The Maths Learning Environment...
How would you describe the maths learning environment in your classroom?
Like most time-pressed teachers, the question may well never have climbed sufficiently far up the priorities list to merit much consideration - if any at all! However, the maths environment can make a huge difference to pupils' attitudes towards the subject and this attitude can affect their progress.
Let's look at each of the questions raised...
Are pupils happy to make mistakes?
While working with a group of pupils, you ask for someone to share their ideas with the rest of the group only to see all eyes lower and then the nervous fidgeting starts; some even start to wipe their work off their whiteboards...
Most of us have experienced teaching pupils who are anxious about maths and it is not necessarily those pupils who have a tendency to struggle. Able pupils, who have excelled at following procedures, can also become anxious when faced with more open-ended problems where they have to first understand the problem in order to work out what procedure is needed to solve it. Maths anxiety even has an entry in Wikipedia but there is no such entry for reading anxiety or writing anxiety even though all teachers have come across pupils who struggle in these areas.
So what can we do?
Pupils need to be explicitly told that making mistakes is important for our learning and that this has been true from when they were born. No baby starts talking in full sentences without babbling first or just stands up and starts walking without falling over sometimes. When learning to read words or to count, children make numerous mistakes before they are fluent. So it is the same with maths. When meeting a new class, it is worth initially rewarding pupils who share their mistakes, and always thank them for their contribution, since it allows us to understand their misconceptions which are also likely to be shared by other pupils. Explain this to the pupils and how mistakes are great - as long as we learn from them! Make a point of rewarding pupils who make progress by learning from their mistakes.
Mistakes don't initially have to come from the pupils within a class. It is worth saving pieces of work containing mistakes from previous pupils or even creating them yourself and asking pupils to work out what the mistake is and why they think it has been made; this encourages pupils to see mistakes as an opportunity for learning. It also gets pupils to focus on how we know an answer must be wrong so it is a useful reasoning activity.
Allow pupils time to fix their mistakes, as we do during writing sessions, so avoid always providing the solution when correcting written maths work. If a pupil has no mistakes, you can provide them with a what if? question to explore... Sometimes we think that we cannot afford the time to do this, due to the pressure of covering a broad curriculum, but it could help reduce the time when revisiting topics if there is a better level of understanding when the topic is introduced.
Encourage pupils to be comfortable verbalising their confusion and uncertainty but discourage (or even ban) the use of, "I don't get it!" or, "I can't do it!" Pupils need to be resilient and learn to persevere and both of these utterances suggest a pupil giving up. Asking pupils what they think a question means or suggesting that they draw a picture or act something out can sometimes help. Also, exploring what parts they think they understand so far or what questions are coming into their heads can help unravel their confusion or, in some cases, their panic!
Primary school classrooms are often wonderfully colourful places with impressive eye-catching displays adorning the walls and corridors. So what subjects do these displays cover? It can be an interesting exercise to walk around your school, taking note of how many walls include any maths content...
Approximately 20% of teaching time is devoted to maths in UK primary schools but would visitors from other countries realise the importance of maths by looking at our classroom walls and other display areas? Perhaps the fact that groups of pupils (or even the whole class) are working on the same activity, which produces very similar results, is one factor that makes the idea of putting maths on the walls less appealing than creative poems that pupils have written. This of course then leads us to consider the purpose of putting content on walls; is it purely to display work or as a resource to aid pupils' learning? Perhaps it is time to think about learning walls...
So what type of maths content can be put on a maths wall?
Thinking of a maths wall as a learning resource rather than a display makes the decision regarding content much easier. Some of the content sections for a maths learning wall (titled Mathematicians' Wall) can be similar to those for a literacy learning wall (titled Authors' Wall). Some of the sections that I have used include:
What we are learning
I didn't think pupils would take much notice of this without me drawing attention to it, until pupils
started to comment if it hadn't been changed at the start of the week in cases where the topic lasted longer than five days.
