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This task comes from YouCubed but I am sure its in many many places. I remember it vaguely from being a student myself.

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Jo has used this task with a lot of success on the first days of school with very hesitant students. Soon after setting the challenge the board area becomes full of students putting up their solutions, then returning to their seat to look for more. For students, it is a very safe and non threatening activity. It builds number sense and is a fun challenge. This task is also a really nice way of helping students become comfortable sharing their work in front of the class. The first appearance of four 4's was in English mathematician: WW Rouse Ball's Mathematical Recreations and Essays in 1892, and then revisited by Constance Reid in 'From Zero to Infinity' in 1955.

Task Instructions

Can you find every number between 1 and 20 using only four 4's and any operation?

or

Can we as a class find every number between 1 and 20 using only four 4's and any operation?

Here's an example √4 + √4 + 4⁄4 =

Give this challenge to students and write all the numbers from 1 to 20 on the board. Then ask students to come to the board and write a solution any time they have one. The more solutions you get for each number the better.

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Jo has used this task with a lot of success on the first days of school with very hesitant students. Soon after setting the challenge the board area becomes full [...]

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A good follow-one from Frogs and Toads as based on similar concept.

https://www.mathsisfun.com/games/towerofhanoi.html

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having readily available extending/enabling prompts

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In a secondary setting, there is so much pressure to get through the syllabus and to keep up with other classes classes ... It's very difficult to find the time to delve deeply into problem solving areas. I'm a relatively new teacher and don't feel confident with the tools in my "kit" to get through all the content AND incorporate many of these imaginative ideas and creative activities. Some colleagues actually deter me from undertaking these sorts of activities because I'm taking up too much time, and not getting through content.

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In a secondary setting, there is so much pressure to get through the syllabus and to keep up with other classes classes ... It's very difficult to find the time to delve deeply into problem solving areas. I'm a relatively new teacher and don't feel confident with the tools in my "kit" to get through all the content AND incorporate many of these imaginative ideas and creative activities. Some colleagues actually deter me from undertaking these sorts of activities because I'm taking up too much time, and not getting through content.

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Providing a variety of concrete materials/manipulatives for students to have that tactile stimulation

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Providing a variety of concrete materials/manipulatives for students to have that tactile stimulation

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For example with the problems today. Toads and Frogs: counters, students on chairs. How many triangles?: a pinboard with the various elastics (although i think the digital version was awesome), perhaps coloured pencils to colour the triangles on grid paper.

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Setting aside the time needed to really delve into problems

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www.maths300.com

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Reminders of visual tools or models that might be helpful.

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Reminders of visual tools or models that might be helpful.

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Examples like:
- use number lines
- draw a diagram
- use a table
- make the problem using concrete materials
etc

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making pedagogies and expectations explicit e.g. "In think lesson I need you to:

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making pedagogies and expectations explicit e.g. "In think lesson I need you to:

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* have a go
* listen to each other's thinking etc...

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Traditional thinking teachers/teachers scared of rich tasks

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Traditional thinking teachers/teachers scared of rich tasks

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Sadly I still have a number of teachers in my team who believe that traditional explicit teaching is best and/or are not confident in running this style of tasks. We do have these tasks in our curriculum but some avoid running them. My hope is to run PL similar to this in my school however being in Melbourne it is on the back burner at the moment..

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Encouraging teachers to use problem solving in their class VS teaching skills through a set of worksheets or algorithms on the whiteboard

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Need to frame the thinking of each lesson, just like Michaela did for us today

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Need to frame the thinking of each lesson, just like Michaela did for us today

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Curriculum constraints and time pressures- we have forgotten to play with maths and it’s become an end in itself

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Building capacity for teachers to Launch a problem and guide students through it.

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Building capacity for teachers to Launch a problem and guide students through it.

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Create a focus on multiple solution paths

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Good questioning to guide students to use what they know to solve what they don't know

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talk about and get students used to the idea of being in the "zone of confusion" zoc it out

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talk about and get students used to the idea of being in the "zone of confusion" zoc it out

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Tell students that it's okay to get it wrong the first time - that it's a sign of their brain growing

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Tell students that it's okay to get it wrong the first time - that it's a sign of their brain growing

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Get students to share as many of their strategies and explain it to their peers.

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Get students to share as many of their strategies and explain it to their peers.

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How do we report/measure growth/ give feedback/ ensure the community values problem-solving

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When sharing back, focus on the process and ideas, more than the answer. 

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When sharing back, focus on the process and ideas, more than the answer. 

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This can be hard at the start, as students often just want to tell you they got the answer. But intentionally asking students who didn't get to the 'end' but had an interesting process is a great way to break this trend.

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Lot's of possible changes and questions that can be asked.

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Creativity and enjoyment- we lose the sense of fun and I liked that Norm said it was great to make mistakes- such a valuable learning tool

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Reframe mathematics to be fundamentally about problem-solving, with all content and skills treated as enabling problem-solving and tied back to problem-solving.

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Reframe mathematics to be fundamentally about problem-solving, with all content and skills treated as enabling problem-solving and tied back to problem-solving.

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Asking students to share their thinking - what do they know. What is their baseline, what are the constraints. If all stuck then model (need to try to only show the start of your thinking - but this is hard)

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Focus on persistence/effort rather than academic ability/ speed

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Fiddle Faddle Flop - Using reasoning and analysis to discover the hidden number

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Fiddle Faddle Flop - Using reasoning and analysis to discover the hidden number

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N-Rich which I think someone mentioned earlier. Lots of great problems at levels from early primary to senior secondary

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N-Rich which I think someone mentioned earlier. Lots of great problems at levels from early primary to senior secondary

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it is important to do regular work - it can be the start of a lesson, it can be once a week, but it can't be a one off wonder lesson once a term.

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Conundrums - you can only answer yes or no to their questions - reasoning and analysis and creating questions

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Conundrums - you can only answer yes or no to their questions - reasoning and analysis and creating questions

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allow students to make mistakes and not rush them

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allow students to make mistakes and not rush them

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Unpack the task first so that you can anticipate misconceptions and how to take the kids to the next stage

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Create relevant tasks that relate to students own lives

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Taken from a previous MTC have had a great time using this with my year 9s and seeing the conversations around strategies.