asset tm 1 and basic number competency maths: class 3€¦ · · 2015-07-274 how do we handle...

Question

Options

A

B

C

D

11 Why was the question asked?

22 What did students answer?

33 Learnings

44 How do we handle this?

Educational INITIATIVES

3 children wrote 3 different number sentences on the board.

All three sentences are correct.

Only Tina and Ali's sentences are correct.

Only Ali's sentence is correct.

Only Tina's sentence is correct.

The question was asked to test whether students are able to understand the meaning and significance of the “equal to” sign - Do they understand just the traditional meaning of 'announcing a result' or do they understand its actual 'relational' meaning of saying that two expressions have the same numerical value? Research shows that a proper understanding of the “equal to” sign is important as students move to higher classes and learn algebra.

Only 21% of the students, at the most, are able to actually understand the meaning of the equal to sign and see that all the three

equations given in the question are correct.Possible reason for choosing B: Students choosing this option

probably feel that “5=2+3” is not a correct number sentence, because they are used to seeing the expression on the left side and its answer on the

right side. As the order is reversed in this case, they conclude that the number sentence cannot be “correct”.

Possible reason for choosing C: Very few students chose this option- so they might just have guessed. Another possibility is that they saw the same number on both sides of

the equal to sign, and eliminated the other two equations as incorrect by seeing different numbers on either side of the “equal to” sign.

Possible reason for choosing D: Almost 50% of students have chosen D which seems to confirm the view that students are going by the 'correct representation' of an equation rather than actually understanding

what an equation (or “equal to” sign) means.

The “equal to” sign is treated by most students as something giving a result rather than as a sign to indicate that the expressions to its left and right are the same. There are other questions like 5 + 3 = ____+ 4 in similar tests. The response data on these questions show that many students answer 8, and some answer 3. Both the above answers indicate the students' inadequate understanding of the “equal to” sign. Research shows that students having problems with understanding of the “equal to” sign also have problems with understanding algebraic expressions and solving algebraic equations, in later classes. Therefore developing an understanding of the “equal to” sign and working with numerical equations is very important from this stage itself.

Problem - Exposure to just the traditional meaning of the sign: Generally, students get exposure to the equal to sign only through the equations like 2 + 3 = 5. No other forms are really introduced to them, in the initial stages. And by the time they are introduced to other forms of usage of the 'equal to' sign, they might have already internalised just the traditional meaning of it deeply. Hence, it becomes difficult for them to appreciate other meanings of the equal to sign.

Possible solution: A good thing would be to start doing balancing activities with numbers - activities demonstrating equations of all types like 5 = 5, 3 + 4 = 7, 3 + 4 = 2 + 5; demonstrating the equality and asking to find the missing number to equate/balance.

Based on this intuitive understanding of the equality, students can be introduced to equating numerical expressions using the equal to sign. This would help students relate the sign to its actual meaning rather than only to its traditional meaning.

Useful resources:

ASSETASSETTM

http://psych.wisc.edu/alibali/files/Knuth_Stephens_McNeil_Alibali_JRME_2006.pdfThis is a research paper describing the types of mistakes that children tend to make in understanding ofthe equal to sign and how it gets reflected in the understanding of algebraic expressions too.

John A. Van de Walle, Elementary and Middle School Mathematics: Teaching Developmentally, 4th ed.(New York: Addison Wesley Longman, 2001), pp. 396-399.These pages in the book describe a few balancing activities to learn number sentences and the meaning of the equal to sign.

Websites: Books:

Ram wrote: 5 = 2 + 3

Tina wrote: 1 + 4 = 5

Ali wrote: 5 = 5Which of the sentencesare correct?

Maths: Class 3

Only 20.9% answered correctlyP

Number sense, related conceptsand basic number competency

No. of students

12022

A. 20.9%

B. 15.2%B. 15.2%

C. 7.6%C. 7.6%

D. 49.8%

P

For any clarifications on this teacher sheet, or to share your feedback, write to us at [email protected], using the feedback format provided to your school. M3-0608-01

Question

Options

A

B

C

D

Measurement, Data Interpretation

A. 12

.8%

B. 36.6%

C. 14.6%

D. 30.1%



33 Learnings



Which of these ribbons is the LONGEST?

Ribbon 1

Ribbon 2

Ribbon 3

All ribbon are of the same length.

The question tries to capture whether students of class 3 have a deeper understanding of what length is. Do they know what the “length” of a curved line means, and the importance of “straightening out” while comparing or measuring lengths of different objects?

Only about 37% of the students have answered correctly.30% have chosen Option D – All ribbons are of the same length!

Possible reason for choosing A: Since very few students chose this option, many of these students probably just guessed. Some of these

students may also believe that Ribbon 1 being thinner than Ribbon 3 and straighter than Ribbon 2, was most qualified to be the answer.

Possible reason for choosing C: Again, since few students chose this option, guesswork is probably involved. Also, some of these students may be confused

between ‘longest’ and ‘most wide’ and chose the widest of the 3 ribbons.

Possible reason for choosing D: About 30% of the students have chosen D as their answer. These students seem to believe that since all the ribbons have the same starting and ending points, the

ribbons must be of the same length.

Most students seem to understand what length is, as long as straight objects are involved. On being faced with a situation where a curved object is involved, they don’t seem to understand that its length is not just the distance between its endpoints, when it is in any “formation”. They need to understand that the length of such a curved object is actually the length of the path along the object. They need to learn to appreciate that the shortest distance between two points is a straight line, and any other “path” between the 2 endpoints has to be longer. While this may seem obvious to adults, and intuitive to some children, it is still necessary to help many children to appreciate this.

• Help children to understand that the length of an object depends on how “straight” it is. For example, ask children how they would stand if they want to check whether they are taller/shorter than their friends.

