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This site aims to bring fresh ideas to the teaching of Mathematics. It has been developed to assist teachers in NSW use the Big Ideas for mathematical understanding identified in the Mindset Mathematics series (Boaler, Munson & Williams. 2018) as the overarching structure for their teaching & learning programmes. It identifies the curriculum outcomes which contribute to an understanding of each Big Idea making it easier to implement the conceptual approach described.
Contemplating the effects of traditional mathematics in "A Mathematicians Lament", Paul Lockhart wrote
"If I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done - I simply wouldn’t have the imagination to come up with the kind of senseless, soul-crushing ideas that constitute contemporary mathematics education.”
The traditional pedagogy of Mathematics encourages students to see the discipline as one that requires them to memorise and recall on demand a set of procedures and isolated facts. Speed and correct answers are overemphasised at the expense of understanding and genuine number fluency. As students focus on learning the procedures they fail to make a connection with the logic behind the methods they are using. They develop fundamental misconceptions and develop a narrow and shallow mathematical knowledge.
Retention of information through rote practice isn't learning; it is training. (Ritchhart, Church & Morrison. 2011)
This traditional pedagogy results in students developing a negative attitude towards mathematics. Many develop a mathematical phobia and believe that they are not a "maths person". When confronted by challenging mathematics they retreat and have no or only poor strategies with which to approach new ideas. This all leads to a decline in the number of students pursuing mathematical learning beyond the years where it is compulsory.
Fortunately there is a growing body of research that shows there is a better way. The research of Stanford's Professor Jo Boaler has resulted in a series of books describing a conceptual approach to mathematics which focuses the learner on understanding the essential concepts of mathematics. These are the Big Ideas described on this site.
It provides only a snapshot of the Big Idea along with links to the NSW curriculum outcomes, background and language along with suggested understanding goals. In the books you find a detailed description of why each Big Idea matters and how the approach taken to it builds mathematical understanding and confidence. Importantly the books describe what a teacher should look for while teaching the suggested activities and offers advice to suit each observation.
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Each Big Idea is presented through a sequence of learning opportunities beginning with a Visualisation activity, then a Play activity and concluding with an Investigation. All activities are low floor, high ceiling by design ensuring they are accessible and appropriately challenging for all students.
A task this is "Low Floor, High Ceiling" is one that is designed to be easily accessible for all learners while offering opportunities for students to extend their thinking. The "Low Floor" invites the learner to engage with the task and to begin exploring possible solutions. Visualising ideas with concrete materials, looking for patterns in sets of friendly numbers, explaining what is there when looking at a resource are all accessible strategies into a task. Students move through these stages of exploration and understanding building at a pace that suits their learning. The teacher provides prompts and nudges that help the student to build an understanding.
Students utilise the "High Ceiling" of a task when they move towards making generalisations, search of elegant explanations for why something works, seek to apply their knowledge in new settings or communicate the logic behind their mathematical thinking to others.
An alternate model adds "Wide Walls". A task is said to have "Wide Walls" when it can be approached in many different ways and where there is a multitude of possible solutions. These tasks are very open ended. Jen Munson offers the following examples for a how a task can be reimagined to make it open-ended as follows:
Question one is very closed and focuses our thinking on doubling, there is one answer and limited strategies are relevant. Question two is more open, but there is only one answer and the thinking required is limited. Question three has many (infinite) answers, can be approached in multiple ways, has a low floor and a high ceiling. This is the sort of task that will build mathematical thinking and meet the needs of a differentiated classroom.
Some of the Big Ideas have been extended to allow greater coverage of the NSW curriculum. These are included as either "Beyonds", where the core concept underlying the Big Idea is extended or as single additional "Visualise", "Play" or "Investigate" activities. Additional activities are indicated with a number e.g. "Investigate Two". You will not find information about these in the Mindset Mathematics books. Additional information has been provided to support teachers implement these activities.
