Example of strong quantitative reasoning
Petya Kan has all her battery scores at or above the national average score of 100. However, she has a particular strength in QR, with a score of 123, significantly above both her verbal (100) and non-verbal (108) scores.
- Students with a strength in QR relative to other batteries are strong in understanding relationships between numbers, in seeing patterns and order in numbers, and in their flexibility in combining and recombining various quantitative elements in meaningful ways.
These profiles tend to be associated with relatively high achievement in mathematics, and in other subjects with a significant quantitative component (such as business studies, science, and statistics), more so than would be expected from their verbal reasoning (VR) scores.
- Conversely, whilst quantitatively biased students may be able to present concise, reasoned arguments using mathematical notation, symbols or diagrams, they may struggle to explain and justify their inferences and deductions verbally.
What does this look like in the classroom?
Effective use of QR in the classroom may be particularly important where students have a positive quantitative bias profile but are less secure in their VR.
Using careful grouping strategies in the classroom (see Alex Honkanen’s mild verbal bias profile on page 50), teachers can create scenarios that allow students to share their learning and so help develop better verbal skills in describing data-based information.
EAL students who appear to have a mild, moderate, or extreme spatial bias often actually have suppressed verbal scores which will develop with the acquisition of the English language. For further help and insight please go to page 77.
If this is the case, then the grammatical strength can be praised and used when asking students to give feedback on each other’s writing. Such group work can, in turn, be linked to the development of higher-level writing skills such as using cross-curricular text-type knowledge to structure writing tasks across the curriculum and develop a wider curriculum understanding of the importance of literacy.
Students with a strength in QR may also find an understanding of computer skills easier than some of their peers, especially procedures such as using text editors, spreadsheets or cut-and-paste facilities.
Students with high QR scores are typically strong in extracting regularities from their experiences and then reasoning with these abstractions. Their abilities in this area can be developed further through mathematical tasks, games and puzzles. If focused on individual achievement, it is preferable that these games have a noncompetitive focus, although in group work a competitive element can bring into focus the valuable asset of the quantitatively strong student with good supporting verbal skills. Teachers can usefully structure groups to take advantage of this.
Examples of strategies for strong quantitative reasoning
The following strategies will be particularly useful in building on the strengths and addressing the weaknesses of students with strong QR.
1. Plenaries and summaries
In addition to a range of different group-work activities, students with this kind of profile across batteries will benefit from a structured approach in one other key area – that of the plenary. The plenary session is an opportunity for students to share their learning, usually at the end of a lesson. The principle is to ensure time for students to share, celebrate and reflect on their learning rather than have it evaluated by the teacher, but the practice is often rather less than this. The issue of timing has always been difficult where the lesson is short – it’s not always easy to find enough time to do justice to this principle. Instead, a plenary can be merely a rushed question from the teacher (OK, what did we learn today – anyone?) rather than an opportunity for deeper thinking. It’s important that the plenary is given the status it deserves, and so it may not be included in some lessons but could be extended in others. It’s the opportunity for the teacher to reflect on whether the objectives of the lesson have been reached and for all students to be drawn towards an evaluation of their learning. The plenary is not designed to be a show or performance. Display elements might feature but this will be incidental: the main point is to enable the teacher and the students to get a sense of the level of achievement across the class. The teacher should be guiding the learning rather than directing it.
Wright has an excellent list of plenary suggestions or challenges in his book How to be a Brilliant English Teacher and, like many guides of this kind, they are not subject specific:
- I didn’t tell you today’s objectives – now, what do you think they were?
- Explain today’s learning in one sentence to a specified audience, such as your mum, a seven-year-old child, a class in the year below yours.
- In pairs, what was the most important thing in your opinion that you learned today?
- Sum up today’s learning in exactly 15 words.
- In pairs, think up a new (better?) activity to teach today’s objective.
- Write an advertisement for a film trailer for today’s lesson.
- Write a two-minute radio news story summing up what happened in today’s lesson.
- That’s the objective, but tell me one other thing you learned today.
- Look back at today’s activities – what was the connection?
- Write a newspaper headline for today’s lesson.
- Write one more example of your own.
- As a class, complete two columns on the whiteboard headed CLEAR and NOT CLEAR about what we’ve learned and what still confuses us.
- Write one more thing from today that needs more explanation.
- What would you guess the next lesson will be about, and why?
- What does the next lesson really need to be about?
It’s easy to see what this kind of plenary activity is trying to do. Wright is encouraging students to ‘speed think’: to try to encapsulate their learning, to reconstruct it in a new form that demonstrates real assimilation. Some of the latter plenary suggestions are encouraging students to see ‘the big picture’ and to contextualise their learning in relation to what has gone before and what is to come. As Wright notes, the learners are “participating in evaluation not so much of their own work or of the teacher’s efforts but of the learning as a joint operation”.