• Ask them to arrange the different objects in the figure in the order of their lengths (Figure 1).• Help them to appreciate the “length along the path” by asking them questions like“If an ant were to walk along the length of the object, in which case does it

have to walk the most?”• Students can also take a string and place it along the length of the object. They can cut these lengths of strings and place them side by side to appreciate the

differences in the lengths.Once students are clear about the concept of length through the above activities, they can be introduced to measurement activities involving a ruler and standard units like centimetres and metres.

Useful resources:

ASSETASSETTM

John A. Van de Walle, Elementary and Middle School Mathematics: TeachingDevelopmentally, 4th ed. (New York: Addison Wesley Longman, 2001), pp. 279-282.These pages in the book describe ways of measuring – starting from informal units to standard formalunits and also estimation etc. The activities used are based on actual understanding of measurement.

http://books.google.co.in/books?id=ZDO8qK6jyqUC&pg=PA127&lpg=PA127&dq=children+understanding+length+of+a+curved+line&source=web&ots=upMHJha9Uh&sig=CkX0TkQF49DiNaXdl6EHhyHORXw&hl=en#PPA128,M1This link talks about the measurement of straight and curved lines within the same two end points.

Websites: Books:

Ribbon 1

Ribbon 2

Ribbon 3

Line 1

Line 2

Line 3

Line 4

Figure 1

Maths: Class 3

Only 36.6% answered correctlyP

No. of students

12022

A. 12.8%A. 12.8%

B. 36.6%

C. 14.6%C. 14.6%

D. 30.1%

P


11Why was the question asked?


33 Learnings



Basic Shapes, Geometry and Visual estimation

Question

Options

Which of these is a triangle?

This question was framed to check if students could recognize a basic shape like a triangle, when presented in a somewhat different context from the “standard” presentation. This level of attainment is expected at this level.

Around 41% of the students gave the correct answer (D). A and B were the most common wrong answers- with around 28% choosing A, and a similar number choosing B.

Possible reason for choosing A: Students selecting this option might be looking at the

two slanting sides pointing upwards and concluding that it is a triangle.

Possible reason for choosing B: Students are probably confused between a cone and a triangle and

feel that it can become a triangle by turning it “upside down”.

Possible reason for choosing C: Since this shape has two slanting sides, a sleeping line at the base, and “equal sides”, it might be

“looking like a triangle”, to some children.

Research done by Dutch educators Pierre van Hiele and Dina van Hiele-Geldof has shown that there is a certain pattern in the development of geometric thought. A certain level of thought has to be mastered before the next level can be developed.

Level 0 (Visualization), begins with thinking about what shapes “look like”, and making a mental model of that shape. It leads to a classification or grouping of shapes that “seem alike”. This allows the person to progress to Level 1 (Analysis), where he can start thinking now in terms of groups of shapes rather than individual shapes. This leads to a progression towards thinking in terms of properties of shapes (even if it is only intuitively).

These are the two basic levels which if crossed by the child will allow her to think in a more organized way about the properties of the shapes, deduce relationships among properties of different shapes and then proceed with advanced geometry. One interesting aspect of this research is that the kind of learning activities children are exposed to is far more important than the age of the child. Experiences of many educators around the world have validated these findings. (Check links in the Useful resources section.)

However it seems that most of the students are stuck at the “visualization” phase. They are unable to disregard irrelevant attributes of the shape like size and orientation and hence fail to analyse those shapes in terms of their properties.

! Try and give cardboard cut-outs of various shapes like circles, squares, triangles etc. to children, and let them feel those shapes first.! Then give different sizes of a single shape. So for example, give triangles of different sizes or squares of different sizes and let them play with them and feel them.! Try and get different types of triangles like scalene, isosceles, equilateral, right-angled, obtuse-angled, acute-angled etc (we don't have to use these names). Ask students to find out similarities among those

different triangles. Then give a shape that is not really a triangle and ask them to check if it's a triangle.! Conduct periodic assessments, using questions such as the above to find out at what level their thinking is, so that you can tailor instruction to the right level.

Useful resources:Websites:

http://www.learner.org/channel/courses/learningmath/geometry/pdfs/session9/vand.pdfOnline version of the chapter in Van de Walle's book mentioned above.

A CB D

Books:

John A. Van de Walle, Elementary and Middle School Mathematics: TeachingDevelopmentally, 4th ed. (New York: Addison Wesley Longman, 2001), pp. 306-11.

http://www.learner.org/channel/courses/learningmath/geometry/session10/part_b/indexk2.htmlA good demonstration of the different levels of progression of geometric thought.

P Only 41.3% answered correctly


Maths: Class 4

A. 10.6%A. 10.6%

B. 19.2%B. 19.2%

C. 27.8%

D. 41.3%No. of students

3811

P



33 Learnings



Question

Options

The length of this pencil is about__________.

Children learn about measuring lengths using a scale right from class 3. However, while they get a lot of practice on converting metres to kilometres or even rarely-used units like decametres, we were curious to know whether they are able to practically measure lengths - an important real-life skill.

Though this was originally expected to be an 'easy question', we find that the majority of students select a wrong answer C. 6 cm. When

asked the reason for this answer, two explanations are mentioned:• that the pencil is 'ending' at that point

• that if we count off the markings from the starting point to the pencil's tip, we get 6.

Some students who choose the right answer (B. 5 cm) say that the pencil should start from 0, and if we shift the pencil to its left to start from 0, the pencil

will end at 5 cm and hence the length of the pencil is 5 cm. Some others count off the markings from the left end of the pencil to the right end, but “without counting

the starting point.”(This suggests that even among the children getting the correct answer, the underlying

concept is not completely clear.)Some students choose the option A; when asked why, some of them said that the eraser or sharpened point

should not be counted as part of the pencil! (Even if the eraser is not measured, the length is still closest to 5 cm.)

