measurement

1Measurement, Rob Whitehorn and Michelle Renda

Presented by:

Rob Whitehorn, 18119003

Michelle Renda, 17676090

Graduate Diploma of Education (Primary)

EDU4PMA Mathematics for Primary Education

Assessment Task 1 - Formal units of measurement


Measurement

“The measurement strand involves the assignment of a numerical value

to an attribute of an object or event” Jorgensen, 2011, p.275

• “Measurement is interwoven throughout all grades in elementary school mathematics; approximately 10 to 15% of the mathematics curriculum at every grade level deals with measurement. In elementary school, measurement has traditionally been presented as procedures and skills. However, a more careful analysis indicates that measurement is a concept.” The Connecting Mathematics for Elementary

Teachers Project, Purdue University http://www.faculty.pnc.edu/dpratt/cmet/chp10.pdf


http://www.faculty.pnc.edu/dpratt/cmet/chp10.pdf

General Measurement

• “A full understanding of numbers, including rational counting, must precede the development of more formal measurement activities using non-standard then standard units.”

Booker, 2010, p. 491

• “In the early years of school, most measurement experiences are associated with comparisons and informal measurement activities.”

Bobis , 2009, p.154.

• “A model commonly used in planning for teaching of any measurement topic delineates five phases:• 1.identifying the attribute• 2. comparing and ordering• 3 .informal measurement using non-standard units• 4.formal measurement using standard units, and• 5.appication to problem solving contexts.”

Jorgensen, 2011, p.282

(explored further in appendix 1.2 ,p. 29)


The curriculum


Strand: Discipline based learning

Domain: Mathematics

Dimension: Measurement and Geometry

Level: 3

Focus: Using units of Measurement

Formal units of measurement

Research into Action Micro Teaching


Proficiency strands

The proficiency strands describe the actions in which students can engage when learning and using the content.

The proficiencies are:

• Understanding

• Fluency

• Problem solving

• Reasoning.

Fluency-

Fluency includes recalling multiplication facts, using familiar metric units to order and compare objects, identifying and describing outcomes of chance experiments, interpreting maps and communicating positions


To engage successfully in understanding measurement children should be able to……

• use informal units of measurement to order objects based on length and capacity. (e.g. Unmarked cups, paperclips and hand span)

• tell time to the half-hour and explain time duration.

• use the language of distance and direction to move from place to place.

• order shapes and objects, using informal units for a range of measures.

• tell time to the quarter hour


Using units of measurement:

• Measure, order and compare objects using familiar metric units of length, mass and capacity (ACMMG061)

Elaborations

• recognising the importance of using common units of measurement

• recognising and using centimetres and metres, grams and kilograms, and millilitres and litres

• Tell time to the minute and investigate the relationship between units of time (ACMMG062)

Elaborations

• recognising there are 60 minutes in an hour and 60 seconds in a minute

AusVELS Level 3


http://ausvels.vcaa.vic.edu.au/Curriculum/ContentDescription/ACMMG061

http://ausvels.vcaa.vic.edu.au/Curriculum/ContentDescription/ACMMG062

Misconceptions/Difficulties

“Curry, Mitchelmore and Outhred (2006) investigated development of children’s understanding of length, area and volume measurement in grades 1–4. They

found that students often rejected the use of different sized units when measuring area or volume but did not see a problem with using different size

units for length.”

https://www.eduweb.vic.gov.au/edulibrary/public/teachlearn/student/mathscontinuum/readmeaslength.pdf



The theory

Length and other dimensions not altered by position : conservation,

Compare length of two objects, using a third : transitivity

Piaget, 1960 – (further investigation in appendix 1.1 and appendix 1.1a, p. 27,28)

Children not ready for linear measurements before they attain the above.

Hiebert, 1981 – appendix 1.1, p.27

Need to compare, but also to estimate, and predict. Pepperell 2009, p.95. appendix 1.2 and appendix 1.3, p.29-37

Space and Time, children confuse their temporal thinking with their spatial thinking.

Piaget, 1960 - appendix 4.1, p.58

“In school it is usual to introduce informal units first for example, matches or straws or digits (thumb widths) or hand or feet lengths are used to measure length. The principle of laying them and counting them is fundamental. It is soon discovered that feet can be of different sizes, so that different numbers can represent the same length. This gives rise to all sorts or problems, so standard units have been established. (Suggate, 2010 p.177)


Informal units of measurement some difficulties

Length - “In the first year of a three-year longitudinal study of children initially in years 1 to 3, found that over 80% of the children saw no problem with mixing two different-length paper clips to measure a pencil.” Lehrer, 1998 (appendix 1.8b, p.43)

Capacity –Children often believe the amount of liquid has changed when a set amount has been poured from one container to another of a different size. Piaget, 1960 (appendix 3.2a, p.54)

Mass – the larger the size of the object the greater the mass. Weight is not visible like Length.Queensland Government Study Authority, 2005(appendix 2.1, p.49)

Time – is a concept in itself, we cannot see time, hence cannot see time pass.12Measurement, Rob Whitehorn and Michelle Renda