We can sometimes overlook how much unfamiliar language we use in maths lessons and it is not just words such as congruent or parallel, which may not be used outside of maths lessons, but also those which are used in a different context such as difference and face.
Language Games can be played, such as:
How many times can pupils use the word in a maths context during playtime, or even at home?
How many sentences can pairs produce, involving the word, in a set amount of minutes?
Create a paragraph where pupils take turns to create the next sentence including the word.
This will depend on the topic but in the case of multiplication, could include a large multiplication grid showing the symmetry involved. If the topic is shape then some selected images that pupils have created of irregular polygons could be used. Anything which helps pupils by providing a visual image can be included in this section and some of these images can be of work created by the pupils.
Encourage pupils to look for links between different areas of maths so when exploring area, do they see the link with grid multiplication? What about division and fractions? Let pupils add any links that they notice.
Why is this a good example?
Rather then displaying all pupils' work at the same time, choose a couple of pieces which are good examples (making sure that all pupils make contributions to the wall over time) and ask pupils to work out why they have been chosen by providing them with post-it notes to add their reasons when they have a spare five minutes.
It is worth mentioning here that when I introduced an Authors' Wall at the same time as using a
Mathematicians' Wall, pupils were far more likely to bring adults into the classroom to show that their work was on their wall; this rarely happened when the writing of the entire class was displayed at the same time...
This section contains examples of where we see or how we use this topic in real-life. This can be a popular homework activity but be aware that a lot of material could be generated if used in this way!
However, surplus creations can always be:
Shared with other classes
Used in a general Real-life Maths display elsewhere in the school
Displayed wherever Maths Club takes place
Incorporated into a pupil presentation for parents
Included in a pupil-led assembly about maths
Hints and Tips
Post-it notes can be left near the wall so that pupils can add hints and tips which others can refer to.
Pupils can be directed to these when unsure, to see if there is anything there to help them.
In our haste to cover the curriculum, we often forget to give pupils an opportunity to ask any burning
questions that they may have about a particular topic. This is useful at both the start and end of topics
but can also be enlightening during a topic. It is interesting to allow other pupils to answer the questions
initially since they can sometimes provide wonderfully lucid answers, while on other occasions their
answer can uncover a common misconception which may have otherwise not come to light.
Different levels of challenge can be offered in this section which pupils can attempt in their own time, if they have finished a particular task, or which can be taken home for homework.
To avoid the problem of not satisfying everybody's expectations regarding both amount and
difficulty levels of homework, it is worth considering providing a core homework and an optional
challenge - both of which may be tailored to specific groups of pupils, as needed.
What can be seen on the walls?
Is there lots of discussion during maths sessions?
Two maths sessions in the same school: in one, pupils sit individually and work mainly in silence; in the other, pupils work in pairs while working at grouped tables and a hum of discussion can be heard in the classroom.
As teachers, we tend to have a specific style of teaching with which we feel most comfortable and when the topic is raised, lively discussions can ensue as to the most effective ways for pupils to work. There are points on which most agree: independent writing and assessments need silence (although I personally favour using certain types of music for the former).
So how is discussion useful during maths sessions?
When pupils discuss maths, they are clarifying their ideas and the act of explaining can highlight loopholes in their thinking or it may develop their understanding. Of course, the act of discussion also involves listening and another pupil's perspective can question or reinforce their ideas. Pupils can be taught to 'debate' their ideas and this can really help to develop their listening skills as well as nurture the confidence to justify their reasoning. Of course, this works most successfully in a classroom where mistakes are welcomed - see above!
Less confident pupils can feel supported when allowed to talk through their ideas with a partner. This is particularly true when working with larger groups or in whole class situations, where allowing time for discussion with a partner before sharing their thoughts can decrease any anxiety.
Listening to pupils' discussions can be a great form of informal assessment since misconceptions can be revealed in situations where pupils are thinking out loud, which may otherwise not have been noticed.
So at what points in the session should discussion be encouraged?