In addition, this kind of evaluative plenary activity provides an opportunity for teachers to reflect on their own practice in the classroom. Bluntly, did the students ‘get it’ or not, and to what extent? Time to reflect is just as important for teachers as it is for learners and this structure is one way to start this process.
2. Active retrieval
After a lesson is over it is worth remembering that students will not retain everything they have covered. Long-term memory is an active process just as short-term memory is, as set out earlier on page 10. This advice relates to the six specific practices the use of which is not supported by research evidence, such that re-reading and highlighting is not enough retrieval practice to facilitate future retrieval.
To support this thinking in the classroom and to help learners, teachers can undertake a number of strategies:
- create learning opportunities that help students to retrieve the new learning, for example, quizzes, low-stakes testing and memory games;
- space the retrieval practice activities over time, for example, daily, weekly and monthly;
- test the retrieval of new information in different situations – vary the context and content to help students transfer and assimilate their learning.
Doddle, part of the GL group that deals with informal low-stakes testing through quizzes, has developed a guide called Making Knowledge Stick: How and Why Retrieval Practice Improves Results which can be accessed at https://www.doddlelearn.co.uk/ retrieval-practice/.
Willingham provides an invaluable list of mnemonic techniques that can be successfully applied in the classroom in his article ‘What Will Improve a Student’s Memory?’ There are also some practical activities for teachers or students that demonstrate the truth of these three principles.
3. Metacognitive skills
Up to now, we have dealt with mainly cognitive skills, for example, preparing students with techniques to approach a set of skills such as reading. One of the most effective skill sets that can be taught at school is for students to know how to know themselves, how to motivate themselves, and explicitly how to improve their own learning processes. Collectively these are known as metacognitive ‘thinking about thinking’, or ‘learning how to learn’ skills.
When you’ve reached your goal, take time to ask, Could I have been better at doing that task? Take a moment or two to think and reflect, Were my thinking strategies correct? Is there any way to make changes to it, so that it’ll be better next time I do it? If you want to make your thinking great, take time to reflect – metacogitate.
This requires time away from the curriculum, and requires students to take greater responsibility for their own learning. However, teaching these skills has been shown to increase progress on the taught curriculum over time. Children as young as eight years old will already have some understanding of their own strengths and weaknesses, and how to regulate their own performance, but these skills can be specifically taught in a school environment.
Modelling from teachers (see page 62) is a very fitting approach for teaching metacognitive skills. The steps that would need to be taught can be as simple as: plan, monitor, evaluate. Ideally these can be tailored to the type of subject matter covered.
- Plan: think about the goal of the task at hand, recall a number of potential techniques that might be relevantly used, filter out the least relevant techniques by process of elimination, and consider how to proceed (maybe selecting the second-best technique for demonstration purposes later).
- Monitor: while carrying out the chosen technique, question whether this is working towards the goal of the task at hand. If not, the technique may be changed for demonstration purposes until the goal is reached.
- Evaluate: actively reflect on how the goal was reached, how the processes were selected and how others might have been better or worse, how to recognise whether the technique is or isn’t working, how difficult the task was, and what might be done differently next time.
With other skills being modelled, the worked example should be followed by guided practice and ultimately work towards independent practice, with teachers adding and then removing ‘scaffolding’ so that students will be able to become independent metacognitive skill users in their own learning in and out of the classroom. To an extent, all modelling implicitly builds metacognitive skills, by allowing the MKO to demonstrate the monitoring that experts do as routine. Teachers can use this consideration to enhance their existing cognitive modelling, but explicitly teaching metacognitive skills will still be advantageous.
Using metacognitive abilities will require the same working memory as the cognitive tasks at hand, and this is one of the key differences between experts and non-experts, as covered at the start of the chapter (see page 103): experts allocate more mental space to monitoring their own performance towards a goal, and find shifting tactics easier as a result. Students with poor reasoning skills are more likely to use trial-and-error approaches to learning, and research has found that low-achieving students are most likely to benefit from explicit tuition in metacognitive skills.
The flip side of this competition for working memory means that developing metacognitive skills is less effective when the material or techniques being selected are unfamiliar, because the attention is taken up by the cognitive rather than metacognitive. This indicates that metacognitive skills are best modelled with material that is well consolidated, rather than immediately after learning a technique.
This does not make metacognitive skills superior or more advanced than cognitive skills: returning to the ideas covered around Bloom’s Taxonomy, metacognitive and cognitive skills compete for the same mental resources and are mutually beneficial processes to develop. Metacognitive skills cannot be considered without a task set by the cognitive domain.