# Further Topics In Primary Mathematics Education Education Essay

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Children can be assessed on five aspects of a topic, namely: concepts (1), procedures (2), processes (problem-solving (3), connections (4) and communication/representation (5)), (Glanfield, Bush & Stenmark, 2003). With reference to the Topic 'Length' as it is presented in Abacus Level 3 (SDM p.3 - 8), comment about the expected assessment tasks presented in the Assessment Book Level 3 (p.60-61) in the light of the afore-mentioned five aspects.

## Concepts

## Definition:

"Measurement involves a comparison of an attribute of an item or situation with a unit that has the same attribute." (Van de Walle, J.A.) Thus, concepts are important for the children to help them distinguish the measurable attributes of the objects.

## Critique of the Assessment Book

Concepts in the topic 'Measurement' deals with the idea of length, size and units. The children in a year 4 class should be able to use vocabulary related to length such as: long, longer, short, shortest, tall, narrow, wide etc. They should also have an idea what estimation, measurement and comparing lengths are all about by using the standard units (cm, m and km). Also, the children should be introduced and asked to suggest suitable units and equipment for any particular measurement. The Assessment book's main concepts do match with the syllabus', but the Assessment book have missed to mention the children's recognition that the longer lengths require a larger unit, that is kilometer, and that the shortest lengths, require a smaller unit, that is the centimeter. Also, the ability to measure and record lengths using formal units and decimal units to the nearest half ex: 50cm = 3.5m. The Assessment book within its limitation of exercises does address most of the listed concepts. Basically all the working exercises and hands on activities in the Assessment Book are a good means of assessing the child's notion of the measurement's main concepts. The oral questions assess the key vocabulary just like in all other activities, and assess the children's notion of 'estimation'. Exercise 1 is a good means of assessing whether the child grasped the idea of comparing different lengths as well as use the suitable units and equipment to measure a particular object. Exercise 2 assess if the children have really savvied the relationship between different standard units, that is, from km and m and m and cm, just like the 'Write the matching measurement' exercise. As the title 'Match the appropriate measurement to each object states, here the children are assessed on identifying the suitable units to match it with the objects taken from real life, and recognizing that longer lengths require a larger unit (km) and smaller lengths require a smaller unit (cm) The last exercise on the Assessment book, again, assesses the child's ability to compare le lengths.

Although the activities in the Assessment Book somehow covers the afore-mentioned mathematical concepts in measurement, there needs to be more exercises which assess harder tasks, concepts which children are likely to grasp and understand by doing one exercise. Concepts such as the relationship and conversion of different units from km to m, m to cm, and vice versa.

## Critique of the Text Book

Coming to analyze the textbook, we think that most mathematical basic concepts are all addressed. Starting off from the textbook's bold headings 'Centimetres' (cm) and 'Metres' (m) are thought to be a good means of a continuous unit abbreviation reminder all through the exercises.

The exercises' titles are very child friendly and assessment friendly. One can note that when the child is asked to do more than one task in an exercise, the steps are written in a separate speech bubble. For instance the first exercise on page 3 invites the child to: first to find one of each object, then estimate its length in centimeters and finally use a ruler to measure it. This help the child to face a task step by step and also help him to analyze a problem bit by bit later on. It's also a good means of helping the teacher analyze and assess the children's weaknesses and strengths by establishing the step which the children failed to accomplish the step which the children failed to accomplish, thus it would be a good way for the teacher to tackle the weak point straight ways. We have noticed that children are not invited to use informal units and also it would have been a good way to introduce measurement. Also, sometimes in one exercise there is more than one concept tackled, which makes the exercises harder.

The first exercise on page 3 makes us wonder what the lessons; objective actually is. Is the lesson based around the estimation of eh measurement of the actual length's concept? These types of exercises which have in them too many objectives are likely to cause confusion. This could have been ideal as a revision exercise rather than the first exercise of the measurement's topic. Also neither of the exercises invite the children to establish when to use standard units. All the exercises in the textbook though cover all the concepts. The first exercise on page covers and addresses the vocabulary related to length, estimation and measurement, whilst the other exercise focuses on choosing the suitable units. Also on page 4, children are assessed on their ability to choose the suitable standard unit and into understanding the fractional units to the nearest half even it this concept is not listed in neither the syllabus not the assessment book and also assessing whether the children have understood the concept that longer lengths require longer units ex. km. Page 5 focuses on the concept of finding the relationship between metres and centimeters ex. 3m 15cm +25cm = 3m 40cm. The rest of the lengths exercises assess the children's ability to solve problems and find the total and the difference of two lengths.