It appears that students are learning about measurement mainly by reading about it in a textbook and without much exposure to actually measuring! Most textbooks illustrate measurement by showing a line being measured by starting from the zero mark (though we have found at least one textbook discussing how a broken scale can be used for measurement!) So students seem to have learnt the strategy - 'look where the pencil ends to determine the answer'. This strategy fails in questions like this.

! The most important thing is to ensure that students spend at least 1 or 2 periods, measuring objects between 1 cm and 30 cm in length. This work can be done in groups, with a set of objects that can move from group to group.! Any kind of project work where children have to measure, cut out cardboard or paper or sticks and use that to make something helps reinforcing the skill of practical measurement.! An interesting activity to strengthen this concept it to get students to compare the lengths of different coloured sticks using a wooden bar (without markings or a zero mark like a scale).! ASSET questions have revealed that the current misconception is a very strong and prevalent one. Questions like these need to be asked in different forms until the underlying concept is understood by all children!

Useful resources:Websites: http://www.aces.mq.edu.au/downloads/crimse/Bragg_P,Outhred_L(2000).pdf

An interesting paper that discusses the finding of a study examining the difficulties students face while learning linear measurement.

A

B

C

D

Only 10.6% answered correctly4 cm

5 cm

6 cm

7 cm

COMMON STRATEGY:The length is the reading at the right - most point

Talks with students reveal that many of them do not completely understand what it means to represent a length with a number. When asked why the length of the given pencil is 6 cm, they explain that counting from 1 to 6, they have counted '6 cm'. In essence, they are counting POINTS instead of actual 1 cm spans. If a 7 cm pencil is placed between the 8 and 15 cm marks, they explain that there are 8 centimetres for the same reason! This misconception is common in class 4, but exists in class 6 also. When students are asked to 'show what is 2cm', they point out the 2cm mark - clearly, some students think that 2cm is the 2cm mark not appreciating that it represents a distance or gap!!

Many children see this and not this as representing 2cm

http://www.aces.mq.edu.au/downloads/crimse/Outhred_L,McPhail_D(2000).pdfAn interesting paper that talks about a framework for teaching early measurement.

Maths: Class 4

P

No. of students5017

B. 10.6%

C. 79.1%

A. 4

.4%

A. 4

.4%

D. 2

.9%

D. 2

.9% P

Measurement, Data Interpretation




33 Learnings



Mensuration - Area and Perimeter, Volume and Surface Area

Question

Options

A square of perimeter 20 cm is cut off from a triangularpiece of paper of perimeter 80 cm as shown.

Children seem to 'learn' concepts like area and perimeter without understanding them well, simply as application of formulae. Yet these are simple and important ideas and a question like this clearly distinguishes between those who have understood perimeter and others who have not understood it.

Only around 13% class 5 students answer this question correctly (C). The most common wrong answer is A - about 57% students

chose that.

Possible reason for A: Students conclude that if a square piece of

perimeter 20 cm is cut off from the triangular piece of perimeter 80 cm, the perimeter of the

cut piece must be 80 - 20 = 60 cm! The answer probably seems 'intuitively' correct.

Possible reason for choosing B: Since the perimeter of the square piece is 20 cm, each side will be 5 cm long. Now,

only 3 sides are left out when the piece is cut and so, they subtract 5+5+5=15 cm from the perimeter of the original figure to get 65 cm.

Possible reason for choosing D: These students are most probably guessing!

choosing

There are many interesting learnings here. Most students who give the wrong answer are able to define perimeter correctly. However they lack the confidence to try and think through or apply their own definition to the problem given. Thus, the misconceptions here seem to be:Confusion between area and perimeter: They seem to know the definitions of area and perimeter. But they do not seem to be able to explain them clearly for a shape like the one above! Maybe since the two concepts are taught in conjunction with one another, they extend the more intuitive concept of area to perimeter. (though they answer many questions about area also wrongly!)"If something is removed, all associated quantities can only decrease": They strongly stick to the fact that when a piece is removed, values of associated quantities must decrease. So whether it is area or perimeter, it will decrease. The belief is so strong that even if we make them work step by step, finding out length of each side of the square first and then find out the perimeter, in this context, it just seems illogical to them that the perimeter can increase.

• Always use cardboard cut-outs or at least paper shapes for teaching area and perimeter.• Make use of questions (like the above) that clearly test for conceptual understanding.• For the above problem, encourage them to reason out giving a real-life context: eg. the triangle is a field that has to be fenced or a deer is walking along its

boundary! Students reach the correct answer and sometimes understand the concept of perimeter in a flash!• Allow students to experiment with shapes - they have to cut out different shapes or join pieces and estimate area and perimeter in different cases to answer the

quesiton: "What happens to the perimeter when a piece is cut out from a shape?"• Finally, ask some questions without using the term perimeter and saying “length of the boundary”. Use of terms sometimes makes students suspend thinking and fall back on 'formulae'.

Useful resources:Websites: http://my.nctm.org/eresources/article_summary.asp?from=B&uri=MTMS1999-10-87a

A good article that talks about how van Hiele's levels of progression of geometric thought can be used to teach area and perimeter.

A

C

Only 12.7% answered correctly

What is the perimeter of the remaining piece?

60 cm B 65 cm

90 cm D 100 cm

Triangular piece After a square piece is cut off

Perimeter = 20 cm

5 cm

5 cm5 cm

Perimeter = 60 cm

Perimeter = 65 cm

Maths: Class 5

P

B. 14.8%B. 14.8%

C. 12.7%

D. 9.5%D. 9.5%

No. of students4841

P

A. 56.6%




33 Learnings



Arithmetic Operations, Order of operations, Properties

Question

Options

Ekta and Zarina each drank a glass of milk. But Ekta

said that Zarina drank less milk than her. Can this be true?