The Mindset Mathematics series offers a unique, research-based visual approach to exploring the big ideas in mathematics, which is essential to future mathematics success. This hands-on resource is for any teacher who wants to engage their students in reasoning and persisting through problems, and provides activities that will engage students' interest and show them the many ways that mathematics is important in their lives.
During their work with tens of thousands of teachers, authors Jo Boaler, Jen Munson, and Cathy Williams heard the same message: Teachers want to incorporate more brain science into their mathematics instruction, but they need guidance in the techniques that work best to promote learning of mathematics concepts. In this much-needed volume, the authors clearly show what the big ideas are at this grade level, why they are important to know, and how students can best learn those big ideas.
The tasks in Mindset Mathematics reflect the lessons from brain science that:
- There is no such thing as a math person - anyone can learn mathematics to high levels.
- Mistakes, struggle and challenge are the most important times for brain growth.
- Speed is unimportant in mathematics.
- Mathematics is a visual and beautiful subject, and our brains want to think visually about mathematics.
This approach to mathematics is structured around a set of core ideas and practices. Each element works with each other element to build Mathematical Confidence for the learner. Mathematical Confidence is achieved when the learner believes that they are able to reason, communicate, problem solve with mathematical concepts. Mathematical confidence requires a deep level of understanding, adaptive expertise, fluency and a growth mindset when confronted with mathematical challenges.
Through Number Talks students develop a rich Number Sense, a capacity to manipulate, de-compose and re-compose numbers, notice patterns and communicate mathematical ideas. By visualising, playing with and investigating "Big Ideas" students develop their understanding of essential concepts in mathematics. Strategies from "Visible Thinking" provide structures for student thinking and make this thinking observable to their teachers. Engagement in real-world, open-ended problems provide students with opportunities to apply their mathematical skills and reveals the relevance of mathematics beyond the classroom. By employing a "Teaching For Understanding" framework to their planning and pedagogy, teachers ensure the focus of mathematics is on developing adaptive expertise and ensures students experience opportunities to demonstrate and refine their understanding. The result is Mathematical Confidence defined by ones ability to bend their mathematical knowledge to new situations, to incorporate new concepts into their existing scheme and embrace challenges with a growth mindset.
Mathematics is a reasoning and creative activity employing abstraction and generalisation to identify, describe and apply patterns and relationships. The symbolic nature of mathematics provides a powerful, precise and concise means of communication.
The study of mathematics provides opportunities for students to appreciate the elegance and power of mathematical reasoning and to apply mathematical understanding creatively and efficiently. The study of the subject enables students to develop a positive self-concept as learners of mathematics, obtain enjoyment from mathematics, and become self-motivated learners through inquiry and active participation in challenging and engaging experiences.
The Aim of Mathematics in K–10 is for students to:
The Big Ideas described in this resource are "Big" because they are the ideas which are at the heart of mathematical understanding; the building block of mathematics. The Big Ideas can be compared to "Big Understandings" in the language of Teaching for Understanding. Big Understandings are "vigorous, flexible, adaptive, and notably proactive" connecting to the world in which we live. David Perkins, describes them has "stretchy" indicating that they are adaptable to many situations. This description suits the "Big Ideas" perfectly as unlike the rote learning of mathematical procedures, the Big Ideas serve the learners well in many situations even those for which they have not been specifically prepared.
From another perspective, "Big Ideas" are Threshold Concepts in that "They elevate learners to a new level of understanding in a domain and open the way to further and deeper learning." By developing a deep, flexible understanding of the "Big Ideas" students are able to approach a broad set of mathematical challenges with confidence.
(The Thresholds) approach builds on the notion that there are certain concepts, or certain learning experiences, which resemble passing through a portal, from which a new perspective opens up, allowing things formerly not perceived to come into view. This permits a new and previously inaccessible way of thinking about something. It represents a transformed way of understanding, or interpreting, or viewing something, without which the learner cannot progress, and results in a reformulation of the learners’ frame of meaning. - (Ray Land, Jan H.F. Meyer and Caroline Baillie)
While the "Big Ideas" are challenging and as a result provide students with opportunities to grow their brains, they are not designed to be extension options. While the "Big Ideas" lead well into open ended, real world learning tasks where students utilise their understanding to solve problems and communicate their ideas, the "Big Ideas" represent the foundational understandings on which mathematics is constructed.
