Measurements, Geometry, Statistics and Probability

Part 1 – Theoretical Background

All three of the subject strands of mathematics—number and algebra, measurement & geometry, & statistics and probability—include the competency strands of comprehension, fluency, dilemma, including reasoning. The proficiencies outline how the subject is studied or developed & underline the importance of using math in the material. They offer the terminology needed to incorporate the mathematical learning process’ developing components. The accomplishment standards cover the competencies and represent the material..

Understanding involves explaining the characteristics of distinct sets of numbers, describing probabilities using fractions and decimals, representing decimals and fractions in different ways including expressing relationships between them, and generating accurate estimates. Doing brackets correctly, translating among fractions & decimals, utilizing procedures with integers, decimals, & percentage, gauging using metric measurements, and comprehending timelines are all examples of fluencyfiguring out real-world issues using arithmetic, fractions, percent, and measures, analysing secondary information displays, and determining the magnitude of oblique angles are all examples of trouble. Describing mental calculating procedures, expressing findings for continuous numerical sequences, describing the conversion of one form into the other, and articulating why and how the actual outcomes of random trials may vary from anticipated outcomes are all examples of reasoning. The following measuring characteristics must be learned by the students: length, liquid limit, end time, force (weight), surface, loudness, and angles.

There are three distinct (but obviously linked) things that students must understand while considering the teaching and learning of measure.

As follows: Knowing the quality being measured (for example, what is length?)

Recognizing the measuring process (for instance, how can you express how lengthy something is?)

gaining knowledge of how to employ measuring devices, such as rulers.

The Following Are the Four Measurement Stages:

Each measurement feature is often taught and learned through four stages: explicit contrast, indirect comparing, comparing using random (non-standard) measures, and finally measuring using standard units. Although though they could devote less time on certain stages when they develop more skill with measuring, they still need to go through these phases with all of the measurable variables.

For instance, students studying solid volumes in will probably devote far less effort to the preliminary three steps than those that did when studying length. However, it is nevertheless crucial that students undergo these phases so they may develop a grasp of the what volume is, how and where to measure it, and exactly how to utilise the appropriate instruments. (Mills, 2022)

A subfield of mathematics called geometry studies the characteristics of space, how objects fit together spatially, and the shapes of particular items..

• Lines, dotted lines, & rays are used to define geometry.The greatest way to teach mathematics is through the show approach.

• This approach makes use of the instructor to illustrate the idea with tangible items. Therefore, utilising this way to illustrate geometry is simple.

• Drill Method: This technique enhances information retention by having students repeatedly practise and repeat an idea.

• Play Method: The emphasis of the play teaching approach is on getting kids to learn via fun games and activities. Thus, sometimes be evident that the presentation approach is the most effective way to teach geometry.

One of the most crucial areas of mathematics, geometries plays a crucial role in schooling. The majority of the things that we frequently see and utilise in our surroundings are made up of geometrical items & forms. Understanding their relationships is necessary for effectively using these items and forms. In addition, we employ geometrical concepts to define the area, manage our business, and solve difficulties (such painting, lining walls, etc.). Our professions and jobs include geometrical items and forms. Definition and comprehension of the relationship between both the item as well as its task are necessary for efficient usage of these items (Altun, 2004:217).

The two main types of geometry are intellectual geometry & pictorial geometry. These two groups call for various methods of instruction. The intellectual components must be seen, or in pictorial terms, turned into experience. According to (Duval, 1998), “their combination is intellectually crucial for mastery in geometry.” He asserts that there are numerous methods for dealing with the graphical components, particularly for explaining, imparting, and understanding the graphs: The operative method is used to identify the comment section for issue solutions; the discourse approach provides a comprehensive overview of the difficulties supplied; and the instantaneous perpetual approach refers to the understanding of the drawings.

Additionally, Fischbein (1993) views the formal & substantive elements, often known as that of the figural & conceptual, as opposite sides of the same coin. Since the instructors will discuss geometrical relationships between objects to one another whereas the figural portion will allude to any of these abstract things, the educators must pay special attention towards both the figural side as well as concept component.

Teacher preparation is a crucial component in the growth of statistics education. Statistics will be gladly included into lessons by well-prepared teachers. With the right training, teachers will feel more comfortable and also be able to motivate students to investigate occurrences, make their possess visualisations, test their own hypotheses, just use appropriate technology tools, as well as spend quality time on reflection and conversation rather than just focusing on teaching them how to do arithmetic and practical knowledge..

Teachers undoubtedly need to master statistics, grow in cognitive statistical inference, and receive didactic statics education. This will allow them to track their instructors’ understanding & thinking and proactively use events in the school environment to help students learn.

