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Teaching Non-Euclidean Geometries in High-School: an experimental study.
Full text linkIn these years it is becoming quite clear that the development of a young student
into an adult citizen requires a solid scientific background. Facing the challenges
of a quickly changing world where political decisions are not only concerned with
economics or ethics, but also with climate sciences, medicine, etc., requires a good
education. Citizens are required to exert logical thinking and know the methods
of science in order to adapt, to understand and to develop as persons.
At the core of all these required skills sits mathematics, with its formal methods to
develop knowledge. Learning the axiomatic method is fundamental to understand
how hard sciences work, and helps in consolidating logical thinking, which will
be useful for the entire life of a student.
In my experience as a secondary school teacher, I have tried to understand how
students perceive mathematics and what difficulties they encounter. One
observation I often made was that the axiomatic study of geometry was a
problematic topic for students, even for those with an interest for mathematics.
For this reason, I decided to focus my PhD work on the teaching and learning of
geometry, focusing explicitly on its axiomatic foundations, in order to concentrate
on those aspects that foster the development of a logical thinking, of the ability of
proving a thesis, etc., which are necessary, as said, for the growth of a modern
citizen.
Axiomatic geometry exposes the students to plural worlds, where the choice of a
few base axioms heavily influences the properties of the objects that can be
observed. The students, who are used to an intuitive study of geometry in the
Euclidean plane, can benefit from the discovery of non-Euclidean geometries in
several regards. First, they are shown that different non-Euclidean geometries
exist, then they discover how these geometries can be developed by slightly
modifying the axioms. Finally, they can be taught how these are useful to model
real problems.
The importance of teaching non-Euclidean geometries in high school has been
debated for decades and several experiences have been conducted with students
in the past. However, all these works were often of qualitative nature, the
experimental protocols were poorly documented, and the statistical data was missing. The main objective of this thesis is to investigate, by means of quantitative
experimental protocols, the viability and effects of teaching an introductory course
on non-Euclidean geometries to high-school students.
The experimental nature of this study required the classroom work to be concise
and limited to a short number of seminars and workshops. Several experiences are
described, involving several high school classes and a total of 154 students and 57
teachers. These have been used to evaluate and refine the teaching tools and topics
that are covered in this thesis and are reported in detail for use in future
experiences. Statistical methods and evaluation questionnaires are discussed to
assess the effectiveness of the approach, which will prove necessary in larger-scale
experiments.
The outline of the thesis follows. In Chapter 1, a more elaborate motivation and
introduction to the topics of the thesis are given. Chapter 2 reports a brief history
of the development of Euclidean and non-Euclidean geometries and then discusses
whether the birth and the development of non-Euclidean geometries constitute a
revolution in mathematics. Chapter 3 discusses the teaching of geometry in
secondary schools, with a specific interest in the Italian education system. Chapter
4 gives an informative introduction to the main aspects of the study and open
questions; summarise and critiques the studies that have been conducted on the
teaching of non-Euclidean geometries; and states the research questions that are
investigated in the present thesis:
- RQ1:What features of a short introductory course in non-Euclidean geometries are
effective in engaging high-school students?
- RQ2: To what extent do students gain a new perspective on the concept of
axiomatic system?
- RQ3: How well do students learn the taught concepts of non-Euclidean
geometries?
- RQ4: To what extent do students’ critical thinking and proof skills improve over
the duration of the course?
- RQ5: Do students’ beliefs about mathematics change over the duration of the
course?
Chapter 5 discusses all the details about the experimental phase. Specifically, this
chapter describes and justifies the research methods and justifies the choice of
adopting an essentially positivist paradigm using quantitative methods; discusses
a preliminary experimentation conducted to investigate a suitable methodology, and – in more detail – a second experimentation. In addition, this chapter contains
two sections dealing, respectively, with topics related to the experimentations: an
experience with high-school teachers; and the description of the necessary
adaptations for the distance learning imposed by Covid-19 restrictions. While
Covid-19 restrictions impaired the possibility of a large-scale experiment, it
allowed me to observe some peculiarities of the distance learning paradigm that
must be accounted for when conducting geometry seminars online. Chapter 6
discusses the results of the data analysis, and provide an interpretation of the data
shown and many conjectures for future developments. Chapter 7 concludes the
doctoral dissertation
PROGRESS REPORT OF AN ONGOING PROJECT ON TEACHING NON-EUCLIDEAN GEOMETRIES AT HIGH SCHOOL
No full textThis paper deals with the progress of an ongoing project, first reported in [1], on teaching non-Euclidean geometry at high school. The main goal of the research work is:
(1) to show whether non-Euclidean geometries can be a tool to allow students to consolidate the knowledge of Euclidean geometry by developing it in a critical way;
(2) to promote the understanding of the modern axiomatic method in geometry; and
(3) to give students a new perspective on mathematics so that they can see it as a creative activity and a widely discussed topic with a historical background. The idea itself is not novel, but one of the main issues related to the state of the art is the lack of experimental studies with students. Specifically, up to the authors’ knowledge, there is only one study that has conducted an extensive experimentation with high-school students, showing data that seem to support the hypothesis (3): teaching non-Euclidean geometries lets students change their view of mathematics [2]. Experimental studies conducted with high-school students related to the first two points seem to be absent. For this reason, our aim is to gather further experimental evidence of the potential benefits of teaching non-Euclidean geometries at high school.