The question tries to test whether students understand that all fractional numbers are relative numbers, and not absolute ones i.e. they are always talked about in relation to a 'whole' or a 'collection'. When we say ‘ ’, it is always ‘ of some quantity', and therefore while comparing fractions, the most important thing is to check whether the 'whole' or the 'collection' is the same or not.

About 40% of the students have answered the question correctly and about 38% of the students have chosen the wrong option D.

Possible reason for choosing Option B: Very few students have chosen this option, indicating that they have probably guessed the

answer. A few students might have made a careless mistake though.

Possible reason for choosing Option C: Very few students have chosen this option, indicating that they have probably guessed the answer. It is possible

that most of these students didn't understand what was being asked.

Possible reason for choosing Option D: About 38% of the students chose D as their answer. These students don't see that “ ” exists in relation to some whole quantity, instead

thinking that is just another number like any whole number.

It is seen from the data (on this question as well as on some other questions) that students do have problems with understanding the meaning of fractions. These problems could be because of the shift from whole numbers to fractions being used to quantities like 2, 6, 10, 25 that mean something in themselves, they now need to appreciate fractions that can be understood only as a part of a whole or a collection. While it is true that , or any other fraction when considered as a decimal number, would have a meaning of its own like a whole number, that part of the concept is irrelevant in the context of this question and at this class level. Given that in Class 5, students are introduced to comparison of fractions, equivalence of fractions, and other more advanced topics in fractions, it is important that the basic grounding be developed by the end of Class 4 at least

• It is important to help students understand the meaning of fractions by laying emphasis on terms like ‘whole’, ‘collection’, and most importantly, ‘dividing the whole into equal parts’. Students should be able to explain what fractions are and should be able to work with them using these terms. The introduction of the notations of fractions and other terms involving the same like numerator, denominator should come only after this basic understanding is clear to students.

• Exposing students to different kinds of problems related to a basic understanding of fractions –i. direct questions like ‘What part of the figure below is shaded?’ ii. extensions of the same question like “Ekta and Zarina each drank half a glass of milk. Is it possible that one of them drank more than the other?” iii. if half of a smaller circle and half of a bigger circle are shaded, are both the same?

Such questions give students an opportunity to think in terms of ‘the whole’ and realise its importance. We can frame similar questions involving the concept of ‘part(s) of a collection’ too.

Useful resources:Websites: http://ncrtl.msu.edu/http/craftp/html/pdf/cp902.pdf

This research paper identifies the problems children face with understanding of fractionsand highlights many misconceptions children have regarding fractions.

A

B

C

D

Yes, it can be true - if Zarina's glass was smaller.

Yes, it can be true - if Zarina's glass was bigger.

Yes, it can be true - even if both glasses are exactly the same.

No, both of them MUST have had the same quantity.

12


Books: John A. Van de Walle, Elementary and Middle School Mathematics: Teaching Developmentally,4th ed. (New York: Addison Wesley Longman, 2001), pp. 218-220.These pages in the book describe how to develop conceptual understanding of fractions using activities.

12

12

12

12

12

Maths: Class 5

P

PA. 39.4%

B. 10.7%B. 10.7%C. 10.1%C. 10.1%

D. 38.1%No. of students

7754




33 Learnings



Arithmetic Operations, Order of operations, Properties

Question

Options

What is value of the missing digit in the addition below?( stands for the same digit both times)

Students, by Class 3, start using the understanding of place value in the standard procedure of addition. This question tries to test if students have the proper understanding of place value and are able to apply it in the given non-standard variation of an addition problem.

Only 42% answered D correctly. B was the most common wrong answer, chosen by 44%.

Possible reason for choosing A: Students selecting this option are most probably just guessing.

Possible reason for choosing B: Students selecting this option are probably just looking at the sum of the digits at the units place without

considering the sum of the digits at the tens place. They probably think that since the sum is 9 and one of the digits is 3, the other 2 will also be 3 (3 +

3 + 3 = 9).

Possible reason for choosing C: Students are probably adding the two digits in the place of the stars, to get 6 (3 + 3 = 6), instead of trying to find the value of the individual

digit. Or maybe, they just see 3 as one of the digits and 9 as the resulting digit, and think that 6 should be added to 3 to get 9.

Following an algorithm without actually understanding how it works : It seems that students have learned a procedure to perform the addition of 2-digit numbers. They seem to be quite competent in this - adding vertically and carrying over to the digit on the left. And this procedural understanding is so deeply rooted that given any such problem, they blindly apply it. However, they seem to be missing the understanding behind carrying over a digit.

In this problem, they aren't realising that the digits at the tens place are adding up to give 14, one less than 15 and so the extra ten is coming by grouping the ones. Hence the digits at the ones place should add up to 19 and not 9. If they understood this, they would be able to answer the problem correctly and verify it too.

• The concept of place value can be taught using physical objects, like base 10 blocks. This has been found to be more effective for many students rather than just explaining on pen and paper. (Find more details in Useful Resources.)

• Teaching addition by grouping and regrouping objects can form a good base for the place value concept. For example, 23 + 38 can be taught by first breaking the two numbers into tens and ones, 20 + 3 and 30 + 8, then grouping the tens and ones, 20 + 30 = 50 and 3 + 8 = 11, again breaking and regrouping, 50 + 10 = 60 and 1, which when added give 61. Such types of exposure can lead to a deeper concept of place value and insights into the addition algorithm.

