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Mathematical mindsets#

See also: mathematical-mindset

Resources: YouTube video - associated with YouCubed - Boaler and students

Following are rough notes from reading

Boaler, J. (2015). Mathematical Mindsets: Unleashing Students’ Potential Through Creative Math, Inspiring Messages and Innovative Teaching. John Wiley & Sons, Incorporated.

Chapter list#

1. The brain and mathematics learning#

2. The power of mistakes and struggle#

  • Find ways to celebrate mistakes
  • Highlight/value conceptual mistakes, rather than numerical errors
  • Abandon testing and grading as much as possible - being marked down for mistakes has consequences

3. The creativity and beauty of mathematics#

  • What do students see as their role in a mathematics class?
    • Mostly it's a solver of problems. Rather than being there to appreciate the beauty of maths, ask deep questions, explore rich connections, learn about the applicability of the subject.

4. Creating mathematical mindsets: The importance of flexibility with numbers#

  • Repetitive and simple ideas (common mathematical teaching approach) fail to move students into disequilibrium - part of learning

5. Rich Mathematical Tasks#

Provides numerous examples and then moves onto offering 6 questions to ask about a mathematics learning task that will help transform them into rich-mathematical-tasks

6. Mathematics and the path to equity#

  • Mathematics (in the US) has the largest achievement differences due to ethnicity, gender and socio-economic levels.
  • Argues that the push for mathematics as a "performance subject" and an elitist take
  • A view that reinforces the fixed mindset view (you are good at math or not) rather than growth (valuing effort and learning)
  • Choices about who proceeds into "higher" math in high school also influences outcomes, and in turn is influenced by perceptions of "good" students (no disciplin problems, homework in on time)
  • Parents equating "procedural skill" with capability, but ignoring conceptual understanding and problem solving
  • Need to move mathematics from an "elitist, performance subject used to rank and sort students (and teachers) to an open, learning subject"

  • Equitable strategies

    • Offer all students high-level content
    • Work to change ideas about who can achieve in mathematics

    • Encourage students to think deeply about mathematics Makes the case for mathematics as a "connected, inquiry-based subject" for addressing inequalities and increasing achievement.

    • Teach student to work together Research showing that working on mathematics collaboratively helps to make connections etc.

    • Give girls and students of color additional encouragement to learn math and science

    • Eliminate (or at least change the nature of) homework

    • PISA (2015) found "homework perpetuates inequalities in education"

    • Numerous other issues

7. From tracking to growth mindset grouping#

  • "Opportunity to learn" key factor in student achievement
  • more access to high-level mathematics means higher achievement
  • tracking seen as greatest reinforcement of fixed mindsets
  • tracking can be seen as being helpful for teachers in being able to deliver learning experiences at the right level, but this argument ignores the fact that even in similarly tracked classes there is a wide range of abilities/needs
  • Focus of this chapter is on modern/effective forms of grouping
  • Growth mindset grouping - i.e. basically heterogenous grouping

  • Teaching heterogeneous groups effectively: the mathematics tasks

    "It is not enough to de-track and then teach through narrow mathematics questions that will be accessible to only a few students"

    • Providing open-ended tasks - i.e. low-floor-high-ceiling-wide-walls#Mathematics

      Point is made that the teacher will engage in explicit teaching, but in response to student working through of the open-ended task. Echoing Fry and Hillman (2018)

    • Offering a choice of tasks

      Where task explained as "going to different places", rather than easy/difficult.

    • Individualised pathways

      Mentions the SMILE cards from the UK.

    • Teaching heterogeneous groups effectively: complex instruction

    "... group work in classrooms can fail when students participate unequally in groups". With perceived status of participants being an issue. Cohen and Lotan developed complex-instruction

Railside approach#

  • Multidimensionality - more student success because there are more ways to be successful

    • Typical math class has 1 dimension - "executing procedures correctly" - one way to be successful
    • Need to encourage engagement in "all the ways to be mathematical": ask good questions; propose ideas; connect different methods; use many different representations; reason through different pathways...
    • student survey "What does it take to be successful in maths" - 97% said "pay careful attention"
    • Responses from RailSide students included: asking good questions; rephrasing problems; explaining; using logic; justifying methods; using manipulatives; connecting ideas; helping others
    • Railside used "rich tasks" that were "groupworthy" when they:
    • illustrate important mathematical concepts;
      • Railside organised curriculum around big ideas e.g. "What is a linear function" (which strikes me more as a driving question) rather than mathematical methods
    • allow for multiple representations;
      • physical manpulatives for algebra
      • encourage students to reprsent solutions/ideas in multiple ways see figure below
    • include tasks that draw effectively on the collective resources of a group; have several possible solution paths

Example of representing mathematical ideas in multiple ways. (adapted from Boaler, 2015)

  • Roles

    • Railside task sheets included explict direction for roles (e.g. facilitator; recorder/reporter; resource manager; team captain)
    • In England role names were changed:
    • organiser - keep the group together and focused on the problem, no talking outside the group;
    • resourcer - only person that can get resources, make sure everyone is ready before you call the teacher;
    • understander - make sure all ideas are explained for understanding for all in group; encourage idea generators to explain if any don't; write down important parts of explanations;
    • includer - make sure everyone's ideas are listened to; invite other people to make suggestions.

