Are you positive you can teach negative numbers

Are you positive you can teach negative numbers? Peter Gates

“Unlike natural numbers, negative numbers do not have natural physical referents” Discuss! Blair, K. , Rosenberg-Lee, M. , Tsang, J. , Daniel Schwartz, D. and Menon, V. (2012) Beyond natural numbers: negative number representation in parietal cortex. In Frontiers in Human Neuroscience. 6

The number line that we use to represent quantities supports a limited form of intuition about numbers. It encodes only positive integers and their proximity relations. Perhaps this is the reason for our intuitive grasp of the meaning of integers, but also for our lack of intuition concern other types of numbers. Stanislas Dehaene, (2011) Number Sense, Oxford: Oxford University Press, (p. 75).

Explain … 5 + 7 7 - 5 7 + 5 -5 + -7 -5 - -7 7 + 5 5 - 7 5 + 7 -7 + -5 -7 - -5

How to understand -3 - -7 (-3) – (-7) is a difference -7 is “smaller” than -3 So the answer is … positive. -7 4 -3

In Nottingham the temperature at 9 am was +3⁰ C By 9 pm it fell to -5 ⁰ C What was the temperature change? Draw a diagram Circle correct ways you work out the answer. 3 -5 3 + (-5) 3 – (-5) – 3 (-5)+ 3

Actually most people do …. 3+5= 8 What would you tell a student about temperature as a metaphor for negative numbers? Does temperature have the properties of a number system? Is temperature a Group: <T, +> Or a Ring: <T, +, x>

Making sense of it all? Year 8 pupil…. . Explain the mathematics Balance the equation Subtract 2 x The binary operation (-3) has become a unitary operator (– 3) With unexplained magic like this …. how can we expect pupils to understand mathematics?

Can you explain … (x + 3) (x + 2) (x + 3) (x – 2) (x – 3) (x – 2) Can you use diagrams

(X + 3)(X + 2) = X(X + 2) + 3(X +2) =Xx. X +2 x. X + 3 x 2 = X 2 +5 X + 6 (X -3)(X - 2) = X(X - 2) - 3(X -2) =Xx. X -2 x. X - 3 x. X - ? ? = X 2 -5 X + 6 “Transfiguramos!”

• How helpful is the Chilver’s model in supporting children to understand the number system? • What messages are there in the Kilhamn article for teaching?

Chilvers Model • How good is the model in representing the mathematical structure? • Is the model likely to clear up pupil misconceptions?

Kilhamn • What might we learn from how pupils understand such mathematics as negative numbers? • “Teachers need alternative models…” (p. 2) • “A metaphor can serve as a vehicle for understanding a concept only by virtue of its experiential basis” (Lakoff & Johnson, 1980 p. 18).

Is the Bar Model any good? • http: //www. greatmathsteachingideas. com/tag/bar-modelling/

Teaching negative numbers 1 Here are some video resources on negative numbers. As you watch note down key points. • Year 4 and the language of mathematics • Diving. . . • OMG This is so awful! • http: //www. bbc. co. uk/skillswise/topic/negative-numbers

Teaching negative numbers 2 Now consider this video and think of key points… • • Algebra Helper Year 10 singing. . . Using integer chips Maths Antics - poor use of language?

Pupil Misconceptions

Children's Understanding … The CSMS Team developed tests into all aspects of Maths understanding in pupils 1116 in the 80 s. The following are taken from their report* and are the results of testing 14 year olds. *Hart, K. (1981) Children's Understanding of Mathematics 11 -16. London: John Murray

Why do learners make mistakes? • • lapses in concentration hasty reasoning memory overload failure to notice important features Mistakes, however, may indicate alternative ways of reasoning. Such ‘misconceptions’ should not be dismissed as ‘wrong thinking’; they may be necessary stages of conceptual development.