The authors have carried out a pilot study based on class activities on non-Euclidean geometries that involved four sets of Italian high-school students (10-18-20-29 students). The pilot study was aimed to understand how to improve the class activities and, most importantly, how to better design useful questionnaires to collect data for the final experimental study. To gather data during the pilot study, the authors used, before and after the class activities, the Van Hiele geometry test as formulated by Usiskin [3] (translated in Italian) and questionnaires previously designed by the authors.
The pilot study made us aware of some students’ misconceptions and difficulties. The paper reports on these findings and on how they led us to change some of the class activities proposed to the students and to elaborate the questionnaires in a different way. Moreover, the paper presents data that seem to support the hypothesis (3) as in [2].
References:
[1] S. Benvenuti, A. Cardinali, "The mental telescope: understanding the geometry of Euclid by learning the non-Euclidean geometry", INTED2018 Proceedings.
[2] P. Schiano, "Convinzioni e cambi di convinzioni degli studenti sugli errori e sullo sviluppo della conoscenza in matematica (studenti di età 14-18)", PhD thesis, Consortium between the University of Bologna, the University of Catania, the University of Pavia, the University “Federico II” of Napoli, the University of Bratislava, the University of Nitra, the University of Alicante, the University of Palermo, the University of Cipro, C.I.R.E “Centro interdipartimentale ricerche educative” Palermo.
[3] Z. Usiskin, "Van Hiele Levels and Achievement in Secondary School Geometry", University of Chicago, 1982
Herbicides runoff in vegetatitive filter strips:evaluation and validation of a recent rainfall return period medel
No full textTHE MENTAL TELESCOPE: UNDERSTANDING THE GEOMETRY OF EUCLID BY LEARNING THE NON-EUCLIDEAN GEOMETRY
Full text linkWe describe a research project about the teaching of non-Euclidean geometries. Our intent is to show
that non-Euclidean geometries can be a tool to promote the understanding of the modern axiomatic
method in mathematics, to stimulate students' aptitudes to logical thinking and to allow students to
consolidate the knowledge of Euclidean geometry by developing it in a critical way
Design of riparian buffer strips affects soil quality parameters
No full textVegetated buffer strips alongside watercourses are commonly used to counteract diffuse pollution from agricultural activities. If properly designed, they can provide multiple environmental benefits by increas- ing wildlife habitats and biodiversity. Little attention has been paid to the effects of buffer strips on soil quality. This study was conducted to determine the impact of different buffer designs on soil biochemi- cal parameters and to define relevant quality parameters for soil monitoring. We compared four buffer arrangements: 3 m wide grass buffer; 3 m grass with one tree row; 6 m grass with one tree row; 6 m grass with two tree rows; plus two controls: an adjacent maize crop field and a plot without buffer. Buffers were established 13 years ago at the Padua University Experimental Farm in the Po Valley, north-east Italy. Studied parameters included soil organic matter composition and soil microbial and enzymatic assays. As expected, control plots showed the lowest values for all the studied parameters. Among buffer designs, 3 m grass and 3 m grass with 1 tree row buffers gave the highest values. Multivariate analy- sis demonstrated that the increase of soil organic carbon content distinguished buffers from controls, whereas soil humic carbon quality parameters such as humic compounds apparent molecular weight, together with acetyl esterase (fluorescein test) enzyme activity, were discriminatory in separating buffer designs. These results are an important contribution to the knowledge base and can help to improve the management of these systems
Effect of vegetative filter strips on herbicide runoff under various types of rainfall
No full textNarrow vegetative filter strips proved to effectively reduce herbicide runoff from cultivated fields mainlydue to the ability of vegetation to delay surface runoff, promote infiltration and adsorb herbicides. A field trial was conducted from 2007 to 2009 in north-east Italy in order to evaluate the effectiveness of various types of vegetative filter strips to reduce spring-summer runoff of the herbicides mesotrione, metolachlor and terbuthylazine, widely used in maize, and to evaluate the effect of the rainfall characteristics on the runoff volume and concentration. Results show that without vegetative filter strip the herbicide load that reaches the surface water is about 5–6 g ha_1 year_1 for metolachlor and terbuthylazine (i.e. 0.5–0.9% of the applied rate), confirming that runoff from flat fields as in the Po Valley can have a minor effect on the water quality, and that most of the risk is posed by a few, or even just one extreme rainfall event with a return period of about 25–27 years, causing runoff with a maximum concentration of 64–77 μg L_1. Mesotrione instead showed rapid soil disappearance and was observed at a concentration of 1.0–3.8 μg L_1 only after one extreme (artificial) rainfall. Vegetative filter strips of any type are generally effective
and can reduce herbicide runoff by 80–88%. Their effectiveness is steady even under severe rainfall conditions, and this supports their implementation in an environmental regulatory scheme at a catchment or regional scale
Fatty Acid Methyl Ester (FAME) Succession in Different Substrates as Affected by the Co-Application of Three Pesticides.