Additionally, it’s crucial to acknowledge the instructors’ worries about omitting any mathematical material by reassuring and demonstrating to them that, in fact, practising statistics involves a lot of algebra.

In order for one to foster the growth of one another, it is additionally vital to link mathematical principles to their implementations in statistics (Dunkels, 1990). By presenting mathematical ideas in relevant and engaging circumstances, statistics can aid in the learning of mathematics.

Points of interaction and conflict between both mathematics and statistics in the classroom include the measurement of phenomena (including such bullying, spare time, fertility, & unemployment), proportionate reasoning & percent, line graphs, average, simulation models, and making inferences (Biehler, 2008).

To learn how to make the potential uncomfortable meeting point of such disparate fields productive, research is required.

This may call for math and statistical teachers to collaborate together, displaying mutual respect and how the two disciplines’ knowledge and ideas may develop harmoniously inside the class (Ottaviani, 2008). Teachers should indeed be introduced to the usage of technology while undergoing statistics education. In reality, technologies may help students “do” statistics, “see” statistics, and “reflect” on data. There are several types of statistical techniques. Some are helpful for quickly visualising and analysing data, some are more suited for learning about information and data exploration, while yet others are better for grasping ideas related to probability density functions. In addition, the The Internet provides a substantial collection of data that may be downloaded to enhance exploratory analysis as well as to help with comprehending variation. (Garfield & Ben-Zvi, 2004).

Since the beginning of the millennium, various models that define crucial factors to take into account have been produced, despite the fact that few models have indeed been created to characterise actual professional knowledge of instructors to teach statistical and probabilities. Moore, for instance, points out that teaching these subjects requires the co-activation of content, teaching methods, as well as technology. The latter plays a crucial role in this co-activation because it directly links content and pedagogy by facilitating activities like visualisation, problem-solving, the utilization of various techniques, simulation models, etc. thru the the application of technology devices. Godino et al. consider the elements of (a) philosophical reflective thinking; (b) transition as well as adaptation of mathematical to a specific level of education; (c) knowledge of the problems, hurdles, and common error schoolchildren face once learning, and of the techniques used to resolve issues; and (d) outstanding example of didactic circumstances, teaching methods, and lesson plans when describing the discipline as well as didactic understanding involved in educating statistics as well as probability. Later, Godino et al. improved this paradigm, developing what they call statistical pedagogic information, which has five parts: teaching resources and methods, epistemology knowledge, cognition understanding, affective understanding, dialogic understanding, & affective understanding. These features could later be incorporated into the instructional knowledge paradigm..

The special characteristics of numerical schooling, that also necessitates unique way of thinking and logic, also calls for the need for a prototype which differentiates these distinctions from mathematical knowledge in general, according to Burgess, who bases his proposal on some of the elements as well as the prototype for trying to describe statistical thinking put forth by Wild and Pfannkuch.

Figure 1 The TKSI model’s constituent parts [18] (p. 4). (Reprinted from reference [18] with consent. (2006) Copyright, ICOTS 7.)

Vásquez et al. display a model to examine the understanding that teachers use when teaching possibility from an early in life (Figure 2), which takes into account five factors: stochastic tasks, decision theory, probabilistic interconnection, communication, as well as probabilistic dialect. This model is based on various models, like those described earlier.

Figure 2 Features and elements of a model of effectively introducing probability to young learners (Adapted from [9] (p. 7)). (Reprinted from reference [9] with permission. 2019 copyright, CIVEEST.)

Part 2 – Concepts / Strategies

Concept One: Measurement

how do you measure something why do you measure something why can’t you measure something in certain kinds of ways but they’re not the only type of tasks that do this so at the end we’re going to talk about the overall questions that well-designed measurement tasks should help students answer and so the first of these is called the button task and the idea here is that you’re giving students pictures of fake students work in measuring sheet of paper with buttons so you literally give students sheets of paper that had pictures like these on them and the hypothetical students should have done a couple of things wrong so in one case they should have had spaced out the buttons they were using and so you know these big gaps in-between each buttoning another one here they measured it diagonally with buttons trying to find the length obviously that’s not going to work here another example they used the buttons to-do like a snake pattern with them and then in one example they correctly did use buttons to measure just a cross to measure the length and while my drawings aren’t fantastic in this case imagine that all the buttons are drawn of equal size the reason the button tasks so strong is because it forces students to really reckon with questions of why measuring works or doesn’t work teachers should really encourage students defend their answers here which is the best way to measure that sheet of paper and why and that forces students to say that the correct answer works because you’re measuring the right attribute you’re measuring the length uh you can’t actually measure with spaces don’t work you can’t measure diagonally because it doesn’t tell you what you want to know things like that you’re really helping students to wrestle with very deep questions and very high level critical thinking questions of why measurement works the way it does the second task is called a foot task and this is when you have students do make a cut out of a tracing of their own foot and that’s the only tool they have to measure with for the purpose of this lesson so with their foot cut out you have them measure various things in the classroom measure their desks measure certain pieces of the floor things like that and by limiting them tone uh sort of tool of measurement you’re again really making them think about what does it mean to measure something when you don’t have a full length of it so and especially let’s say that there aren’t any lines let’s say they’re not measuring they’re measuring the descend the desk doesn’t have a lot nice lines in it like the floor tiles might what do you do with that how do you actually measure in that situation and why what are the challenges things like that.