Pupils need individual thinking time so that they have the opportunity to start forming their own ideas before listening to others' thoughts. The amount of time needed can vary a great deal between pupils, although all pupils benefit from sharing for some of the time since being able to explain their ideas clearly to others is an important skill and verbalising their understanding prior to writing can help. There will of course be times when all pupils need to be silent in order to listen to an adult or fellow pupil explaining something or when formal or informal assessments are taking place.
How many pupils should be involved in a discussion?
Flexibility in grouping is required in maths as it is for other subjects. One group of four may work brilliantly together while another group will be totally unproductive. My preferred mode is for pupils to work in pairs and then join up with another pair to discuss results. Of course there should also be opportunities for whole class discussion, which can happen at any time during a maths session, as needed. As always, pupils benefit from working with a range of pupils although it is worth allowing a period of time with the same people before changing groupings.
What about recording their work?
Longer stints of shared work can be recorded on one sheet of A3 paper, on which you give feedback, and then it can be copied and each pupil then has their own record of work on which they can respond to your feedback or suggested next steps.
Do pupils help themselves to visual resources and dictionaries?
"I don't use those resources - they're for younger kids!" For some reason, pupils can have a tendency to avoid using visual resources once they get to a certain age, even though it can really help understanding and embed the learning when they are not quite secure within a topic. In the case of more able pupils, visual resources can also help when working on some challenging problems.
Perhaps we do not model the use of visual resources sufficiently as pupils move through the school and even if we do, maybe they are not easily accessible to pupils or it is unclear which resources would be helpful in particular situations. Being able to represent maths visually can help pupils enormously so this is an area which may need some focus. However, make sure to allow older pupils time to play with the resources for a short time initially, if they haven't seen them for a long time!
So which resources can help?
Numbered number lines can be used to help pupils see how adjustments can make calculations easier: the difference between 28 and 36 is the same number of jumps as between 22 and 30, with the second being easier to calculate. Vertical number lines are worth introducing from a young age to help with the concept of negative numbers and also so that pupils associate the axes of graphs with number lines, which some older pupils fail to do.
Cubes can be useful when looking at volume; building cubes can help pupils understand why we measure volume in cubic units. They can also be used with any pupils for working with operations and smaller numbers.
Hundred Squares aren't just for helping younger pupils with addition, subtraction and place value: they are also helpful for revealing multiple patterns and other types of number.
Base 10 materials are useful for working on place value and for initially working with algorithms involving larger numbers. They can also be very effective when introducing percentages, where the hundred square becomes one unit split into one hundred parts. It would be worth having different coloured materials from those used for algorithm work to avoid confusion if the pupils have recently worked on algorithms.
Laminated place value charts and sliders are effective for pupils being able to visualise that the digits are moving positions which affects their value while the decimal point stays in position.
Coloured multiplication grids which show the symmetry within the grid can help pupils to see the commutativity which reduces the number of facts to be learned.
Measuring equipment such as rulers and metre sticks, tape measures, jugs, water bottles and packaging are all useful for helping pupils to develop a feel for quantities which can help to avoid unrealistic answers when working with measures and assist pupils to use sensible estimates, which are frequently used in real life.
Fraction circles, fraction squares and fraction towers can all help pupils to compare fractions and the squares and circles parts can be effectively used on a number line to show, for example, that the increasing denominator in unit fractions reduces the size of the number!
Geostrips and geoboards can be effectively used when exploring shape ideas.
This is not an exhaustive list and teachers will build up favourites over time but it is always worth having an open mind towards manipulatives which you may not have used in the past, since they may prove more effective than those you are currently using.
Dictionaries should be available for pupils to check terms that they might have forgotten or are still unsure about. Having a range is useful (including access to online dictionaries) since slightly different definitions and explanations may be found which can often add insight.
Multi-sided dice, numbered cards and playing cards are an essential resource to me in maths classrooms, for game playing and number generation and this includes ten-sided and twenty-sided, as well as the usual six-sided dice.
Squared paper (both 1cm and 2cm squares) for topics such as: area, arrays and grid multiplication.
