been said that if a student understands fractions, then they can understand any mathematics concept. it is then very important for every math teacher to know how to teach fractions in the most approachable way possible. The NCTM Curriculum and Evaluation Standards (1989) promotes the use of physical materials and other representations to help children develop their understanding of fraction concepts. the three commonly used representations are area models (eg. fraction circles, paper folding, geo boards), linear models (eg. fraction strips, Cuisenaire rods, number lines) and discrete models (eg. counters, sets).
DeleteRecommendations that has worked:
1) Use manipulatives and visual models.
2) Explain the numerator and denominator until every student understands.
3) Draw lots of pictures of fractions.
4) Ensure that you teach fraction simplification slowly and thoroughly.
5) Give students lots of hands on experiences with fraction pictures they draw themselves.
Using these recommendations, will be a useful guide for the maths teacher.
Group members:
DeleteRaheem, Chrisann, Eunice, Yoshoda.
Utilising the NCTM principles into the curriculum will prove benefical to the teacher and student. Moreover, I agree with Cssandra's and Kelsey group view of how to integrate visual manipulatives and model into the teaching of fractions.