EDUC4731Catherine Day 2095749

Instruction for Students with Special Education Needs – Implementation of an Instructional Program

Introduction

The concept of backwards design is like choosing the holiday destination before planning the travel to get there. The

end result remains the same; like enjoying the sunny beaches of Surfers Paradise, regardless of the travel route or

means of transportation taken to get there. Planning a unit of work is the same; all effort should be geared toward

reaching the final destination - understanding the concept or process. Just like transportation, some methods are

better suited to certain travellers than others, meaning that research and care should be taken when planning a

variety of learning activities and lessons. Unlike the holiday analogy, there are various ways of assessing knowledge

which should also be taken into account during the planning stages.

The following paper discusses an instructional program implemented in a Year 8/9 Special Class. All names have

been changed for confidentiality reasons. This paper explores a different instruction program than one previously

discussed, as the pre-service teacher (author) made a professional judgement that the students were not ready for

that content (commas, full stops, exclamation and question marks). The following plan regards teaching

multiplication with elements of division.

Curriculum Area: Knowledge, Skills, Understandings

Know:

(Declarative

Knowledge)

1, 2, 3, and 4 multiplicative facts up to 12

x is the symbol for multiplication

÷ is the symbol for division

Multiplication can be arranged in groups and arrays

Terms: half, double, third of, triple, quarter, quadruple, array, columns and rows

Understand:

(Conditional

Knowledge)

Multiplication and division involves equal groups of a number

Multiplication in needed when making the number bigger; more ‘groups of’ the

original number

Division is separating one number into smaller but equal parts; the number will

become smaller

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There is a variety of language used for particular operations

Distinguish when and how to use multiplication and division

Do:

(Procedural

Knowledge)

Complete written multiplication questions

Arrange groups of numbers in arrays

Solve two and one digit numeral multiplication and division questions.

Use a calculator to solve multiplication and division questions

Achievement Standards

o Recognise and represent multiplication as repeated addition, groups and arrays (ACMNA031)

o Recall multiplication facts of two, three, five and ten and related division facts (ACMNA056)

o Solve word problems by using number sentences involving multiplication or division where there is

no remainder (ACMNA082)

o Develop efficient mental and written strategies and use appropriate digital technologies for

multiplication and for division where there is no remainder (ACMNA076) (ACARA, 2015)

Performance Objectives

1. Students will answer 6 random numeric multiplicative equations up to 10x10, five out of six times on a

worksheet.

2. Students will answer worded 2-digit double, triple, and quadruple questions, with or without manipulatives,

3 out of 3 times when given a worksheet.

3. Students will answer written 2-digit half, third, and quarter questions, with or without manipulatives, 3 out

of 3 times when given a worksheet.

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Rationale

Of a class of 12 students, two were absent on the day of this assessment, and only six students returned their

consent form. The class are of Year 8 and 9, but learning objectives reflect challenging objectives for the class. There

is variation in skill levels, however, at the commencement of the unit, no students were able to meet all of the

objectives. These curriculum areas are a logical next step in developing the student’s mathematical skills, and have

been prioritised as most relevant for their futures. ‘Double’, ‘triple’, ‘half’ and other such words are quite common

in daily life, so it is important to develop a good understanding to avoid future confusion.

Ideally, more than three questions would be used to gauge understanding of a concept. This particular class tends to

respond negatively when more than 12 or so questions are asked on any given worksheet. To combat this, short,

regular tests focussing on one particular concept could be given, although time spent on tests takes away valuable

teaching time (Salvia, Ysseldyke, & Bolt, 2013). However, when testing all of the above objectives together, the

alternative is to give fewer questions testing each objective, with a higher success rate (3 out of 3 times). The pre-

assessment and post-assessment deliberately share the same format and content, so that it is easy to measure

learning once the unit is finished.