At the heart of learning are essential questions which act as guides for our efforts to build understanding. When we consider essential questions we explore what matters most within a discipline, concept, belief, value or idea. Essential questions take us beyond knowledge and bare recall of facts towards meaningful understandings.
“What is an essential question? An essential question is—well, essential: important, vital, at the heart of the matter—the essence of the issue. Think of questions in your life that fit this definition—but don’t just yet think about it like a teacher; consider the question as a thoughtful adult. What kinds of questions come to mind? What is a question that any thoughtful and intellectually-alive person ponders and should keep pondering? (McTighe & Wiggins, 1999 Understanding by Design)
An Essential Question:
Jay McTighe describes Essential Questions as follows and this description offers sound advice for how they may be used in our classrooms.
These are questions that are not answerable with finality in a single lesson or a brief sentence—and that's the point. Their aim is to stimulate thought, to provoke inquiry, and to spark more questions, including thoughtful student questions, not just pat answers. They are provocative and generative. (McTighe, 2013 p3)
For additional guidance in writing and using Essential Questions refer to "Essential Questions" by Jay McTighe
McTighe, Jay. (2013) Essential Questions: Opening Doors to Student Understanding Association for Supervision & Curriculum Development. Kindle Edition.
Typically, the lesson begins with the teacher presenting the required method to the students. The teacher begins with step one being demonstrated on the board. Once step one is complete, the teacher demonstrates step two, and then step three and sometimes steps four and five. With triumphant zeal the teacher indicates the correct answer with a flourish of whiteboard marker and perhaps a double underline for effect. In phase two the students copy the process they have been shown with the teacher looking on to ensure the steps have been followed accurately. Naturally there are some bugs and errors that require correction. By the end of the lesson most students are able to accurately follow the steps and arrive at a desirable answer even if some of the numbers are changed.
Compare this to how a computer is programmed. The ‘coder' determines the steps to be completed and enters them into the machine ensuring accuracy; this equates closely to phase one of our lesson although with our students the coding occurs visually and aurally rather than via keyboard. The coder then runs the code on the computer and looks for bugs in the code which may cause unwelcome results; this is phase two of our lesson. Finally, having checked the code and feeling confident that it is bug free and fit for purpose the coder releases their programme into the world where it runs on a range of subtly different systems and with a mix of inputs; a very near comparison to phase three of our lesson.
The focus is on mimicry and memorization rather than deep mathematical thinking and understanding, flexible use of mathematical concepts, communication of mathematical arguments and justifications, and developing a positive disposition that values connections between mathematics and students’ identities beyond the classroom. I think it is important that mathematics teachers use instructional routines that not only build procedural fluency through conceptual understanding but also support strategic competence, adaptive reasoning, and productive dispositions. (NCTM President - Robert Berry)
There is a better way, one that supports conceptual understanding.
The alternative approach is to present content in a sequence that allows students to develop their number sense while also engaging in exploration of concepts or Big Ideas (through visualisation, play & investigation), with opportunities to problem-solve and problem-find as ways of strengthening their capacity for working mathematically. At the point of need in this cycle, new content including mathematical methods can be taught through targeted instruction with the goal to always build understanding.
This approach to organising the presentation of content is built on three research based beliefs:
1. Unless mathematics is to be viewed as a discipline defined by rules, where success is determined by how accurately one follows those rules, learning must begin with the underlying concepts (Big Ideas). This is backed by research from Australia's Chief Scientist that examined the approach taken to mathematics in 619 Australian schools achieving significantly above expected growth. "87% of case study schools had a classroom focus on mastery (i.e. developing conceptual understanding) rather than just procedural fluency."