• Allow students to develop the algorithm instead of teaching it to them. For example, in this case, if students are clear with the concept of place value when addition is taught in the context of place value, they would perhaps “see” how this algorithm for vertical addition works, for themselves.

• Instead of asking only standard questions like 'Add 29 + 41' or 'Add 30 + 33 + 69', expose students to various different kinds of questions where they actually need to apply their understanding apart from the algorithm. For example: '3* + *3 = 132. What could be the value of *?' or '39 + 47 = 80 + ?'

Useful resources:

Websites: http://www.susancanthony.com/Resources/base10ideas.html A good resource that talks about various activities to teach place value and operations using base 10 blocks.

A

B

C

D

1

3

6

8

3 4 7

15 93

Maths: Class 6


P

A. 3

.3%

A. 3

.3%

B. 43.9%

C. 9.4%C. 9.4%

D. 42.5% No. of students 3974

P




33 Learnings



Websites:

Fractions, Decimals, Ratios and Percentages: concepts, applications

Question

Students learn various things about fractions by class 5, like comparison and equivalence of fractions, converting fractions into decimals and vice versa. Therefore by class 6, they are expected to have the understanding of what a fraction is. This question was designed to test if students can identify a fraction represented on a number line.

Only about 14% of class 6 students could answer this question correctly. Around 52% of the students chose the wrong option B

and another 24% chose the wrong option A.

Possible reason for choosing A: Students selecting this option are probably just seeing the point R as the midpoint of two numbers and

relating it to half. They seem to understand that midpoint represents half, but they are ignoring the two numbers of which the marked point is

the midpoint.

Possible reason for choosing B: Students selecting this option are probably just looking at the two numbers between which the point R is marked. They don't seem to

have understood what fractions mean.

Possible reason for choosing C: Students selecting this option might either be guessing or selecting just looking at the two numbers 3 and 4. It is possible that they have learnt the fact that is less

than 1 and since this number is more than 1, they might have decided that it could be since it cannot be .

The data indicates that a majority of the class 6 students are not able to identify a fraction given on a number line. They seem to be having a limited understanding of fractions and the inability to see fractions as a quantity. Students, who by this level ought to have known various things about fractions don't seem to understand the very basic fact of what a fraction is. The fact that they are going for the wrong option or probably indicates that they do not consider a fraction as a single numerical quantity. Many of these students probably see the numerator and the denominator of a fraction, as two different whole numbers.

Clearly most students have failed to make the connection between the informal understanding of fractions, the quantity a fraction represents, and its symbolic representation. The bottom line is that they aren't able to see the number represented as three and a half which when represented as a fraction would be 3 or .

• As seen above, it is important that students make the connection between the quantity and the symbolic representation of a fraction. Teachers should introduce the symbolic representation of fractions only once students are clear about the fraction-quantity relationship. This could probably help to prevent many more misconceptions that they may develop regarding fractions.

• They should understand the meaning of fractions as equal divisions of a whole. Having understood this they should be able to relate fractions to the physical quantities that they represent. For this, physical objects should be used and students should be allowed to manipulate them to represent a given fraction, compare equal and unequal parts and reason out whether they can represent a given fraction or not based on the quantity that they represent.

Useful resources:Books: Teaching Fractions and Ratios for Understanding

An interesting book that talks about content as well as instructional strategies regarding fractions and ratios.http://www.emis.de/proceedings/PME29/PME29RRPapers/PME29Vol2Amato.pdfA good resource that talks about the effect of certain methodology of teaching fractions on developing students' understanding of concept of fractions as numbers.

43

34

34

43

34

43

12

72

What number is R likely to represent on the number line?

Options

A

B

C

D

0 1 2 3 4

R

12

34

43

72P

Maths: Class 6


No of students 3974

A. 23.9%A. 23.9%

B. 52.2%

C. 8.5%C. 8.5%

D. 13.8%

P



C


33 Learnings



Number sense, related conceptsand basic number competency

Question

Options

A

B

D

The concept of place value is one of the most important ones to be learnt and internalised. This question was asked to check whether students are able to split/combine the groups of tens/hundreds to identify the number given. The diagram below represents what children are expected to do in this question:

About 50% of the students tested, answered the question correctly. But about 35% of the students chose option C.

Possible reason for choosing A: Only a few students chose this option. These students probably didn’t understand what is asked for and just added the given

numbers by seeing the addition sign. Some students may have guessed the answer too.Possible reason for choosing B: Again, very few students chose this option. These students

probably saw 23 tens and made a careless mistake of interpreting 19 hundreds as 190 and hence got the answer 420. Some students may have guessed the answer too.

Possible reason for choosing C: About 35% students chose this option. These students may have thought that since hundreds precede tens in a number, 19 should come before 23 and chose 1923 as their answer. A few students may

have made a careless mistake of seeing 23 ones and got the answer as 1923.

Many students seem to have missed out that 23 tens makes 2 hundreds and 3 tens, and hence the place value of some digits is changing and regrouping is required. This fact, emphasizes that the place value concept is taught in much lower classes but students are not able to understand and internalise it as per the expectations, even in higher classes. Techniques or algorithms help in solving some problems but blind learning of just the algorithms without an understanding of the concept, can lead to misconceptions in many other related concepts too.

! Students not having the understanding of place value might not be able to distinguish between the two numbers say, 53 and 35 and may have difficulty in understanding many concepts like arithmetic operations.! Various activities related to grouping and splitting, as given below, asking children to work to establish the equivalence between different types of groupings of the same number, would help a lot in developing a

good and proper understanding of place value. An example of 2 types of groupings of a number is given below:

Useful resources:

Books: Burns, Marilyn, About teaching mathematics, A K-8 resource (pp. 173-182). Sausalito: CA: Math Solutions Publications. This chapter on place value describes the ways to teach the concept and also different ways of assessing the child's understanding of the same.