Formalising Units : The Metric System

“Units of measurement have been agreed by mathematicians and scientists in order to make measurement consistent throughout the world. In the UK imperial and metric units are used, but metric units, first introduced in eighteenth-century France, are the system taught in schools. Metric units are in common usage as they are based on powers of 10 and so are easier to work with. Metric units are sometimes referred to as SI units (Système Internationale d’Unités).” Cotton, 2010 ,p.156.

https://www.hope.ac.uk/media/liverpoolhope/contentassets/documents/education/media,25539,en.pdf

http://www.bbc.co.uk/skillswise/factsheet/ma21impe-e2-f-metric-measurements


https://www.hope.ac.uk/media/liverpoolhope/contentassets/documents/education/media,25539,en.pdf

http://www.bbc.co.uk/skillswise/factsheet/ma21impe-e2-f-metric-measurements

The right tool, for the right job

First Stage Choose likely unit needed

Other considerations Choose Instrument

LENGTH Estimate mm, cm, metres, km Is length being measured a straight line distance, do I need to iterate

Ruler, tape measure (dressmakers, or builders), trundle wheel

MASS Estimate Mg, grams, kilograms Resetting apparatus, reading of scales

Kitchen scales, Bathroom scales

CAPACITY Estimate Litres, cm^3 Straight line or other Ruler, tape measure, measuring jug

TIME Estimate Seconds, minutes, hours, days, months, years, ....

The passing of time (a Race, the school year), or the telling of time (it is three o clock)

Stopwatch, analogue clock, digital clock

Children need to first be comfortable with first what they need to measure, what units they need to express the ‘answer’ in, and how they are going to measure ....


Practical Difficulties “A litre is too large for practical purposes in Year 2 or 3, so millilitres might be used, but this means the

numbers become larger than some children can handle with confidence. The Numeracy Strategy suggests using a scale marked in hundreds of millilitres, but this can be difficult to read. Similarly the cubic metre is much too large, so the cubic centimetre (cc or cm^3) might be appropriate.”

Suggate, 2010, p.203

Commonly in the real world use litres and millilitres to express a measurement of the capacity of an object (a car has an engine capacity of x litres , or a kitchen measuring jug has a capacity of y millilitres) ; in the classroom a child’s first introduction to capacity (and volume) is maybe using cm^3, so a further step, is needed to convert. (1m^3 = 1000litres)

“There is a problem about which standard unit to introduce first. The kilogram is the basic SI unit, but this is really too heavy for children to handle safely. If grams are used the numbers tend to be large and may well be out of range which Key Stage 1 children can handle with confidence. Usually 10,20, and 50 gram masses (or weights) are used in the classroom” Suggate, 2010, p.209

The base words of metres, grams and litres are at the core of words of measurement but as we have seen are rarely used in the classroom, so what do the centi, milli, kilo prefixes mean to children ?

http://www.lessontutor.com/lw_decimals2.html15Measurement, Rob Whitehorn and Michelle Renda

http://www.lessontutor.com/lw_decimals2.html

Formal Units – Difficulties : Geometric Length

How long is the pencil? (question posed to US fourth grade children)

75% of the children asked answered incorrectly, giving the answer of either 6 or 8. This was due to tick counting and not understanding the meaning of the hash marks.

Purdue University, p.218 – further explanation in Appendix 1.8a, p.42

Children fail to align the start of the unit of measurement being used i.e. starting the measurement at zero when using a ruler. Lehrer (2003) notes that many children start with one rather than zero on a ruler. Some rulers don't start at 0 !!


Formal Units – Difficulties : Geometric Length

• In the first instance children use centimetres, if they are not introduced to millimetres also then these ‘sub ticks’ may confuse and in turn counted by the child

• A further problem is termed iteration, that is using the same instrument over and over again to measure something that is longer than the instrument being used. Overlapping and leaving gaps are common errors here. Lehrer, 2003 , appendix 1.4, p.38

• That straight line distance is always shorter than actual distance (a walk from one end of the CBD to the next) Battista 2006. appendix 1.10 and 1.10a, p.44,45

• Length is the same no matter the effort required (sitting in the car as opposed to walking to school) Battista 2006. appendix 1.9, p. 44


Capacity is the amount of space available, volume is the amount of space taken up

Language difficulty (we do not use comparison words ‘capaciouser’ in the same way as ‘longer’). Pepperell, 2009, p.98.

We can use cm^3 and litres in different situations

Fullness : “Harrison (1987) raised an interesting observation with young children when he saw that they seemed to be focussing on fullness rather than capacity. He gave an example of a girl who thought that a cup ‘held more’ than a teapot. Harrison suggests that she was focussing on the fact that she had just filled the cup.” appendix 3.4 ,p.56

Formal Units – Difficulties : Capacity


Regarding Mass “some children find it difficult to understand and read the gradations on the dials of a kitchen scales” Suggate, 2010, p.209

Resetting the weight to 0 after use fof weighing scales, re-checking to get consistent weight

When reading weighing scales, children may be exposed to different weighing scales with differing ‘scale’. Kitchen scales going to 1kg, Bathroom to, say, 100kg, both analogue use a 360 degree dial.

Children use Mass and weight interchangeably, may cause confusion later on . “By the time they are 11 years old, children should be aware of the difference between mass and weight” Suggate, 2010, p.209.

Formal Units – Difficulties : Mass


Time does not use base 10.

The different areas of the clock face confuse.