Concept Two: Geometry (including explanation of an appropriate strategy)

The meet the expectation approach is one we use as children to learn from our surroundings. Their folks did not explain to you what a dog needs to survive. They used the term “dog” to describe a number of different creatures. These examples taught you how to construct the idea of “dog.” You could have mistaken a cat for a dog when you initially saw one since they share a lot of traits. With practise and a lot of instances, you now can glance at a dog & instantly “know” it’s a dog, irrespective of whether you’ve seen the whole breed of dog there is, how big or little the dog is, what colour it is, if it has four legs, etcetera.

Any new idea, including equality, geometric forms, art genres, etc., can indeed be explored using the meet the expectation technique.

The Method:

Preparation\s>

First, choose an idea.

Triangle, for instance:

Step 2: List the concept’s essential characteristics

For instance, triangles are closed figures with three straight sides and three vertices.

Step 3: Compile a list of illustrations and opposition.

Provide at least 10 triangular instances & 10 non-triangle examples.

Ensure that the complexity is raised.

Quasi of the rule: polygons with four sides, three sides, one curved line, and three sides that are not completed.

The Method: Working with Students

Make a “Yes” & “No” column in the first step.

If you’re using visuals, write “Yes” and “No” on the board. If you’re using physical items, set up “Yes” and “No” tent flags on a tabletop.

Step 2: Present two or three examples that are definitely “Yes.”

Show pupils an illustration of such an equilateral triangle as an illustration.

Put it in the “Yes” area. “This is a powerful yes example,” you say. Put this into the affirmative. Make an isosceles triangle visible. Put the answer in the affirmative and tell the pupils it’s a resounding yes. For a scalene triangular, continue.

Step 3: Give 2 or 3 concrete examples of “No”

Step 4: Present further illustrations and contraventions

Step 5: Have students debate and offer perhaps an instance or a quasi. Show a fresh quasi in step six.

Step 7: Rerun the previous steps. Students exchange, debate, and improve their ideas in step eight.

Concept Three: Statistics

The key technique Jean Piaget’s work served as the inspiration for the so-called operational approach. The foundation of something like the operative method is made up of his notion of “operation,” both the formal and real intervention, as well as the mental structure of “grouping.” This was created in a non-formal, non-deductive manner by Aebli (1963) & Fricke and Besuden (1970). The methodological concepts of Fricke have been further refined by certain friends just at Padagogische Hochschule Heidelberg (n.d.) who’ve already kept working on this level to make them more manageable for use at the elementary level. They had created the subsequent five ideas as a result.: The idea of associativity is when a job or issue is presented with the possibility of several solutions, or at the absolute least multiple approaches to finding a solution.

The concept of composition states that when two or more mathematical operations are combined, such as when multiplying by two followed by another two, the result is equal to multiplying by four.

The concept of reversal is when you double one side and cut the other half to transform a square into something like a rectangle with the identical area. The transitivity concept is employed when you make the following arguments: 5 + 3 = 8, 15 + 3 = 18, 25 + 3 = 28, etc.

a variational principle. Of something like the five pillars, this one is the broadest. It is frequently demonstrated by the development of easier neighbouring tasks as a method of completing increasingly harder tasks, or by the use of variety in the design of a work series, such as in the ones that follow: A trip is taken on bicycle by four pals. Uwe completes 30 km 2 Slow learners with statistical ideas 147 hours; Peter travels 30 kilometres in 14 hours, Jens 40 kilometres in 2 hours, Bernd 40 kilometres in l+ hours, and Peter’s parents drive after them. They were 3 times as quick as Peter, thus they’ll complete 60 kilometres in.

