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Carol Dweck: The power of believing that you can improve

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Carol Dweck researches "growth mindset" - the idea that we can grow our brain's capacity to learn and to solve problems. In this talk, she describes two ways to think about a problem that's slightly too hard for you to solve. Are you not smart enough to solve it ...

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3 Act Math, Formative Instructional Practices, Number Talks

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How you can be good at math, and other surprising facts about learning | Jo Boaler | TEDxStanford

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The Mathematics Assessment Project is part of the Math Design Collaborative initiated by the Bill & Melinda Gates Foundation. The project set out to design and develop well-engineered tools for formative and summative assessment that expose students' mathematical knowledge and reasoning, helping teachers guide them towards improvement and monitor progress. The tools are relevant to any curriculum that seeks to deepen students' understanding of mathematical concepts and develop their ability to apply that knowledge to non-routine problems.

More about the Math Assessment Project

100 lessons for formative assessment, some focused on developing math concepts, others on solving non-routine problems. A Brief Guide for teachers and administrators (PDF) is recommended reading before using these lessons for the first time.

A set of 94 exemplar summative assessment tasks to illustrate the range of performance goals required by CCSSM. The tasks come with scoring rubrics and examples of scored student work.

Complete summative test forms and rubrics designed to help teachers and students monitor their progress using a range of task types similar to the 'Tasks' section.

5 Prototype modules that encourage groups of teachers to explore the practical and pedagogical concepts behind the materials, such as formative assessment, collaborative learning and the use of unstructured problems.

The Teaching for Robust Understanding of Mathematics (TRU Math) suite is a set of tools with applications in Professional Development and research based around a framework for characterizing powerful learning environments.

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The Mathematics Assessment Project is part of the Math Design Collaborative initiated by the Bill & Melinda Gates Foundation. The project set out to design and develop well-engineered tools for formative and summative assessment that expose students' mathematical knowledge and reasoning, helping teachers guide them towards improvement and monitor progress.

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Why is algebra so hard? | Emmanuel Schanzer | TEDxBeaconStreet

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Five Principles of Extraordinary Math Teaching | Dan Finkel | TEDxRainier

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Depth of Knowledge Matrix Secondary Math: Robert Kaplinsky

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DOK Elementary/Secondary: Robert Kaplinsky

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Why Math Education in the U.S. Doesn’t Add Up

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Research shows that an emphasis on memorization, rote procedures and speed impairs learning and achievement 

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Brain Science

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This page includes evidence of 3 key ideas:

1) Anyone can learn to high levels
2) Mistakes and struggle are good for brain growth and brain strengthening
3) Visual mathematics helps brain connections and is really important for students' learning of mathematics - more evidence and ideas on this, including evidence of the importance of finger use, is provided on the visual mathematics page.

The first idea - that everyone can learn to high levels is described in the paper with this name, and the second idea - concerning the importance of mistakes and struggle is in the paper on mistakes. We include a few papers on the importance of visual connections, including the research paper co-authored with neuroscientist Lang Chen, and our teacher paper: Seeing as Understanding.

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This page includes evidence of 3 key ideas: 1) Anyone can learn to high levels 2) Mistakes and struggle are good for brain growth and brain strengthening 3) Visual mathematics helps brain connections and is really important for students' learning of mathematics - more evidence and ideas on this, including evidence of the importance of finger use, is provided on the visual mathematics page.

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vcmc, Ventura county math council is an afffiliate of CMC and NCTM

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National Council of Teachers of Mathematics

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National Council of Teachers of Mathematics

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Index - National Council of Teachers of Mathematics

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I train mathematics educators who want their students to be better problem solvers.

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Dr. Carol Dweck, Stanford
Mindsetworks

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Grit: The power of passion and perseverance

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Leaving a high-flying job in consulting, Angela Lee Duckworth took a job teaching math to seventh graders in a New York public school. She quickly realized that IQ wasn’t the only thing separating the successful students from those who struggled. Here, she explains her theory of “grit” as a predictor of success.

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"Learning from failure is a very intuitive and compelling idea that's been around for ages, but teachers may not know how to use it," Kapur said. He has run enough experiments both in lab settings and in real classrooms to have a fairly good idea of how to structure lessons that include failure up front, followed by consolidation of understanding through instruction.

The general idea is to develop tasks that students will not be able to solve, but require them to call upon their preexisting knowledge to try to solve the problem. That knowledge can be of the subject itself, as well as the informal insights students bring from their lives. The students will inevitably fail -- as the teacher expects them to -- but that failure is framed as part of learning and so is not seen as shameful. This process primes students' brains to learn the new concept from their instructor after the initial failure.

"It is failure-based activation of knowledge to prepare them to learn," Kapur said.

It might seem like this process would frustrate kids until they stop trying, but Kapur's studies found that instead of feeling bad about their inability to solve the problem, students' interest in the concept spiked. "I think that's a great place to get students to before we teach them something," Kapur said.

After students experience failure in their own discovery and problem solving process, the teacher facilitates a discussion that highlights various student attempts and teaches the new concept, consolidating students' understanding of the processes required to complete the task.
PRINCIPLES OF PRODUCTIVE FAILURE LESSON DESIGN

Tasks must be challenging enough to engage learners, but not so challenging they give up.

