COMMON MISCONCEPTIONS in Space, Measurement, and Chance & Data Adrian Berenger 24 August 2010 Teaching & Learning Coach Moreland Network
Primary Mathematics Teachers • The focus of today’s professional learning is on misconceptions in mathematics in dimensions other than number. • Warm-up Activity: Meg’s Number • Explore: Misconceptions in • Space • Measurement • Chance & Data • Summary
Introduction • ‘Misconceptions’ has become a word to describe misunderstandings in mathematics. What do children do incorrectly all the time and why? Often our lack of modeling and representations lead to misconceptions. The language and symbolic nature of mathematics can cause several problems. • Teachers need to uncover misconceptions in classes in order to understand what students are thinking. Strategically and deliberately making mistakes or leading students to make mistakes are some ways of identifying common misconceptions.
Meg’s number 1 Meg’s number 4 The sum of the digits of Meg’s number is greater than four. Help your group find Meg’s number on the Hundred chart. The difference between the two digits of Meg’s number is greater than four. Help your group find Meg’s number on the Hundred chart. Meg’s number 2 Meg’s number 3 The first digit of Meg’s number is larger than the second. Help your group find Meg’s number on the Hundred chart. Meg’s number is not odd. Help your group find Meg’s number on the Hundred chart. Meg’s number 5 Both digits in Meg’s number are even. Help your group find Meg’s number on the Hundred chart.
SPACE • Language • Orientation & Modeling • Limited Definitions • Mathematical Tools
Language • Imprecise – words in mathematics take on different meanings • What does volume mean? • What’s the difference between volume and capacity? • What is a solid in mathematics? • Van Hiele Levels (recognition, analysis, ordering, deduction, rigor) • Language is developmental and so therefore it is vastly different at levels 1 & 2 than at levels 3 & 4
Recognition, Analysis, Ordering, Deduction, Rigor Students at level 1&2 Students at level 3&4 angles rectangular rhombus or kite a square has four equal sides and at least one R.A. • corners, pointy • like a square • diamond • a square has four sides Mathematical language is and should be different at different developmental levels.
Orientation & Modeling • Coordinates in mapping are commonly reversed, so the order of ‘horizontal then vertical’ needs to be consistently represented and emphasised. • Map reading depends on the orientation of the viewer with respect to the map itself. • Difficulties with perspective representations • Hidden surfaces in 3D representations
Limited Definitions But most would not agree that the following shapes are also hexagons What is a hexagon? • - The first figure is only a regular hexagon. What is a polygon? • - any closed shape made up of straight lines • 3 straight lines (triangles) • 4 straight lines (quadrilaterals) • 5 (penta), 6 (hexa), 7 (septa), 8 (octa), nine (nona), • 10 (deca), 12 (dodeca), 20 (icosa)
Definitions that matter • What is a solid? • Mathematically, stability or rigidity does not define a solid. A solid is a region of space enclosed by a 3-D figure. It may be a rigid structure but need not be. It may be open or closed. It may be regular or irregular. It may have curved surfaces. • What is a shape? • A shape is the appearance of something especially its outline that is not dependent on size, position or orientation. It need not be 2-dimensional. • A regular shape is not simply one that is common. A regular shape is one where all its sides and angles are equal. REGULAR REGULAR IRREGULAR
Symmetry • Students find more lines of symmetry than actually exist. • Simple or convenient definitions that lines of symmetry ‘chop’ shapes into half do not necessarily imply that these lines must also create one half the exact mirror image of the other • How can ICT be used to overcome this misconception?
Angles • Angles should be defined as the ‘amount of turn’. • Modeling with two sticks joined together of different lengths is important to overcome this common misconceptions with angles. • What other terms can be used to develop completeness of definition in relation to angles? eg. pivoting, rotation • What other modeling can be used? eg. clock hands
Mathematical Tools • Reading and using the scale of a protractor • Many children believe a protractor measures the length along curved lines • They often confuse the position or alignment of the centre. • Many students misread protractors as there are two sets of numbers on most protractors.
60 Kindergarten children… • The children were shown 5 quadrilaterals • They had to colour all the shapes that were the same 1 2 3 4 5
Results… • 56% coloured figures 1 & 4 • 30% coloured figures 1, 4 & 5 • 7% coloured figures 1, 2 & 4 • 7% coloured figures 1, 2, 4 & 5 1 2 3 4 5
One student’s response… • “Even though that shapes maybe turned around, they’re still the same shape that they were… • “1, 2, 4 are identical but are changed. 5 if you’re talking about the same shape even though they might be small or large, 5 is the same as 1, 2 & 4… • “Shape 3 is just like a square – stretched” 1 2 3 4 5
Summary • ‘…a rectangle is a long shape…’ • ‘…a square is not a rectangle…’ • This is a square, this is a diamond. • angle is smaller than • a right angle and left angle Children associate the word right with directional language. a b
Summary • A triangle is not a polygon. • Using a protractor. • The diagonal of a square is the same length as its side. • 3D shapes have diagonal lines. • All lines which divide a shape into two equal 'halves' are lines of symmetry.
