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Rief & Heimburge, (2006) indicates that establishing a profile of the student strengths, weaknesses and capitalising on these strengths all should be supported by planning tasks that will be better suited to that child's ability. The student used in this report is having difficulties with the concepts, of patterning and its functions. Identifying objects or numbers that can be described in sequence, order, arrangement, number patterns and sequences based rules that involve repetition, and addition or subtraction rules. E.g. identify and continue the pattern in 2, 4, .6,, and 10. A series of activities will be conducted with the students to help in this area of need. "Solving problems in everyday activities could be considered a main aim of mathematics." (Foreman, 2008, pg.304). Numeracy is the understanding of mathematical knowledge, concepts and ability so that the individual can apply it to real life situations.
Formative assessment refers to a number of ways that we can uncover student ideas or knowledge about concepts important being taught therefore teachers can adjust instructions to the needs of the students. Gathering evidence of students' understandings and knowledge, teachers use a combination of methods including student observation, listening to student discussions and student responses from questioning. The purpose of this technique is to improve quality of student learning and can also provide important information to examine if the learning goals and objectives are met in all sections of the content. Effective teachers constantly monitor for engagement and when they identify that students are not engaged, they do something to change it, which is constantly applying varies feedback through praise. Summative assessment gathers evidence of a student's deep understanding about a unit of work and it is vital that teachers know if this has been grasped so students can go further into understanding skills, rather than just making sense out of something. To develop students into critical thinkers they need to have 'deepened their understanding to eventually use information in new ways' (McMillan, 2007. Pg.194). Assessment is a term used to 'describe the activities undertaken by a teacher to obtain information about the knowledge, skills and attitudes of students' (Reys et.al 2009, pg. 66). Gathering evidence of a students' understanding and knowledge, teachers use a combination of methods including student observation, listening to student discussions and student responses from questioning. "Assessment is to gauge students' learning, diagnose weaknesses in understandings, and adjust instructions as needed." (McMillan, 2007, p.117). The purpose of this technique is to improve quality of student learning it and can also provide important information to examine if the learning goals and objectives are met. Developing confidence in using patterns is essential. Because the knowledge of patterns is fundamental to mathematics, it is
important for the students to experience additional activities that teach the same concepts before moving on. (National Council of Teachers of Mathematics, 2010).
Teachers should identify these principles when they plan to meet with the diverse learning needs of students. Department of Education Queensland (2010) states that planning intervention for the diverse learning needs of students, teachers should consider the: relevance of the learning, interest of student in the topic and/or activity, engage students (intellectually, physically, emotionally and socially) and sustain their attention and participation for deep learning to occur potential of the activities to actively and ways in which students can experience success.
Profile of the student.
Child Abbey WALKER AGE: 8
An interview with Abbey's father, Mr. Mathew WALKER discussed they have moved from interstate, Abbey has become very shy and over time had lost focus on her work at school. Recently Abbey's vision was also affecting her learning. After seeing an ophthalmologist she was diagnosed with astigmatism. Astigmatism is a vision problem that causes objects to appear distorted and blurry. By the time Abbey was diagnosed she had already started school and still found learning challenging, she became frustrated angry and confused. A short informal conversation was conducted to get a better picture of how the student feels about school, home life, and other interests. "The affective traits component consists on a feeling associated with an object or persons." (McMillan, J.2007, p298). This sort of information will allow the teacher to get a better understanding of likes and dislikes, behaviour and other traits, which will help when planning activities for this student. McMillan (2007) discusses how teachers need to identify positive and negative attitudes that can be related to current and future behaviours. This is an important step because; knowing the profile of the student and identifying their traits, assists with the planning of tasks that are better suited to the child's ability. The student lives in a stable and caring environment, plays tennis, dances and occasionally plays soccer. Student enjoys art and craft, but does not like mathematics. Loves going on camping trips with family, and likes cooking yummy cakes with mum.
How the student learning difficulty was identified
Some students have trouble making meaningful connections with mathematical experiences. Westwood (2000) expresses that assessments should help uncover the reason why students are having difficulties with a particular concepts, processes or strategies. To discover what knowledge and understanding the student has about patterns, the student and the teacher had a conversation on what patterns were and how patterns can be found in everyday items, like puzzles or making a beaded necklace. We investigated together the different ways patterns and functions could be used, these included the sequence or order of objects that grow or repeat. After discussion the student made patterns from various materials (See appendix 0 activity B- perfect patterns). During the assessment task the observer had to identify any errors or difficulties, what strategies the student used to demonstrate understanding. Westwood (2000) states" not only do students dislike school mathematics they lose confidence in their own abilities". The student lacked some confidence in performing the tasks. She kept asking if what she was" doing it right" looking for reassurance.