## Compare and Contrast the Assessment Book with the Textbook.

Having to compare and contrast the assessment book with the textbook we realized that the vocabulary related to length is absent. Also the assessment book does not mention that one of the concepts learnt by the end of the topic is decimals for metres and centimeters even though there happen to be an exercise to 'Write each length in metres using a decimal point' ex. 2.48m. Adding to this, even the textbook contains exercises which concepts mainly are: choosing suitable units and using fractional units to the nearest whole. Also, we think the textbook is more child and teacher friendly as regards to its content. Surely the textbook has got quite more pages with measurements exercises than the assessment book, thus scaffolded learning is exercised even more. Also, children have more opportunities to grasp the topic's concepts through the multiple exercises. On the other hand, the assessment book includes practical hands on activities which help into engaging the children more into the topic whilst creating a richer learning environment and were concepts are easily understood and applied.

## References:

## Main Reference:

Van de Walle, J.A (2007) Elementary and Middle School Mathematics. United States of America.

## Other References

Northern Territor Government (2009) Mathematics Measurement Retrieved on ___________ from http://www.det.nt.gov.au/__data/assets/pdf_file/0018/2376/ntcf_maths_measurement.pdf

## Procedures

## Definition:

Procedures basically refers to the scaffolding of a lesson so as the children comprehend the mathematical concept easier. Scaffolding mainly consists of questioning and listening both from the teacher and the children. Group work, pair work, class discussions, and hands on activities are ways which the teacher can utilize for scaffolding.

## Critique of Assessment book:

The Assessment Book offers a variety of procedures that children can use to learn measurement. Teachers can find oral questions which can be discussed as group activity in class. This encourages the children to share their own result and clarify any problems. After such verbal exercises, the Assessment Book moves on to written exercises. Here, scaffolding is quite evident since after the children work together in class, they will later be invited to work individually.

Activity 2 of the Assessment book (Practice Activities) children are also cognitively challenged since they are asked to think, estimate and then, measure to assess themselves whether they were got it right or wrong.

However, not all activities in the Assessment Book aim to assess procedures relevant to a year 4 level. For instance, question (number 1) 'Do you think this line is 3cms, 30cms, or 300cms long' does not require any particular thinking skills since here we are dealing with eight year olds. A more appropriate activity would have been if children were given three measurements to choose from, but with closer digits, for examples 30cms, 4cms, and 50cms. The rest of the questions (2, 3 and 4) are appropriate for the year 4 level.

Adding to this, we think the 'Match the appropriate measurement to each object' exercise is too simple for children in year 4, although one can still utilize it as a mental warm up it helps children visualize and differentiate between the two basic units of measurement (cm and m).

The 'Write the matching measurement' exercise is not scaffolded since it does not build on the previous exercise. It would have been wiser if there was a table or simple exercises which help in assessing the child's understanding of how many centimeters there are in a meter, and how many metres there are in a kilometer. Hence, this exercise would have been more effective if it was given as the topic's revision activity in the Assessment Book as it covers all the 3 basic measurements.

## Critique of Text book:

The textbook offers various procedures for children to learn and grasp the length's concept. Some exercises invite the children to estimate and measure length, finding a distance in the classroom to match the length given, finding objects to measure, writing how many metres or centimeters and finding the difference or adding up metres and centimeters. Therefore, as it can be seen from the afore-mentioned activities, the exercises involve written and verbal tasks and also activities where children are provoked to use their thinking skills.

Children are first introduced to centimeters, then to metres and then, a combination of the two. Hence, we think that the basic units of measurement are scaffolded for the children to learn and have better comprehend the topic.