2. We are more likely to engage with new ideas when we see how they will help us achieve a short term goal. Telling students you will need this in the future, or worse, you will need this for the test is unlikely to increase motivation. When what we are learning has immediate relevance, when we are clear on the purpose of what we are learning our intrinsic motivation is triggered according to Daniel Pink, author of Drive. (Learn more about 'Drive' in this Video)
Jo Boaler et al., shares the following research that reveals the power of teaching mathematical processes at the point of need, rather than in isolation of need and in advance of application to contexts that matter to the learner.
A really interesting research study (Schwartz & Bransford, 1998) showed that students learned more when they worked, using their intuition, on problems that needed methods, before they learned the methods. They did not learn the methods until they had encountered a need for them. This caused the students' brains to be primed to learn them.
Boaler, Munson, Williams. (2018) Mindset Mathematics: Visualizing and Investigating Big Ideas, Grade 5 (p. 249). Wiley.This approach of teaching to big ideas and teaching smaller ideas when they arise, has the advantage of students always wanting to learn the smaller methods as they have a need for them to help solve problems. (Boaler, Munson, Williams. What is Mathematical Beauty? Teaching through Big Ideas and Connections YouCubed )
3. Manu Kapur describes an instructional order "that reverses the (traditional) sequence, that is, engages students in problem solving first and then teaches them the concept and procedures. (He) call(s) this sequence of problem solving followed by instruction productive failure". Kapur's research shows that "Productive failure students, in spite of reporting greater mental effort than Direct Instruction (DI) students,significantly outperformed DI students on conceptual understanding and transfer without compromising procedural knowledge". (Kapur)
One pedagogical method commonly utilised in mathematics is known as "I do, We do, You do". It is a model that begins with teacher instruction and modelling of a method, opportunities for whole class repetition of the method and finally students use the method with independence. The danger with this method as it is most typically used is that places an emphasis on the rote memorisation of a method. Students may learn the method but they do not develop an understanding of how or why it works. Even when confronted by a similar set of circumstances, students may lack the sufficiently flexible understanding they require.
"An adaptation to the “I do—we do—you do” instructional routine that attempts to address this concern is “You do—we do—I do.” McCaffrey’s (2016) blog post Rethinking the Gradual Release of Responsibility Model explains “You do—we do—I do.”
- “You do”—the teacher gives students a task to see what students know and understand. The task should have multiple entry points, have varied solution paths, and focus on mathematical processes. The teacher monitors the classroom for strategies and asks probing questions.
- “We do”—after working on the task independently, students collaborate with peers in pairs or small groups. On the basis of the monitoring of the classroom while students worked independently, the teacher is purposeful in putting students in pairs or small groups. Additionally, this might be an opportunity to orchestrate a productive mathematical discussion.
- “I do”—the teacher engages in instruction, pulling together the mathematical ideas that arose during “you do” and “we do.” Also, the teacher’s instruction connects and deepens the mathematical understanding of students.
“You do—we do—I do” provides opportunities for students to engage in the mathematical practices that deepen their understanding of mathematical content and the practices." (NCTM President - Robert Berry)
People who love playing a sport do so because they love to play the game; the whole game. People who succeed at sport do so because they are good at playing the game; the whole game.
Every sport has a particular set of skills which the player combines to achieve their purpose and there are many drills used during practice sessions to develop and refine these skills. The player who is keen to improve their match day performance values these drills and commits the required time and energy to maximise the benefits. A good coach will know what drills are going to help the developing player most and will aim to make the drills enjoyable. But, the player and the coach know that the drills are a means to an end. In Football, they player needs to be able to control the ball. During practice you can expect the team to spend some of their time developing the skills of 'dribbling' and passing. Some players will do this well, some need more practice. At the end of the session when the team plays a game, everyone gets to play, even the players yet to master passing. If young players of a sport never had the opportunity to play the whole game, if they spent all of their time on drills, they would never become old players; they will turn away from what might have become a passion.