Counters for arrays, grouping, representation of types of number and ratio. Given that some pupils can struggle with laying out counters, the 2cm squared paper helps by allocating a counter to a square.
Large chalk for drawing on the playground.
Large pieces of rope and long pieces of elastic for working in a larger space.
Straws and pipe cleaners are great for even young pupils to construct shapes without the use of sticky tape.
Where does maths happen?
Is maths always undertaken while pupils sit at their assigned tables or does it happen in various parts of the classroom, in the assembly hall, along the corridors and on the playground or the field? Do groups of pupils ever visit the local garden centre or supermarket (if applicable)?
Once you start thinking about it, there are lots of opportunities for undertaking some maths activities in places other than the classroom. Remember that it doesn't have to be the whole class - groups could be elsewhere for some of the time.
So what maths topics lend themselves to pupils working away from their desks?
The playground can be a great resource when you are working on shape and measures. Apart from the obvious tasks of spotting shapes or tessellations in the environment and taking pictures for the learning wall and other activities, teaching a session on symmetry outside for example, can be very effective. Armed with large thick chalks and some metre sticks (which can double as mirror lines), pupils can be asked to create half of a large 2D shape which then has to be reflected in the mirror line. This really emphasises the fact that all points have matching points which are equidistant from the line of symmetry on either side.
Measuring the perimeter of the netball or tennis court or painted markings on the ground helps pupils to get a feel for larger measures. Asking for estimates beforehand can be revealing! Human Venn and Carroll diagrams can be created outside and questions created that involve pupils moving to the correct part of the diagram.
Teachers of younger pupils often use the outside area for maths activities and this should not necessarily stop as pupils get older; it can be much easier to work on capacity with actual fluids outside, rather than making a mess in the classroom!
Using PE is a great way to incorporate some maths into other subjects and a whole range of maths topics can be covered. Gymnastics is an obvious choice for shape work where pairs and groups can be asked to incorporate some evidence of parallel and perpendicular lines as well as symmetry into their sequences. Videos and photographs of this can make an eye-catching presentation or display. Warm-ups can involve different types of movements according to certain criteria such as multiples of six or seven, or whether the number is odd or even for younger pupils.
Are pupils reluctant to finish the session?
You look at the clock and see that you should be wrapping up the session because break starts in ten minutes. As you mention this to the class, there are some groans, some cries of, "Can we just carry on after break?" and some simply don't notice since they are absorbed in what they are doing.
Now I am certainly not advocating missing break times here because all pupils need to get out of the classroom, and my preferred response in these situations would be to continue for some time after break rather than rushing through a plenary.
So why would pupils be reluctant to finish a session?
Sufficiently challenging activities will allow pupils to make progress with them while holding their interest. Low threshold, high ceiling activities allow all pupils to get started and have various achievement points.
The type of activity makes a huge difference so lots of consolidation work can be achieved through games where the interaction with other pupils will potentially challenge and eventually strengthen newly forming concepts. The element of competition also tends to absorb pupils and this can even be achieved through simple How Many Ways? activities such as How Many Ways can you add two numbers to make ten?
Active activities are not always applicable but measures is an obvious area in which pupils can be solving actual problems such as working out how much floor covering would be needed for all the corridors in the school. Another example is ratio and proportion, where pupils can be challenged to make the tastiest drink by using varying proportions of ingredients. Sometimes acting out a context problem helps pupils to get started.
Working at a level in which they are comfortable for that particular topic so pupils can move between flexible groups depending on the focus for that session and how much support they feel they need. This can work particularly well where there is additional teaching support in the classroom or where there are several classes within a year group. Even those pupils who may be classed as able in maths, may struggle with certain concepts, such as shape work, so flexible grouping allows them to get some extra support for this.
Of course, all these ways of working are only successful where pupils are used to an environment where:
mistakes are seen as learning opportunities; it's normal for pupils to say that they are not sure about something; discussion is encouraged to clarify and justify ideas and pupils are given some independence. All of this takes place within clearly agreed boundaries; working with others and being active still means that pupils are expected to focus on the maths task!