## Compare and Contrast the Assessment Book with the Textbook.

There is a sense of procedure in both books as the questions posed to the students are scaffolded according to their difficulties, thus every questions builds on each other.

math.ecnu.edu.cn/.../EARCOME3_LAU_NGEE%20KIONG_TSG406().doc -

the website cannot be viewed on the internet since it file extention is .doc, however if you want to view the website we kindly ask you to access this link http://www.google.com.mt/search?hl=mt&source=hp&biw=1259&bih=569&q=Furthermore%2C+ideas+are+not+isolated+in+memory+but+are+organized+and+associated+with+the+natural+language+that+one+uses+and+the+situations+one+has+encountered+in+the+past.%E2%80%99+%28NCTM%2C+1989%2C+p.+10%29, and access the link titled Scaffolding Students' learning

## Ryan.M (2003). Classroom Tips For Teaching Measurement. Cited on: ____________ from http://www.primarymaths.ie/files/measurementtips.pdf

## Processes (Problem-Solving)

## Definition:

"Allowing the subject to be problematic means allowing students to wonder why things are, to inquire, to search for solutions, and to resolve incongruities. It means that both the curriculum and instruction should begin with problems, dilemmas, and questions for students."

Hiebert er al (1996, p.12) Van de Walle

## Or

## "Good problems can inspire the exploration of important mathematical ideas, nurture persistence, and reinforce the need to understand and use various strategies, mathematical properties, and relationships. Such habits are of value not only in the mathematics classroom, but also in formal and informal learning and work environments throughout life."

(NCTM, 2000, p. 182)

## Critique of Assessment Book:

The Assessment book lack problem solving questions. Looking carefully at the questions in the assessment book we noticed that there are only two problem solving questions. These are found under the heading 'Oral questions' and are questions number 3 and 4. Here, the children are required to find the total length of two pieces of strings and to find the difference between the heights of two children. Since these two questions are under the heading of 'Oral questions' we thought that it would have been better if these questions were presented in a written manner as the children might find it hard to remember the lengths when working it out. Afterwards the teacher can switch the lesson to an oral way where they can discuss the reasons for the methods used.

## Critique of Text Book:

There are about five story sums in the textbook, which we think are a bit challenging for eight year old students. For instance, on page 5, the children have to explore which routes from school they will need to run, so as to run 1Km. We think, that this question is misleading and the children would find it challenging if the teacher will not explain it well.

On page 7, there is another story sum about a puppy and how much did it grow each week. We also think that this is challenging for the children since they can get confused from where they are going to start. We thought that this would be a possible way, how the teacher can present this story sum to the children:

Teacher: What information was given?

Student 1: That the puppy is 55cm long. It was 15cm long when it was born. That it is more than 2 weeks old but less than 10 weeks old and that it grows the same amount each week.

Teacher: How can you know how much the puppy grew weekly?

Student 1: First I subtracted the puppy's present length (55cm) with the length when it was born (15cm). Then I divided the answer (40cm) by 4 by trial and error since I know the puppy is more than 2 weeks old but less than 10 weeks old, and got an answer of 10cm growth every week.

Teacher: Who has a different answer?

Student 2: I divided the puppy's growth by 5cm and got an answer of 8cm growth per week.

Teacher: How did you check your answer?

Student 2: I multiplied the puppy's weekly growth (5cm) by the number of weeks (8weeks) and got 40cm (the puppy's present length).

Teacher: How else can you check the answer?

Student 3: You can add 5cm + 5cm + 5cm + 5cm + 5cm + 5cm + 5cm + 5cm which makes 40cm (the puppy's present length)

Teacher: Who agrees? Why?

Students: I do/I don't

OTHER PROBLEMS FROM PAGE 8

## Compare and Contrast the Assessment Book with the Textbook.

Even though problem solving is seen in both the Assessment book and the Text book, we think that they are not scaffolded since they do not increase in difficulty at every question. The Assessment book has two problem solving sums which are not difficult for the children to understand. However, story sums found in the textbook are quite challenging for a year 4 level since they require a lot of thinking skills.

