Visible Learning for Mathematics: What Works Best to Optimise Student Learning – Hattie, Fisher, Frey

Some notes:

According to The National Council of Teachers of Mathematics (NCTM) publication ‘Principles to Action: Ensuring Mathematical Success for All’ (2014) the 8 high leverage teaching practices that support meaningful learning are:

  1. Establish mathematics goals to focus learning
  2. Implement taks that promote reasoning and problem solving
  3. Use and connect mathematical representations
  4. Facilitate meaningful mathematical discourse
  5. Pose purposeful questions
  6. Build procedural fluency from conceptual understanding
  7. Support productive struggle in learning mathematics
  8. Elicit and use evidence of student thinking

Similarly the 2012 National Research Council report Education for Life and Work identifies the following essential features of instruction:

  • Engaging learners in challenging tasks, with supportive guidance and feedback
  • Using multiple and varied representations of concepts and tasks
  • Encouraging elaboration, questioning, and self-explanation
  • Teaching with examples and cases
  • Priming student motivation
  • Using formative assessment

Focus on rigor defined as a balance among conceptual understanding, procedural skills and fluency, and application with equal intensity.

Mathematics instruction should be intentionally designed and carefully orchestrated in the classroom, and should always focus on impacting student learning. Start with appropriately challenging learning intentions and success criteria. Teachers need to be clear about where their students are, where they need to go, and what achievement of learning milestones looks like. Good mathematics learning is embedded in discourse and collaboration – both with teachers and among peers – and is orchestrated around appropriately challenging tasks. Students should be doing more of the thinking and talking than the teacher. Must be partners in understanding with metacognition (thinking about their own thinking).

Surface, deep and transfer learning

Surface – initial development of conceptual understanding, procedural skills, and vocabulary of a new topic

Deep – begin to make connections among conceptual ideas, and practice and apply procedural skills with greater fluency. Plan, investigate, elaborate on their conceptual understandings and begin to make generalisations. Can facilitate transfer.

Transfer – ability to more independently apply deeply understood concepts and skills to new and novel situations

 

 

Quick checks for understanding

These strategies come from an article: Assessment for Understanding by Janelle Wills, published in Assessment into Practice: Understanding assessment practice to improve students’ literacy learning. PETAA edited by Heather Fehring, 2017

3-2-1
Students complete a 3-2-1 exit slip before leaving the room or at the end of the learning episode. For example 3 things I learned, 2 connections I made, 1 question I still have.

3-Minute pause
The 3-minutes pause provides a chance for students to stop, reflect on the concepts and ideas that have just been introduced, make connections to prior knowledge or experience, and seek clarification:

  • I changed my attitude about…
  • I became more aware of …
  • I was surprised about…
  • I felt…
  • I related to…
  • I emphasised with…by…

ABC summaries

Each student in the class is assigned a different letter of the alphabet and they must select a word starting with that letter that is related to the topic being studied.

Quick Writes

Pose a question to a group or class of students and pause for 30-60 seconds. Ask students to write an answer to the question using note cards, sticky notes or scrap paper. Students share responses with each other using a process such as Give One, Get One (Lipton &Wellman, 2010)

Synectics

Present students with a analogy prompt: (A certain concept, principle or process) is like  _____ because __________.

Choral response

In response to a cue, all students respond verbally at the same time. The response can be either to answer a question or to repeat something the teacher has said.

Self-check

At the end of a lesson, students respond to the following questions in a daily journal log:

  • Was this a productive lesson for me?
  • What did I learn?
  • What did I do to help myself learn?
  • What interfered with my learning?
  • What will I do differently next time to help me learn better?

Compare and Contrast

Identify the theory or idea the author is advancing. Then identify an opposite theory. What are the similarities and differences between these ideas? This process demonstrates the depth of understanding a student has attained.

Compare and Contrast

Create a two-column table. Use the left column to write down 5-8 important quotations/ points. Use the right column to record reactions/ explanations to the quotations/ points.

Cornell Notes

Create a two-column table. Use the left column to write down 5-8 important points from a learning episode or text. Use the right column to create a non-linguist representation of the key idea or concept.

Duelling charts

1. Select a topic students have been studying (ie. democracy)

2. Write the topic on two charts

3. Divide the students into two teams

4. Each team lines up behind a chart

5. On signal, a student from each team goes to the chart and write a phrase associated with the topic. The phrase must start with the 1st letter of the word (ie. ‘d’ for democracy – distribution of political power in the hands of the public. Then, the next letter ‘e’ – eligible citizens participate equally and ‘m’ and so on).

