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8.3: Categories of Diversity - Mathematics


Figure (PageIndex{12}): Writer Malcolm Gladwell’s racial expression has impacted his treatment by others and his everyday experiences. (Credit: Kris Krug, Pop!Tech / Flickr / Attribution 2.0 Generic (CC-BY 2.0))

Figure (PageIndex{13}): Asia Kate Dillon is a non-binary actor best known for their roles on Orange Is the New Black and Billions. (Credit: Billions Official Youtube Channel / Wikimedia Commons / Attribution 3.0 Unported (CC-BY 3.0))

Table 9.1 The website Transstudent.org provides educational resources such as the above graphic for anyone seeking clarity on gender identity. Note that these are only examples of some gender pronouns, not a complete list.
Table Gender Pronoun Examples
SubjectiveObjectivePossessiveReflexiveExample
SheHerHersHerself

She is speaking.

I listened to her.

The backpack is hers.

HeHimHisHimself

He is speaking.

I listened to him.

The backpack is his.

TheyThemTheirsThemself

They are speaking.

I listened to them.

The backpack is theirs.

ZeHir/ZirHirs/ZirsHirself/Zirself

Ze is speaking.

I listened to hir.

The backpack is zirs.

Figure (PageIndex{14}): Our identities are formed by dozens of factors, sometimes represented in intersection wheels. Consider the subset of identity elements represented here. Generally, the outer ring are elements that may change relatively often, while the inner circle are often considered more permanent. (There are certainly exceptions.) How does each contribute to who you are, and how would possible change alter your self-defined identity?


8.3: Categories of Diversity - Mathematics

In radio, multiple-input and multiple-output, or MIMO ( / ˈ m aɪ m oʊ , ˈ m iː m oʊ / ), is a method for multiplying the capacity of a radio link using multiple transmission and receiving antennas to exploit multipath propagation. [1] MIMO has become an essential element of wireless communication standards including IEEE 802.11n (Wi-Fi), IEEE 802.11ac (Wi-Fi), HSPA+ (3G), WiMAX, and Long Term Evolution (LTE). More recently, MIMO has been applied to power-line communication for three-wire installations as part of the ITU G.hn standard and of the HomePlug AV2 specification. [2] [3]

At one time, in wireless the term "MIMO" referred to the use of multiple antennas at the transmitter and the receiver. In modern usage, "MIMO" specifically refers to a practical technique for sending and receiving more than one data signal simultaneously over the same radio channel by exploiting multipath propagation. Although this "multipath" phenomena may be interesting, it's the use of orthogonal frequency division multiplexing to encode the channels that's responsible for the increase in data capacity. MIMO is fundamentally different from smart antenna techniques developed to enhance the performance of a single data signal, such as beamforming and diversity.


July 20, 2014

The Place of Diversity in Pure Mathematics

Posted by Tom Leinster

Nope, this isn’t about gender or social balance in math departments, important as those are. On Friday, Glasgow’s interdisciplinary Boyd Orr Centre for Population and Ecosystem Health — named after the whirlwind of Nobel-Peace-Prize-winning scientific energy that was John Boyd Orr — held a day of conference on diversity in multiple biological senses, from the large scale of rainforest ecosystems right down to the microscopic scale of pathogens in your blood.

An Australian Curriculum for all students

The Alice Springs (Mparntwe) Education Declaration (Education Council, 2019) affirms the goals of the Melbourne Declaration (2008). The Melbourne Declaration’s goals provide the policy framework for the Australian Curriculum, to promote excellence and equity and enable successful learning opportunities for all students.

The ways in which the Australian Curriculum addresses these goals are detailed in The Shape of the Australian Curriculum Version 4 (ACARA, 2012). The propositions that continue to shape the development of the Australian Curriculum establish expectations that the Australian Curriculum is appropriate for all students. These propositions include:


October 23, 2011

Measuring Diversity

Posted by Tom Leinster

Christina Cobbold and I wrote a paper on measuring biological diversity:

As the name of the journal suggests, our paper was written for ecologists — but mathematicians should find it pretty accessible too.

While I’m at it, I’ll mention that I’m coordinating a five-week research programme on The Mathematics of Biodiversity at the Centre de Recerca Matemàtica, Barcelona, next summer. It includes a one-week exploratory conference (2𠄶 July 2012), to which everyone interested is warmly welcome.

In a moment, I’ll start talking about organisms and species. But don’t be fooled: mathematically, none of this is intrinsically about biology. That’s why this post is called “Measuring diversity”, not “Measuring biological diversity”. You could apply it in many other ways, or not apply it at all, as you’ll see.

It’s an example of what Jordan Ellenberg has amusingly called applied pure math. I think that’s a joke in slightly poor taste, because I don’t want to surrender the term 𠇊pplied math” to those who basically use it to mean 𠇊pplied differential equations”. Nevertheless, I suspect we’re on the same side.