Chapin, Suzanne and Johnson, Art (2000), Understanding the Math You Teach, Grades K-6 (pp. 17-18). Sausalito: CA: Math Solutions Publications. These pages in the book describe grouping activities to understand different combinations of numbers.

42

420

1923

2130


23 tens + 19 hundreds is equal to 23 tens 20 tens = 20 x 10 = 200

3 tens = 3 x 10 = 30

(2 hundreds)200 + 30 = 230

19hundreds

10 hundred = 10 x 100 = 1000

9 hundreds = 9 x 100 = 900 1000 + 900 = 1900

And add 1900 and 230 to get 2130.

Activity 1: Ask students to -a. split 160 boys into 1 group of hundred boys and rest of ten boys eachb. split the 160 boys into groups of ten boys each

Activity 2: Ask students to -a. split 1670 boys into some groups of ten boys and some groups of hundred boysb. different students can come up with different types of groupings – two different ways are 16 hundreds and 7 tens, 15 hundreds and 170 tens

The above activities can be done using place value blocks or any other articles.

Maths: Class 7

P

D. 50.8%PC. 34.8%

A. 7

.7%

A. 7

.7%

B. 6.5

%B.

6.5%

No of students 5743



C


33 Learnings



Websites:

Factors, Multiples, Primes and related concepts

Question

Options

A

B

D

Division is a concept that students are introduced to, around class 2, and they are able to divide a number by another correctly by class 4. This question was asked to check whether the concept of repeated subtraction (that if the divisor is 4, we can take out 4 repeatedly from the dividend and what would remain is any of the 4 whole numbers 0, 1, 2, 3) is understood by the students.

Only about 16% of the students answered the question correctly. And about 76% of the students have chosen D as their correct answer.

Possible reason for choosing B: Just about 4% of the students chose B as their correct answer. These students probably just guessed the answer

as 4 by seeing 4 in the question.

Possible reason for choosing C: Again, just about 3% of the students chose C as their correct answer. These students probably just guessed the answer without

understanding what was asked.

Possible reason for choosing D: It is interesting to see that almost 76% of the students chose option D as their answer. These students probably have the misconception that the

remainder depends on the dividend and there is no upper limit on the possible remainder based on the divisor.

Many children keep doing different types of problems involving division, but don't seem to understand that the maximum remainder one can get in a division is fixed by the divisor, no matter what the dividend is. This could be because children very often deal with sums like 'find the quotient and the remainder when 3764 is divided by 4', and hence are completely used to actually dividing and finding out the remainder using the algorithm they are taught, rather than thinking of what could be the possible remainder, in the given division.A part of the problem also lies in the teaching methodology. Normally, while teaching division, we introduce the terms involved, and the procedure/algorithm to find out the quotient and remainder. Generally students are just told that division is nothing but repeated subtraction and they are asked to solve problems using algorithms of both of them. They are also probably asked to check whether they are getting the same remainder using both the methods. But, this is probably not enough for them to understand why the remainder should be the same in both the methods.

Giving algorithms is different from making the children understand why it happens. The conditions on the remainder can probably be better understood through the repeated subtraction procedure.While introducing division initially using small numbers (eg. sums like 9 ÷ 2), a student can be given 9 pencils, and can be asked to keep giving 2 pencils to each of his friends and see how many are left out. Once the student answers that 1 pencil is left out, he can be asked questions like:

! How many more pencils would he need to be able to give 2 pencils to another friend?! Is it possible that he will remain with 3 pencils if he keeps distributing 2 pencils whenever he can to his friends? Why?

Such questions help the students realise the importance of remainder instead of blindly ‘finding them out using the procedures they have learnt’.

Useful resources:Books: John A. Van de Walle, Elementary and Middle School Mathematics: Teaching Developmentally, 4th ed.

(New York: Addison Wesley Longman, 2001), pp. 120-127, pp.143-147.These pages in the book describe the division problems along with its meaning and also activities.

http://www.learner.org/channel/courses/learningmath/number/session4/part_a/division.htmlApart from the description on the web page, there are materials available for download and a video (that may be viewed online using a free login) explaining how division concept can be explained to students.

3

4

5

We can't say, it depends on how large the number is.

Maths: Class 7


B. 3.7%

C. 3.4%

Amrita is dividing a very large number by 4.What is the maximum remainder that she can get?

P

D. 76.1%

A. 16.1%

No of students 9725

P



C

33 Learnings



Websites:

Fractions, Decimals, Ratios and Percentages: concepts, applications Maths: Class 8

Question

Which of these is closest to 0.015?

Options

A

B

D

Students start learning about the concept of decimals from class 5 onwards. They learn various concepts related to decimals by class 7 and so are expected to have a good grasp of decimal numbers by then. Understanding of place value is integral to understanding decimal numbers. This question was designed to test if students are able to apply their decimal number sense to find out the number closest to a given number.

Only 18% of the students who attempted this question got it correct. 70% of the students selected the wrong option B.

Possible reason for choosing A: Students selecting this option most probably don’t understand the question and are just guessing.

Possible reason for choosing B: It seems students are just taking the numbers excluding the zeroes and the decimal point and comparing them. So,

they might be considering 0.015, 0.010, 0.016, 0.0115 and 0.0145 as 15, 10, 16, 115 and 145 respectively. And since 16 is the closest to 15 amongst them, they might be

selecting it.

Possible reason for choosing C: Students selecting this option most probably don’t understand the question and are just guessing.