This diagram above is from a national sample of (US) 8-year olds

and how they perceived the time from the clock face shown.

Samantha puts a cake in the oven at this time : 9:20. The cake was taken out of the oven 50 minutes later. At what time did the cake come out of the oven ?’. In this case 16 per cent of the 8 year olds made the expected error of 9:70, involving the treatment of the hours and minutes as separate entities, or as a decimal.”

Ryan 2007. see appendix 4.4 , p. 61

On a clock face the minutes are superimposed on the hours.

Children do not understand the hands rotate and are interconnected.

The Australian Association of Mathematics. see appendix 4.3, p.50

The formal units of seconds, minutes, hours are seen to be independent of one another,

Burny, appendix 4.4 p.62

Language – children use ‘where’ questions, before they use ‘when’ questioning – spatial versus temporal thinking. (Piaget, 1960 - appendix 4.1, p. 58 )

Formal Units – Difficulties : Time


End


References (1)

Battista, M. T. ((2006). Understanding the development of students’ thinking about length. Teaching Children Mathematics, October, 13(3), 140–147.

Bladen E, Wildish K, Cox J, ‘The development of linear measurement in elementary school children’ , George Fox University, US

Bobis J, Mulligan J, Lowrie T, (2004) Mathematics for Children 2nd edition, Pearson Education, Australia.

Booker G, Bond D, Sparrow L, Swan P, ‘Teaching primary mathematics’ 4th edition, Pearson Education, Australia.

Burny E, Martin Valcke M, Desoet A, ‘2011, Clock Reading: an Underestimated Topic in Children with Mathematics Difficulties’ , Ghent University (Belgium)

Cotton T, 2010, Understanding and Teaching Primary Mathematics, Leeds Metropolitan University, Pearson Education, UK.

Hiebert J, National Council of Teachers of Mathematics ‘The arithmetic teacher’, vol 31, No. 7, March 1984 (US)

Hiebert, J. (1981). Cognitive development and learning linear measurement. Journal for Research in Mathematics Education, 12(3), 197–211., (US).

Horvath, J., & Leherer, R. (2000). The design of a case-based hypermedia teaching tool. International Journal of Computers for Mathematical Learning, 5, 115-141., Kluwer Academic Publishers, Netherlands.

Kamii, C., & Clark, F. B. (1997). Measurement of length: the need for a better approach to teaching. School Science and Mathematics, 97, 116-121. (US)

Kilpatrick, J., Martin, W. G., & Schifter, D., A research companion to Principles and Standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics. (US)


References (2)

Lehrer, R., Jenkins, M., & Osana, H. (1998). Longitudinal study of children’s reasoning about space and geometry. In R. Lehrer & D. Chazan, Designing learning environments for developing understanding of geometry and space, pp. 137–167. Mahwah, NJ: Lawrence Erlbaum.

Lehrer, R. (2003). Developing understanding of measurement. In Kilpatrick, J., Martin, W. G., & Schifter, D., A research companion to Principles and Standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.

Jorgensen R & Dole S, 2011, ‘Teaching Mathematics in Primary Schools’, 2nd edition, Allen & Unwin, Sydney, Australia

Pepperell S, Hopkins C, Gifford S, Tallant P (2009), Mathematics in the Primary School 3rd edition, Routledge UK

Piaget J, Inhelder B and Szeminska A (1960), The Child's Conception of Geometry, Routledge .

Suggate J, Davis A, Goulding M (2010), Mathematical knowledge for primary teachers 4th edition, Routledge, UK.

Ryan J, Williams, J , 2007, ‘Childrens mathematics 4-15 : Learning from errors and misconceptions’, McGraw Hill, New York, US


New South Wales Department of Education and Communities

http://www.numeracycontinuum.com/index.php/aspects-of-the-continuum/aspect7

Purdue University , The Connecting Mathematics for Elementary Teachers Project,


Victorian Department of Education and Early Childhood development , 2009, ‘Measuring Length’


National Science Foundation (USA) 2012 , John Smith, Michigan State University

http://www.nsf.gov/discoveries/disc_summ.jsp?cntn_id=123020

Math Coach Corner Blog

http://mathcoachscorner.blogspot.com.au/2012/05/measurement-misconceptions.html

Queensland Government Study Authority, 2005

http://www.qsa.qld.edu.au/downloads/p_10/kla_maths_info_measurement.pdf

BBC Resources, UK

http://www.bbc.co.uk/skillswise/factsheet/ma23capa-l1-f-capacity

National Centre for excellence in the teaching of Mathematics UK

www.ncetm.org.uk/resources/22757

Casanto et al, 2010, ‘Space and Time in the Child’s Mind’, Cognitive Science Society , US.

http://lera.ucsd.edu/papers/space-time-child.pdf

The Australian Association of Mathematics Teachers, Inc. ; Gale, Cengage Learning ; 2008

http://www.thefreelibrary.com/It's+about+time%3A+difficulties+in+developing+time+concepts%3A+Sally...-a017781176624Measurement, Rob Whitehorn and Michelle Renda








http://www.ncetm.org.uk/resources/22757


http://www.thefreelibrary.com/It's+about+time:+difficulties+in+developing+time+concepts:+Sally...-a0177811766

Appendices

Appendix 1 : LengthAppendix 2 : MassAppendix 3 : CapacityAppendix 4 : TimeAppendix 5 : General Measurement


Appendix 1 : Length


APPENDIX 1.1 - LENGTHConservation and Transitivity

“Piaget, Inhelder and Szeminska (1960) identified the reasoning concepts of conservation (for example, understanding that the length of an object is not changed by changing its position) and transitivity (the ability to infer that one object is longer than another by making direct comparisons of each with a third object)”

“Hiebert (1981) notes that it is commonly believed that children are not ready to learn certain linear measurement concepts until they have developed these basic logical reasoning processes of conservation and transitivity.”