MEASUREMENT • Language • Confusion with rules, units and conversions • Measuring tools - time and temperature
MEASUREMENT • Students perceive volume as a solid measurement and capacity as a liquid measurement. • Mathematical rules for calculating perimeter, area and volume and their units get confused. They often believe that rulers can be used to measure area. • Children are shown to fill a space with other units and simply count these as a measure for area. They often believe that it doesn’t matter if these other units are of equal size as long as they don’t violate the boundary. • Squared units for shapes that are not square.
Language • Volume and Capacity • Volume is the amount of space an object takes up. • Capacity is the amount a container can hold. • Sometimes words have inaccurate associations • Bigger = larger, taller, longer • Smaller = lighter
Time • Students confuse the minute and hour hands. • They have difficulty in estimating the duration of a given length of time. • Digital clock and timers have a number scale based on 60 not 100.
My Open-Ended Problem A rectangular fence has a perimeter of 240 m. What might the area be? • How can this problem be modified to fit a primary classroom? • What problem-solving strategies would help solve this problem?
Perimeter & Area • Students believe area is bigger than perimeter since it involves multiplying. This misconception results from a misunderstanding of how perimeter and area are different and cannot be compared, as well as a misconception from Number that multiplying numbers gives a larger result.
Mathematical Tools • Students misread or misunderstand measuring tools such as clocks and thermometers. • Some students believe that rulers can be used to measure area. • Students often misuse rulers by not beginning to measure a length from the zero mark. • They use the edge of the ruler or start at 1. • When measuring lengths longer than the ruler, some students flip the ruler over and over.
Chance & Data • Language & Symbols • Limited Definitions • Modeling and Representations • more simulated activities
Language & Symbols • Students have imprecise meaning for the following terms • fair, chance, luck, odds, likelihood, randomness • Meaning is normally based on subjective reasoning rather than quantitative reasoning • Some symbols and words cause confusion • and, or, , , n(X=3), Pr(Y >2)
Quick Task • The Venn Diagram opposite shows the number of students that like particular ice-cream flavours. CHOC STRAW 5 3 6 2 • Find the following • n(STRAW) = • N(CHOC) = • How many like both STRAW and CHOC? • How many like STRAW or CHOC?
Limited Definitions • Most students believe that data is about drawing graphs. • This comes from an over-emphasis on displaying data in different ways rather than interpreting these displays • Statistics involves using a range of tools and summary statistics (mean, mode, median) to represent, interpret, analysis and summarise data. • This means that teaching and learning activities needs to be centred on students developing a set of questions that they want to know answers to. • Student may be able to calculate summary statistics but what do they mean?
Modeling and Representations • Students have general difficulties in organising, displaying and analysing data especially if it is not their own • When a tally in required they sometimes group in 6’s not 5’s ie. instead of • They often misread scales and axes on graphs
Chance Experiments • Spinners are always balanced. • Expressing probabilities as fractions, understanding proportion and ratio present great problems in this area of mathematics. • example: Prob of selecting an apple =
Modeling • Getting a 6 on a die is harder than getting any other number. • When experiments do not have enough trials to match theoretical probabilities then many students believe experimental probabilities (what they have seen, what they have witnessed) to be more precise or more reliable than theoretical probabilities. • The Roulette Phenomenon • HHHHHH? 1 15 17 30 1 36 23 16 18 32
Resources • Beesey, C. (1997). Jigsaw: Ideas for Assessment in Mathematics Level 5. Macmillan Aust • Bobis et. al. (2004). Mathematics for children: Challenging children to think mathematically, p.105 • Booker, G.B. (2004). Teaching Primary Mathematics 3rd Edition. Pearson Aust • Leung, A. (2001). Learning Study 5: P.4 Mathematics Lesson on Perimeter and Area. Last viewed 13th July 2010. http://iediis4.ied.edu.hk/cidv/webdata/documents/medward102p4math/medward102p4mereport.pdf • Teachernet. Maths Misconceptions Last viewed12th July 2010.www.teachernet.gov.uk • VELS. Mathematics Level 2. Last viewed 12th July 2010. http://vels.vcaa.vic.edu.au/vels/levels.html