Tasks must have multiple ideas, solutions or ways to solve so that students generate a multitude of ideas. It cannot be a closed task with only one path to finding a correct answer.

The task must activate prior knowledge, and not just formal learning from a previous lesson. "If you design a task where a student only displays their prior class learning it's not good because then you aren't tapping into their intuitive reasoning," Kapur said. Intuitive reasoning is a big part of how students transfer knowledge to new situations.

While the task should activate knowledge, it should be designed so that the knowledge students have is not sufficient to solve the problem. They should hit a roadblock that they can't get around. "It makes the child aware of what he or she knows, and the limits of what he or she knows, and that creates a motivation to figure out what it is they need to know to solve this problem," Kapur said.

It helps if that task as an "affective draw," in that it's related to something students care about or is concerns something with which they identify.

Kapur has tested productive failure teaching strategies with students of varying abilities in Singapore and has found it to work with all students, regardless of ability. "Initial pre-existing conditions between students do not predict how much they learn," Kapur said. "How they solve the initial problem is what predicts how much they learn."

'We won't make the assumption that you're prepared to learn yet; what we will do is activate your formal and informal knowledge systems.'

Singapore tracks students into ability-based schools after primary school, which makes it easy to conduct research that compares low, middle and high achievers. However, Kapur has also tested productive failure in Indian schools in which students were not grouped by ability. He saw good results there as well. "The task is open enough that kids from different abilities can work together," Kapur said.

Part of Kapur's research has been to show that teaching with productive failure doesn't harm students' ability to perform on tests, but does improve knowledge transfer and conceptual understanding. In the process he's discovered an interesting element of creative thinking in math that appears to disprove the generally held notion that students need basic content knowledge before they can move on to more creative uses of the information.

"We've found that creativity actually suffers if you teach kids something too early," Kapur said. When students who have been taught with direct instruction are later asked to generate as many ways of solving the problem as they can, many can't go beyond the method they have already been taught.

"They were locked into that way of thinking," Kapur said. "When we start with generating or exploring we find that students still learn the material later on, but the knowledge was more flexible." This finding tells Kapur that creativity is itself a function of how students' acquire information.
SINGAPORE TAKES IT TO SCALE
Kapur's research on productive failure has convinced Singapore's Ministry of Education to use the pedagogical model for the statistical portion of it's A-level curriculum. Statistics make up about one third of the Cambridge A-level exam, Kapur said. All university-track junior-college students in Singapore are in school to pass that exam (junior-college in Singapore is like high school in the US).

Although Singapore's education system is very test-based, its Ministry of Education is interested in research-proven pedagogical approaches that lead to lasting learning beyond the test. "There is a very strong policy emphasis on changing how we teach," Kapur said of Singapore. "Just because there are tests does not mean we can't teach in ways that lead to very deep learning while doing well on the tests."

Kapur was able to show that productive failure worked well with students at the least prestigious of Singapore's 20 junior colleges, which provided a compelling proof of concept to scale up to all students studying for the Cambridge A-levels. Kapur and his team have designed a curriculum of tasks that use productive failure, and are training Singapore's teachers in the method.

The concept is new to many Singaporean teachers and Kapur says the first part of his training focuses on helping teachers understand the problems with direct instruction. He uses the analogy of watching a film. The average viewer focuses on plot, and perhaps pays some attention to acting ability or cinematography. When a director watches the same film, on the other hand, she is likely noticing nuances of camera placement, shot selection, and much more. That's the difference between what a novice sees and what an expert sees.

"No matter how engaging, entertaining or logically structured the new information is, the novice by definition is not going to see the same thing as the expert in the presentation," Kapur said. He works to help teachers understand the flawed assumption that students will understand after a concept has been told to them, explaining that direct instruction doesn't prime students' brains to process the new information.

"We won't make the assumption that you're prepared to learn yet; what we will do is activate your formal and informal knowledge systems," Kapur said.

The teacher training program also focuses on improving teachers' content knowledge. Working with student ideas and misconceptions requires the instructor have a deep understanding of the subject matter. Finally, Kapur helps teachers improve on important pedagogical aspects of this model like facilitating group work and consolidating ideas after students have grappled with a problem and failed.

"Your job as a teacher is to first prepare them, to give them the proverbial eyes to be able to see what is important, and then show them what is important in interesting and engaging ways," Kapur said.

Singapore's Ministry of Education has agreed to give Kapur's team four years to build teachers' capacity in this new style of teaching before evaluating its effectiveness. Kapur sees this as a huge gift, knowing that the effectiveness of any program lies in its implementation and that it takes time to get people up to speed.

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How 'Productive Failure' In Math Class Helps Make Lessons Stick

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"Learning from failure is a very intuitive and compelling idea that's been around for ages, but teachers may not know how to use it," Kapur said. He has run enough experiments both in lab settings and in real classrooms to have a fairly good idea of how to structure lessons that include failure up front, followed by consolidation of understanding through instruction.

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Teaching Tools | Resources for Teachers from Scholastic

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Choose your own maths adventure with our interactive tools that allow you to build a custom playlist of inspirational activities!

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Jo Teaching a Visual Dot Card Number Talk - YouCubed

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