Inhelder, B., Piaget, J., & Szeminska, A. (1960). The child's conception of geometry, New York: W.W. Norton & Company.)

Hiebert, J. (1981). Cognitive development and learning linear measurement. Journal for Research in Mathematics Education, 12(3), 197–211.



APPENDIX 1.1a - LENGTHPiaget : conservation, transitivity and symmetry

“Piaget studied the development of the concept of measurement in children. Two of his best known works, The Child's Construction of Space, and The Child's Conception of Geometry include these notions. Piaget suggested that "the conservation of length is a necessary postulate for measurement" (measurement is dependent on the fundamental concept that length is conserved (not altered by spatial arrangement). He also proposed that comparison between two objects does not involve measurement but is purely perceptual if the person does not possess a spatial coordinate system. His research involved students' construction of a map and then reconstruction of a 180-degree reversal of that map to assess the presence of a spatial coordinate system in the child's thinking. The combination of these two (conservation of length and a spatial coordinate system) is necessary or ‘there can be no operational transitivity’.”

http://www.piaget.org/GE/2000/GE-28-3.html

Bladen E, Wildish K, Cox J, 2000, ‘The development of linear measurement in elementary school children’ , George Fox University, US



APPENDIX 1.2 - LENGTH5 sequences for Measurement-

• Identifying the attribute• “unless the student can isolate the attribute being taught from all other attributes , it is unlikely they will be able to

understand any further developments of the concept”

• Comparing and Ordering• “Comparisons initially are made directly, and then through an intermediary tool (indirect comparison). The use of indirect

measures creates a need for informal measurement.”

• Non standard units• “Using informal measurement tools, students have access to experiences that allow them to recognise that different units

can be used for different contexts.”

• Standard units• “In concert with the standard units being used, students need to develop the appropriate language for such

measurement, along with the correct symbolic representation for units of measure. Estimation of quantities is also central.”

• Applications• “Once students have developed and are confident with formal units of measure, they can be introduced to applications of

the units to enable the development of formulae.”

Jorgensen R & Dole S, ‘Teaching Mathematics in Primary Schools’, 2nd edition, Allen & Unwin, Sydney, 2011, pp.282 - 286


We have a block of wood

...then we turn it so it is ....

Like this

Is the block now longer?

APPENDIX 1.2 - LENGTH5 sequences for Measurement- Identifying the object


Now we have 4 different blocks of wood

block A block B block C block D

Which is the 1st longest, 2nd longest, 3rd longest, and 4th longest?

Are they the same size ?

APPENDIX 1.2 - LENGTH5 sequences for Measurement-Comparing and ordering


We may be able to see by eye that block B is 1st longest, and block C is 4th

longest ... Maybe not everyone looks at the picture in the same way..


How can we prove this, and how can we prove which is the order ?

We could use a system of measurement, lets use a hand length, is everyoneshand length the same ?

APPENDIX 1.2 - LENGTH5 sequences for Measurement- Informal Measures


So block B is 1st longest, and block C is the 4th longest, but what of block A and block D ?Lets place the two blocks A and D next to each other and see which is bigger

block A block D

... No two peoples hands are the sameIf Rob says block A is 1.5 hands long, is it the same as Michelle saying block D is 1.5 hands long, are they the same length ?

To my eye they look the same length !!

APPENDIX 1.2 - LENGTH5 sequences for Measurement- Informal Measures


15---------------------------------------------------------------------------------------------------------------------------------------------14---------------------------------------------------------------------------------------------------------------------------------------------13---------------------------------------------------------------------------------------------------------------------------------------------12---------------------------------------------------------------------------------------------------------------------------------------------11---------------------------------------------------------------------------------------------------------------------------------------------10---------------------------------------------------------------------------------------------------------------------------------------------9---------------------------------------------------------------------------------------------------------------------------------------------8---------------------------------------------------------------------------------------------------------------------------------------------7---------------------------------------------------------------------------------------------------------------------------------------------6---------------------------------------------------------------------------------------------------------------------------------------------5---------------------------------------------------------------------------------------------------------------------------------------------4---------------------------------------------------------------------------------------------------------------------------------------------3---------------------------------------------------------------------------------------------------------------------------------------------2---------------------------------------------------------------------------------------------------------------------------------------------1---------------------------------------------------------------------------------------------------------------------------------------------0 --------------------------------------------------------------------------------------------------------------------------------------------

Now the question is still ....


.... which is the 1st longest, 2nd longest, 3rd longest, and 4th longest?

