MathEd Studies

The mathematics of fountain design: a multiple-centres activity by Marshall Gordon describes a study about a variation of differentiated instruction in mathematics which caters to students of all ability needs. The lesson is about transforming quadratic equation. The article was published online in January 4, 2013 in the journal Teaching Mathematics and its Application. Here’s the abstract of the article. Use the email below to get access to the journal article.

Teachers of mathematics recognize the difficulty of reaching every student when the range of student abilities puts a considerable strain on the classroom discussion and time. In a response to the problem, students are grouped so that those with greater mathematical aptitude help those who have difficulties. While this approach is to be appreciated, it tends to mean that the more able students have less opportunity to explore further their own initiatives in mathematics, while those who have more difficulties find themselves on the receiving end with little opportunity to be in the role of enriching the mathematics experience for everyone, including themselves. A ‘multiple-centres’ approach is designed to overcome these problems. In this variation of differentiated instruction, all students get the chance to engage the material from a vantage point and at a level they find interesting and challenging as a consequence of their selecting extensions of the teacher’s initial focus problem. This article will present some findings of 11th year (roughly Fifth Form) average mathematics students at a US Independent School in transforming the standard quadratic equation to represent fountain parabolic trajectories, which was the teacher’s focus problem, along with some multiple-centre investigations they chose. A further set of opportunities with commentaries providing additional centres for student inquiry are included.

© The Author 2013. Published by Oxford University Press on behalf of The Institute of Mathematics and its Applications. All rights reserved. For permissions, please email: journals.permissions@oup.com

Learning theorists have provided us with a set of ideas about how people learn that have practical implications for teaching.

1. The Role of our Brain

Our brains are set up to process information coming in from the outside world, to make sense of these stimuli, and to draw connections.
Different parts of the brain perform different functions, and that learning actually changes the physical structure of the brain by expanding the brain’s capacity.
The development of the brain is lifelong and not predetermined at birth or even within the first three years of life.
Individual learners process information differently. This influences how each of us handles visual, aural, verbal, or tactile information. Information is easier to understand and use when it is introduced through learning pathways that are better developed.

2. Learning Is Based on Associations

Learning is a process of drawing connections between new information and what is already known.
For learning to occur, new ideas must be related to old ones: People make connections and draw conclusions on the basis of what they already know and have experienced.
Teachers can influence this process by organizing information and helping learners access their prior knowledge and draw connections to new material.
Becoming aware of the central ideas in a subject area, and of how they relate to each other, can also help learners to make sense of information and use it more flexibly.

3. The Learning Environment Makes a Difference

Learning involves both internal development and external interactions with the environment. People learn by making sense of the environment and stimuli around them.
Greater perceptual development and learning occur in environments that are rich with stimuli and provide feedback in response to a learner’s efforts.
Learning is also enhanced when content is relevant to students’ lives, when teachers provide opportunities to build understandings and practice skills, and when students have choices that are interesting to them.
Access to teachers and peers who can model, explain, discuss, or critique shapes the learning process.

4. Learning Occurs in Cultural and Social Contexts

Sociocultural theorists have shown us that the way people make sense of information depends on what they encounter and what is valued at home, in the community, and in the classroom.
Culture influences the experiences people bring to the classroom, how they communicate, how they expect to learn, and what they think is worth learning.
Learning is facilitated when efforts are made to connect classroom activities to students’ experiences at home. How teachers organize the culture of the classroom—how they communicate, define students’ roles, and support or discourage collaboration—all influence what is learned.

5. People Learn in Different Ways

Understanding individual differences in how people learn can help us to understand and guide the learning process better.
Learners possess spatial, musical, and kinesthetic abilities.
Students’ sense of their own abilities and expectations for success will influence how willing they are to engage in learning.
Not only do people learn in different ways, but there are also different kinds of learning. Some learning is simple and repeatable, other kinds of learning require more complex thinking and strategizing.

6. People Think About Their Own Learning, and Their Feelings Matter

Both thoughts and emotions shape the learning process. Being able to think about and monitor their own thinking enables learners to direct their own learning.
Reflective learners are also good intentional learners. They are able to redirect the normal frustration that occurs when tasks are confusing, or when their efforts are not initially productive, into further learning.
Emotions also play a role. Students who are fearful, anxious, depressed, or distracted cannot focus to process information. Positive emotions—feelings of confidence and willingness to exert effort—can help students to think, perform a learning task, and process new knowledge.
The ability to recognize and manage their emotions, to solve conflicts, to motivate themselves, and to persevere in the face of difficulty can help students become lifelong learners.

You may want to read the whole article How People Learn: Introduction to Learning Theories. I also recommend the book How People Learn: Bridging Research and Practice by S. Donovan and J. Bransford for a deeper understanding of how we learn.