As students start developing decimal number sense, it is important that they realise that decimal numbers are obtained on division. So, a tenth, 0.1, is obtained by dividing 1 into 10 equal parts and a hundredth, 0.01, is obtained by dividing 1 into 100 equal parts or dividing 0.1 into 10 equal parts. They should be able to make the decimal-fraction linkage and understand the basic decimal place value system.

However, the most common wrong answer selected by students indicates that they are probably still going by the whole number logic, and have not internalized the decimal place value system. The meaning and significance of a decimal point and the value of digits at places after the decimal point are not clear to them. They probably think that a decimal is just another way to write certain numbers but when operations have to be performed, it is only the digits that play a role.

! Use a number line to represent decimal numbers. Help students to understand the difference between whole numbers, which cannot be found between two consecutive whole numbers, and decimals which can be found between any two consecutive numbers, whether they are whole numbers or decimals.

! Try and use physical objects or models like the base-10 blocks to teach the concept of place value. Students should be asked to make decimal numbers using those objects.

! Incorporate questions like these that go beyond testing the mechanical understanding students have regarding operations on decimals, and test the basic concepts underlying decimals.

Useful resources:http://extranet.edfac.unimelb.edu.au/DSME/decimals/SLIMversion/tests/miscon.shtml#longerA very good website that talks about various kinds of misconceptions students have about decimal numbers.

0.010

0.016

0.0115

0.0145

http://www.education.vic.gov.au/studentlearning/teachingresources/maths/mathscontinuum/number/N40001P.htmA good resource that talks about certain misconceptions students have about decimal numbers and ways to handle it.

P



No of students 6532

A. 7

.7%

A. 7

.7%

B. 69.2%

C. 4.8%C. 4.8%

D. 17.8%

P



C


33 Learnings



Websites:

Pre-algebra and Algebra skills

Question

The weight of a bottle of jam when it is FULL and when it is HALF full is shown below.

Options

A

B

D

Students are familiar with formulating and solving equations, given a word problem. This question was designed to test if students can do the same, given a visual problem.

Only 26% students were able to answer the question correctly. 50% of the students selected the wrong option D and another 24% selected one of the other wrong options.

Possible reason for choosing A: Students selecting this option most probably don't understand the question. These might be

trying to match some words in the question to the options and seeing 'half' in both, might be selecting this.

Possible reason for choosing B: Students selecting this option are probably thinking that the difference between the two weights shown

will be the weight of the empty bottle, but making a careless mistake thereafter, computing 800 – 550 as 250.

Possible reason for choosing D: Students selecting this option are also calculating the difference between the two weights shown to arrive at their answer,

except that they are doing the subtraction 800 – 550 correctly.

As the data indicate, a majority of the students are perhaps not able to form the required equation and solve it.

They are expected to do the following:

! interpret the problem: interpret the visuals correctly and see that each balance in the visual represents an equation:Balance 1: Weight of the empty bottle + Weight of full jam = 800 g Balance 2: Weight of the empty bottle + Weight of half the jam = 550 g

! break the problem into smaller steps: see that the weight of the empty bottle is one variable, and the weight of the jam is another variable and then represent them as an equation describing their total weight

! solve the mathematical equations: having framed the equations properly, they should be able to solve them simultaneously to find out the value of the desired quantity, weight of the empty bottle in this case

The failure of the majority of students to solve this problem indicates that they have a difficulty in identifying the correct variables and formulating the equation. The problem being presented as a visual might have also posed a further challenge for some of them.

A systematic, patient and step-wise approach would help students to handle such problems, word or visual, easily.

An initial step could be to allow them to frame equations to show simple relationships between a variable and a constant like 'A number is 5 more than 8. What is that number?' Later, the same thing can be done for 1 variable like "Section A has 5 boys more than that in Section B." and then for 2 variables like “A weighing scale showing weight of 2 boys standing on it. And we can ask students to translate the information in the problem into mathematical form, an equation representing the weights of the two boys.” Thereafter, we can move on to complex relationships like the one shown in this question.

While word problems are important and should be used to develop a good mathematical sense, even visuals like these are necessary. It is important that students come across a variety of such problems, word and visual, and get used to formulating equations, as much as solving them.

Useful resources:Books: http://books.google.co.in/books?id=MNxOjzWYlPsC&pg=PA1&lpg=PA1&dq=teaching+and+learning+algebra&sou

rce=web&ots=vYgHG6s9N1&sig=T-olzB6HcDC6cRU4Wqq1UaRyRiE&hl=en“Teaching and Learning Algebra” A good book on teaching algebra.

http://www.partnership.mmu.ac.uk/cme/Student_Writings/CDAE/NaomiBartholomew.htmlAn interesting resource that talks about teaching algebra to 10 year olds. It has an interesting section on Forming Equations.

750g

50g

500g

50g

kg

350 g

300 g

250 g

12

What is the weight of the empty bottle?

P

Maths: Class 8


No of students 3093

A. 12.1%A. 12.1%

B. 10.5%B. 10.5%

C. 25.8%

D. 50.1%

P




33 Learnings

44 How do we handle this


Integers and Rational numbers, Powers and bases

Question

Options

Which of the following statements is true?

Students start learning the laws of powers from class 7. By class 9, it is expected that students know these laws thoroughly and are able to work with them in different ways to find the values of expressions, and compare them. This particular question tries to test whether students are able to apply the known laws to check whether two expressions give the same value.

Only about 26% of the students could get the correct answer given in option D. About 46% of the students chose option C as

their answer.Possible reason for choosing Option A: These students probably

saw the ‘+’ sign between the two terms on the left hand side and also between the two powers on the right hand side and chose this option.

Possible reason for choosing Option B: These students probably applied a similar logic as those who chose option A, for the ‘x’ operation (multiplication).

