## Differences between numeracy and mathematics

Discuss about the Report for Numeracy and Mathematics of Communicating Ideas.

Mathematics is concerned with communicating ideas, searching for patterns, and solving problems. It involves the ability to apply logical and abstract reasoning to answer certain kinds of problems. On other words, mathematics is a language that assists in relaying complex concepts and ideas in a concise and precise manner. The language used in mathematics is symbolic and results in the emergence of exciting discoveries by manipulating the statements. Mathematics extensively applies conjectures and patterns in studying change, space, structure, and quantity in an attempt to establish truth using suitable definitions and axioms. Therefore, mathematics is studied as a body of knowledge that entails statistical analysis, quadratic equations, and calculus.

Numeracy on the other hand, is a concrete concept that involves realistic approach to mathematics and concentrate on functional addresses as opposed to mathematics that is platonic and abstract by giving absolute truths concerning relationships among ideal objects. It is the disposition, confidence, and capacity to the application of mathematics to meet the needs of the civic, community, work, home, school, and learning life (National Curriculum Board, 2009). It goes beyond the confines of mere computation and includes fundamental skills such as interpretation of diagrams, chats, and data (Perso, 2011). It also involves logical reasoning and thinking, understanding and explanation of solutions and problem solving. Perso (2011) cites that numeracy entails the disposition to the application in content a permutation of various skills such as algebraic, statistical, graphical, spatial, and numerical skills and the underpinning mathematical concepts. Other concepts applied in numeracy include solid appreciation of context, general thinking skills, and mathematical strategies, and thinking. Therefore, numeracy is a capability meaning, a learner is either numerate or not. A numerate learner possesses the disposition and capacity to apply mathematical concepts in a wide range of contexts other than the confines of mathematics classroom. This means that numeracy is contextual and concrete and offers contingent solutions to real life problems.

Competence is a state or quality of being adequate in terms of functionality or possessing adequate skill, strength, and knowledge (Kendal & Stacey, 2003). The authors cite that Differentiation Competency Framework is a technological tool that supports teaching of calculus by working with several representations of functions. To assist the instructors in teaching differentiation, the multiple representations are employed in computing gradients of tangents and curves, difference quotients, and the application of symbolic differentiation principles. Learners however, require a broad range of skills to link and apply the representations in differentiation.

## Differentiation Competency Framework

According to the Kendal and Stacey (2003), Differentiation Competency Framework can be used for three functions. To start with, Differentiation Competency Framework can be used for numerical representation in a case where the differentiation has both rate of change and quotient. Such a differentiation involves tabular presentation of a given differential function by obtaining the respective rate of change that can be found by determining the average of change over a suitable interval of x [x, x+h] = ( f (x + h) − f( x))/ h. Therefore, numerical differentiation representation entails the computation of arbitrary infinitesimal quotients known as numerical derivatives.

Secondly, Differentiation Competency Framework can be used for graphical differentiation. In a general sense, differentiation involves the determination of the gradient of a given curve. The gradient or slope can be interpreted to mean the undulation of the line represents the given curve at the highest resolution. In most cases, the gradient of the tangent or derivative to the curve is an estimation rather than precise. Thus, cannot be detected in a normal introductory problem. Thirdly, Differentiation Competency Framework can be used in symbolic differentiation. The derivative of a function from the perspective of symbolic representation can be determined by standard rules by using CAS or by hand.

Brendefur (2011), a mathematics professor at the Boise State University notes that one of the most effective forecasters of the later success in mathematics is the capacity to manipulate stimuli visually. The professor also believes that this concept of spatial reasoning is of unparalleled importance in mathematics in which 3-D, 2-D shapes are broken apart and put together, and the ideas are combined together, twisted, and turned where there are changes in the orientation of the object. The author mentions that such a model can effectively be used as a predictor of the afterward success in mathematics and can be acquired through practices and methods. The most important aspect of this model is that is assists the learners in developing fluency in regards to arithmetical operations and is therefore, necessary in strengthening the various measurement concepts. In his explanation, Brendefur reiterates that mathematics is not merely about notations, symbols, and numbers but stretches to encompass spatial reasoning. According to Uttal and Cohen (2012), spatial thinking is a class of reasoning skills that encompasses the ability to think about objects in 3-D and to draw inferences about such objects from partial information available. A learner that possesses good spatial qualities might also possess remarkable thinking qualities about how an object/body will appear when given a rotation.

## Brendefur and the Spatial Reasoning

There are five levels of Van Hiele’s Geometric Thinking.

**Level One (Visualization)**

In this level, learners identify and recognize figures by appearance only, usually by comparing such figures to a familiar prototype (Genz, 2006). The attributes of the figure are not comprehended. In this level, learners make decisions on the basis of perception as opposed to reasoning. Examples include flipping, sliding, and rotation.

