MathPPT Analysis and Reflection
Numbersand operations are the most fundamental focus at every level oflearning. It forms the core foundation for other mathematical areas.The development of number concepts and skills is developed over time.The skills are adopted for the day to day class routines as well asin other subjects. Just like any other mathematical process, itbegins with a basic concept of math standard and moves to moresophisticated as a child move from one grade to the next. Childrenfocus on counting and comparison At preK. Counting entails recitingsequence of number names. At kindergarten, they focus on representingand ordering the whole number, this entails the basic paths tomastering numbers and functions, they also do simple addition andsubtraction.At 1stgrade, children develop a more complex understanding and strategiesof addition and subtraction. At this level, they also begin to groupnumbers in 10s and 1s as a core concept of place value. At 2ndgrade,they learn more sophisticated facts and procedures of addition andsubtraction, they are also inclined to full understanding of thedecimal number system for place value. General learning paths fornumbers and operations are sub-categorised into five specific areas,namely subitizing, counting, comparing and ordering, early additionand subtraction, and composing numbers and place value. Skills andconcepts involved in Number and operation standard are Subitizing,Counting, Quantity, Change Operations, Comparison, Recognising andWriting Numbers, and Place Value.
Subitizingis ‘recognising the numerosity of a group quickly’ this isknowing how many quantity they are in a group with just one glancewithout counting. Subitizing is developed from preK to 2ndgrade.Counting is a skill of reciting the sequence at which the numbersoccur. It involves memorising just like alphabets. Counting isdeveloped as early as the age of 1 where they repeat some countingwords but has little meaning and sequence. At the age of 2-5, theylearn verbal sequence as they can do counting from 1-10, backwardsand beyond. At age 6, they develop skip counting and count usingpatterns to 100. They, however, count to 200 and count forward andbackward using their knowledge of place value. By age 4, childrenstart comparing by matching objects one-to-one with other objectswhere they can compare and mentally order small sets of 1-5 objects.At age 5 and 6, they count to compare and order sets of 1-10. At age7, they develop a concept of place value and can compare and ordernumbers to thousands. Addition and subtraction are portrayed as earlyas the age of 2 and 3, by age 4, children results using directmodelling and counting to find sums and differences. As from age 4-7,they develop certain strategies find results, the strategies becomemore diverse and can solve problems using combinations. Quantity isthe ability to know the amount of objects that are contained in a setof items by looking at the last number in a counting sequence.Comparison is a skill use comparison terms such as morethan, fewer than, greater than, less than, smaller than, bigger than,andsameas tohelp children in relating to or more values. Change operation is askill that produces new values from different input values todetermine a part or whole. There are two common change operations,and they are addto (join)and takeaway from (separate).Recognition and Writing Numerals is a skill where a child identifiesthe symbol and connects the symbol to the meaning and value of thenumber and being able to represent it on paper. Place Value isanother way of identifying part-part-whole relationships andunderstanding the base 10 number system by recognising the value ofeach digit in the number. It involves 1s, 10s and so on.
Measurementis an essential aspect in mathematics. It entails the ideas ofcomparing the following attributes, volume, area, and length. Justlike adults, the young also have an understanding of measurement andcomparison. At pre-K and kindergarten, children make comparisons ofobjects and recognise particular attributes of measurements. At 2ndgrade,children understand the linear measurements with standard units.General learning paths and developments are categorised by length,area and volume. Children begin to identify length and capacityattributes at age 3. At age 4, children compare lengths indirectlyusing strings, they start understanding area, and they directlycompare different volumes. At age 8, children can calculate volumes,and they can use a ruler to measure the length. Important skills andconcepts have been developed in the following areas in measurementcomparison and ordering capacity and volume length and areaweight time temperature conservation transitive reasoning unitmeasurement processes and procedures and estimation.
Withrespect to the different measurement attributes, children can developthe skill and concept of comparison and ordering by comparing things.Length is the linear measurement of an object and is measured usingrulers, tape measure, and meter sticks. Area is derived from itslength, it is obtained from more than one dimension and explains thecovered space within the linear dimensions. Capacity and Volume moreor less means the same in that capacity is the maximum amount thatfill a container while volume is the space occupied by an object.It’s measured in cubic units. Weight is the gravitational effect ona body it highly depends on mass which is the amount of matter thebody has. Time concept is developed in children and is learnt throughtheir daily routine and conversations. Temperature concept inchildren is enhanced by the use of these words hot, warm, cold, orfreezing to describe temperatures of common things surrounding thechild. Children can understand that particular objects might changetheir shape, but its relevant attributes remain the same thisability is known as Conservation. Transitive reasoning is theunderstanding of measurements such as when A is larger than B and Bis larger than C, definitely A is greater than C. Unit is used todefine particular measurement, can either be standard or nonstandard.Accuracy of the unit size is very critical in measurement.
