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Fractions: Teaching for Understanding
How did you work that out?
Use this to diagnose misconceptions/ current understanding.


Practical hints for the classroom teacher:
1. Give a greater emphasis to the meaning of fractions than on procedures for manipulating them.
2. Develop a general rule for explaining the numerator and denominator of a fraction. In the fraction a/b, b is the name or size of the part (e.g., fifths have this name because 5 equal parts can fill a whole) and a is the number of parts of that name or size
3. Emphasise that fractions are numbers, making extensive use of number lines in representing fractions and decimals.
4. Take opportunities early to focus on improper fractions and equivalences.
5. Provide a variety of models to represent fractions. Student made ones will help them to understand the importance of congruency when comparing fractional parts.
6. Link fractions to key benchmarks, and encourage estimation. Innovative strategy – residual thinking (when comparing 5/6 and 7/8 students may conclude that the first fraction requires 1/6 more to make the whole (“the residual”), while the second requires only 1/8 to make the whole (a smaller piece of number), so 7/8 is larger. Students sharing strategies is an important teaching strategy to encourage.
7. Give emphasis to fractions as division. Students need lots of opportunities to partition objects in sharing and other contexts to help build this notion.
8. Link fractions, decimals and percentages wherever possible. Students will choose to convert it to decimals or percentages, in order to make sense of it. This flexibility is to be encouraged, as percentages particularly seem to make sense to many students intuitively.
9. Take the opportunity to interview several students one-to-one to gain awareness of their thinking and strategies.
10. Look for examples and activities which can engage students in thinking about fractions in particular and rational number ideas in general. E.g., Cuisenaire rods:
What fraction of brown rod is the red rod?
If the purple is 2/3, which rod is the whole?
If the brown rod is 4/3, which rod is one?
If the blue rod is 1 1/2, which rod is 2/3?
Your brilliant question for another group!

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Understanding by Design 2nd Edition

Understanding by Design 2nd Edition
Teaching for understanding! Not coverage! How many times have I had this argument with secondary maths teachers!
“…conventional teaching abets the three “pathologies of mislearning: we forget, we don’t understand that we misunderstood, and we are unable to use what we learned. I have dubbed these conditions amnesia, Fantasia, and inertia” (Shulman, 1999).
Three types of “uncoverage” in designing and teaching for understanding:
• Uncovering students’ potential misunderstandings (through focused questions, feedback, diagnostic assessment)
• Uncovering the question, issues, assumptions, and grey areas lurking underneath the black and white of surface accounts
• Uncovering the core ideas at the heart of understanding a subject, ideas that are not obvious – and perhaps are counterintuitive or baffling – to the novice

What should students come away understanding?
What will count as evidence of that understanding?

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Assessment for Teaching

One reason for focus on evidence is that this can be impacted upon through explicit teaching, evidence being what students do, make, write and say. It is through improvements in what students do, make, say and write that can infer what they have improved in terms of knowledge and understanding.

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Assessment for Teaching – Patrick Griffin

Professional Learning Team
1 identifies the student’s Zone of Proximal Development (ZPD) and describes the evidence on which this is based
2 records on the log the specific skill or concept being targeted. This is described as the learning intention(s). Depending on the developmental level being targeted, there may be one or more of these.
3 considers what the student will be able to do, say, make or write to demonstrate that the learning goal has been achieved
4 makes a clear distinction between intervention strategy and learning activity. The emphasis here is on teacher action: what the teacher will do, say, make or write to facilitate teaching.
5 considers the nature of the learning activity to ensure that it aligns with the learning intention and selected strategies
6 identifies resources, asking what support the teacher needs and whether any resources need to be sourced from elsewhere. If materials have been developed, a sample of each should be attached to the log for later reference by the PLT
7 checks whether or not the intervention resulted in improvement in the student’s ZPD, reviewing the intervention strategies and their impact. The review is at the heart of the PLT Inquiry and learning cycle.


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Algebra – Roots of Routes to Algebra

The four roots of algebra are:

  1. Generalised arithmetic
  2. Expressing generality
  3. Possibilities and Constraints
  4. Rearranging and Manipulating

Some tasks to uncover misconceptions:

Activity 1

Invite a class to write down an arithmetic expression such as 13=3*4+1  and then rearrange it without actually performing the operations. Calculators will be most helpful for discussions about whether a rearrangement is correct.




and 3 can be isolated,


²Engage the class or group in discussion about how to write down the rearrangement. You can expect to expose all sorts of weaknesses and confusions about the four operations and their rules!

When a class has caught onto the idea, get pairs of pupils to produce arithmetic all expressions and then challenge other pairs to rearrange them with a particular number isolated on the left-hand side. often it is the case that a number which appears more than once cannot be isolated. Can the class come to any conclusions about when a number can be isolated?

Activity 2

Here are 3 consecutive numbers: 10,11,12

Multiply the first and third numbers: 10 *12=

Now square the middle number: 11*11=

What is the difference between the numbers?

Does this always work for consecutive whole numbers?


(A formal algebraic solution to this problem is:

Let x, x+1,x+2 represent three consecutive numbers.

I want to prove that the difference between the square of the Middle number and the product of the first and third, is one. In symbols, I want (x+1)²-x(x+2)=1

Activity 3

Express algebraically the statement that “There are three feet in a yard.”

One probable response is 3f=1y (f=feet and y=yards). This conveys the sense of the relationship between feet and yards, but it isn’t clear exactly what the letters f and y stand for.

The annotation would be a shorthand rather than an algebraic statement. As the f and y are being seen as units (feet and yards) rather than as generalised numbers.

When the letters are used in a more algebraic sense the appropriate formulation becomes:

3y = 1f (three times the number of yards…)

many children never properly grasp how a letter can be used to represent a generalised number. They tend to think of 2a + 3b as 2 apples plus 3 bananas, using the letters as units of measure rather than things which can take on different values ( twice the number of apples…)


Activity 4

Question a) If a + b = 43, a + b + 2 =

Question b) If e + f = 8, e + f + g =







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Text Dependent Questions – Grades 6-12

“Classroom discussions allow for the co construction of knowledge. Discussion elevates the act of reading deeply from a private one to a public one.”
Necessary features of discussion:
* sustained dialogue! not just short questioning cycles
* Uptake, such that the teacher poses new questions derived from the comments of students
* Authentic questions that do not always have a single correct answer

“Questions focused on the literal level of meaning of a text will not accomplish the kind of critical thinking we seek from our students. But they are the start of the journey.”

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Planning Leaning Sequences – Kath Murdoch

Ask questions such as:
What are the ways of exploring the concept or concepts that are engaging and relevant to my students?
What understandings, dispositions and skills will students be learning?
What experiences will be used to ensure that students learn?
What evidence will be gathered to inform learning and understanding?
What is a current, real-life issue for this particular cohort of students?

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A New Year, A New School, A New Beginning

Wow, what a difference I have noticed this year.
Eager, appreciative students who want to learn.
Professional teachers, who are used to collaborating and have systems established to help their work.
Welcoming parents.
A calm atmosphere.
High expectations from leadership in a supportive and appreciative environment.
I am looking forward to the next few years.

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Hard Choices – Hillary Clinton

“No matter who you are or where you are from, if you work hard and play by the rules, you should have the opportunity to build a good life for yourself and your family.”

“…had to know both how to find common ground and how to stand our ground.”

“…a healthy society = a three legged stool, supported by a responsible government, an open economy, and a vibrant civil society.”

“It is true that clamping down on political expression and maintaining a tight grip on what people read or say or see can create the illusion of security, but illusions fade, while people’s yearning for liberty does not.”

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