**Level Two (Analysis)**

In this level, learners view figures as collections of characteristics. The learners can distinguish and name attributes of geometric figures although they cannot establish the relationships between such attributes. In describing an object, a learner in this level may enumerate all the attributes that he knows although he may not tell apart which attributes are crucial and the ones that are adequate to describe the object/shape (Genz, 2006). Examples include translation, and reflection.

**Level Three (Abstraction)**

According to Genz (2006), in this level, learners can identify the relationships between attributes and between figures. This implies that learners can establish momentous definitions and offer informal arguments to rationalize their thoughts. This is because they understand the class inclusions and logical implications like square as a form of a rectangle. The learners however, do not understand the function and implication of formal deduction. A good example is the ability to calculate the area of a triangle having mastered how to calculate the area of a rectangle.

**Level Four (Deduction)**

In this level, learners are able to assemble proofs, appreciate the function of axioms and definitions, and discern the importance of essential and adequate conditions. This implies that learners are able to develop proofs similar and typical to the ones found in the geometry class in a high school. A good example is establishing the similarity and congruence in triangles.

**Level Five (Rigor)**

In this level, learners are able to comprehend the formal facets of deduction, for examples, in constituting and comparing mathematical models and systems. The learners can comprehend the application of oblique proof and proof by contra positive analysis. Further, the learners are able to appreciate the non-Euclidean models at this level. A good example is being able to develop theorems without referring to figures.

According to Brown (2009), the state of quantitative disciplines like the mathematical sciences in Australia has been deteriorating to a precarious degree and continues to deteriorate. The deterioration can be attributed to the deterioration in the quantitative skills in Australian curriculum. The quantitative skills entail the numbers and how they are applied for data analysis, recording, and measuring. The concept also implies the performance of statistical or mathematical calculations. The quantitative skills are necessary as a foundation for higher learning such as in mathematics, engineering, biology, chemistry, and physics. Unfortunately, the skills have not been improved although identifying and developing functional science programs in Australia could achieve this. The reason is that most learning institutions in Australia continue to struggle to comprehend the process of integrating quantitative skills within the curriculum especially in modern science to reflect their quantitative and interdisciplinary nature. Another reason is that the Australian curriculum is not future looking and innovative. This greatly hinders the implementation of quantitative skills in the learning programs and models designed for the various levels.

## Van Hiele’s Levels of Geometric Thinking

There are several advantages of learning mathematics with technology. To start with, technology makes the process of learning mathematics more interactive. This allows for creating learning experiences that are dynamic. Through technology, learners can receive and share information among themselves and with their teachers such as the use of computer and mobile phone. In this manner, the learners are able to get feedback from their instructors even when outside the classroom setup. The incorporation of technology in learning mathematics boosts communication between the teachers and the learners. For instance, the use of computers in classroom allows learners to collaborate, interact, and present their learning outcomes. This trains the learners to be publishers, editors, writes, and readers (Borovik, 2011).

Another reason for incorporation of technology in learning mathematics is increased adaptability and flexibility to differentiated learning using devices like vodcasts and podcasts. This helps the learners to study at their own pace in private. Besides, the slow learners and those with learning disability can also study at their preferred pace without undue pressure from their faster counterparts.

**Part 1**

- The student did not follow the necessary mathematical logic
- The student directly multiplied the corresponding numerator and denominator yet this is a problem involving division.
- I would help the student understand that division involving fractions requires the multiplication of the first term by the reciprocal of the second term for instance,

2/3 ¸ ¾ = 2/3 x 4/3 = 8/9

**Part 2**

- The argument about the conversion is wrong
- The student misrepresented the fact about currency conversion
- I would take the student through the correct steps

$1 USD = $ 1.25 CAD

Therefore, $ 10 CAD = 10/1.25 = $ 8.0 USD

**Part 3**

- The student got the wrong answer
- The logic of converting square meters to square kilometers is wrong
- I would take the student through the following steps

A =250 x 500 = 125000 m^{2}

But 10^{6} m^{2 }= 1 km^{2}

Therefore 125000 m^{2 }= 125000/1000000 = 0.125 km^{2}

**Part 4**

- The answer is wrong
- The reasoning is not accurate since this is a case of probability
- The students should use the probability logic P (winning) = number of chances/ total number of lotto plays

**Part 5**

- The answer is wrong
- The student failed to follow a mathematical principle
- The student should use the identity (a – b)
^{2 }= a^{2}– 2ab + b^{2 }

Where a = x and b = 2

Therefore, the solution is x^{2 }– 4x + 4

**Part 6**

- The answer is wrong
- The student failed to factor in the gender in the calculation
- The student should include the gender in computing the solution

**Part 7**

- The answer is wrong
- The probability logic is inaccurate
- The probability is a definite number. In this case it is 80%

**Part 8**

- The answer is wrong
- The product of an odd number and an even number is always even
- I would give the student example like

3 x 2 = 6 (even)

5 x 4 = 20 (even)