Measurementprocesses and procedures entail choosing an appropriate tool and thenusing it to attain measurements which are accurate. Young childrenare taught about the appropriate tool to use for measuring a certaintribute. The children then learn the complicated procedure ofmeasuring using the appropriate tools. Teachers, therefore, have thetask of introducing the children to length measurements at first,then down to more complicated types of measurement. Estimation is a“good guess” based on the already known. Children use estimationin maths and other measuring activities. Words such as toomuch, way too high, close to, almost, orwaytoo high areamong the words used to make meaning during estimation. Estimationconcept is used on a day to day routine. Learning measurement forchildren is fundamental. Therefore they should grow in amathematic-rich environment. They are given opportunity to explorethe various tools of measurements such as balances, weights, clocks,scales, rulers, meter sticks, grid paper, measuring tapes,thermometers, containers, graduated cylinders and more. Childrenperformance is assessed by their teachers in the areas of measurementskills and understanding.
Patterns,Functions, and Algebra
It’sfundamental to teach the concepts of patterns, functions, and algebrato young children. Pattern is a part of nature, and it is a regulararrangement of objects, numbers and shape. Children experiencepatterns even before setting foot in a school. Although they can’trepresent patterns verbally or with symbols therefore teachers helpthem to recognise, define and enhance their understanding. Patternshelp children to understand mathematics better they help thempredict that what is about to happen and make connection betweendifferent mathematical concepts. Function is a special relationshipbetween different items. It builds on the understanding of patterns.Algebra is a mathematic branch that uses symbols to express rulesabout numbers and their relationship. It’s the fundamentallanguage of mathematic that uses symbols and numerals together inrepresenting relationship between objects and quantity. In children,understanding patterns, functions, and algebra is a continuousprocess as portrayed in the general learning paths and development.As early as age 2, children start recognising patterns. They begin toextend and duplicate simple patterns at age 4-5. Later on, theyexpress growing patterns numerically. Algebraic thinking helpchildren recognize patterns, make generalizations, and then usesymbols to represent problems and their solutions. Teaching ofpatterns occurs in a natural progression as follows Finding andrecognizing patterns, copying patterns, extending patterns, creatingpatterns.
Childrencan easily find and identify patterns. Teachers engage children in‘reading’ the patterns using simple vocabularies. Copyingpatterns might be a problem for children however teachers are thereto provide with familiar materials to copy the existing patterns.Extending patterns begins with children copying an existing patternthen extending them from the point at which they stopped. Teachershelp them by providing experience on how to extend the existingpatterns. Later on, children learn to make new patterns, this comeswith a deeper understanding and experience of patterns. Growingpatterns are ones that change from one value to another in a mannerthat is predictable. Symbols are used to describe mathematics, theyprovide a way to condense mathematical sentences into a logicalunderstanding of numbers and their relationship. Teachers are taskedwith introducing symbols to children and giving them meanings. Withtime and experience, children can make generalizations about numbersand properties. Teachers help them by promoting their recognition ofthese properties by strategizing to make a principle. Change can bedescribed qualitatively or quantitatively. Using comparison wordssuch as bigger,smaller, more, less, heavier, lighter, colder,orwarmer helpsdescribe qualitative changes. Children can also describe quantitativechanges using numbers, thus developing algebraic thinking as theydescribe and analyze the changes that occur on various objects. Manymaterials are made available in classrooms, this provides amathematical-rich environment to help children learn patterns.Teachers help children in recognising and working with patterns togive them a more understanding about numbers. Several activities arecarried out to promote patterns, functions, and algebraic thinking inthe classrooms.