Long-time patrons of the Café may remember a pair of posts in 2008 on entropy, diversity and cardinality. But those were long posts, a long time ago, and there’s a lot about them that I𠆝 change now. So I’ll start afresh.

We then record two things about the community. First:

The second thing we record is:

What we have to do now is take this data and turn it into a single number, measuring the diversity of the community. Actually, it’s not going to be quite as simple as that… but let’s take it one step at a time.

The average ordinariness of an individual in the community is, then,

This is greatest if the community is concentrated into a few very similar species. Economists have used the word concentration for quantities like this. Now, we’re after a measure of diversity, which should be inversely related to concentration. So we could define the diversity of the community as the reciprocal of the concentration:

This turns out to be a good measure of diversity. But it’s not the only good one.

Why not? I’ll give two explanations: one mathematical, one ecological.

This is a measure of concentration. Its reciprocal is

Technical note: in order for everything to be well-defined, you have to take the sums and max to be over only those values of i i for which p i > 0 p_i gt 0 (that is, over only the species that are actually present).

So, we’ve got not just one measure of diversity, but a one-parameter family of them:

Ecologically, this spectrum of diversity measures corresponds to a spectrum of viewpoints on what diversity is. Consider two bird communities. The first looks like this:


When datasets are graphed they form a picture that can aid in the interpretation of the information. The most commonly referred to type of distribution is called a normal distribution or normal curve and is often referred to as the bell shaped curve because it looks like a bell. A normal distribution is symmetrical, meaning the distribution and frequency of scores on the left side matches the distribution and frequency of scores on the right side.

The Normal Curve

Many distributions fall on a normal curve, especially when large samples of data are considered. These normal distributions include height, weight, IQ, SAT Scores, GRE and GMAT Scores, among many others. This is important to understand because if a distribution is normal, there are certain qualities that are consistent and help in quickly understanding the scores within the distribution

The mean, median, and mode of a normal distribution are identical and fall exactly in the center of the curve. This means that any score below the mean falls in the lower 50% of the distribution of scores and any score above the mean falls in the upper 50%. Also, the shape of the curve allows for a simple breakdown of sections. For instance, we know that 68% of the population fall between one and two standard deviations (See Measures of Variability Below) from the mean and that 95% of the population fall between two standard deviations from the mean. Figure 8.1 shows the percentage of scores that fall between each standard deviation.

IQ Score Distributions

As an example, lets look at the normal curve associated with IQ Scores (see the figure above). The mean, median, and mode of a Wechsler’s IQ Score is 100, which means that 50% of IQs fall at 100 or below and 50% fall at 100 or above. Since 68% of scores on a normal curve fall within one standard deviation and since an IQ score has a standard deviation of 15, we know that 68% of IQs fall between 85 and 115. Comparing the estimated percentages on the normal curve with the IQ scores, you can determine the percentile rank of scores merely by looking at the normal curve. For example, a person who scores at 115 performed better than 87% of the population, meaning that a score of 115 falls at the 87th percentile. Add up the percentages below a score of 115 and you will see how this percentile rank was determined. See if you can find the percentile rank of a score of 70.

Skew. The skew of a distribution refers to how the curve leans. When a curve has extreme scores on the right hand side of the distribution, it is said to be positively skewed. In other words, when high numbers are added to an otherwise normal distribution, the curve gets pulled in an upward or positive direction. When the curve is pulled downward by extreme low scores, it is said to be negatively skewed. The more skewed a distribution is, the more difficult it is to interpret.

Kurtosis

Kurtosis. Kurtosis refers to the tails of a distribution. A normal distribution or normal curve is considered a perfect mesokurtic distribution. Curves that have more extreme tails than a normal curve are referred to as leptokurtic. Curves that have less extreme tails than a normal curve are said to be platykurtic.

Statistical procedures are designed specifically to be used with certain types of data, namely parametric and non-parametric. Parametric data consists of any data set that is of the ratio or interval type and which falls on a normally distributed curve. Non-parametric data consists of ordinal or ratio data that may or may not fall on a normal curve. When evaluating which statistic to use, it is important to keep this in mind. Using a parametric test (See Summary of Statistics in the Appendices) on non-parametric data can result in inaccurate results because of the difference in the quality of this data. Remember, in the ideal world, ratio, or at least interval data, is preferred and the tests designed for parametric data such as this tend to be the most powerful.


DIVERSITY, DIFFERENCES, & SIMILARITIES CURRICULUM

The following is a list of diversity, differences, and similarities curriculum topics exploring race, gender, color, ethnicity, sexuality, religion, culture, ability, learning style, and all the diversity or our individual experience through exploration with our senses. We have divided it into 2 categories: “Relational Diversity, Differences & Similarities” and “Experiential Diversity, Differences, & Similarities.” If you have an idea or section that you believe should be included here, please send it to us using our Suggestions Page. Visit our Lesson Plans Page for lesson plans designed to integrate this curriculum.

