This text is intended for a brief introductory course in plane geometry. It covers the topics from elementary geometry that are most likely to be required for more. In particular, I would like to acknowledge the influence of L. Chosid,. ~. Graber, S, Katoni, F. Parisi and E. Stern. Henry Africk. New York City Technical College. Elementary College Geometry. Henry Africk O bv Henrv Africk ft covers the toplcs flom elenent*qy geometry that a:ce nost l1ke1y to be requlred. for.
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Elementary College Geometry by Henry Africk. Publisher: CUNY Academic Works Number of pages: Description: This text is intended for a brief. It covers the topics from elementary geometry that are most likely to be required for Elementary College Geometry by Henry Africk free book. Elementary Geometry for College Students,. Fifth Edition. Daniel C. Alexander and Geralyn M. Koeberlein. Acquisitions Editor: Marc Bove. Assistant Editor.
The population was of which two hundred and ninety- eight were used as the sample. The sample composed of The Anas study recorded the following Van Hiele Level 0 Anas concluded that only His study again concluded that The sample used were PSTs in the second-year. From the results analyzed, it was clear that The following were also recorded for the Journal of Education and Practice www. The conclusion based on Van Hiele geometry test levels was that It was clear from the quantitative analysis of students who did not meet any Van Hiele Geometric thinking level was The rests are 1.
The sample students came from the Winneba metropolis. Their study was conducted when the students were four weeks old in the SHS campus level. College of Education, Bimbilla-Ghana. From the population Out of the sample one hundred and thirty-three Convenient and simple random sampling were as adopted in selecting the college and the general programme PSTs respectively.
The duration of the test was 45 minutes. Before administering the test a written permission request was sent to Prof.
Zalman Usiskin for the use of his test items. He gave his approval and advice through an email. The VHGT questions are organized chronologically into five 5 subgroups such that it starts from the very cheap to the most difficult. Each subgroup covered the Van Hiele geometric thinking levels. Two experienced mathematics tutors of the researchers department subjected the VHGT to both face and content validity.
Reliability of the test items were tested when thirty 30 PSTs were involved in the writing the pilot test. The 30 subjects of the pilot test were not part of the actual study sample.
Kuder-Richardson formula 20 methods were utilized to determine the reliability coefficient of the instruments. A reliability coefficient of 0. It aims to determine whether the students recognize the shape by looking at the shape of the figure. They identify students who respond correctly to questions in this group and have proven that they have knowledge of axioms. In these questions, it is determined whether the students are at a level of understanding and writing. Source: Hurma, , p.
So the expected maximum mark was 25 while a minimum mark expected was 0. The PSTs were assigned a weighted sum as follows as proposed by Usiskin Usiskin reiterated a student who satisfies the criterion at levels 1, 2 and 5 would be assigned the Modified Van Hiele Level 2.
Also, student who satisfies the criterion for only level 3 only would not be assigned modified Van Hiele level. College of Education Bimbilla. The researcher organized the results of the study by employing descriptive statistics and presented it in a bar chart.
Dev Maximum Minimum 8. The mean and standard deviation of the PSTs score are 8. This represents Also, 20 5. Also, 33 PSTs representing 9. It was amazing to see that the maximum mark scored in the test was16 out of 25, which was obtained by only one PST. A close look at the performance of PSTs was weak.
Question 3 was correctly answered by PST representing Figure 1: sample item in subtest 1. Figure 1 is question 5 of the VHGT. The reason for the massive inability of the students to get the question right was that they are not aware that all quadrilaterals are parallelograms. The overall correct response rate in subtest 1 of the VHGT level1 is Out of PSTs, representing Contrary to this, PSTs performance on items 6, 8, and 10 were discouraging since representing This question was poorly answered by PSTs since out of , representing This is an indication that PSTs lacked knowledge of properties of a rhombus.
The overall correct response rate in subtest 2 of the VHGT level 2 is PSTs performed woefully at this level.