6. After the first student finishes, the next student comes to the chart etc.

7. When both teams are done the charts are compared and shared.

 

Evidence card

Similiar to an exit card. The ticket out the door is to list at least two important ideas they have learned from the lesson and specific EVIDENCE regarding this learning.

 

Exit card

Exit cards are written responses to questions posed at the end of a class or learning activity or at the end of a day.

Find someone who

Students circulate to find others who can contribute to answers on their worksheet. They give answers and receive answers for purposes of review and showing gaps in ethic learning. (Kagan & Kagan, 2009)

Gallery Walk

Students rotate around the room stopping at posted posed questions, or pieces of learning, quotes, concepts, etc. As they stop at each chart, students have discussions with each other, write responses on poster charts or sticky notes or they pose questions that they have as a result of viewing the gallery walk material. (Lipton and Wellan 2011)

Generic question to guide self-assessment

  • What did you do well? What are the reasons for this?
  • What Could you do better? How could you do this?
  • What did you enjoy about your learning? Why?
  • What would you like me to know about your learning?
  • What would you like me to notice about your work? Why?
  • What sort of thinking did you use to help you with this?
  • What was the most important thing you learned?
  • How do you feel when things get hard?
  • What is your greatest improvement/ achievement?
  • What helped you achieve this?

Quick Draw

Students quickly draw pictures to show what they know. They then explain their drawing to a partner.

Get One, Give One

Students respond to a prompt by writing on a sticky note or card. They then take their card/ note and find a partner; share the information or ideas and then exchange cards. After two or three exchanges, hey return to their table group and share information on their last card. Table groups identify themes and patterns to share with full group. ( Lipton and Wellman 2010)

Hand signals

Ask students to display a designated hand signal to indicate their understanding of a specific concept, principle, or process: I understand____ and can explain it (ie., thumbs up). I do not yet understand (ie., thumbs down). I’m not completely sure about _____ (ie., hand wave)

 

I have, who has?

Reviw questions and responses are handed out to students on cards. The student with number one card begins the review by reading their question (ie. I have what are words that mean exactly what you say.) and then reads the question also contained on their card ( Who has the definition for figurative language?) and the review continues until all cards are used. (Kalgan and Kagan, 2009)

Idea spinner

The teacher creates a spinner marked with four quadrants and labelled “Predict, Explain, Summarise and Evaluate.” After new material is presented, the teacher spins the spinner and asks students to answer a question based on the location of the spinner.

Index card summaries and questions

Distribute index cards and ask students to write both sides with these instructions: (Side 1) Based on our study of (unit/topic) list a big idea that you understand and word it as a summary statement. (Side 2) identify something about (unit/ topic) that you do not fully understand and word it as a statement or question.

Inside/Outside circle

Students in concentric circles rotate to face partner to answer the teacher’s questions or those of a partner (via cue cards).

Instruct, insight, internalise

Teacher provides instruction to the students for 5-7 minutes, then says: Take a minute to think and record the key ideas or points you’ve heard so far or any questions you have. Teacher then continues instruction to the next stopping point and repeats the previous directions. When instruction is complete, students pair up and share their insights, key ideas, questions and summaries of what they heard.

Jigsaw

Students read different passages of the same text or selection. After reading the passage, they take on the role of an expert for their specified piece of text. The “experts” then share the information from their reading with a specific rotating group or the entire class.

K-W-L

Use a K-W-L chart as a preview activity. Prior to instruction, students complete the ‘K’ (know) and W (want to know) columns. When instruction is complete, students complete the L (learned) column. Collect the organisers and check for understanding (Ogle, 1968)

Misconception Check

Pr sent student with common or predictable misconceptions about designated concept, principle or process. Ask students whether they agree or disagree and explain why. (Science resource 2016, assessment.aaas.org/topics

Numbered Heads Together

Each student is assigned a number. Members of a group work together to agree on an answer. The teacher randomly selects one number. The student with that number answers for the group.