- Correct answer

Gradient m, = Ã¢Ë†â€ y/Ã¢Ë†â€ x = (y_{2} – y_{1})/ (x_{2} – x_{1})

But (x_{1}, y_{1}) = (0, 2)

(x_{2}, y_{2}) = (4, 0)

Therefore, m = (0 – 2)/(4 – 0) = - 0.5

- Three possible wrong answers by the student
- -2 by dividing Ã¢Ë†â€ x/Ã¢Ë†â€ y instead of Ã¢Ë†â€ y/Ã¢Ë†â€ x
- 0.5 the student failing to include the –ve sign

iii 0.5 the student dividing 2/4 directly instead of computing the difference

**Part 3.2**

- Correct answer

Speed S = distance D/ Time T

S = D/T = 0.87 km/h

Convert distance into kilometers

1000 m = 1 km

Therefore, 350 m = 0.35 km

Time = Distance/Speed = 0.35 km/0.87 km/h = 0.4023 hours = 24 minutes

Possible answers by the student

- 402 .3 hours by failing to convert the distance to km
- 0025 hours by reversing the order of division (dividing speed by distance)

- 5 by directly multiplying speed and distance

**Part 3.3**

- Correct answer

Let the number of people be p

Let the number of tables be t

In one table there are 8 people

In 2 tables, there are 14 people,

The difference is 6,

Meaning, for every increase in the number of tables by 1, there is additional 6 people

Therefore,

1 table = 8

2 tables = 8 + 6 = 8 + 6(1)

3 tables = 8 + 6 +6 = 8 + 6(2)

4 tables = 8 + 6 + 6 + 6 = 8 + 6(3)

Meaning,

p = 8 + 6(t-1)

- Possible student answer
- p = 8 + (2t-2) since the difference is the first multiplied by 2 less 2
- p = t + 6(t-1) assuming the first tables carries 8 people

- p= 8t -2 since the difference between resultant product is less by 2

**Part 3.4**

- correct answer

angle sum of a triangle = 180

let the 3^{rd} angle be y

53 + y + 90 = 180

y = 37

but the triangles are the same and six

therefore, 37 x 6 = 222

angle sum at a point = 360

x = 360 – 222 =138^{0}

- possible answers
- 42 degrees by taking 53, multiplying by 6 and subtracting from 360
- 222 by multiplying 37 by 6

iii 37 by failing to recognize the 5 triangles

**Part 3.6**

- correct answer

P (even and multiple of 3) = chances of picking an even number that is a multiple of 3/ total number of chips

Even numbers that are a multiple of 3 are 6, 12, and 18 = 3 numbers out of 12

= 3/12 = ¼

- possible student answers
- 8/12 = 2/3 by considering all the even numbers
- 5/12 by considering all the multiples of 3

iii 4/12 = 1/3 by considering all the odd numbers

**Part 3.7**

- correct answer

let the short tree be x m tall

the tall tree will be (x + 2) m

ratio of the shorter to the taller is 5:9

x: x + 2 = 5:9

x/5 = x + 2/9

9x = 5x + 10

x = 2.5 m ( the shorter tree)

x + 2 = 4.5 m (the taller tree)

- possible student answers
- 6.5 m by reversing the numerator and the denominator
- -6.5m by misrepresenting the tall and short tree

iii 3.8 m by dividing 9/5 and adding 2

**References**

Borovik, A. (2011). Information technology in university-level mathematics teaching and learning: a mathematician's point of view. Research In Learning Technology, 19(1). https://dx.doi.org/10.3402/rlt.v19i1.17106

Brendefur, J. (2011). Spatial Reasoning and the Mathematical Mind | Beyond the Blue.Beyondtheblue.boisestate.edu. Retrieved 27 September 2016, from https://beyondtheblue.boisestate.edu/blog/2011/11/28/jonathan-brendefur/

Brown, G. (2009). Review of Education in Mathematics, Data Science and Quantitative Disciplines: Report to the Group of Eight Universities. Group Of Eight (NJ1). Retrieved from https://eric.ed.gov/?id=ED539393

Genz, R. (2006). Determining high school geometry students' geometric understanding using van Hiele levels.

Kendal, M. & Stacey, K. (2003). Tracing learning of three representations with the differentiation competency framework. Mathematics Education Research Journal, 15(1), 22-41. https://dx.doi.org/10.1007/bf03217367

National Curriculum Board,. (2009). Shape of the Australian curriculum. Carlton, Vic.: National Curriculum Board.

Perso, T. (2011). Assessing Numeracy and NAPLAN. Australian Mathematics Teacher, 67(4), 32-35. Retrieved from https://file:///C:/Users/gg/Downloads/amt67_4_perso.pdf

Uttal, D. & Cohen, C. (2012). Spatial Thinking and STEM Education. The Psychology Of Learning And Motivation, 57, 147-181. https://dx.doi.org/10.1016/b978-0-12-394293-7.00004-2

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