Published September 2011,July 2011.

We're not sure if we invented it*, but here at NRICH the phrase 'Low threshold, high ceiling' (LTHC) has certainly become one of our favourites. We like to think that many of our tasks can be used in this way, and in many of the projects where we work with schools we take LTHC as the title. We thought therefore it would be a good idea to explore what LTHC means, and indicate which activities on the site are particularly good examples.

**What does LTHC mean?**

Imagine a room. The way in is unproblematic - a few small steps and you're in. Once inside there are lots of possibilities for activity, many of them unproblematic too, others more challenging. In fact the only limit on your choice of activity is the space in the room, and the height to which you can rise.

Well the analogy may be a bit contrived, but that's exactly what LTHC means. A LTHC mathematical activity is one which pretty well everyone in the group can begin, and then work on at their own level of engagement, but which has lots of possibilities for the participants to do much more challenging mathematics.

**Why do we like LTHC?**

We like LTHC activities because they promote a positive classroom culture - one where the whole class does maths together. We know that for many learners, and particularly for those at the ends of the attainment spectrum, maths can be a lonely subject. When we visit ordinary classrooms we find, usually, two different types of organisation. The first is where the class is split into groups (often by attainment), each of which is engaged in a different activity. Sometimes all the groups are working on different parts of the same mathematical topic, but not always. It can often be difficult to feel part of a classroom community when you have no idea what the others are doing.

The other format is where everyone does the same, usually closed, activity. This time everyone knows what they are supposed to be doing, but those learners at either end of the spectrum either finish first and are bored, or get left behind and feel excluded. LTHC tasks offer activities which are suitable for pretty well everyone in an ordinary classroom, allowing the high flyers to explore and challenge themselves whilst the less confident can stay close to the original task and consolidate their learning. Everyone has a sense of what is going on, plenaries are much more meaningful and as they hear how others in the classroom have worked on the activity, learners often raise their game.

So another reason we like LTHC tasks is that they allow learners to show what they can do, not 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.

The third reason we like LTHC tasks is that they offer many possibilities for learners to focus on more sophisticated process skills rather than more knowledge. It's often mistakenly thought that the only way to challenge learners is to offer them *content* at a higher level; in LTHC tasks the content often remains quite simple but the *level of thinking* required can become very
sophisticated. If you don't believe it, try finding a winning strategy for Strike it Out, below.

We always write our tasks with an eye on the high ceiling. Often these ideas are offered in the task itself, but in the teachers' notes you'll find even more.

Below are some of our favourites. You might like to skip through to the age range examples appropriate to your class (although you might find that the others are interesting too ...).

**Primary classrooms**

Strike it Out is a LTHC game. The threshold activity is **number bonds to 20** and some children will stay at this level without engaging much with how you might win. Others will become very involved in **conjecturing** and **testing** possible winning strategies and there is even the possibility of some elegant
**proving** that the challenge of crossing off all the numbers 0-20 (and 1-20) is not possible. Others can extend the line to **bigger numbers** or use **more advanced** **number systems** - decimal fractions, or directed numbers for example.

Magic Vs is another favourite primary task. The threshold is **counting to 15** as children try to make both sides of the V add up to the same number. Others will start to **recognise patterns** and begin to make **generalisations** about them. The high flyers will make up their own questions (**what
if** ...) and **test their hypotheses**.

*Other primary LTHC tasks we like*

Number Differences (as a 'suite' of three)

*Secondary classrooms*

Take a look at Arithmagons. At the threshold students use **trial and improvement** to arrive at a solution. As they collect more evidence, they start to **look for patterns** which they can then **generalise**. They may use an **algebraic notation** to arrive at, or confirm, a conjecture about
ways of solving the problem. And once the confident students have satisfied themselves that they can solve any triangle arithmagon, they can try out some different shaped ones which involve **multi-step solutions**.

Odds and Evens has been a favourite of ours for some time. At the threshold students **work systematically** to derive all possible results, **recording** their work in the most useful way. They use their results to **justify** which version of the game is fairest. Others can move on to working together to
generate lots of data which they can **sort and analyse.** They may use it to form a **conjecture** about the balls needed for a fair game, and **predict** which other combinations would do so too, **justifying** their statements. The activity moves from experimental probability to theoretical probability (and a discussion of the difference) to
sophisticated analysis.

*Other secondary LTHC tasks we like*

Opposite Vertices (and Vector Journeys as follow-up)

What's Possible (this links also to Plus Minus)

Number Pyramids (again with follow-ups)

*Post-16 classrooms*

Post-16 classrooms are often the ones where everyone does the same A level questions, perhaps at different speeds. It's good to be able to offer tasks which highly able students find a challenging alternative to racing through the exercises, but which less confident members of the group can still access.

Painting by Numbers is a practical activity which introduces students to the idea of topology, and hence the Four Colour Theorem. At the threshold students will be using **trial and improvement** to clarify their ideas about what it is that makes two apparently different diagrams have the same topology. More confident students will be
able to use their new knowledge to **engage with the theorem** and some will even be able to understand it fully - which takes them up to **university level** pure mathematics.

The problem Whose Line Graph is it Anyway? has some inbuilt support and extension and is one of a new set of tasks we are developing which make links across and between the sciences and mathematics. Students have to match the statement, equation and graphical representation of various physical processes. At the threshold they will be
**consolidating** existing knowledge as they match sets of which they are certain. More confident students will **generalise** what they know and **apply it** to new specific situations, looking for similarities between the physical processes. For the high flyers there are links to other activities which use similar content, and coming soon will be links to
readings about the maths and science involved.

*Other Post-16 LTHC tasks we like*

We enjoy hearing how teachers use our tasks. Do get in touch if you have ideas about other tasks that you have used in a Low Threshold High Ceiling way.

**You can download a pdf version of this article, focusing specifically on primary tasks,** here**.**

*In the context of learning technologies, the phrase LTHC or 'low-threshold high-ceiling' (also sometimes referred to as 'low-threshold-no-ceiling') originates in Seymour Papert's description of the central design principle of the Logo programming language (Papert, 1980). Another example of a widely used LTHC platform is NetLogo, a modeling environment. In the context
of programming languages 'low threshold' means that new users, including those who never programmed before, should find it easy to get started, whereas 'no ceiling' (or 'high-ceiling') means the language shouldn't be limiting for advanced users.