Types of Knowledge Required

The teacher must have good declarative knowledge (Hoy, Woolfolk, Margetts, 2010) of the multiplication facts, to

quickly reinforce student responses. Other relevant parts of declarative knowledge the teacher must know are

concepts and principles. The multiplication concept is much easier to understand when the teacher can explain it in

other ways, such as counting rows and columns, or number of groups x how many in each group. Using these visuals

enhances students understanding as they are not solely relying on numerals and symbols. The principles of

multiplication and division need to be well understood by the teacher, so that the students are able to learn them

both together.

The content of this unit should be reflective of real-life problems the students may face, which require division or

multiplication to solve. A strategy used is asking questions such as:

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Tom has 30 chocolate freddos to share between 15 friends. How many freddos will each friend receive?

Or: iced coffees are on special at Coles for $3 each. How much would it cost if Jamie bought 5?

Not only are the questions attention-grabbing because of the use of popular treats, but also, using student’s names

to gain their attention is a good strategy for encouraging active participation. A teacher who has good declarative

knowledge should be able to come up with lots of relevant and interesting questions similar to those.

Of course, the teacher must have great procedural knowledge to teach how to multiply or divide numbers, and how

to use the calculator to solve numeric and written questions. If the teacher has not developed understanding of how

to do this, they should not expect their students to know, either.

Background knowledge Required

An awareness of the addition and subtraction operations is helpful, however, if this is not fully understood, explicit

instruction as well as games for automaticity are a component of this unit, along with the multiplication and division

operations. Calculator skills and interpreting results are also helpful to know, but are practiced and can be explicitly

taught if the need arises.

Students should begin this unit with a basic to proficient understanding of addition, to make connections to

multiplication through repeated addition (3+3+3+3 is the same as 4x3). This can later be expanded upon through the

use of arrays.

Basic English language skills are crucial to this unit. There are many different ways of describing each operation, and

some of these terms are used within the lessons and assessments. In written questions where the operation is not

specified, students need to be able to read and make sense of what is being asked.

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Pre-assessment

Pre-assessments are extremely useful to gauge learner’s readiness before beginning a topic (Tomlinson, & Moon,

2013). The pre-assessment used for this program (Appendix A) was adapted from a diagnostic assessment

developed by Booker (2011). The questions were altered to reflect the learning outcomes of the teacher. When

administering the test, it was found that the students were of a much lower skill level than originally estimated. The

learning objectives were then altered to reflect the theme of the unit (half/doubles, thirds/triples,

quarters/quadruples). The calculator skills and deciphering worded problem objectives remained unchanged

through this process. It is important to be flexible with the unit plans, to cater to the skill levels of the students.

Teaching prime numbers and indexes would have been too challenging for the students, who did not originally have

a clear understanding of multiplication or division.

The Learning Environment

All of the students in the special class have an intellectual disability, and it can be a difficult challenge to maintain

attention throughout the lesson. This should be aided with relevant and interesting activities, as well as a number of

effective active participation strategies. “One of the most effective strategies to use in the opening of a lesson or

activity is to tell students directly what they will learn and why” (Price & Nelson, 2014). Giving an explanation of why

the students would want to know the content is a simple yet extremely helpful strategy. Another fun way of

encouraging student responses is giving everyone a mini-whiteboard to write answers individually before showing

the class. Think, pair, share, unison responses, and symbolic answers like using coins takes away some of the stress

of quick verbal responses. Asking the students to move into groups (ie. 10 students divided by 5 means how many in

each group?) is a good way of involving everyone while taking a break from sitting. Another favourite strategy was

the teacher answering questions on the board incorrectly, to spark students to point it out silly mistakes. This shows

who is paying attention and sometimes can lead to ‘votes’ on who thinks the teacher is right vs the student being

right, so the teacher can monitor answers. These strategies are adapted from Price & Nelson (2014) and should be

chosen as the teacher gauges how the lesson is going.

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Quick lesson closure assessments, such as asking each student a question before leaving the classroom, are

extremely valuable to use to ensure students understand the big idea of the lesson (Wiliam, 2011). It is vital that

these concepts are discussed and checked at a later date to ensure concepts are understood and remembered.