  • Assigning competence

    • teachers raise status of students who think may be lower status in a group
    • feedback that is public, intellectual, specific and relevant to the group task - from (Cohen, 1994) - designer of complex instruction

  • Shared student responsibility

    • Railside - first 10 weeks explicitly taught students who to work in groups

    • tactics

      • "do like and don't like group members do" lists generated by students
      • reinforcing what the teacher values
      • in group work, ask one member of the group a conceptual follow-up question, if not able to answer encourage the group to ensure everyone understands, teacher leaves, and returns later to ask the same student
      • group tests - taken individually, but only one random test from a group marked

8. Assessment for a growth mindset#

Reducing the diversity of ways individual students understand mathematics to a number/grade si problematic - "not only fail to adequately describe children's knowledge, in many cases they misrepresent it"

Finland - reportedly - does not test students. Instead relying on teachers "rich understanding" of students

Students with no experience of examinations and tests can score at high levels because the most important preparation we can give students is a growth mindset, positive beliefs about their own ability, and problem-solving mathematical tools that they are prepared to use in any mathematical situation

Regular testing and grading lead to various problems: ego comparison, fixed mindset etc. Links made to intrinsic/extrinsic motivation.

Over reliance on summative assessment and associated practices, including for formative purposes. Which is problematic.

Instead, recommends and discusses at length assessment-for-learning. In part, because it teaches students responsibility for their own learning.

Developing student self-awareness and responsibility#

Most powerful learners are: reflective, engage in metacognition; and take control of their own learning.

In mathematics classes students often don't know what area of mathematics they're working on.

Nine strategies for "encouraging students to become more aware of the the mathematics they are learning and their place in the learning process" i.e. the first two parts of the 3-part assessment-for-learning process.

  1. Self-assessment

    Provide students with clear statement of the math they're learning. They are given criteria and time to use those criteria to self-assess

    For example: I can do independently and explain to other; I can independently; I need more time. I need to see an example to help me. More detail example for high school

  2. Peer assessment

    Similar to self-assessment, but use peers. e.g. two stars and a wish

  3. Reflection time

  4. Traffic lighting

    Students use red, yellow, green "objects" to indicate their understanding of a concept. Giving teacher visible feedback, but also involving self-assessment.

  5. Jigsaw groups

    Particular groups become experts on a topic. Then groups split up and each expert joins a group to teach them about their topic.

  6. Exit ticket

    Each student writes about their learning in the class and hands it in before they leave. reflection and feedback for teacher. Variation on minute-paper. Boaler's example: 3 things I learned today; 2 things I found interesting; 1 question I have.

  7. Online forms

    Students completing online forms during class. e.g. comments/thoughts on the lesson. Echoes of the course barometer

  8. Doodling

    Ask students to represent their understanding of the lesson's content through doodling/drawing. Using something like a Vi Hart video as an example

  9. Students write questions and tests

Diagnostic comments#

i.e. helps students understand how to close the gap between where they are and what the need to be.

Main story is about a teacher who eventually stopped grading entirely and focused on diagnostic feedback and using a "rubric" focused on multi-dimensional mathematical activities (perserve - ask questions; reason; justify...) and completion.

Advice on grading#

  1. Always allow students to resubmit any work or test for a higher grade.
  2. Share grades with school administrators but not with the students.
  3. Use multidimensional grading.
  4. Do not use 100-point scale. (One alternative is to use 4-point scale)
  5. Do not include early assignments from math class in the end-of-class grade.
  6. Do not include homework, if given, as any part of grading

9. Teaching mathematics for a growth mindset#

A summary of the book

Encourage all students#

Norms#

Both teacher and student have opportunity to establish norms. Students in groups identify what they do and do not appreciate in terms of behviour from other students.

Suggested norms to promote

Norm Description
Everyone Can Learn Math to the Highest Levels Encourage students to believe in themselves. There is no such thing as a “math person.” Everyone can reach the highest levels they want to, with hard work.
Mistakes Are Valuable Mistakes grow your brain! It is good to struggle and make mistakes.
Questions are really important Always ask questions. Ask: why does this make sense?
Math is about creativity and making sense At its core about visualising patterns and creating solution paths for others to see, discuss and critique
Math Is about Connections and Communicating. Math is a connected subject, and a form of communication. Represent math in different forms—such as words, a picture, a graph, an equation—and link them. Color code!
Depth Is Much More Important Than Speed Top mathematicians, such as Laurent Schwartz, think slowly and deeply.
Math Class Is about Learning, Not Performing. Math is a growth subject; it takes time to learn, and it is all about effort.
Participation quiz#

For encouraging good group work. complex-instruction suggests that participation quiz be graded. Graded at the group level. Helping reinforce to students behaviour/participation in group work is important and the nature of the expected work.