Geometryand Spatial Sense
Geometryis the study of shape, size, direction, position and movement of ourworld. Spatial sense is the awareness of oneself in relation to thesurrounding environment. Knowledge and understanding these two arecritical steps in mathematics, therefore, it should remain a mainfocus during learning of maths in childhood years. They helpchildren understand their world and other mathematical standards.These concepts are also exhibited in other fields of study such asscience, social studies, movement and music, and reading andwriting. Shapes definitions are the core areas of geometry inclassrooms. Children work with shapes in activities involving art andpuzzles they construct shapes with blocks which offer great insightin exploration of geometry and spatial relationships. At preK,children recognise shapes and uses spatial vocabulary in describingtheir orientation and relationship with other shapes and objects.Atkindergarten, identification of shapes by their attributes andexistence in space and making simple directional steps. At 1stGrade,geometry studies involve dismantling and making up of shapes.Children also begin to comprehend such terms as symmetryand congruence.Connection of geometry to measurements and numbers are attained at2ndGradethis enables the measuring of these shapes’ dimensions. Duringchildrens’ development of geometry, the following Levels ofGeometric Thinking are considered Level 0, Level 1, and Level 2. Atlevel 0, children learn to identify the basic shapes to theirappearance and also relate shapes with those basic shapes. They startlearning the attributes of the shapes at level 1. At level 2,children can make connections between attributes of different shapes.Teachers play a crucial role in facilitating learning of geometry andspatial sense. They do this by relating the child’s informalknowledge to formal school mathematics. Teachers give the childrenopportunity to engage in creations using blocks for betterunderstanding of geometry. General learning paths and development ofgeometry and spatial sense include the following five areasrecognition and comparison of shapes composition and decompositionof 3D shapes and 2D shapes spatial orientation and spatialvisualization.
Childrenstart to compare shapes in their surrounding and match identicalshapes at an early age. They can construct whole shape from parts asthey get older and also recognise and group most common shapes withtheir geometric attributes. Children can make up 3D shapes, first byarranging blocks then their building skills develop further byrepresenting actual structures with models made by blocks. They alsoexperiment with individual shapes and can draw 2D representation ofthe real objects. Young children can learn the meaning of landmarks.They can also learn spatial orientation vocabulary such as on,inside, under, over, next to, between, with thisthey can locate objects using landmarks and with time they can followsimple routes indicated on maps. Spatial visualization is the abilityto recreate shapes after looking at them. Children can duplicate asimple structure and make shapes based on their memories of imagepatially.There are several Essential categories that children need to learnand improve skills and concepts on they include shapes concepts,spatial sense, transformations, and line symmetry. Spatial senseconcepts entail thinking spatialy where children visualize objects indifferent positions and imagining how they move in space. Line ofsymmetry divides an image into two equal parts so that one looks likethe image of the other. Transformation is an extreme change that toan object or a shape, changing its form and appearance.
DataAnalysis and Probability
Dataanalysis uses analytical and logical reasoning in evaluating data toexamine every component of the provided data. Probability is thelikeliness of something will occur it is represented as fraction,ratio, and the percentage. It uses terms such as impossible,maybe, and certain todescribe the chance of occurrence. Data analysis and probability arenot a key focus in child’s education and should not be treated asother mathematical standards explained above. However, there areplaces where they can be introduced in early child’s education, andthis includes collecting data, organising data, and displaying data.This engages the children with data analysis in a variety of contextsincluding daily graphing activities, classroom routines, questionbox, teachers questions and data from home and classrooms.
Generallearning paths and development of data analysis and probability isnot fully researched on children of young age. However, there is adevelopment sequence for sorting, this is an important skill neededin organizing and analyzing data. The sequence is categorised indifferent levels as shown. In level 1, children start sorting byseparating objects from a pool on the basis that they share aparticular attribute. At level 2, children can sort an entirecollection using one characteristic. At level 3, children sort acollection of objects in multiple ways using different attributes. Atlevel 4, children can give an explanation of rules of how they sortthe different objects giving evidence on how they did the sorting.Skills and concepts for the development of this area include thefollowing as shown. Posing questions and gathering data to answerthem organizing data, including sorting and classifyingrepresenting data using concrete objects, pictures and graphsdescribing and comparing data beginning to grasps concepts andlanguage of probability. Data analysis and probability skills andconcepts can be used in connection with other mathematical areas inthe curriculum such as social studies, science, literacy and finearts.
Childrenare curious creatures who pose endless questions. Teacher’s role isto nature the curiosity by engaging them in data analysis and surveysso that they get to answer themselves. This also helps the childrenunderstand better the process of data collection. Sorting andclassifying are fundamental processes to promote reasoning skills.Teachers determine how data should be organised prior directing howthe children should go to organising data. Visual displays are alsoused in the representation of data. Materials provided by teachers,enable the children to show organised information. The children canlabel and describe parts of what they are presenting. Childrendevelop comparisons skills by the help of teachers who asks questionsthat require children to make comparisons based on data given.
Teachersare tasked in providing a variety of materials useful in sorting andclassifying, data collection and analysis, and probability. Thisprovides the children with mathematic –rich environment.
Copley,J. V. (2000). The young child and mathematics. Washington, DC:National Association for the Education of young Children.