NOTE: Curriculum with asterisks is curriculum we consider “core curriculum” that is central to the lesson plan design process. To identify these subjects we adopted the California Department of Education guidelines (because they are some of the most comprehensive and strict in the nation), added in the national SAT/ACT expectations for grade 7 and beyond, referenced the US Common Core State Standards Documents, and then double checked our curriculum and goals against the best selling and #1 teacher recommended Spectrum Test Prep workbook series. Curriculum with “*” is curriculum identified through these methods as foundational to 1st-6th grade learning. Curriculum with “**” is curriculum identified through these methods as foundational to 7th-12th learning.

Relational Diversity, Differences, & Similarities

PEOPLE AND THEIR RELATIONAL DIVERSITY, DIFFERENCES, AND SIMILARITIES
  • What is diversity? Value of diversity*
  • Learning about similarities through experiencing differences
  • Learning about differences through experiencing similarities
  • Reasons for diversity in the world*
  • Differences and similarities: the need for both*
  • Differences and similarities of the races of the world*
  • Differences and similarities of ethnicities of the world
  • Differences and similarities of cultures of the world*
  • Differences and similarities of world religions*
  • Differences and similarities in appearances and biological reasons for them
  • Differences and similarities of gender
  • Differences and similarities of sexual orientation
  • Differences and similarities of communication style (see also Communication Curriculum)
  • Self-identity and how we identify others and with others (see also Community Curriculum)
  • Learning about YOUR own uniqueness through understanding relational diversity
  • Differences and similarities in individual values and beliefs – all perspectives are right in the eyes of the individual with the perspective
  • Diversity awareness and celebrating differences and diversity (see also Community Curriculum)
  • Stereotypes -why do people produce them?
  • Learning to identify biases and overcoming them
  • Caring about expanding diversity while preserving uniqueness
  • Forming groups while preserving and promoting diversity
  • See also Freedom and Celebrating Other Perspectives curriculum !

DIVERSITY, DIFFERENCES, AND SIMILARITY RELATIONS IN NATURE
  • What are species diversity, ecosystem diversity, genetic diversity, etc.?*
  • Differences and similarities in animals and other living creatures*
  • Differences and similarities in plants and other living things*
  • Differences and similarities in earth, minerals, and other non-living things*
  • Promoting and preserving diversity in nature (see also Interconnectedness Curriculum)

Experiential Diversity, Differences, & Similarities

PERSONAL DIVERSITY, DIFFERENCES, AND SIMILARITIES TOPICS
  • Exploration of individual biological rhythms
  • Differences and similarities of abilities, personal talents, gifts, etc.
  • Differences and similarities of learning style (see also Multi-Intelligence)
  • Learning about YOUR own uniqueness through understanding diversity
  • Diversity integration training – how to learn faster by mirroring something that already works
  • Exploration of the nature of our reality and past individual life experiences that help us define it
  • Personal preferences, personality traits, and individuality

EXPERIENTIAL DIVERSITY AND DEVELOPMENT OF THE SENSES

Exploring the Diversity, Differences, and Similarities of Seeing Things:

  • Learning about vision and ways to develop and improve it
  • Developing the ability to see shades and hues of colors
  • Exploration of health modalities involving colors
  • Developing visual memory

Exploring the Diversity, Differences, and Similarities of Sound and Hearing Things:

  • Learning about hearing and ways to develop and improve it
  • Training and developing absolute hearing/absolute pitch
  • Exploration of health modalities involving sounds
  • Developing an ear for music

Exploring the Diversity, Differences, and Similarities of Touch and the Feel of Things:

  • Learning about the sense of touch and ways to develop and improve it
  • Exploration of health modalities involving touch
  • Touching and relationships

Exploring the Diversity, Differences, and Similarities of Taste and the Tasting of Things:

  • Learning about taste and ways to develop and improve it
  • Exploration of health modalities involving food
  • Developing the pallet and a taste for food

Exploring the Diversity, Differences, and Similarities of Smelling and the Smell of Things:

  • Learning about smell and ways to develop and improve your sense of smell
  • Exploration of health modalities involving the sense of smell
  • Individual smells and perfume choices

Exploring the Diversity, Differences, and Similarities of Equilibrium and Balance:

Exploring the Diversity, Differences, and Similarities of Intuition and Intuitive Sense:

  • Learning about intuition and ways to develop and improve it
  • Learning about the differences in logical and intuitive approach while practicing both

8.3: Categories of Diversity - Mathematics

Learning mathematics enriches the lives and creates opportunities for all individuals. It develops the numeracy capabilities that all individuals need in their personal, work and civic life, and provides the fundamentals on which mathematical specialties and professional applications of mathematics are built (Australian Curriculum Assessment and Reporting Authorities , n.d. ). It is important that individuals know and understand more than just the basic procedural skills of mathematics but also the concepts behind it. The following paragraphs elaborate on and discuss the two approaches of teaching mathematics, the learning ideas used in behind them and implications they place on teachers for teaching mathematics.