The learning environment has to feel safe and non-judgemental to encourage students to have a go (Price & Nelson,

2014). Therefore, teasing and laughing at any incorrect answers has to be completely unacceptable. The more the

students feel comfortable, the greater opportunity they have to communicate their understanding with the teacher

and class.

Pre-Assessment Results

Pre-Assessment

Student PO1 (out of 6) PO2 (out of 3) PO3 (out of 3)

A 5 0 0

B 4 1 0

C 4 0 0

D 6 0 1

E 6 0 0

F 3 0 1

Table 1: Results of pre-assessment. Green indicates where performance objective has been achieved, red indicates 0.

Observation

It is clear that the computations are not fully clear to the students. Performance Standard 1 is the most correctly

answered, that is:

1. Students will answer 6 random numeric multiplicative equations up to 10x10, five out of six times on a

worksheet.

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Most students were able to achieve this to some level, with three students achieving the objective. Students had the

most trouble with:

2. Students will answer worded 2-digit double, triple, and quadruple questions, with or without manipulatives,

3 out of 3 times when given a worksheet.

It is the worded questions that the students struggled the most with. This could be because their skills of

interpreting the questions are basic at the beginning of the program. This indicates that there should be a focus

on this area. Performance objective 3 is also quite underachieved, however, this shows that this area, too,

should be focused on.

3. Students will answer written 2-digit half, third, and quarter questions, with or without manipulatives, 3 out

of 3 times when given a worksheet.

Assessment Results

Post-Assessment

Student PO1 (out of 6) PO2 (out of 3) PO3 (out of 3)

A 5 3 2

B 6 1 2

C 5 3 2

D 4 0 1

E 6 3 3

F 6 2 3

Table 2: Results of end of unit assessment. Green indicates where performance objective has been achieved, red

indicates 0 or where results have declined since pre-assessment.

Student D results actually reduced at the end of this program. This particular student had missed quite a few lessons,

however, these results were below expectation for that student. A more targeted approach for this student may be

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Student E has responded well to the program, and is the most improved student, achieving all three performance

objectives.

Only two students were able to reach the third performance objective. The teacher should continue teaching and

reviewing these words and concepts, even if moving to another unit.

Reflection

Overall, the instructional program could have been improved to ensure the students met the objectives. While some

good learning occurred, more care could have been taken to target the objectives. The Six Principles for Designing

Instruction for Diverse Learners (Price & Nelson, 2014) would have been an excellent framework for responding to

the needs of the students, and should have been followed more closely. In particular, Universal Design for Learning,

should have been utilised, where there is greater variation in student responses, presentation, and engagement

(Price & Nelson, 2014). Allowing students to show their learning in other ways than through a written test may have

shown better results. While manipulatives were allowed for working out, one student, during the assessment, wrote

incorrect answers while displaying correct results. This means that despite a good understanding of multiplication,

other unexpected factors impacted his final result. The teacher should work on developing assessments where that,

and other student’s knowledge is presented in a way that is best suited for them.

Conclusion

This class has almost reached their destination, but seem to need more fuel for their journey. A new method of

instruction or a choice of presenting methods may be the energy that they need. More explicit instruction would also

be recommended, especially for Student D, while the achievers could expand and deepen their knowledge of

multiplication. The performance objectives should be reviewed and updated for these students so that they are

always challenged and are working toward improving. While some of the students were able to meet the

performance objectives, the aim is for everyone to arrive safely, which is likely to occur with more time and practice.

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References

Australian Curriculum, Assessment and Reporting Authority, (2015) Australian Curriculum Foundation to Year 12,

Accessed at: http://www.acara.edu.au/curriculum.html

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Booker, G. (2011). Building Numeracy: moving from diagnosis to intervention (pp. 27-37). South Melbourne, Vic: Oxford University Press.

Hoy, A., Woolfolk & Margetts, K., (2010) Cognitive views of learning, Hoy, Anita Woolfolk & Margetts,

Kay, Educational psychology, 2nd edn Australian, Pearson, Frenchs Forest, N.S.W.