Full text linkIntroductionIn intensive agriculture areas the use of pesticides can alter soil properties and microbial community structure with the risk of reducing soil quality.Materials and methodsIn this study the fatty acid methyl esters (FAMEs) evolution has been studied in a factorial lab experiment combining five substrates (a soil, two aged composts and their mixtures) treated with a co-application of three pesticides (azoxystrobin, chlorotoluron and epoxiconazole), with two extraction methods, and two incubation times (0 and 58 days). FAMEs extraction followed the microbial identification system (MIDI) and ester-linked method (EL).Results and discussionThe pesticides showed high persistence, as revealed by half-life (t1/2) values ranging from 168 to 298 days, which confirms their recalcitrance to degradation. However, t1/2 values were affected by substrate and compost age down to 8 days for chlorotoluron in S and up to 453 days for epoxiconazole in 12M. Fifty-six FAMEs were detected. Analysis of variance (ANOVA) showed that the EL method detected a higher number of FAMEs and unique FAMEs than the MIDI one, whereas principal component analysis (PCA) highlighted that the monosaturated 18:1ω9c and cyclopropane 19:0ω10c/19ω6 were the most significant FAMEs grouping by extraction method. The cyclopropyl to monoenoic acids ratio evidenced higher stress conditions when pesticides were applied to compost and compost+soil than solely soil, as well as with final time.ConclusionOverall, FAMEs profiles showed the importance of the extraction method for both substrate and incubation time, the t1/2 values highlighted the effectiveness of solely soil and the less mature compost in reducing the persistence of pesticides
Effect of sewage amendment on the dissipation of terbuthylazine, its degradation compound desethyl-terbuthylazine, and S-metolachlor in a field study
No full textThis study evaluates the effect of sewage amendment (SA) on the dissipation of terbuthylazine, its degradation compound desethyl-terbuthylazine, and S-metolachlor in the soil. The experiment was conducted at Padua Experimental Farm (Italy). Herbicides dissipation was evaluated in soils differently fertilized for three years: with inorganic fertilizer, with sewage sludge, and with a combination of them. Terbuthylazine and S-metolachlor were applied on sorghum as a formulated product at a dose of 2.8 L ha-1, and their dissipation was followed for 2.5 months. The concentrations of herbicides and one metabolite in soil were analyzed by liquid chromatography-mass spectrometry. The dissipation of terbuthylazine and S-metolachlor followed a pseudo first order kinetics; they dissipated faster in soil amended only with inorganic fertilizer than in soils amended with sewage or sewage + inorganic fertilizer. The reduction in mineralization of the herbicides after sewage addition can be attributed to the reduced herbicide availability to microorganisms. The degradation of terbuthylazine led to the formation of desethyl-terbuthylazine. SA slowed down the formation and the degradation of desethyl-terbuthylazine, leading to a higher amount measured at the end of the incubation. These findings have practical implications for the assessment of the environmental fate of terbuthylazine and S-metolachlor in agricultural areas
Effect of Vegetative Buffer Strips on Herbicide Runoff from a Nontilled Soil
Full text linkBuffer strips can reduce herbicide runoff fromcultivated fields due to the ability of vegetation to impede the flow of surface runoff, promote infiltration,
and adsorb herbicides. Conservation agriculture hasmany environmental advantages, but the transition phase froma conventional to a conservation
systemis a critical period, especially for surface runoff. In 2015, a field in Italy that was transitioning from conventional to conservation agriculture was
tested to analyze the efficacy of different types of buffer strips in reducing the runoff of three herbicides compared with no-buffer plots. At each runoff
event, water volume was measured and terbuthylazine,metolachlor, and mesotrione concentrations were determined. Buffer strips were able to reduce
the number of runoff events by 63% to 83%. The runoff volumes were significantly reduced in the presence of buffer strips, whereas no differences were
found between different types of buffer strips. Among herbicides, mesotrione was not detected in runoff water samples. The highest losses of
terbuthylazine and metolachlor were from plots without buffer strips during the first three runoff events. All types of buffer strips significantly reduced
the total losses of terbuthylazine and metolachlor in the monitored runoff events, with a reduction of more than 99%. This study confirmed that buffer
strips are a very effective mitigation system against surface water pollution by herbicides, even in agronomic situations that promote runoff
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