We need a formal ‘unit of measurement’(a system that is the same for everyone)

APPENDIX 1.2 - LENGTH5 sequences for Measurement- Formal Measures


15---------------------------------------------------------------------------------------------------------------------------------------------14---------------------------------------------------------------------------------------------------------------------------------------------13---------------------------------------------------------------------------------------------------------------------------------------------12---------------------------------------------------------------------------------------------------------------------------------------------11---------------------------------------------------------------------------------------------------------------------------------------------10---------------------------------------------------------------------------------------------------------------------------------------------9---------------------------------------------------------------------------------------------------------------------------------------------8---------------------------------------------------------------------------------------------------------------------------------------------7---------------------------------------------------------------------------------------------------------------------------------------------6---------------------------------------------------------------------------------------------------------------------------------------------5---------------------------------------------------------------------------------------------------------------------------------------------4---------------------------------------------------------------------------------------------------------------------------------------------3---------------------------------------------------------------------------------------------------------------------------------------------2---------------------------------------------------------------------------------------------------------------------------------------------1---------------------------------------------------------------------------------------------------------------------------------------------0 --------------------------------------------------------------------------------------------------------------------------------------------

Now the question is still ......


.... which is the 1st longest, 2nd longest, 3rd longest, and 4th longest?

We need a formal ‘unit of measurement’

Block A = 10.50 Block B = 13.25Block C = 5.25Block D = 10.00

APPENDIX 1.2 - LENGTH5 sequences for Measurement- Formal Measures


15---------------------------------------------------------------------------------------------------------------------------------------------14---------------------------------------------------------------------------------------------------------------------------------------------13---------------------------------------------------------------------------------------------------------------------------------------------12---------------------------------------------------------------------------------------------------------------------------------------------11---------------------------------------------------------------------------------------------------------------------------------------------10---------------------------------------------------------------------------------------------------------------------------------------------9---------------------------------------------------------------------------------------------------------------------------------------------8---------------------------------------------------------------------------------------------------------------------------------------------7---------------------------------------------------------------------------------------------------------------------------------------------6---------------------------------------------------------------------------------------------------------------------------------------------5---------------------------------------------------------------------------------------------------------------------------------------------4---------------------------------------------------------------------------------------------------------------------------------------------3---------------------------------------------------------------------------------------------------------------------------------------------2---------------------------------------------------------------------------------------------------------------------------------------------1---------------------------------------------------------------------------------------------------------------------------------------------0 --------------------------------------------------------------------------------------------------------------------------------------------

Now from our measured blocks we can ORDER them


Block A = 10.50 Block B = 13.25Block C = 5.25Block D = 10.00

ORDERBlock B = 13.25 Block A = 10.50Block D = 10.00Block C = 5.25 COMPARE : So we can state that

block B is 8 longer than block Cblock A is 0.5 longer than block D.

We can also COMPARE them

APPENDIX 1.2 - LENGTH5 sequences for Measurement- Compare and Order


APPENDIX 1.3 : Estimation

• “Estimation often appears in teacher resources books as a natural part of measurement work and children are routinely asked to estimate, or to make a good guess at the result, before actually measuring to check. Children may not be clear about the point of estimation in this situation and are often reluctant to make estimates which are immediately shown to be inaccurate.” Pepperell, 2009, p.95.


APPENDIX 1.4 –LENGTHIteration errors

“In addition to having difficulty understanding how to repeat units, children are also simply

unaware of the consequences of leaving “cracks” in their measuring. First and second grade

children frequently leave “spaces” between units without noticing (Horvath & Lehrer, 2000).

Also, many young children begin measuring with 1 rather than zero.

To sum up, when it comes to learning the concept of iteration, children may:

Not understand how to repeat units, e.g. how to measure a room with only one meter stick

Leave gaps or overlap the units when physically measuring an object (Lehrer et al., 1998)

Start measuring from 1 on the ruler rather than the zero point – the point on the ruler

where measurement begins! “


The Connecting Mathematics for Elementary Teachers Project, Purdue University



APPENDIX 1.4 –LENGTHIteration errors

“Children given meter or yard sticks to measure the room will say they cannot measure the room because they do to have enough meter sticks to go all the way across the room. “


The Connecting Mathematics for Elementary Teachers Project, Purdue University

“To measure a continuous quantity, such as the length of a desk, the length has to be partitioned into units that can be counted by either repeating the unit along the length, or subdividing the length into units of a given size. ....We call using the unit repeatedly end to end iterating the unit.”

Iteration example video :


New South Wales Department of Education and Communities

“Lehrer (2003) notes that children may often iterate a unit leaving gaps between the units or

even overlap the units.”


Lehrer, R. (2003). Developing understanding of measurement. From : Kilpatrick, J., Martin, W. G., & Schifter, D., A research companion to Principles and Standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.





“Johnny is good at measuring. He usually gets all his worksheet problems right, especially when the problems involve using his ruler to measure things. But one day Johnny’s second grade teacher gave him a ruler that had 0 marked a short distance from the end of the ruler, and Johnny had trouble. “...