Possible reason for choosing Option C: About 46% of the students chose this option; these students seem to believe that ‘the power of a sum of two terms is the sum

of the powers of the two terms’, probably relating the distributive law a (m+n) = am + an to exponents wrongly. This appears to be quite a common misconception among children, as even

direct experience with children shows.

4Many of the students don’t seem to have understood the meaning of the power of a number that for example 3 means 3 x 3 x 3 x 3.

Students, by this age are exposed to many laws which they are asked to remember. Most often, students try to ‘remember’ the laws and try to apply these to problems, blindly. This sometimes leads to errors due m n m+nto their inability to recollect what they had memorised. For example, in the context of this question, students choosing option A may have tried to ‘memorise’ the law a x a = a without ‘understanding’ why it

holds and hence, on seeing “similar looking” expressions like in Option C, ended up choosing these blindly.

This brings out the need to ensure that students understand how the laws have been derived and why they hold true. If they have also developed a sense for the relative magnitudes of numbers “added”, “multiplied”, or “raised to power of”, they are less likely to make such mistakes.

It is important to realise that laws are not arbitrary rules without proofs, and to appreciate that they are tools that make working with some problems simple.

One way of handling this is to get students to derive the laws themselves, and get them to generalise through examples, rather than just “telling” them the law and giving practice problems based on the same.

Once the laws are understood, an activity to develop a deeper understanding of powers and their laws, would be to give an expression to the students and ask them to write all possible expressions that give the same value. In this activity, different students will come up with different expressions and every student can validate the expression written by the other and so on. Further to this, students can be exposed to similar problems as this question on which the teacher sheet is based.

Useful resources:Books: John A. Van de Walle, Elementary and Middle School Mathematics: Teaching Developmentally,

4th ed. (New York: Addison Wesley Longman, 2001), pp. 419-422. These pages in the book describe how to develop the understandingof exponents and various ways of working with them with actual understanding rather than just memorising.

B

C

D

A40 20 40 + 20

3 + 3 = 3

12 10 12 x103 x 3 = 3

14 14 143 + 4 = (3 + 4)

20 2 2 11 3 x 3 = (3 )

Websites: http://books.google.com/books?id=lLeo33Tvz9UC&pg=PA361&dq=teaching+exponents+and+laws+of+exponents&as_brr=3&rview=1&sig=N3fkcMQwoVRqqwX3WfYktekp390#PPA361,M1

Maths: Class 9

Only 26% answered correctly

P

A. 15.7%A. 15.7%

B. 11.5%B. 11.5%

C. 46.0%

D. 26.0%

No. of students 2785

P


Why was the question asked?


33 Learnings

44 How do we handle this


Mensuration - Area and Perimeter, Volume and Surface Area

Question

Options

A gardener can trim a square lawn of side 50 metres in 4 hours. At the same rate, how long will it take him to trim a square lawn of side 25 metres?

Students start learning about the concept of area from class 5 and the concept of proportion from class 6. However, sometimes they learn certain concepts in isolation and are not able to use multiple concepts together to solve a problem. This question was designed to test if students are able to apply the concept of proportion, along with the understanding of area of a square and solve the given problem systematically.

Surprisingly only around 13% of the class 9 students, who ought to have understood the two concepts being tested, were able to

answer this question correctly. Most of the students (around 81%) chose the wrong option B.

Possible reason for choosing B: Students selecting this option are probably just applying the unitary method on the numbers that appear in the

question and thus calculating it as: 50 tends to 4 and so 25 should tend to 2.

Possible reason for choosing C and D: Students selecting these options are most probably just guessing.

Students were expected to understand this problem by breaking it down into its parts, and approach it in a systematic way to solve it. However, a surprisingly high percentage of students chose the wrong answer B. This indicates that while they understand

the application of proportions, they seem to be failing in solving the problem itself. This could be because of either of the following reasons -

Lack of understanding of area relationships: Inadequate understanding of the concept of area could be a reason for students getting this incorrect. They might be able to calculate the area of a square given its side lengths but when asked the question in this way, they perhaps do not realise that when side-length is halved, area is not halved but instead reduces by four times.

Inability to gather all the relevant information required to solve a problem: For a problem like the one given above, what students have to do is just see the factor that is varying in the two different squares, area, and then apply the understanding of proportion. Even if students do not appreciate the relationship between side length and area, they could have solved this problem even by computing the actual areas of the 2 squares in question and performing the division. However, they don't seem to have done this- probably because they are unable to approach the problem systematically and fail to interpret and extract the required information from the question.

! Expose students to problems that allow them to apply more than one concept in a given context. They should understand the systematic way of solving a problem - interpret the problem correctly, understand what exactly the question is asking, rephrasing the question in one's own words, drawing a picture where required, and then solving it taking one factor at a time. The website in the “useful resources” section elaborates this further.

! To help children appreciate area vs side-length relationships, they can be asked to draw a square with side length 50 cm and a square with side length 25 cm. Visually they will start understanding that the area of the smaller square is ¼ that of the bigger square and get an indication of the factor that is varying. They can also be asked to cover these squares with small unit squares, and confirm the factor by which they vary. Actual divisions of calculated areas can also be performed to help internalize this concept, and the discussion could be extended to other plane shapes or even volumes and 3D shapes.

Useful resources:

Websites: http://mathforum.org/dr.math/faq/faq.word.problems.htmlA good resource that talks about how to handle a problem.

B

C

D

A 1 hour

2 hours

2 hours 30 minutes

3 hours

11Maths: Class 9


P

A. 13.1%

B. 81.1%

C. 4

.2%

C. 4

.2%

D. 1.0%

No. of students 2986

P


asset tm 1 and basic number competency maths: class 3€¦ · · 2015-07-274 how do we handle...

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