Concept Four: Probability (including explanation of an appropriate strategy)

The cause of the issue The United States’ state of Tennessee recently mandated that children must study arithmetic for at minimum two academic years in order to finish high school. Those 2 years will typically be made up of either 2 years of algebra or one year since mathematics and one year of geometry for many pupils. For those who are less advanced, two years of “applied maths” will enough to meet the criteria. As is to be expected that more “generic mathematics,” which places a focus on fundamental computing abilities, won’t be covered in the two years..

The chance of the entrance

It is impossible to include all the subjects and all of the experiences from dealing with both the resources and also the pupils in a little chapter. The following is a list of important subjects, with an emphasis on those that appeared to spark the most interest and zeal. a traffic issue It’s a good idea to introduce a challenge while conducting statistics surveys with pupils. Should the school put in a traffic light? That was the original quandary. The issue was a serious one because none of the other three high schools included in the research had traffic signals.

This involved measuring the number of vehicles at the start and end of the school, as well as on weekends and holidays, as well as the number of occupants in each vehicle. Questions about how to swiftly capture data and what it would cost to install traffic signals inevitably came up. Similarly, it became necessary to distinguish “truth” from “opinion” in order to make informed decisions, and the mechanics of conducting and interpreting surveys inevitably came up for discussion. It must have been subsequently decided that the necessity for traffic lights could not totally be justified, especially considering that they were only required during school hours because all three secondary schools are situated in rural or suburban regions. It appeared more effective to utilise county traffic officials to guide drivers.

Part 3 -Lesson Plan and Intervention Plan Digital Resource One:

Intervention Plan

Misconceptions in math are now referred to that as misunderstandings. What else do students consistently do wrong, and why? Our lack of role models and representations frequently results in misunderstandings. Numerous issues might arise as a result of the & symbolic nature underlying arithmetic. To comprehend what pupils are thinking, teachers must dispel myths in the classroom. Common misunderstandings might be found, for example, by strategically and purposefully making mistakes or encouraging learners to make mistakes.

Dallas (2015) Students think of capacity as just a fluid measure & volume as nothing more than a solid measurement. Determining perimeter, area, & volumes mathematically and determining their units can be challenging. They frequently think that measuring area could be done using rulers. They are taught to simply fill a place with the other unit & count them to determine the area. They frequently think that as long as those that don’t cross the boundary, it doesn’t issue if these additional units are equivalent in size units which are squared for non-square forms.

References

1. Norma Dennis, 2015 COMMON MISCONCEPTIONS in Space, Measurement, and Chance & Dat https://slideplayer.com/slide/6184965/

2. Mills, 2022 The Lesson Study Group https://lessonresearch.net/content-resource/measurement-2/

3. Altun, M. (2004). Matematik Öğretimi. İstanbul: Alfa Yayıncılık https://www.researchgate.net/publication/323373285_Perspectives_on_the_Teaching_of_Geometry_Teaching_and_Learning_Methods

4. Duval, R. (1998). Geometry from a cognitive point of view. In C. Mammana & V. Villani (Eds.), Perspectives on the Teaching of Geometry for The 21st Century (37–52). Dordrecht, The Netherlands: Kluwer.

5. Fischbein, E. (1993). The theory of figural concepts. Educational Studies in Mathematics, 24(2), 139–162.

6. Dunkels, A. (1990). Examples from the in-service classroom (age group 7-12). In A. Hawkins (Ed.), Training teachers to teach statistics: Proceedings of the International Statistical Institute Round Table Conference (pp. 102-109). Voorburg, The Netherlands: International Statistical Institute.

7. Biehler, R. (2008). From statistical literacy to fundamental ideas in mathematics: How can we bridge the tension in order to support teachers of statistics. In C. Batanero, G. Burrill, C. Reading & A. Rossman (2008).

8. Ottaviani, M. G., Peck, R., Pfannkuch, M., & Rossman, A. (2005). Working group report on teacher preparation for statistics education. In G. Burrill & M. Camden (Eds.), Curricular development in statistics education: International Association for Statistical Education 2004 Roundtable. Voorburg, The Netherlands: International Statistical Institute. Online www.stat.auckland.ac.nz/~iase/publications.

9. Garfield, J., & Ben-Zvi, D. (2008). Developing students’ statistical reasoning: connecting research and teaching practice. Dordrecht: Springer.

10. Bruner, J. S., & Anglin, J. M. (2010). Beyond the information given: studies in the psychology of knowing. Abingdon, Oxon: Routledge.

11. Bruner, J.S., Goodnow, J.J. & Austin, G.A. (1956) A Study of Thinking. Chapman & Hall, Limited. London.

12. Dean, C. B., Hubbell, E. R., Pitler, H., Stone, B., & Marzano, R. J. (2012). Classroom instruction that works: research-based strategies for increasing student achievement. Alexandria, VA: ASCD.