From table 7 it is clear that out of PSTs, 60 representing The best item answered correctly by PSTs was 12 and the least performance was in item Figure 3: sample item in subtest 3 Journal of Education and Practice www.
From the subtest 3, 26 out the representing 7. This performance was abysmal. This performance is as a result of lack of knowledge on the properties of rectangles and squares. The overall correct response rate at this level was Subtest 4 is meant to test PSTs ability to exhibit deductive geometric proofs, understanding the role definitions, and theories.
PSTs are expected at this level to be able to give reasons for a statement in formal proofs. Table 8 indicates that 72 At this level, PSTs had their best performance in item 20 and worst performance in item Meanwhile students at SHSs are supposed to be at this level before completion.
Question 23 was best answered and the worst item answered was Figure 5: sample item in subtest 5 At this level, PST were expected to work with a variety of axiomatic systems.
That is being able to study Journal of Education and Practice www.
Further analysis indicates that, Pictorial representation of the various levels attained by the PSTs is shown in figure 6 below.
College of Education, Bimbilla — Ghana. It is clear from the descriptive analysis of the overall performance of the PSTs that the lowest score was 3. The mean score is 8. The correct response rate for the various VHGT levels are Also, the overall correct response rate for the entire 25 items is This is a clear indication that PSTs performed woefully similar to Anas It revealed from the results of the VHGT that Apart from that 28 8.
Also This is disturbing because with similar sample within southern Ghana by Armah et al only This is an indicator that majority of the level PST in E.
P College of Education, Bimbilla cannot teach their prospective basic school pupils since the pupils are supposed to be at level 3. These findings are in line with previous students by Anas , ; Halat and Sahin, ; Pandisco and knight , ; Ndlove, ; and Armah, et al, The findings also reveal that a little above half of level PST in E.
P College of Education , Bimbilla were found to be operating at van Hiela level 1,2,3 and 4 Also only a few were at level 3 and 4 which is consistent with other studies by Anas, ; Armah, et al, ; Halat , ; Erdogan and Durmus, ; Halat and Sahin, ; Pandisco and Knight who also recorded very low performance in van Hiele levels 3, 4 and 5. If van Hiela level 4 is the pre-requisite for students leaving SHS and majority of level PSTs are below this level, then something serious needs to be done about teaching geometry at the SHS.
Basic school teachers need to be at level 3 and above if they should function well at the basic school Crowley, ; Usiskin, ; and Armah, et al, Further analysis has proven that 33 out of the level PSTs representing 2. This is to suggest that all level PSTs who are below the van Hiele level 3 will have extreme difficulty in teaching students at the basic school level.
Item by item analysis also revealed that level PSTs could easily visualize and identify Journal of Education and Practice www. It also revealed from item to item analysis that level PSTs lacked understanding of the relationship between different geometrical shapes. The PST also lacked the knowledge to create meaningful definitions and give informal arguments to justify their reasoning.
Lastly level PST lacked understanding in logical implication and class inclusion. These findings concur with Baffoe and Mereku and Armah, et al who observed that Van Hiele level 3 is most difficult for learners because class inclusion such as square being types of rectangles was not understood by PST only a few PST understood geometry reasoning at level 4 of the VGHT. It is discovered from this study that geometry teaching is essentially focused on level 1 and 2 and very little or no emphasis on the advance levels 3, 4 and 5.
No PST attained level 5. The analysis revealed that the minimum score was 3, maximum 16 , and a mode 8 The mean and standard deviation of the PSTs score are 8. The correct response rate for level 1 was PSTs who scored less than half of the total score was While those who scored above half of the total score was 33 9. Adolphus, T. Retrieved 25th February, Modern Mathematics Curricular, Design and Teaching.
Amman: Dar Hmad for publishing and distribution. Alex, J. Anas, S. Armah, R. Higher Education Research. International journal of Reaserch in Education and Science , 4 1 , DOI: International Journal of Mathematics and Statistics Studies, 5 3 ,pp.
Atebe, H. Unpublished Ph. D Thesis. Van Hiele levels of geometric thinking of Nigerian and South African mathematics learners. Polaki, T. Nyabanyala Eds.
