Age
3 to 18
NRICH is leading a series of six linked face-to-face PD sessions with teaching staff at the Evelina Hospital School, London. These are taking place during the academic year 2018-19 and focus on developing rich mathematical curriculum-linked resources for their learners.
**Next session Monday 13th May**
Here's a recap of our sessions so far, with links to the resources. (Note: sessions 1 and 2 focused on problem-solving; sessions 3 and 4 focused on reasoning; and sessions 5 and 6 will focus on fluency).
Session 1: Monday 3rd September 2018
In session 1, we explored the stated aims of the school mathematics curriculum and compared them with the additional expectations represented by the rope model for teaching mathematics, especially the focus on developing positive attitudes towards mathematics as well as the importance of nurturing resilient learners.
The session focused on the fluency aspect of the curriculum, exploring four NRICH activities from across the key stages, and identifying ways to adapt them for pupils and students attending a school in a hospital setting:
Frogs
and the linked task of
Tower of Hanoi
Totality
Estimating Angles
Four Triangles (and an EYFS interactive display created for the activity)
Gap task
To use one of the above activities with pupils in conjunction with the Teachers' Resources.
Then to make notes to supplement the existing Teachers' Resources for the task, which reflect supporting the additional needs of pupils in a hospital setting.
Session 2: Monday 5th November 2018
We enjoyed an incredibly useful session sharing feedback about the various activities which colleagues had tried out on the wards with students - many thanks to everyone who shared their ideas.
Both Frogs and Estimating Angles were popular choices for the gap task activity. Here's a photo from one of Argon's sessions where he was working with a student who was trying out Frogs using three blue frogs and three red frogs. Do you think they have a winning strategy? Can you explain why? Which key questions could you ask? Where would you take the activity next with this student?
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Here's another student trying out the interactive version of Frogs. What difference does using the interactive make for the students? How did its facility for keeping count of their moves affect their resilience?
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How has working systematically enabled this student to generalise? Looking at the number of steps taken for each set of frogs, how did they arrive at their equation? Hint: remember to look at the difference between the numbers. Don't forget to look back at the Teacher Notes with the Frogs activity - there's planning advice but also student solutions you can use or adapt for your own students.
After everyone had shared their feedback, we moved on to the main focus of the session: exploring the four stages of problem-solving.
Stage One: Getting Started
Encouraging students to tackle an activity is often the biggest challenge. We looked at an activity, One Big Triangle, which could be tackled in numerous ways. We discussed some of the ways adopted around the room - trial and improvement, making it simpler by finding pairs first of all, working top down so there were fewer options and so on. Whichever way we tried seemed to work, we did reach a solution. A follow up activity with students could involve exploring which were the most effiicient approaches as well as reflecting on the student solutions on NRICH too. Remember to encourage students to consider more than one solution. Have they found all possible solutions? How do they know?
Stage Two: Working on the Problem
Once we'd explored ways to get started, we moved on to another activity called Two-digit Targets. This offered plenty of opportunities to explore an activity in different ways. To begin with, the most popular approach was finding a number which satisfied the criteria as we worked our way down the list. As we began to reflect on the challenge, we realised that we could get nearer, much nearer, to some of the target numbers by reviewing our initial answers. We became even more focused when groups decided to award penalty points for how far their answers were away from the actual target. Again, it is worth exploring the Teacher Notes for ways to adapt this activity as well as key questions for the students.
Stage Three: Digging Deeper
This was the stage where we looked at the 6 Beads activity. Although finding an answer is straightforward, ensuring that students have found all possibilities is a little more challenging. One popular way of knowing that we had found all the possible answers was using proof by exhaustion. By tring out different numbers of beads we were also able to test out our pattern spotting skills and begin to generalise about our findings.
How could this activity be adapted for younger or older students?
Stage Four: Reflecting
This stage is so important for the students because it allows them to reflect on their learning and communicate their findings too. We tried out the activity How Would We Count? and soon found out there were many ways to calculate the total, some probably a little more efficient than others!
Don't forget the RSPB organise a national bird watch for schools every January which offers an ideal context for students to apply their counting skills and contribute towards a live research project too.
Gap Tasks
Visit this 'Evelina' page on NRICH - explore at least one of today's resources with students. Take careful note about the ways in which you work through the four problem-solving stages. When did the students seem to struggle the most? How did you support them? If you chose Two-digit Targets, don't forget that it's a Live Problem this term so their solutions can be submitted to NRICH.
Please remember to share your photos and examples of work with Kate before the next meeting so that she can send them to Ems who will upload them to this page.
Session 3: Tuesday 22nd January
In this session we changed our focus from problem-solving to reasoning for the next two sessions. In a very hands-on session we explore some of NRICH's favourite problems for promoting reasoning.