Price, K., & Nelson, Karna L. (2014). Planning effective instruction: Diversity responsive methods and

management (5th ed.). Belmont, CA: Wadsworth/Cengage Learning.

Salvia, J., Ysseldyke, J. E, & Bolt, S. (2013). Assessment in special and inclusive education (12th ed.). Belmont, CA:

Wadsworth/Cengage Learning

Tomlinson, CA & Moon, TR (2013). 'Assessment, grading and differentiation', in Tomlinson, Carol A & Moon, Tonya R,

Assessment and student success in a differentiated classroom, ASCD, Alexandria, Va., pp. 120-140.

Wiliam, D. (2011). Embedded formative assessment. Bloomington, IN: Solution Tree Press.

Differentiated Unit Plan using Backwards DesignTopic of Unit: MultiplicationCurriculum Area: MathematicsLinks to other Learning Areas: Home Economics, English10

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Year Level/s: Year 8/9 Special Ed class with 12 students of varying readiness and skills

Strandso Number and Place Valueo Number and Algebra

General CapabilitiesLiteracy Numeracy

Critical and creative thinking

AimsWhat relevant curriculum aims will this design address?The aim is to develop skills in simple computations with practice enough to reach a level of automaticity.

o Recognise and represent multiplication   as repeated addition, groups and arrays(ACMNA031)

o Recall multiplication   facts of two, three, five and ten and related division facts (ACMNA056)

o Solve word problems by using number sentences involving multiplication   or division where there is no remainder (ACMNA082)

o Develop efficient mental and written strategies and use appropriate digital technologies for multiplication   and for division where there is no remainder (ACMNA076)

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The Big IdeaWhat are the big ideas?Multiplication and division can be used to solve real every day math problems regarding calculating numbers within groups, sharing (division) measurement, and time.Division and Multiplication involve equal groups of a numberArrays are an organised way to display large numbers and groupsKnowledge & UnderstandingsWhat students will KNOW and need to UNDERSTAND?Students will know the terms:Double (x2) Half (÷2) Triple (x3) Thirds (÷3)Quadruple (x4) Quarters/Fourths (÷4)Multiply DivideArray Columns, rows

- How to format questions, written and numerical, into a calculator- Multiplying is used when working out equal groups of a number- Dividing is used when separating a group into equal parts

Types of Tasks involved:

Pre-assessment: Screening Test A adapted from George Booker (2011) (Appendix A)Post-assessment: Screening Test B adapted from George Booker (2011) (Appendix B)

Math games for lesson openings and automaticity (see below)Question sheets FlashcardsProblem solving group activitiesUsing materials (connector blocks, dice, calculators, coins, buttons) for symbolic learningPresenting information, answering questions on the smart board Math games on iPads

Essential Questions:When is multiplication needed?

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Accommodations:Extra verbal instruction and encouragementExtra time if neededRead or reread instructions if necessaryGive definitions of subject-specific and mathematical wordsChunk tasksAdditional practice activities if neededColour code instructions or stepsHighlight or bold important wordsGive short, written instructions Allow verbal answers rather than writtenUse materials to symbolize answers

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When is division needed?How can a calculator be used to answer worded questions?

Lesson Sequence: Each Lesson has a ‘key idea’ that is presented on the smart board or whiteboard. After the first 10 minutes of math games, the idea is discussed, along with exploring the language used. i.e. Halving means dividing a number into 2 equal groups. It looks like: 4 ÷2 = 2Students then use materials and calculators to solve focus questions and worksheets (set by the teacher) involving the key idea.

Lesson 1: Pre-assessment/OverviewGain an understanding of where the students are at in their learning through the use of diagnostic screening test adapted from Booker (2011). Teacher to walk around the class and ask questions about how the students reached the answer.