The student's body language and facial expressions changed when she was unable to explain or identify the meaning of a pattern and its functions. While observing this change it communicated that the student did not understand the concepts completely. "Like facial expressions, voice, body language, communicates messages. By paying attention to these gestures, teachers are able to confirm whether students have a complete or partial understanding of something." (McMillian, J 2007, pg.124).
Also established that the student had language difficulties with explaining and communicating answers and questions, Student has a basic concept and understanding of patterns, has the ability to identify repeating patterns, but was unable to express the patterns rules or functions. (See appendix 0 Activity A and C titled house numbers and growing patterns).
Westwood (2000) explains that good assessment practices begin with establishing appropriate standards and learning targets. Learning targets are derived from the content that students should know, understand and be able to do, as well as criteria that indicate performance. Patterns and functions form the base of mathematics,
because recognising patterns and describing their relationships mathematically by number sequences, and functions, helps us interact with and make sense of our world. (The National Council of Teachers of Mathematics, Inc, 1999).
After gathering information from activities and identifying the students strengths and weaknesses, it is time to plan appropriate teaching, learning and assessment methods that will further the students current ability level in algebraic number patterns and functions. "Good quality teaching of numeracy requires a skill blend of student centered activities, enquiry, discovery, discussions, relevant practice and meaningful application" (Westwood, 2000, pg. 31). As teachers, the need to communicate the purpose of each learning activity is so that student understands what is expected to do as well as build on new concepts.The core content of this plan is for the student to create patterns into number patterns, identify and describe the relationships between input and output function machine rules, identify and explain the missing parts in repeating and growing patterns, as well as use the rule to determine the next, other, or missing number in a pattern.
The lesson plans to do to help the student learn.
(Algebra- Number Patterns sequencing and Functions)
Objective is for the student to identify,
The relationship between objects or numbers can be described using patterns and simple rules. At Year 3, Students use rules to create and describe number patterns based on addition and subtraction. They identify number sequences that are not patterns. They complete missing parts of, or continue, a number pattern when given the rule. (Queensland Studies Authority, 2010).
Number patterns and sequences based on simple rules involve repetition, order and regular increases or decreases
Simple relationships between objects or numbers, including equivalence, can be represented using concrete and pictorial materials
Simple relationships between objects or numbers can be described in terms of order, sequence and arrangements.
1. Learning Activity one.
Students will identify repeating patterns and recognise and create elements of repeating patterns
Create a repeating pattern, using a variety of concrete materials.
Developing confidence in using patterns is essential. Because, the knowledge of patterns is fundamental to mathematics, it is important for the students to experience additional activities that teach the same concepts before moving on. (National Council of Teachers of Mathematics 2010).
Teacher: "Ask the student to describe why their item makes a repeating pattern?"
Student "The pattern is the same thing over and over again."
Teacher: "When it is the same thing over and over again we call that a special word the starts with R, it is called repeating." "Great job."
Teacher "Can you show me which part of the pattern is repeating?
Student- "The yellow and red ones."
As seen in Appendix 1 and 2, she clearly has demonstrated how a pattern sequences based on simple rules involving repetition. Teacher is modeling language to ensure the student develop on their language skills.
2.Learning Activity Two
Create and continue repeating and growing patterns, using concrete materials, describe the growing pattern. Write/say the function (rule).
Discuss a growing pattern and show an example using concrete material to demonstrate how growing patterns requires rules to make it grow. Model how using addition makes my pattern grow, (talk about what rules) so by adding on the rule for making my pattern to grow is adding two more. Student experiments with buttons to create their own growing pattern. Discuss and write down what the rule would be (E.g. add two buttons on to make the triangle shape grow.) As student creates their own number patterns, it will reinforce their understanding of the concept of patterns.
Discussion with student on what she had noticed about her pattern. See Appendix 3
Teacher: "What did you do to make your pattern grow?"
Student, "You need to add more buttons to make it longer each time"
Teacher: Can you tell me what the rule is for this pattern? How do you work it out?
Student, "I need to add 3 more, I can just see it, I will show you!" The student demonstrated that her pattern started at the three buttons. "I have three; I add three more on what I already have." Student explained what rules were needed to make the pattern grow.
The form of assessment used for this activity is both formal and summative. Observation and questioning techniques will be used to question how the student came up with the answers. Observing what the student does to work out specific answers and seeing what questions the student requires assistance with.
3. Learning Activity Three
Represent patterns by skip counting, and to be able to count in steps of different sizes. Counting on by 2's, 5's 10's rule and complete a number line identifying missing numbers.