However, Euclid's reasoning from assumptions to conclusions remains valid independent of their physical reality. To produce extend a finite straight line continuously in a straight line. To describe a circle with any centre and distance radius.
That all right angles are equal to one another.
Although Euclid only explicitly asserts the existence of the constructed objects, in his reasoning they are implicitly assumed to be unique. The Elements also include the following five "common notions": Things that are equal to the same thing are also equal to one another the Transitive property of a Euclidean relation.
If equals are added to equals, then the wholes are equal Addition property of equality.
If equals are subtracted from equals, then the differences are equal Subtraction property of equality. Things that coincide with one another are equal to one another Reflexive property.
The whole is greater than the part. Modern scholars agree that Euclid's postulates do not provide the complete logical foundation that Euclid required for his presentation. Main article: Parallel postulate To the ancients, the parallel postulate seemed less obvious than the others. They aspired to create a system of absolutely certain propositions, and to them it seemed as if the parallel line postulate required proof from simpler statements.
It is now known that such a proof is impossible, since one can construct consistent systems of geometry obeying the other axioms in which the parallel postulate is true, and others in which it is false.
Many alternative axioms can be formulated which are logically equivalent to the parallel postulate in the context of the other axioms. For example, Playfair's axiom states: In a plane , through a point not on a given straight line, at most one line can be drawn that never meets the given line. The "at most" clause is all that is needed since it can be proved from the remaining axioms that at least one parallel line exists.
Methods of proof[ edit ] Euclidean Geometry is constructive. Postulates 1, 2, 3, and 5 assert the existence and uniqueness of certain geometric figures, and these assertions are of a constructive nature: that is, we are not only told that certain things exist, but are also given methods for creating them with no more than a compass and an unmarked straightedge. For example, a Euclidean straight line has no width, but any real drawn line will. Though nearly all modern mathematicians consider nonconstructive methods just as sound as constructive ones, Euclid's constructive proofs often supplanted fallacious nonconstructive ones—e.
Euclidean geometry also allows the method of superposition, in which a figure is transferred to another point in space. For example, proposition I. Some modern treatments add a sixth postulate, the rigidity of the triangle, which can be used as an alternative to superposition. The angle scale is absolute, and Euclid uses the right angle as his basic unit, so that, for example, a degree angle would be referred to as half of a right angle.
The distance scale is relative; one arbitrarily picks a line segment with a certain nonzero length as the unit, and other distances are expressed in relation to it.
Addition of distances is represented by a construction in which one line segment is copied onto the end of another line segment to extend its length, and similarly for subtraction. Measurements of area and volume are derived from distances. For example, a rectangle with a width of 3 and a length of 4 has an area that represents the product, Because this geometrical interpretation of multiplication was limited to three dimensions, there was no direct way of interpreting the product of four or more numbers, and Euclid avoided such products, although they are implied, for example in the proof of book IX, proposition An example of congruence.
The two figures on the left are congruent, while the third is similar to them. The last figure is neither. Congruences alter some properties, such as location and orientation, but leave others unchanged, like distance and angles. The latter sort of properties are called invariants and studying them is the essence of geometry.
The stronger term " congruent " refers to the idea that an entire figure is the same size and shape as another figure.Basic school teachers need to be at level 3 and above if they should function well at the basic school Crowley, ; Usiskin, ; and Armah, et al, The best item answered correctly by PSTs was 12 and the least performance was in item Also, 33 PSTs representing 9.
They aspired to create a system of absolutely certain propositions, and to them it seemed as if the parallel line postulate required proof from simpler statements. Apart from that 28 8. This is an indication that PSTs lacked knowledge of properties of a rhombus. In the present ICMI study, the whole spectrum of teaching and learning of geometry is analysed. Triangle angle sum[ edit ] The sum of the angles of a triangle is equal to a straight angle degrees.
That is being able to study Journal of Education and Practice www. Cape Coast: University of Cape Coast.