The same is true of mathematics, and learning in general. David Perkins compares his experience of learning to play Baseball and the sense that he developed early on about the nature of the whole game to his learning of mathematics:
I’m simply stunned when I think how rarely formal learning gives us a chance to learn the whole game from early on. When I and my buddies studied basic arithmetic, we had no real idea what the whole game of mathematics was about. . . . It was kind of like batting practice without knowing the whole game. Why would anyone want to do that? (Perkins, 2009 p2)
There are certain skills and pieces of knowledge that the young mathematician needs to engage with, but if we deny them opportunities to engage with mathematics that is challenging, meaningful, relevant and beautiful, they will turn away from what might have been a passion and what is undoubtedly a discipline that will play a part in their future success. In mathematics perhaps more than any other discipline there is a belief that students need to master the basics (this mostly translates to multiplication facts) before they can move on to problem solving and working mathematically. This is the equivalent of deciding that a student is not capable of attempting to write a story until they have mastered the spelling of all the words they are likely to need.
Perkins cautions against two dangers to learning that are both seen frequently in mathematics education. The first is 'elementitis' which occurs when we focus on teaching the part of the whole without an opportunity to bring them all together and see how they fit. Elementitis occurs when we spend great chunks of time drilling isolated facts without the chance to apply that knowledge in a meaningful context. The damaging power of 'elementitis' is diminished when we start our learning journey with a meaningful context and teach the required elements when they are needed and then take the time to show the learner how the bits fit together.
The second danger is named 'aboutitis' by Perkins. In mathematics 'aboutitis' occurs when we learn about polygons, the cartesian plane, prime numbers and Pythagoras' theorem but never have the opportunity to develop this knowledge into 'an empowering and enlightening body of understanding'. (Perkins, 2009 p6)
Perkins offers the following advice for teachers looking to make the "whole game" a part of their learners experience:
This thinking is reflected in the Rationale of the NSW Mathematics K-10 Syllabus:
The study of mathematics provides opportunities for students to appreciate the elegance and power of mathematical reasoning and to apply mathematical understanding creatively and efficiently. The study of the subject enables students to develop a positive self-concept as learners of mathematics, obtain enjoyment from mathematics, and become self-motivated learners through inquiry and active participation in challenging and engaging experiences.
Perkins, David. (2009) Making Learning Whole: How Seven Principles of Teaching Can Transform Education. Wiley.
Visible Thinking is a flexible and systematic research-based conceptual framework, which aims to integrate the development of students' thinking with content learning across subject matters. - Project Zero - Visible Thinking
At the heart of Visible Thinking are two closely connected questions that challenge our thinking as teachers. “What kinds of thinking do you value and want to promote in your classroom?" And the associated question, "What kinds of thinking does this lesson force students to do?” (Ritchhart, Church & Morrison. 2011 p9). From a different perspective we might ask, what kinds of thinking will my students require in my classroom?
When we begin with these questions in mind we force ourselves to consider how I might make the type of thinking I desire and that my students will require a routine part of their learning and then how might I make their thinking visible so that together we can refine it.
"To develop understanding of a subject area, one has to engage in authentic intellectual activity. That means solving problems, making decisions, and developing new understanding using the methods and tools of the discipline." (Ritchhart, Church & Morrison. 2011 p10)
In Mathematics the thinking tools of the discipline overlap with those of other disciplines but include some that take on a particular meaning and importance for the mathematician. Our list of thinking tools would include, but is not limited to:
Having identified the thinking students will require the teacher using a Visible Thinking approach utilises a variety of routines to scaffold and make visible the thinking of their students. These routines offer guidance to the thinker and unlock a deeper understanding through their use of prompts which have been shown through research to promote the desired thinking.
"Just as routines for lining up or handing in homework become engrained, thinking routines also become part of the fabric of the classroom over time. . . .When thinking routines are used regularly in classrooms and become part of the pattern of the classroom, students internalize messages about what learning is and how it happens."
(Ritchhart, Church & Morrison. 2011 p45)
Below are some starting points for teachers looking for a Thinking Routine:
Credit: Boaler, Munson & Williams (2018) - Mindset Mathematics: Visualizing and investigating big ideas Grade 3, 4, 5, & 6
NESA - NSW Educational Standards Authority
