# Teacher knowledge

Effective teachers develop and use sound knowledge as a basis for initiating learning and responding to the mathematical needs of all their students.

#### Guidance for effective practice:

1. Continually develop your personal deep and connected knowledge of mathematics and statistics using a variety of sources, including communities of teaching colleagues, consultants/facilitators, online resources, and books, and by actively listening to the ideas of your ākonga.
Kasie is new to teaching at Year 8. Her previous experience was in Junior classes. She would like to teach in the same investigative and interactive way with her older students.

Kasie is aware that her own personal knowledge about concepts like decimals, percentages, fractions, and integers is a bit ‘rusty’ and she is concerned about being able to understand the ideas that her students suggest. After talking to colleagues Kasie finds that she is not alone. Many other teachers in her school find mathematics challenging to teach.

To improve her personal knowledge, and that of her colleagues, Kasie sets up a study group.
Every two weeks four colleagues, Kasie, Moses, Ella, and Lelani, meet to plan their next unit together. This week the unit is about patterns and relationships (Algebra). The teachers start by working through an e-ako module from PLD360, hosted on nzmaths. The module, called Relations, graphs and linear equations, develops their own personal knowledge of algebra. It gives them feedback on their answers and provides many examples of patterns they can use with students in their upcoming unit.

Using the long term plan for Level 4 they locate a unit called Solving Linear Equations that is suitable for students in their class. Collectively the study group work through the unit plan and discuss the important mathematical ideas that are developed through each lesson.

2. Use learning progressions to inform your planning, teaching, and assessment.
Mia has a class of very energetic Year 2 students. The class has a range in current levels of achievement so Mia is conscious of how she can cater for this range. She looks for information about how the ability of children to classify shapes develops. The achievement objectives at Levels One and Two describe sorting shapes by features, with justification, and identifying plane shapes in other objects.

Mia finds that the Learning Progressions Framework for Geometric Thinking, at Steps 1-3, provides clear examples of students’ problem solving to illustrate how classification develops. From her pre-service study, and professional inquiry, she knows that the following phases generally occur:
• Children start classifying when they notice shapes look alike.
• They learn to put alike shapes together based on global appearance.
• They begin to use personal words to describe features, like pointy, straight, and curvy.
• They learn to use mathematical words, like side and corner, to describe why given shapes do or do not belong.
• They learn to sort the same set of objects in different ways, by attending to different features.
Mia notes also the students are often confused by changes in size, and orientation. For example, they may not see a square as the same as a ‘diamond.’ Children also need to learn how to disembed flat shapes from their environment, such as see rectangles in a building, or circles in a playground.

Looking online Mia finds many useful activities that use Attribute (Logic) Blocks, her favourite piece of equipment. She chooses Shape Soup, The Difference Game, and Silhouette Puzzles. Mia notices that all the activities can be altered easily to cater for the learning progression she identified.

3. Acquire and create a repertoire of adaptable rich examples for important mathematics topics.
Callum is very discerning about the problems he uses with his students. Over his six years of teaching Callum has collected and made up a lot of examples. Some survive and many are discarded. But he is always on the hunt for more, and he has a filing system on his computer to store them. Many of the best problems are made up by previous students and he has books of student-made problems about topics, like probability, symmetry, and fractions.

One of Callum’s favourite ways to create group worthy examples is to start with a regulation textbook problem and ‘open it up.’ In a Figure It Out book, he finds this activity called Getting in Shape.

The problem requires students to recreate the whole shape, given one quarter of it. There are already plenty of possible shapes that can be made. To open the problem further and move beyond quarters, Callum changes the problem to this:
“Imagine there are many species of fraction bugs, like the two thirds bug, the three fifths bug, and the five eighths bug.
Choose a shape from the set of pattern blocks. If that is what each bug leaves behind, what does the whole shape look like. Find as many answers as you can.”
Callum expects that students will need to make sense of less common fractions to solve the problem. The demands of the problem will encourage students to connect fractions and compositions of shapes.