There are two teaching approaches to mathematics. They are behavioural and constructivist. The behavioural approach or behaviourism refers to a theory of learning that is focused on external events as the cause of changes in observable behaviours of students (McInerney & McInerney, 2010). Learning occurs from classical conditioning which means that any stimulus provided will lead to a particular response and operant conditioning is learning in which a voluntary behaviour is strengthened or weakened by consequences or antecedents (McInerney & McInerney, 2010). Students are taught in teacher-centred lessons or with direct instruction. There is a large possibility that the students will learn the procedural content and not the concept content. Students are also extrinsically motivated in this form of teaching. Constructivism is the opposite. Students actively engage in the lessons by asking questions based on prior knowledge to construct new knowledge and understanding. The knowledge they develop will have a contextual element that will allow it to be more meaningful to the students. Siemon, Beswick, Brady, Clark, Faragher, & Warren, (2011) define the constructivist approach as 'envisaging learners actively interacting with their environment: physical, social and psychological,' therefore the focus is on the individual as an active agent in the construction of mathematical meaning on the basis of the prior knowledge and experience they have. Inquiry or problem solving allows students to view content in a more realistic way as they analyse and create resolutions to the problems (McInerney & McInerney, 2010). The teacher becomes a facilitator of learning in this approach in contrast to being central to learning that is occuring. Both approaches can be linked to mathematics as they are useful in different ways. Direct instruction is useful for teaching the order of operations, new procedures and revising those procedures which have been taught previously. Inquiry is used for problem solving based questions where students are using prior knowledge to work their way to a resolution.

Applying these approaches in a mathematics classroom can be done through either explicit teaching or an inquiry lesson. To begin with, explicit teaching (also known as direct instruction) involves teachers beginning the lesson by spending time modelling the desired learning for the lesson or introduction that clearly states the procedure of the lesson. The teacher then guides students through classroom problems by applying instructional problem solving methods that involves ‘the problem to be analysed and interpreted relative to context’ (Siemon et al, 2013). This is consistent with the behaviourist approach in the way that students are conversing with mathematical concepts and developing their own conceptual knowledge and understanding, while the teacher observes students’ reaction to content. Inquiry lessons on the other hand are based on the principles of the constructivists approach. It usually involves an introductory activity that makes connections between student’s prior knowledge of mathematical content and strategies so that teachers can clarify the level of learning expected of students. Also, an inquiry lesson involves student reflection that provides another opportunity for teacher assessment of student development. Although both approaches are different, it is beneficial for teachers to use a combination of these approaches to attain optimal results from students and successfully teach a diverse class.

However there are some issues associated with these approaches when used in the classroom. The behaviorists approach for one, although effective for teacher-centered lessons, there are issues in regards for student learning. In the behaviorists approach, direct instruction plays a vital role in teaching. It is imperative that there is good communication within the classroom. If the teacher does not effectively communicate with the students, then there will be a lacking in student understanding and they in turn become disengaged and bored (Killen, 2003). Another key issue with the behaviorists approach is that it is very difficult to cater to the various learning demands of students during direct instruction as each student learns in a different way to their peers.

The constructivists approach on the other hand is more learner-focused and can cater to these issues. As noted previously, the constructivist approach is a learner-centered approach that allows students to engage and expand on their own knowledge. Consequently this means that students are learning at their own pace and to their own learning interests, even if this means not answering to content descriptions. It also means that students might develop their own conceptual understanding of concepts and ideas. Students in turn become focused on their own interests instead of what the task is asking. This then means that student outcomes might not be met, or will not be achieved at a high academic level. As this approach is primarily learner-centered, there are ample opportunities for students to work in pairs, groups or in a whole-class discussion that can lead to ‘lack of student involvement and boredom’ (Marsh, 2010, p.137). Also, it is very easy for unconfident students to be dominated by confident students during group work or activities. As viewed in this paper, the implications for these approaches are both positive and negative.

In conclusion, there are benefits to using both direct instruction and an inquiry lesson in a mathematics classroom. In these lessons there are some issues associated with the learners, learning styles and learning outcomes. Looking at the deconstruction of the behavioural and constructivists approaches, it is beneficial to have aspects of both to maximise the learner engagement and achievement. To accomplish this, there were several strategies that teachers can use in their own pedagogies outlined in this paper to achieve high levels of engagement and achievement.


Watch the video: Perform Quality Control Process. Project Quality Management Knowledge Area (December 2021).