Clapping hands counting in twos, as well as playing a quick game of hop scotch to recap what is meant by skipping numbers. Number lines drawn on the ground with chalk with missing numbers, student has to identify what the missing number should be. See Appendix 4
Part of formative assessment is using oral questioning to assess the students progress during this activity. Encourage lots of discussion to ensure students understand the concept of skipping or missing numbers,' 'Discussions are use to promote student questioning and exchange ideas and opinions to clarify issues, promote thinking, generate idea or solve a problem' (McMillan, 2007, pg.129). The teacher will allow sufficient wait time for answering questions Specific follow up questions will be used so the teacher can get a better understanding of how the student got the answer (McMillan, 2007).
4. Learning Activity Four & Five
Identify functions based on input and output numbers
Look for the pattern and use it to work out the function. Show a function machine with an addition function and input number, but no out number.
To play the input and output machine, identifying patterns in numbers. E.g. 5 is the input number, pattern spins requires you to add on 2 and needs to find the output number relating to that spin. (See Appendix 5) Using a wall number chart with reusable stickers to help assist with complete worksheet determining rules (write the rule of the missing number in the in and out boxes). Student is to write in the numbers that have a secret rule. To ensure complete understanding of the pattern rule functions (See Appendix 6 )
This task seemed to be a little more complicated for the student to complete on her own. Without the wall chart 1- 100, the student was unsure how to tackle the function machine task. Allowing the student to refer to the number wall chart assisted with her finding the answers. This task was readjusted to lower numbers to build Abbeys confidence. Encouraging her for her efforts, also built her confidence. Learning is about making sense of the activity, by providing a challenge that is appropriate to the learners' ability and experiences. Learners get better with practice, with continual repetition of the same method; can develop a deeper conceptual understanding. (Willis, 2004).
5. Final activity tasks
The final assessment tasks were given to determine relevant and purposeful mathematical learning experiences. (See Appendix 7 part A, B, C). The final tasks were to determine if the student developed deeper concepts, but also to identify if the student is achieving expected outcomes. The student in most cases was able to, describe and record number patterns using a variety of strategies and complete simple number sentences by calculating missing values. Student was able work mathematically when applying mathematical knowledge and skills in a variety of contexts. The final assessment task was to check the efficiency of the lesson plans, identify any concepts or procedures that may need to be adjusted.
Describe how the intervention went, and how well the student coped with the tasks.
The "Characteristics of assessment of learning, for learning and as learning." (McMillan, 2007 p.16). The goal and objective were to focus on building on the student's weakness of patterns and number functions. The activity experiences used manipulative material to assist with the learners mathematical concepts. Students construct a range of mathematical processes that requires several facets of learning these include problem solving; logical reasoning, and various thinking processes. "Assessment is to gauge students' learning, diagnose weaknesses in understandings, and adjust instructions as needed." (McMillan, 2007 p. 117). The students learning ability with particular tasks were successful and in others the learner had difficulties understanding several steps in a task. Although, in activity 4 and 5 the learner understood some aspects of the problem, she still had not fully mastered the strategy to solve these activities. The student became a little frustrated, but by encouraging and readapting activities to ensure that the learners related different strategies to assist with understanding and development. An example is identifying that number patterns require using either addition or subtraction. Once this strategy was recognised it built on the learner's confidence, it also assisted to expand new concepts. The learner explained and demonstrated her understanding of activity 1, 2, 3 which indicated that she reached the specific objectives.
The learning that occurred
The student developed new knowledge by connecting to various mathematical activities and investigations. Developing an understanding related mathematics is a way of thinking, reasoning and working. An example of this is in activity 2, the learner confidently showed her learning ability when questioned.
Question : What is the rule for this pattern?
Response, to make each new triangles to grow I need to add 3 more.
Language communicates ideas, describes concepts; student is developing mathematical terms through language. Language used in learning experiences, to build on a student's vocabulary. "Language is crucial to learning mathematics
because through discussions learners can come to term with mathematical ideas, develop ways of expressing concepts and processes and take on new ways of thinking." (Booker, G. Bond, D. Sparrow, L. & Swan, P. 2004, p. 23) Example is asking the student how much more do you need, or that pattern is repeating.
After teaching many activities a final assessment was taken to ensure that the full concept of pattern and number functions was understood.(See appendix 7 part c). The learner has successfully completed these tasks demonstrating that she understood this unit.