“If children do not understand the concept it may become difficult for them to apply the skill to a new situation where some adjustments are needed”

James Hiebert in National Council of Teachers of Mathematics ‘The arithmetic teacher’, vol 31, No. 7, March 1984http://www.jstor.org/discover/10.2307/41192320?uid=3737536&uid=2&uid=4&sid=21103878188983

APPENDIX 1.7 –LENGTHRulers don’t start at 0


http://www.jstor.org/discover/10.2307/41192320?uid=3737536&uid=2&uid=4&sid=21103878188983

APPENDIX 1.8 –LENGTHNot starting at 0, Tick counting

• “In a first or second grade classroom, a teacher asks students to take a ruler and measure (in inches) the length of a rectangular block. A student aligns the "0 inch" mark of the ruler with the end of the block, and counts the number of inches from the end of the ruler to where the block ends.

• "It's three inches," the student says.

• In reality, the block is two inches. The student counted the 0 inch mark as part of the measurement, instead of starting at the 1 inch mark. The child moved from one end of the object to the other, but counted the inch marks on the ruler, instead of the intervals of space between them.

• This is just one of common misconceptions that elementary-school-aged children make when learning how to measure various objects.”

National Science Foundation (USA) 2012 , John Smith, Michigan State University http://www.nsf.gov/discoveries/disc_summ.jsp?cntn_id=123020

• “One of the big misconceptions when children measure with a ruler is what is called tick counting. What that means is that instead of counting the units between the ticks on the ruler, kids count the tick marks instead. “





http://www.faculty.pnc.edu/dpratt/cmet/chp10.pdfThe Connecting Mathematics for Elementary Teachers Project, Purdue University, p.218

APPENDIX 1.8a –LENGTHmis-reading a ruler



APPENDIX 1.8b –LENGTHSize of measuring device

“Another key understanding is that the units must be equal in size. In the first year of a three-year longitudinal study of children initially in years 1 to 3, Lehrer, Jenkins and Osana (1998) found that over 80% of the children saw no problem with mixing two different-length paper clips. Over time, however, 80% of children in grades 4 and 5 said that the units needed to be the same. Unless students understand length measurement they will not have the basis for developing area and volume concepts.”

http://www.numeracycontinuum.com/aspects-of-the-continuum/aspect7/14-aspect-7/87-why-bother-with-informal-units

NSW Government Education and Communities

Lehrer, R., Jenkins, M., & Osana, H. (1998). Longitudinal study of children’s reasoning about space and geometry. In R. Lehrer & D. Chazan (Eds.), Designing learning environments for developing understanding of geometry and space (pp. 137-167)

Mahwah, NJ: Erlbaum.


http://www.numeracycontinuum.com/aspects-of-the-continuum/aspect7/14-aspect-7/87-why-bother-with-informal-units

APPENDIX 1.9 and 1.10 –LENGTHDistance or effort involved, and as the crow flies

1.9 : Distance versus effort. (Length in time versus Length in Distance)

Battista (2006) claims that the concept of length, while seemingly simple for adults, can be difficult for young children tounderstand. The length of a journey may be interpreted in terms of the journey time or the effort involved. Walkingfrom home to school, for example, may seem to the child a greater distance than driving from home to the next town.In fact the expression “how long does it take” may add to a student’s difficulty in grasping the concept of length as adistance between two points. “

1.10 : Straight line distance and Actual Distance

Children sometime get confused with straight line distance, and actual distance travelled ......“Then there is the distinction between the straight line distance between two points (asindicated by the dotted line between the ends of a piece of bent wire), and the distance along the wire, that is, thelength of the bent wire. Children may initially consider these as the same. It is important that this distinction is madebetween the length of an object and distance between two points. “

Both from :-


Battista, M. T. ((2006). Understanding the development of students’ thinking about length.

Teaching Children Mathematics, October, 13(3), 140–147.



APPENDIX 1.10a –LENGTHStraight line distance example

• Tommy walks to school, it takes him 15minutes, and Mummy tells him it is 900 metres from his house to the school gates. When they go in the car it only takes 2 minutes, but Mummy says it is just as long(???)

Tommy’s House

Tommy’s School

900m Mummy had told Tommy the STRAIGHT LINE distance ... The ACTUAL distance was 1200m

school road

home lane

main

road

• One day Daddy walks with Tommy to school, they take exactly the same route as Mummy does, Daddy explores on Google Maps the exact route they take ...

• It turns out poor Tommy had been walking 300 metres more than Mummy thought every trip, now wonder he ‘s tired !!!! Why is this ?


APPENDIX 1.11 –LENGTHTeaching linear measurement

“ Kamii and Clark (1997) conducted research on the age at which children construct unit iteration out of transitive reasoning (proposed by Piaget) in order to discover implications this has for teaching linear measurement in the schools. These two cognitive abilities (iteration of unit and transitivity) are necessary for measurement. As an example, they proposed that rulers are useless for children who have not constructed transitive reasoning because they cannot compare two lengths that are not placed next to each other. ”http://www.piaget.org/GE/2000/GE-28-3.htmlKamii, C., & Clark, F. B. (1997). Measurement of length: the need for a better approach to teaching. School Science and Mathematics, 97, 116-121



• Would you measure the length of the classroom with a 15cm ruler ?

• Would you measure the length of a text book with a trundle wheel ?

• How would you measure a curved surface ?