For a given group task

  1. Show the ways of working you value
  2. Students start working

  3. Teacher circulates observing and writing down comments

    Take notes, quote student works (when noteworthy)

  4. Assign the groups a grade, or give feedback without the grade

See also: PQuiz page on MindsetKit

Believe in all your students#

Cohen & Garcia (2014) - 1 experience/message (I am giving you this feedback because I believe in you) can change everything for students

Grasping mathematical concepts quickly is not indicative of mathematical ability.

Give growth praise and help#

Standard stuff about praise, but also focused on help

Help should not remove the cognitive difficulty from a task. Rather than breaking the problem down for students, help restate/revisualise the problem in a different way.

One teacher responses with a question "Do you want my brain to grow, or your brain to grow?"

Opening mathematics#

Teaching math as an open, growth, learning subject#

Narrow, procedural questions require students to perform a calculation. Are right or wrong. Are closed, fixed. Some are reasonable. But should be the minority.

Design activities that provide students the opportunity to explore, create and grow.

  1. Rule doesn't matter, make sense of your answer, explain why your solution makes sense

    e.g. for questions like $1 \div \dfrac{2}{3} = $

  2. Find all the ways you can represent an algebraic expression

    e.g. $\dfrac{1}{3} (2x + 15) + 8 $

Encourage students to be mathematicians#

They see math as creative, beautiful and aesthetic.

Encourage students/class to work on conjectures, reasoning and proof.

Teach mathematics as a subject of patterns and connections#

Mathematics is all about the study of patterns

Mathematical methods are teaching a pattern - something that happens all of the time. it is general.

mathematics is the classification and study of all possible patterns - W. W. Sawyer (1955, p. 12)

  • Encourage students to use different methods; make connections between methods; discuss connections
Teach creative and visual mathematics#
  • color coding: geometry, fractions, algebra, and division
Encourage intuition and freedom of thought#

Ask students what they think might work.

Value depth over speed#

In lessons/questioning

  • avoid procedural questions - with a single possible answer
  • work to challenge conceptual understanding through statements where the students respond with conjectures and reasons
Connect mathematics to the world using mathematical modelling#
  • Avoid "pseudo contexts" - fake real world problems
  • Use mathematics that model the world (and exist in it)
  • Recognise that some mathematics is without a real world context.
  • use real data and situations
Encourage students to pose questions, reason, justify and be skeptical#
  • Gives example of a "We wonder/want to investigate" handout
Teach with cool technology and manipulatives#

To follow up#

Resources#

  • MindSetKit free set of online lessons and practices

Teaching and impact#

  • Cohen & Garcia (2014) - 1 experience/message can change everything for students
  • Beilock, Gunderson, Ramirez & Levine (2009) - primary school teachers' negative emotions on math predict achievements of girls in their classes, not boys
  • Silver (1994) - give students opportunities to pose mathematical problems, to consider a situation and think of mathematical questions to ask - the essence of real math - they become more engaged and perform at a higher level
  • Boaler (2015a) - What's math got to do with it - describes a classroom approach based on posing mathematical questions
  • Schwartz & Bransford (1998) - comparison of three ways of teaching mathematics - related to productive-failure

  • Wang, 1998; Elmore & Fuhrman, 1995 - OTL most important condition for learning

  • Good, Rattan & Dweck (2012) - studetns sense of belonging in math

Value of mathematics#

  • Rose & Betts (2004) and Boaler (2005) - more math classes == higher earnings 10 years later - as much as 19.5%
  • In the US the high school maths courses done from year 9 influence what people do for the rest of their life, is this the case in Australia?

Rich tasks#

  • Horn 2005 -
  • https://nrich.maths.org
  • Nasir et al (2014) Railside activities?
  • (Picciotto, 1995) - physical manipulative for algebraic understanding
  • (Ball, 1993) on conjectures, reasoning and proof

Visuals

  • division quilt (Lupton, Pratt, and Richardson, 2014)

Assessment#

  • Pulfrey, Buchs, and Butera (2011) replication of study on impact of assessment feedback as: grades, diagnostic comment, or both. Grades always negative
  • Reeves (2006) non-mathematical nature of US-based 100% scales
  • Need much more on making diagnostic comments/feedback

Not quite - [x] Gustein, Lipman, Hernandez & de los Reyes (1997) - curriculum designed to raise issues of gender, culture or class

What is mathematics#

  • Devlin (1997) - mathematicians describe math as "the study of patterns; that it is an aesthetic, creative, and beautiful subject"
  • Devlin (2006) - mathematics used by animals, and humans as natural mathematics users
  • Reuben Hersh (1999) - What is mathematics, really?

Misc#

  • Boaler (2008) - concept of relational equity

References#

Fry, K., & Hillman, J. (2018). The Explicitness of Teaching in Guided Inquiry. In Mathematics Education Research Group of Australasia. Mathematics Education Research Group of Australasia. https://eric.ed.gov/?id=ED592484

Sawyer, W. W. (1955). Prelude to Mathematics. Penguin Books.