Published 2013 Revised 2019

**What do we mean by "Low Threshold High Ceiling"?**

Here at NRICH, we've been using the phrase Low Threshold High Ceiling (LTHC) for many years to describe attributes shared by lots of our tasks. In the context of learning technologies, the phrase LTHC (also sometimes referred to as 'low-threshold-no-ceiling', or 'low-floor high-ceiling') originates in Seymour Papert's description of the central design principle of the Logo programming
language. The 'low threshold' means that new users, including those who have never programmed before, should find it easy to get started, and 'no ceiling' (or 'high-ceiling') means the programming language shouldn't be limiting for advanced users. We use the term in a similar way when talking about our resources. It can be summed up as follows: a Low Threshold High Ceiling task
means **everyone can get started**, and **everyone can get stuck**.

**Everyone can get started**

If an activity is considered to be LTHC for a particular group of learners, pretty much everyone in the group should be able to make a start on it. The threshold needs to be mathematically accessible for all the students in the group, that is, everyone needs to have the prior mathematical knowledge required to start working on the problem. This will of course vary depending on the age and prior
attainment of the learners in the group - a task that has a low threshold for 17 year olds might be far out of the reach of most 7 year olds!

As well as the mathematical threshold, it's important to consider the psychological threshold of the task - how resourceful and resilient will students need to be? Some problems may require a very basic understanding of the mathematical content to solve, and yet still be extremely challenging (see for example the task Shopping Basket). When faced with such a
task, learners might struggle to know where to start, or to find any small step that they can take towards solving the problem. This places a high level of demand on learners at the start of the task. This doesn't mean such tasks are without value! They can still be rich tasks, worthwhile for students to spend time struggling with, but they are not LTHC. On NRICH, the Challenge Level of a task
indicated by one, two or three stars might give an indication of the mathematical and psychological threshold. LTHC tasks are likely to have a challenge level of one star - you can see some examples of such tasks in the article Creating a Low Threshold High Ceiling Classroom. (Shopping Basket has a challenge level of three stars!)

**Everyone can get stuck**

Part of becoming a resilient mathematician is learning to recognise what it feels like to be stuck, and what strategies can be useful in getting yourself unstuck. Some undergraduate mathematics students report that they found maths lessons at school very straightforward until the very end of school or the beginning of university, and the culture shock of finding maths difficult for the first time
was very daunting! One way to mitigate against that is to make sure that all students experience mathematical struggle throughout their time in school, and that being stuck is normalised and recognised as an important part of mathematical problem solving. Low Threshold High Ceiling tasks are designed to have lots of built-in extension opportunities, so that there are harder questions to be asked
and more challenging problems to solve. This means that all learners can potentially reach a point where they don't immediately know what to do next, and they can start to develop their resilience and learn fruitful strategies for making progress when they feel as if they've come up against a brick wall.

**Why do we like Low Threshold High Ceiling tasks?**

At NRICH, we promote the use of Low Threshold High Ceiling tasks because we believe that working on such tasks helps learners develop into competent, confident mathematicians. The Low Threshold may mitigate against the development of maths anxiety by making sure that learners do not fail at the first hurdle. The High Ceiling offers everyone the opportunity to develop their resilience.
Furthermore, LTHC tasks give learners a sense of agency as there is more than one pathway available to them. Rather than a classroom situation where different children are given different activities to work on, the whole class is working together on the same task. There is no predetermination as to who will work on extension work and who will stick to the support material, and everyone has a
sense of what is going on, with plenaries giving everyone a chance to hear how others in the classroom have worked on the same activity in different ways. LTHC tasks allow learners to demonstrate what they can do, rather than what they can't. As teachers it's very easy to predict how well our learners will cope with a particular piece of mathematics, and sometimes that prediction can be a
self-fulfilling prophecy. When the ceiling is raised it can be surprising what heights learners can achieve.

*To find out more about effective teaching using Low Threshold High Ceiling tasks, read the article Creating a Low Threshold High Ceiling Classroom.*