 

Maths – Staff Meeting

Reflect on the things you do not like people to say and do when you are working on maths in a group

  • This is easy
  • This isn’t useful
  • Someone dominating the group
  • Hurry up!
  • I can’t do it
  • This is boring
  • That’s not right
  • I do not want to do this
  • Put downs
  • Competitiveness
  • Distracting
  • Off task behaviour
  • Sitting in rows
  • Silence
  • Worksheets
  • This is too easy
  • Talking about other things – off task
  • That’s wrong
  • I don’t want to work with him/ her
  • Not engaged
  • I don’t understand
  • I don’t like this
  • This isn’t useful
  • It’s easy
  • Rather than take risks, students use their group as a resource for answers
  • Someone else doing everything
  • Too complicated task

Reflect on the things you do like people to say and do when you are working on maths in a group

  • I’ve worked this out
  • I’ll give you more time
  • If you need help I’m happy to lend a hand
  • Time to consider and then work collaboratively
  • Keep trying
  • Let’s work it out together
  • I love Maths
  • This is fun
  • Keep trying
  • Supportive comments
  • I like a challenge
  • Asking questions
  • Collaborating
  • Fun
  • Interactive
  • Lots of talk
  • Groups
  • Tricky
  • I have to think about that
  • Lots of chatter
  • Work with buddies
  • A challenge
  • Building off others’ ideas
  • That’s a good idea
  • Mistakes seen as opportunities to grow
  • On task
  • I like that idea
  • Could you try
  • Can I help you?
  • This is so much fun
  • I like going in this pit
  • You can do it
  • The strategy I used was…
  • Busy
  • Engaging conversations about how to solve the problem

Reflection:

What stuck with you today?

There is more than one way to do the task.

2,3,4 heads are better than just 1

Everyone can learn maths

Low Floor, High Ceiling – entry points and differentiation for all

Maths needs to be hands on – not sheets – to give children the opportunity to experiment

Planning for differentiation is much easier than we sometimes make it (when the task is well-designed)

Youcubed lesson plans and resources, easily accessible and ‘ready to use’

Great ‘Emoji sorting’ lesson – Will definitely try it out!

How many different ways there are to understand a simple concept like ‘what is half?’ Along with encouraging students to take those risks and make mistakes.

Who would have thought that a simple concept like ‘halving’ would result in rich conversations – growing our learning together.

I’m going to look up Jo Boaler’s Youcubed, Daily Inspiration and try some new tasks as feeling a little like the meerkat in Maths at the moment.

Youcubed week of Inspiration site excellent ideas using Emoji’s. So many things you can do with this one activity, also very relevant for kids current fads.

Importance of developing positive attitudes in maths

Open ended, collaborative maths tasks.

Finding ways of exposing misconceptions/ highlighting understandings, and things we do/ say as teachers which might create misunderstandings.

During today’s activities reinforced how everyone sees the same problem differently. Allowing students to explore and work collaboratively is extremely valuable.

The Youcubed Week of Inspiration in Maths – lesson ideas, videos and resources to use

A great Emoji task to do.

Collaborative dialogue is the key to group feeling successful.

Fun working together if group collaborative

Thanks for sharing – inspiring indeed!

Look at and explore Youcubed, Inspirational maths lessons, continue to inspire students in maths.

I have good ideas but my group has great ideas!

Once you convince yourself …convince others.

 

 

 

Good Maths Tasks – Reference Points

Low Floor – High Ceiling Tasks:

1. Open up the task so that there are multiple methods, pathways, and representations.
2. Include inquiry opportunities.
3. Ask the problem before teaching the method.
4. Add a visual component and ask students how they see the mathematics.
5. Extend the task to make it lower floor and higher ceiling.
6. Ask students to convince and reason; be skeptical.

Jo Boaler recommends these sites:

Youcubed: www.youcubed.org

NCTM: www.nctm.org (membership required to access some of the resources

NCTM Illuminations: http://illuminations.ntcm.org

Balanced Assessment: http://balancedassessment.concord.org

Math Forum: www.mathforum.org

Shell Centre: http://map.mathshell.org/materials/index.php

Dan Meyer’s resources: http://blog.mrmeyer.com

Geogebra: http://geogebra.org/cms/

Video Mosaic project: http://videomosaic.org/

NRich: http://nrich.maths.org/

Estimation 180: http://www.estimation180.com

Visual Patterns; grades K-12: http://www.visualpatterns.org

Number Strings: http://numberstrings.com

Mathalicious, grades 6-12; real-world lessons for middle and high school: http://mathalicious.com

Good Teamwork and Bad

This is a great little video to begin a discussion about good and bad teamwork. I would probably stop the video at the polar bears, because of the advertising.