We began with an activity that we found could be approached in several different ways but which all lead to the correct answer. Eggs in Baskets (2002) is a terrific resource for encouraging learners to share their reasoning. During the session, we found that the team took several different approaches to this low threshold high ceiling problem including:
trial and error,
working systematically
using a table,
and, using algebra too.
We also considered adopting a bar model approach to the problem since growing numbers of schools are now embedding bar models in their teaching.
The Eggs in Baskets resource features a number of learners' solutions too (these can be accessed by clicking on the 'solutions' tab on the left hand side of the screen). We discussed how these solutions could be used in different ways to promote reasoning: they offer learners an opportunity to compare their answers with others but they also offer prompts if a learner gets stuck too.
After exploring Eggs in Baskets we looked at Poly Plug Rectangles (7511) as another example of a resource which promotes reasoning; this is a resource can be be explored in different ways so it is very flexible for learners with different needs. We started off by exploring a paper-and-pencil version of the activity a s agroup, then explored it further working in pairs. This activity also has an interactive version too which learners can play in pairs or with an adult - we looked at how the settings could be changed to alter the level the challenge.
Having reasoned our way through Poly Plug Rectangles, we tried our skills at Rectangle Tangle (1048). Again, this resource is another ideal opportunity to set a task which offers plenty of opportunitie for learners to share their reasoning; this time, they needed to work out the fractions of each shape within the larger rectangle. Across our room, colleagues adopted several different approaches, and cseveral choose different starting points too, we everyone managed to find the solution too. Another useful aspect of this activity is its shape focus - it enables adults to ensure learners are using mathematical vocabulary to describe shapes and notice their similarities and differences.
We finished this session by looking at one of our Live Problems: Name that Triangle (14042). This activity has been designed to offer supports for individuals or pairs of learners to help build their resilience at problem-solving. The low threshold approach enbales most, if not all learners, to engage with the problem by identifying the different types of rectangles, before challenging them to draw examples of different types to match the criteria - this activity enables the learners to draw on their reasoning skills to justity their choices and convince others about their answers. As a Live Problem, they could also send in their solutions to the NRICH team for possible publication.
Gap task
Please explore one or more of today's activities and come prepared to discuss your thoughts on ways to maximise their potential with learners in your setting at our next meeting. If a learner explores a live problem, please do consider sending their work into the NRICH team for possible publication.
Session Four: Tuesday 12th March
In our second of our two sessions exploring reasoning, we moved our focus to from exploring activities which promote reasoning to considering a progression in reasoning and ways to support learners make progress along it. We were also joined in this session by NRICH's resource designer Oscar Gillespie.
We began by exploring the activity Sealed Solution (1177). This is another example of a low threshold high ceilig activity which offers learners an opportunity to reason and reflect on their approach. Using examples of learners' solution to this problem, we identified a progression as follows:
Describing - learners simply recall the process they adopted but do not give reasons for their decision-making
Explaining - in this next step, the learners do explain some of their thinking
Convincing - although they may not yet have reached a correct solution (being convinced, and convincing others, does not mean you are right), they are able to explain the reaosns for their choices using words such as 'because'
Justifying - the learners have a watertight, correct explanation for their solution
We also looked at the NRICH page Reasoning: From Novice to Expert (11336) which offers a commentary on the solutions for Sealed Solution, drawing out the reasons for palcing them at different points along the progression. Crucially, the progression also enables learners to identify their next steps as well as using examples of solutions to see 'what a good one looks like.'
Today's session continued by exploring the progression in reasoning for some of our other favourite NRICH resources. Which Scripts? (774) is another NRICH activity which can be explored using the interactivity or by matching pieces printed off on paper. As we found,=in the session, there is more than one way to approach this problem which requires both reasoning and resilience to arrive at the answer, as well as attempt the follow-up question. Looking at the solutions revealed a number of useful approaches suggested by learners which would be helpful for others struggling to get started or make further progress when their first attempt was unsuccessful.
The activity Largest Even (7431) is a lovely ineractive activity where the settings can be altered to adjust the level of challenge, such as altering the number of digits. It is also a flexible task which can be played with a partner using a set of digit card, making it ideal for a variety of younger learners if you're focusing on their reasoning skills. Again, do look at the solutions tab for examples of other learners' work.
We finished this session by exploring an activity for the older learners whose low threshold high ceiling approach also makes it accessible to young learners too. Steel Cables (7760) can be used in a number of ways. We began by trying to work out how many cables are needed for a size 5, but without counting each strand! Sharing our approaches, we found that there were a number of ways of working this out by spotting patterns within the shape. Then we looked at some examples of stduent's work, which are provided with the resource, and tried to explain how they reached their answers. Again, this is a lovely resource for expalining reasonign which is popular with older students. It enables them to try different approaches and reflect on their efficiency too.
Gap task
Please continue to explore today's resources with your learners and share any feedback, photos etc with Ems before the next session so she can upload them to this webpage.