Lesson 2: HalvingLesson 3: Halving

Lesson 4: DoublingLesson 5: Doubling

Lesson 6: Overview/Check inA quiz/ practice test is given to see where students are in their learning. Students correct their tests with the class, using the smart board to write their answers and display their way of thinking to the class. Lesson 7: ArraysStudents use arrays to organize numbers. Vocab: columns and rowsCount how many rows, columns, regardless of which you count first, same number overall. Connection to __x__= because when multiplying, numbers can switch places with the same answer

Lesson 8: Thirds

Notes/Resources Needed:All materials are stored in the classroom for easy access.Pens, connector block materials Buttons, coins, countersCalculators Number lineiPadsClear time table grid

Format of Lessons:Each lesson will begin with 10-15 minutes of math games to gain attention and get students thinking about numbers.

Game 1: roll the dice, times number on dice by the lesson focus number (i.e. triples= x3), add to a running total. First person to designated number (or whoever has the highest at the end of the 10 minutes) wins. For more advanced students, use 12 sided dice or 2 die added together.

Game 2: Divide class into 2 groups. Choose focus number. Pass a basketball around the group, students say multiplication or division facts as you pass. I.e. x2 = 2, 4, 6, 8 or ÷3 = 30, 27, 24, etc. First group to get to designated number wins.

Game 3: Students roll a dice once. This is their starting number. Then flip a coin. If they guess heads/tails correctly, they multiply their starting number. If they guess incorrectly, they divide by focus number of the lesson (2, 3, or 4). First student to designated number (or whoever has the highest at the end of the 10 minutes) wins.

Array Game: Each student has a blank laminated times table frame. Roll 2 die and

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Lesson 9: Thirds

Lesson 10: TriplingLesson 11: Tripling

Lesson 12: Worded problems and operations - words to symbolsUse questions of halves, doubles and thirds to reinforce learning and making connections from numerals and operation symbols to words. Students create a graphic organizer for vocabulary/symbol reference

Lesson 13: Worded Problems

Lesson 14: Quartering

Lesson 15: Quartering

Lesson 16: Quadrupling

Lesson 17: Quadrupling

Lesson 18: Post-assessmentUse adapted version of Booker’s Test B to gage student learning over the unit.

times number together. Colour in the square respective square. (i.e. 3, 5= colour in both squares where the 3 and 5 meet. More advanced students write in the number, and/or use 12 sided die.

Lesson Closure – Check for understandingContent should be reflected upon at the start and end of every lesson. Before the students leave the lesson, they must verbally answer a question from the teacher. The teacher will take note of students who are unable to answer, or are answering incorrectly-to see how they are working questions out when not given much time. Teacher to shape thinking to develop automaticity.

References:

Booker, G. (2011). Building Numeracy: moving from diagnosis to intervention (pp. 27-37). South Melbourne, Vic: Oxford University Press.

Appendix A: Adapted Screening Test A from Booker (2011)Appendix B: Adapted Screening Test B from Booker (2011)

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Appendix A:

Multiplication Pre-assessment Name: ___________

You may use counters to find your answer.

4 x 5 = 7 x 3 = 2 x 9 =

3 x 0 = 6 x 9 =8 x 1=

A tap leaks 3 litres of water each day. How much water will leak in one week?

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Simone planted 4 rows of strawberries in the morning and 3 rows in the afternoon. Each row had 8 plants. How many strawberry plants did Simone put in her garden?

Joshua packed 5 boxes of books in the morning and 4 boxes in the afternoon. Each box has 36 books. How many books did he pack for the bookshop?

Triple 15

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Quadruple 7

A third of 90

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Appendix B:

Multiplication Assessment Name: ____________

Put a C next to the question if you used a calculator.

2 x 4 = 1 x 9 = 3 x 7 =_____

6 x 0 = 5 x 2 = 6 x 10 =____

There are 15 students in the class. They are all given 4 lollies. How many lollies were given overall?

A chocolate milk costs $4. If Dylan bought 100 chocolate milks, how much would it

cost? 🍫

Panae has a quarter (1/4) measuring cup, but she needs to use 1 full cup of flour for her recipe. How many quarter cups will she need?

What is...

Double 6

Half of 40

A quarter of 8