Forum for Science, Industry and Business

Sponsored by:     3M 
Search our Site:

 

The importance of clarifying language in mathematics education

22.03.2011
The way in which teachers and textbooks use language and different metaphors in mathematics education determines how pupils develop their number sense.

This is shown in a thesis from the University of Gothenburg, Sweden.

When school pupils encounter numbers that cannot be conceptualised as quantities in an obvious way, clarity is required in terms of the language used by both teachers and textbooks. One significant risk is that the explanatory models and metaphors used by teachers are overgeneralised by pupils or create contradictions.

In her thesis, Cecilia Kilhamn shows how the difficulties experienced by pupils in understanding negative numbers – numbers that are less than zero – are similar to the difficulties that mathematicians have experienced historically. This suggests that better knowledge about the history of mathematics would provide a better understanding of pupils’ problems.

Concretising the abstract
“A reluctance to accept negative numbers is closely linked to our desire to be able to concretise that which is abstract and understand negative numbers in terms of concepts such as debts, lifts or temperatures,” explains Kilhamn.

However, many of the concrete explanatory models used in school mathematics cannot deal with subtraction, multiplication or division using negative numbers. A transition to a clearer mathematical language is therefore needed when the number domain is expanded from natural numbers to signed numbers, i.e. positive and negative numbers.

The study in question is a longitudinal case study in which pupils in a school class were followed over a period of three years. The results show that pupils’ ability to accept and make sense of negative numbers depends on how well developed their sense of natural numbers is.

Clear explanations
Insights such as being able to visualise zero as a number and not just a representation of nothing, understanding how subtraction works and being able to deal with the number line are important prerequisites for negative numbers. Another crucial factor is how clear teachers and textbooks are in their explanations. Numbers can be seen metaphorically as quantities, points, distances or operations, as constructed objects and as relations.

“But no individual metaphor for numbers can make negative numbers fully comprehensible,” continues Kilhamn. “It is therefore important that the deficiencies and limitations of these metaphors are also made clear in teaching, and that logical mathematical reasoning is used in parallel with concretised models.”

Contradictory properties.
Her study also highlights a number of problems relating to the fact that the mathematical language used in Swedish schools is a little ambiguous or inadequate. For example, no distinction is made between subtracting the number x and the negative number x if both are referred to as “minus x”. There is also no word in the Swedish language corresponding to the English term “signed number”.

“Swedish textbooks introduce negative numbers without making it clear that all the natural numbers change at the same time and become positive numbers,” she adds. “Another difficulty is the size of negative numbers, which have two contradictory properties that are distinguished in mathematics by separating absolute value (magnitude) from real value (position). A large negative number has a smaller value than a small negative number. This distinction also needs to be made clear to pupils.”

The thesis was completed within the framework of the Graduate Research School in Educational Sciences at the Centre for Educational Sciences and Teacher Research (CUL) at the University of Gothenburg.

For further information, please contact:
Cecilia Kilhamn, tel.: +46(0)31 786 2037, cecilia.kilhamn@ped.gu.se

Helena Aaberg | idw
Further information:
http://www.gu.se
http://hdl.handle.net/2077/24151

More articles from Science Education:

nachricht Starting school boosts development
11.05.2017 | Max-Planck-Institut für Bildungsforschung

nachricht New Master’s programme: University of Kaiserslautern educates experts in quantum technology
15.03.2017 | Technische Universität Kaiserslautern

All articles from Science Education >>>

The most recent press releases about innovation >>>

Die letzten 5 Focus-News des innovations-reports im Überblick:

Im Focus: The pyrenoid is a carbon-fixing liquid droplet

Plants and algae use the enzyme Rubisco to fix carbon dioxide, removing it from the atmosphere and converting it into biomass. Algae have figured out a way to increase the efficiency of carbon fixation. They gather most of their Rubisco into a ball-shaped microcompartment called the pyrenoid, which they flood with a high local concentration of carbon dioxide. A team of scientists at Princeton University, the Carnegie Institution for Science, Stanford University and the Max Plank Institute of Biochemistry have unravelled the mysteries of how the pyrenoid is assembled. These insights can help to engineer crops that remove more carbon dioxide from the atmosphere while producing more food.

A warming planet

Im Focus: Highly precise wiring in the Cerebral Cortex

Our brains house extremely complex neuronal circuits, whose detailed structures are still largely unknown. This is especially true for the so-called cerebral cortex of mammals, where among other things vision, thoughts or spatial orientation are being computed. Here the rules by which nerve cells are connected to each other are only partly understood. A team of scientists around Moritz Helmstaedter at the Frankfiurt Max Planck Institute for Brain Research and Helene Schmidt (Humboldt University in Berlin) have now discovered a surprisingly precise nerve cell connectivity pattern in the part of the cerebral cortex that is responsible for orienting the individual animal or human in space.

The researchers report online in Nature (Schmidt et al., 2017. Axonal synapse sorting in medial entorhinal cortex, DOI: 10.1038/nature24005) that synapses in...

Im Focus: Tiny lasers from a gallery of whispers

New technique promises tunable laser devices

Whispering gallery mode (WGM) resonators are used to make tiny micro-lasers, sensors, switches, routers and other devices. These tiny structures rely on a...

Im Focus: Ultrafast snapshots of relaxing electrons in solids

Using ultrafast flashes of laser and x-ray radiation, scientists at the Max Planck Institute of Quantum Optics (Garching, Germany) took snapshots of the briefest electron motion inside a solid material to date. The electron motion lasted only 750 billionths of the billionth of a second before it fainted, setting a new record of human capability to capture ultrafast processes inside solids!

When x-rays shine onto solid materials or large molecules, an electron is pushed away from its original place near the nucleus of the atom, leaving a hole...

Im Focus: Quantum Sensors Decipher Magnetic Ordering in a New Semiconducting Material

For the first time, physicists have successfully imaged spiral magnetic ordering in a multiferroic material. These materials are considered highly promising candidates for future data storage media. The researchers were able to prove their findings using unique quantum sensors that were developed at Basel University and that can analyze electromagnetic fields on the nanometer scale. The results – obtained by scientists from the University of Basel’s Department of Physics, the Swiss Nanoscience Institute, the University of Montpellier and several laboratories from University Paris-Saclay – were recently published in the journal Nature.

Multiferroics are materials that simultaneously react to electric and magnetic fields. These two properties are rarely found together, and their combined...

All Focus news of the innovation-report >>>

Anzeige

Anzeige

Event News

“Lasers in Composites Symposium” in Aachen – from Science to Application

19.09.2017 | Event News

I-ESA 2018 – Call for Papers

12.09.2017 | Event News

EMBO at Basel Life, a new conference on current and emerging life science research

06.09.2017 | Event News

 
Latest News

Rainbow colors reveal cell history: Uncovering β-cell heterogeneity

22.09.2017 | Life Sciences

Penn first in world to treat patient with new radiation technology

22.09.2017 | Medical Engineering

Calculating quietness

22.09.2017 | Physics and Astronomy

VideoLinks
B2B-VideoLinks
More VideoLinks >>>