APPENDIX 1.12 –LENGTHThe right tool for the right job


Mass Appendix


APPENDIX 2.1- Mass misconceptions:sight versus touch and quantity

“Mass poses particular problems for many students because they confuse it with volume, size or quantity. A common misconception is that the larger the size of an object, the greater the mass. This occurs when the judgement made is based on sight (a transfer from learning about length) rather than on feel. For example students may think that a tennis ball is heavier than a golf ball because it is larger. In relation to quantity, students may think that three foam balls are heavier than a soccer ball because there are ore of them.”

http://www.qsa.qld.edu.au/downloads/p_10/kla_maths_info_measurement.pdfQueensland Government Study Authority, 2005



• “Weight is a force – the gravitational pull exerted on an object – whereas mass is the amount of matter in an object. The weight, or downward force on an object, can change depending on where the measurement is made. The mass of an object, in contrast, remains the same no matter where the measurement is made. With young children there might seem to be little point in differentiating between the two and there may be more familiarity with the term ‘weight’.”

“The difficulty with weight is that it is not visible in the same way that length is.”

Pepperell S, Hopkins C, Gifford S, Tallant P, 2009, ‘Mathematics in the primary school : a sense of progression’, Routledge publishing, London UK, p. 97.

APPENDIX 2.1- Mass misconceptions:Mass is not visible in same way


Appendix 3 : Capacity


APPENDIX 3.1 - CAPACITYCapacity different to volume

• http://www.bbc.co.uk/skillswise/factsheet/ma23capa-l1-f-capacity



APPENDIX 3.2 - CAPACITYCapacity in Estimation

• Which watering can hold more water ?

• http://www.abc.net.au/countusin/games/game15.htm

a. b.


http://www.abc.net.au/countusin/games/game15.htm

APPENDIX 3.2a - CAPACITYCapacity : ‘conservation’ of liquid (Piaget)

• “Piaget determined that children in the concrete operational stage were fairly good at the use of inductive logic. Inductive logic involves going from a specific experience to a general principle. On the other hand, children at this age have difficulty using deductive logic, which involves using a general principle to determine the outcome of a specific event.”

http://psychology.about.com/od/piagetstheory/p/concreteop.htm

www.ncetm.org.uk/resources/22757

National Centre for excellence in the teaching of Mathematics UK


http://psychology.about.com/od/piagetstheory/p/concreteop.htm

http://www.ncetm.org.uk/resources/22757

• We want to create some spell potions for some of our mystical classmates, each for a cauldron of capacity 500ml, with 4 elements :

A. Wicked Witch

B. Wizard of Oz

C. Harry Potter

D. Merlin E. The Goblin King

F. The sorcerers apprentice

Hubble 200ml 150 25

Bubble 250 100 200 25 250

Toil 100ml 100 150

Trouble 50ml 100 50 400

G. Which element have we used most in making our 6 spells ?

Fill in the table below, so that each total equals 500ml.Merlin needs the same amount of Hubble, Toil and Trouble.The sorcerers apprentice needs equal amounts Hubble and Trouble.

Hubble, Bubble, Toil and Trouble

Total

575ml

975ml

700ml

750ml

150ml

50

200

100

100

100

50

50

50

APPENDIX 3.3 - CAPACITYCapacity game


APPENDIX 3.4 - CAPACITYLanguage confusion

• “Harrison (1987) raised an interesting observation with young children when he saw that they seemed to be focussing on fullness rather than capacity. He gave an example of a girl who thought that a cup ‘held more’ than a teapot. Harrison suggests that she was focussing on the fact that she had just filled the cup. The teacher needs to look for such phenomena and consider also the language that we use. Harrison suggests that since there is not an equivalent to ‘longer’, when dealing with the comparison of lengths, (for example, ‘is capaciouser than’), it is helpful to use phrases such as ‘larger/smaller than’, or ‘can hold more than’ in the context of capacity.”

• Pepperell S, Hopkins C, Gifford S, Tallant P 2009, ‘Mathematics in the primary school : a sense of progression’ 3rd edition, Routledge publishing London UK, p. 97.

• Harrison R (1987), ‘On fullness’, Mathematics Teaching, 119.


Appendix 4 : Time


APPENDIX 4.1 - TIMETime and Space, Space and Time

• “Piaget studied children's conceptions of time and space extensively, and observed their close relationship. He emphasized that time and space for an inseparable whole’ in the children’s mind, suggesting a symmetric relationship.”

• “Results of Piaget’s experiments on time, motion, and speed suggest that children mistakenly use spatial information for time more often than the other way round”

• “Young children use the word in spatially (e.g., in the box) far more than they use it temporally (e.g., in a minute), even though temporal uses of in are common in adult speech. Children use here and there to designate points in space before they use now and then for points in time. They produce where questions earlier than when questions, and sometimes misinterpret whenas where”


Casanto et al, 2010, ‘Space and Time in the Child’s Mind’, Cognitive Science Society

• http://www.ucl.ac.uk/~uctyibp/Attention%20to%20the%20passage%20of%20time.pdf



http://www.ucl.ac.uk/~uctyibp/Attention to the passage of time.pdf

APPENDIX 4.2 - TIMETime can not be seen

• “For years, students have become frustrated with the task of learning to tell the time and teachers have become frustrated at not fully understanding why this task is such a difficult one .... you can't go into a shop and buy a dozen minutes, or stub your toe on midday, or an hour. Considering that the clock system is "probably the least studied of the major symbol systems that confront children, and that it is "interconnected with almost everything" else in the wider mathematics curriculum it is easy to understand this frustration. ”

• “Most children can read a digital clock with relative ease because you just read the numbers. There is a general consensus that children should be taught digital time first as this is less difficult to learn.