Rules that expire http: //www. isk. ac. ke/uploaded/Parents/Resources/ Parents_Math/13_Rules_that_expire. . elementary_m ath_2. pdf e. g. • Multiply by 10 put a 0 on • Addition makes numbers bigger

Misconceptions -4 + ? = -10 ? = 6 ? = 14 ? = -14

Misconceptions -8 + 6 = ? ? = 2 ? = 14

Deconstructing Textbooks Key Maths Year 7 and Year 8 WARNING Reading Key Maths can seriously damage a love of mathematics

Teacher misconceptions • • Writing 3 + -2 Saying “two minuses make a plus” Saying “negative numbers are just like a debt” Assuming negative numbers are just like temperatures* *Temperature is not a true ratio scale! You can’t multiply two temperatures, and one temperature is not twice as hot as another.

Didactics of Mathematics Misconceptions and errors Key mathematics What common errors do pupils make in this topic? What are the key mathematical ideas in the topic Language and vocabulary What sort of language and technical vocabulary is used Imagery and visualisation What images, pictures, diagrams are involved? Negative Numbers Techniques and methods What sort of things do you do in this topic Contexts and settings Where is this topic used? What sorts of questions?

Before being introduced formally and systematically to negative numbers, children as young as 7 are able to have some understanding of this complex concept. Their understanding is, however, limited by the form of representation used to solve problems involving negative transformations. Not every meaning for negative numbers is clearly understood and, thus, whilst introducing negative numbers formally the most familiar meanings should initially be used and also it must be made clear the necessity of distinct markings for positive and negative numbers. Borba, Rute. & Nunes Terezinha (1999) Young children's representations of negative numbers. In Bills, L. (Ed. ) Proceedings of the British Society for Research into Learning Mathematics 19(2) June 1999, 7 -12.

Implications … • Young children can have some intuitive awareness of direction in numbers. So find out what pupils know. • Thus, understanding is limited by the model used. So be careful of the models you use • Distinction in sign should be clear. So take care with symbols used

Because most individuals have considerably less experience with negative numbers, the representations for negative numbers may be less-refined than those of positive numbers. By this account, negative numbers take longer to compare than positives because they have less resolution. Negative numbers appear less well differentiated than positive numbers in the intra parietal sulcus and that greater differentiation within negative number problems is associated with faster reaction time on negative problems. These findings support the proposal that people develop facility with negative numbers by creating a new representation that incorporates magnitude properties while remaining distinct from the natural numbers. Blair, K. , Rosenberg-Lee, M. , Tsang, J. , Daniel Schwartz, D. and Menon, V. (2012) Beyond natural numbers: negative number representation in parietal cortex. In Frontiers in Human Neuroscience. 6

Implications … • Children have less experience with negative numbers. So develop their experience in a variety of ways • Negative numbers take longer to resolve So give them time to work out strategies and models • In the brain negative numbers are handled differently meaning they are more difficult to sort out. Think of different ways of representing negative numbers

This study highlights the importance of understanding limitations and conditions of use for different metaphors, something that is not explicitly brought up during the lessons or in the textbook. Findings also indicate that students are less apt to make explicit use of metaphorical reasoning than the teacher. Although metaphors initially help students to make sense of negative numbers, extended and inconsistent metaphors can create confusion. This suggests that the goal to give metaphorical meaning to specific tasks with negative numbers can be counteractive to the transition from intuitive to formal mathematics. Comparing and contrasting different metaphors could give more insight to the meaning embodied in mathematical structures than trying to fit the mathematical structure into any particular embodied metaphor.

Implications … • Different representations, models and metaphors need to be explored So do not just depend on one model • Children think in terms of metaphors differently to the teacher So do not just assume they can grasp your metaphor • Metaphors can be counterproductive • Some do not use metaphors too freely • Working on metaphors is better than just working with metaphors So let pupils explicitly examine the metaphor

Participants in the study showed quite different learning trajectories concerning their development of number sense. Problems that students had were often related to similar problems in the historical evolution of negative numbers, suggesting that teachers and students could benefit from deeper knowledge of the history of mathematics. Students with a highly developed number sense for positive numbers seemed to incorporate negatives more easily than students with a poorly developed numbers sense, implying that more time should be spent on number sense issues in the earlier years, particularly with respect to subtraction and to the number zero. Kilhamn, C. (2011) Making Sense of Negative Numbers. Unpublished Ph. D Thesis. University of Gothenburg. Faculty of Education. http: //hdl. handle. net/2077/24151; https: //gupea. ub. gu. se/handle/2077/24151