## References:

Thomas R. Post (1988)

Teaching Mathematics in Grades K-8

Allyn and Bacon, Inc

Printed in the United States of America

## Main Reference:

Post, T.R (1988) Teaching Mathematics in Grades K-8. United States of America. Allyn and Bacon, Inc

## Other References:

Annenberg Foundation (2011). Observing Student Problem solving. Cited on _________from http://www.learner.org/courses/teachingmath/grades3_5/session_03/section_01_b.html

## Connections

## Defintion:

" connecting mathematical ideas includes linking new ideas to related ideas considered previously. These connections help students see mathematics as a unified body of knowledge rather than as a set of complexâ€¦concepts, procedures and processes."

(ktieb ref)

## Critique of Assessment Book:

Mathematics should not be seen as a set of separated topics but as a "web of closely connected ideas" (book ref). Such comparabilites amongst these connected ideas are not easily perceived by the children, and so they prompt for more explanation to answer the 'why'/ Such equivalences in the various mathematical content fields ma create a sense of correctedness and linkage in the mathematical studies. The assessment book presents such connections really well. Exercises like exercise 1 under the heading ' Practice activities', and the "Match the appropriate measurement to each object" exercise both provide opportunities for children with exercises involving real life situations. This helps the children to view what they are learning with their own real life situations and environments, and thus it further enhance their learning on what is being taught. The oral questions in the assessment book also help in encouraging and challenging children for further explanation of their new ideas, from which they could develop new schemas grounded on previous mathematical knowledge.

## Critique of Textbook:

The integrity between various areas and mathematics is also evident in the textbook. It is full of real life situations to which the children can relate such mathematical concepts. Such opportunities help children to enrich learning in various areas. The 'snake' activity offers the opportunity to discover if the children have understood that a fraction has an equivalent decimal presentation. Other exercises also help students to draw their town with various routes they might use to walk from home to school during a social studies lesson. Children are then invited to calculate the distance traveled. Since Mathematics is not a standalone subject, teachers need to pre-plan such activities so that from such connections, more opportunities for the enrichment of learning are provided.

## Compare and contrast Assessment book with textbook:

Both books make a smooth connection with other areas so that Mathematical areas are not seen isolated from others. Through the various problem solving exercises found on both books but mainly in the textbook which involve investigation, prediction and measurement exercises, children are enabled to build such processes and incorporate them with science. Both books connect really well with other disciplines, but both books could have provided exercises with further investigation and exploration of such mathematical concepts and also provide more opportunities where children are asked to explain their strategies when estimating lengths

## Communication / Representations

## Definition:

Communication is an essential tool for children to understand mathematical concepts and for problem-solving. Communication consists of enriching the mathematical vocabulary so that the children will find it easier to communicate with each other in order to clarify questions, discuss a solution or discuss a problem. Communication should not only be between the teacher and the student, but also between the children themselves.

Representation assists the children to develop and communicate their thinking skills and represent their own solutions in various ways. Models and pictures aids the children to understand and communicate better the results.

## Critique if the Assessment Book:

The 'Oral questions' (numbers 1-4) allows communication to discuss the answer for each questions. Children can share their ideas of how they worked the question out, to the rest of the class. However, since they are 'oral questions' the children do not have the opportunity to represent their working or diagrams that they used to get the answer. Hence, it lack representation.

Page 2 of the Assessment book, involves written exercises. Here, the children do not have the opportunity to discuss the answers as a group since they are required to write down their answers in the space provided.

## Critique of the text book:

The textbook offers quite engaging illustrations that will help the children to understand the measurement concept well. Some of the exercises, for instance page 3 and 4 could be worked out in pairs or groups. This will also allow sharing of ideas through communication. Here, they are required to find objects, estimate and measure them.

The activity found on page 5 where the children has to see where they can run from school so as to reach 1 Km has quite an engaging picture which the children can follow, even though the question is a bit challenging for LOW ABILITY STUDENTS .

The story sums found on page 7 and 8 allow the children to draw diagrams or write information in their own way so as to find the answer. The answers can then be discussed as whole group.

## Compare and Contrast the Assessment Book with the Textbook.

Both the Assessment book and the Text book offer the opportunity to the children to discuss answers as a whole class, if they are used carefully by the teacher. However the Assessment books lacks a bit Communication since it only has 4 questions to be discussed.

Apart of this, both books lack using models to represent their answer.