This is an easy solution if the aim is merely time telling. ”

• http://www.thefreelibrary.com/It's+about+time%3A+difficulties+in+developing+time+concepts%3A+Sally...-a0177811766

The Australian Association of Mathematics Teachers, 2008, Gale, Cengage Learning



APPENDIX 4.3 - TIMEAnalogue clock

• “Tasks that students find difficult in telling the time on an analogue clock include:

* ignoring the second hand that is usually present

* understanding that hands rotate

* identifying the hour hand and understanding its meaning at, after, and before the number

* identifying the minute hand and learning to count minutes by skip counting fives to fifty five,

* learning that the minutes are superimposed on the hours and that "1" means five and "5" means twenty five

* understanding that "o'clock" follows fifty nine / fifty five”

http://www.thefreelibrary.com/It's+about+time%3A+difficulties+in+developing+time+concepts%3A+Sally...-a0177811766

The Australian Association of Mathematics Teachers, Inc. ; Gale, Cengage Learning ; 2008



APPENDIX 4.4 - TIMETime : clock face and base 10

• “The national sample of 8 year olds was asked to select the correct time from a list containing 3:45, 2:45 ; 9:23, 2:40 ; 9:15 and 2:15. The most common errors are 9:15 and 2:15. The most common errors are 9:15 and 3:45 and have sensible reasoning behind them. Nearly one-third of the 8-year old children confuse the hour and minute hands on the clock and another 11 per cent of the children confuse the hour involved. The children who read this time as 9:15 have mastered the invisible quarters (or units of 15) but not the convention for the ‘big hand’ and the ‘little hand’. The children who read this time as 3:45 have similarly mastered the invisible quarters and think that the little hand should point to the hours, but have not mastered the subtlety of the fractional position the hour hand holds between the 2 and the 3 here.”

• “The non-decimal nature of hours and minutes is also a problem for children. Another time task that proved equally difficult for 8-year-olds was : ‘Samantha puts a cake in the oven at this time : 9:20. The cake was taken out of the oven 50 minutes later. At what time did the cake come out of the oven ?’. In this case 16 per cent of the 8 year olds made the expected error of 9:70, involving the treatment of the hours and minutes as separate entities, or as a decimal.”

• Ryan, Julie ; Williams, Julian , ‘Childrens mathematics 4-15 : Learning from errors and misconceptions’, McGraw Hill, New York, 2007

http://books.google.com.au/books?id=Lvhj4mjOFLAC&pg=PA99&lpg=PA99&dq=time+misconceptions+primary+school+children&source=bl&ots=FzBsifRQt8&sig=QgyeEyz_IvMn91_lw84BPIg26iY&hl=en&sa=X&ei=SKczU6nMHcnBkwX6hYDoAQ&ved=0CEMQ6AEwAw#v=onepage&q=time%20misconceptions%20primary%20school%20children&f=false


APPENDIX 4.5 - TIMEReading analogue clocks

“The interpretation of numbers is especially difficult in clock reading, as

• (1) the clock does not make use of the base-10 structure,

• (2) the meaning of the numbers depends from which clock hand is pointing at it, and

• (3) it involves understanding of spatial, clockwise movements.

• Children with spatial deficits might therefore experience problems with the interpretation of analogclock times, where they have to interpret the upper and below part of the clock differently (in dutch ‘10 past 8’ (8:10) versus in dutch ‘10 to half 8’ (8:20) and where they also have to differentiate left from right (‘to’ versus ‘past’)”

“Considering mathematical facts in clock reading, it can be argued that children have to acquire a set of facts in order to understand the basics of clock reading. For example, one should be aware of the fact that one hour consists of sixty minutes, that there is a scale for hours (1-12) and a scale for minutes (1-60) on a clock face, etc”

http://users.ugent.be/~mvalcke/CV/TIME_JLD_2011.pdf

Elise Burny, Martin Valcke and Annemie Desoet, 2011, ‘Clock Reading: an Underestimated Topic in Children with Mathematics Difficulties’ , Ghent University (Belgium)


http://users.ugent.be/~mvalcke/CV/TIME_JLD_2011.pdf

Appendix 5 : General Measurement


Appendix 5.1 : General Measurement

• “ Informal measurement and comparison are used more extensively in our daily lives than formal, standardised measures. In the early years of school, most measurement experiences are associated with comparisons and informal measurement activities. Students are encouraged to identify and distinguish between the attributes of length, mass, area and volume. They are encouraged to select an attribute to make comparisons between objects. As their understandings become more sophisticated, they describe these comparisons using appropriate language – with the language becoming more precise and contextualised as their comparisons become more accurate. In the first three years of school, students generally rely on non-standard units when they estimate length, mass, area and volume.”

“By Year 3, students increasingly begin to recognise the need for accuracy when measuring length (moving from informal to formal units) and have the capacity to sequence everyday events using concepts of time.”

Bobis J, Mulligan J, Lowrie T, 2009, ‘Mathematics for Children : Challenging children to think mathematically’ 3rd edition, Pearson Education, Australia, p.154.


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