Implications … • Children's problems mirror the conceptual development So learn more about the nature and history of number • A strong sense of number helped the later incorporation of negative numbers So spend time strengthening positive number sense

In addition to the three previously shown aspects of understanding negative numbers these results shed light on a fourth aspect that might create difficulties and cause incorrect calculations; relying on metaphorical reasoning using a model that is insufficient. Maybe that is a sign of a very poorly developed net of mental representations. The results of this study suggest that teachers should make use of the grounding metaphors and experience based models but emphasize their limitations concerning negative numbers. •

Implications … • Some metaphors are insufficiently robust So do not depend on a simplistic model, but examine their limitations

Knowing the potentials and constraints of a model is necessary if it is to function as a conceptual metaphor and for the learner to be creative in striving to understand. Results suggest that the debate should not concern which model to use and why one model is better than another but rather what are the consequences of our use of metaphors and how do we deal with these consequences. Kilhamn, C. (2008) Making sense of negative numbers through metaphorical reasoning. Paper presented to The Sixth, Swedish Mathematics Education Research Seminar, Stockholm, Sweden January 29 -30, 2008. http: //www. mai. liu. se/SMDF/madif 6/mad 6 eng. htm.

This phylogenetic hypothesis of the origin of our comprehension of numerosities implies that the cognitive representation referred to as the mental number line cannot easily represent negative numbers because it is not possible to experience negative numerosities. Hence, negative numbers may not become associated with space in the same way that positive numbers are. (…evolution of ideas) Alternatively, the ontogenetic hypothesis suggests negative numbers might become associated with the left side of space as a result of experience with them. (…development of an individual) Fischer, M. (2003) Cognitive Representation of Negative Numbers. Psychological Science 14(3), 277 -282, http: //pss. sagepub. com/content/14/3/278, DOI: 10. 1111/1467 -9280. 03435

12 Rules of Negative Number Pedagogy • • • Find out what pupils know. Take care with symbols used Be careful of the models you use Do not just depend on one model Let pupils explicitly examine the metaphor Develop their experience in a variety of ways Spend time strengthening positive number sense Do not just assume they can grasp your metaphor Give them time to work out strategies and models Learn more about the nature and history of number Think of different ways of representing negative numbers Do not depend on a simplistic model, but examine their limitations

Now organise those into some structure and wrote down the implications for teaching negative numbers.

You can play a C major scale without understanding the history of western music or the relative frequencies of intervals in the major scale. Children don't learn mathematics, they just get used to it….

4 4 8 12 16 20 3 3 6 9 12 15 2 2 4 6 8 10 1 1 2 3 4 5 0 1 2 3 4 5

4 4 8 12 16 20 3 3 6 9 12 15 2 2 4 6 8 10 1 1 2 3 4 5 0 1 2 3 4 5 -1 -1 -2 -3 -4 -5 -2 -2 -4 -6 -8 -10 -3 -3 -6 -9 -12 -15 -4 -4 -8 -12 -16 -20 -5 -5 -10 -15 -20 -25

-20 -16 -12 -8 -4 4 4 8 12 16 20 -15 -12 -9 -6 -3 3 3 6 9 12 15 -10 -8 -6 -4 -2 2 2 4 6 8 10 -5 -4 -3 -2 -1 1 1 2 3 4 5 -5 -4 -3 -2 -1 0 1 2 3 4 5 -1 -1 -2 -3 -4 -5 -2 -2 -4 -6 -8 -10 -3 -3 -6 -9 -12 -15 -4 -4 -8 -12 -16 -20 -5 -5 -10 -15 -20 -25

Play a game? • http: //nrich. maths. org/content/id/5865/Prim ary. Connect. Three. swf