Following the Queensland curriculum the learning that occurred relates to the Queensland Studies authority curriculum (2010) states that by the end of year 3 engaging in relevant and purposeful mathematical learning experiences, students will:
â€¢ Develop the knowledge, skills and positive dispositions required to operate
confidently and competently in the areas of number, algebra,
â€¢ Think reason and work mathematically when applying mathematical knowledge and
skills in a variety of contexts.
â€¢ Patterns repeating and non-repeating patterns both number patterns and pictorial
â€¢Recognises the pattern
Evaluation/Reflection of your intervention.
"Evaluation involves professional judgements of the value or worth of the measured performance." (McMillan, 2007, p 23). Evaluating is taking the appropriate measures to understand the importance of students' performances and reflecting and analysing the learning process and learning outcomes. Westwood (2000) affirms that assessments should help uncover the reason why students are having difficulties with a particular concept, process or strategy. The student has demonstrated an acceptable level of competence in creating; continuing and recording number patterns operations. (See Appendix 7 A, B and C) A variety of strategies have been used to sequence, order, arrange number patterns and sequences based rules that involve repetition, and addition or subtraction. As students learn these strategies through practice, the teacher models less and students gradually take over the responsibility of determining which strategy to use. Students will become more independent learners (The Access Center:, n.d.). Strategies that include the use of manipulative materials to enhance learning concepts, related learning to real life experiences, an example of this is playing hop scotch to recap missing numbers activity. Using mathematical terms and language may help students understand the relationship between numbers and the words describing them.
Ensuring that learner is progressing toward stated objectives. Brookhart (2009) points out that multiple measures are characterised by using more than one score before making any judgements. We should not base our decisions onto one measure alone. Multiple assessment methods provide a broader widespread view of whether students have achieved the learning outcomes. Planning for individuals through the curriculum key learning areas and assessment should focus on their particular learning needs. Identifying and assessing several different traits requires a great deal of experience and practice as supported by .McMillian (2007, p 296). discusses how assessment of affective targets is filled with difficulties, and student affect involve both emotion and cognitive qualities. Reflecting on tasks allows teachers to re-adjusting activities to suit the learning level, but also provide challenging tasks. Killen (2005) article discusses how the results of assessments are used to reflect on what students have achieved and can be used to modify teaching programs to improve students' learning.
Providing progressive feedback allows the learners to achieve challenges at their ability. Fisher and Frey (2009) suggest effective feedback must begin with clarity about learning targets, tasks, and desired outcomes. Teachers should adjust and re-adjust programs to raise expectations of student achievement as students' progress. Effective teaching plans should meet the learning needs of students.
What might you change?
More scaffolding and modeling examples of various learning activities to demonstrate what is required, support the learner to understand tasks by rephrasing directions and questions, modifying the language in different mathematical terms. Example of this is when discussing the mathematical problems use language that relates to mathematics terms e.g. addition, plus, add or more. All these terms can be related to and develop a student's mathematical language. Have student summarise directions and questions, to ensure that complete understanding of what is required. Design an assessment tool for structured observation, a scale or checklist. Provide more quality feedback during the task activity.
Were the assessment tools effective?
(See appendix 8)
Assessing student's gives an indication what targets need to be addressed. An assessment tool allows a teacher to identify with a specific target, whether it is collaborative skills, cooperative skills or peer relationships that need addressing. Using different assessment tools measures are used to construct the aspect of students' achievements. 'It is essential that teachers explore students thinking, because assessments are a bridge between teaching and learning." (Wiliam, 2005 p.22).
During the time spent with the student assessing the student traits was a little difficult. A structured observation tool should have been prepared to record what was observed. An example of a structured tool would include a checklist or a rating scale. Unstructured observations which are making judgments through summative
assessments were effective during this experience, because you knew what you wanted to work on. McMillian (2007) states that "unstructured observations is usually open-ended, there are no checklists or scales for recording observations." The student and parent interview were effective for the teacher to identify and understand the student's needs. This is supported by McMillan (2007, p308) if you have an opportunity to directly interview students it allows students to express their feeling more openly. Another assessment tool included observations, and recording information was made immediately.
The future- What to do next with the student. What needs to be learned next?
Students use their existing understandings of mathematical concepts and processes. Queensland Studies Authority (2010) states that the learning and assessment focus for a learner is to understand that mathematics is a way of thinking, reasoning and working and they construct new knowledge by engaging in a range of purposeful mathematical activities and investigations. Under the scope a sequences of algebra the next lessons that are to addressed for this student, is how to
Record patterns in tables and on graphs and use relationships between the objects and/or numbers to identify the functions used to create the patterns
To improve, student could learn more complex number patterns involving multiplication and division or odds and evens.
Further develop on mathematical language.
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