In 2012, Barack Rosenshine published the Principles of Instruction: a set of 10 research-based principles of instruction, along with suggestions for classroom practice. The principles come from three sources: (a) research in cognitive scienceThe study of the human mind, such as the processes of though... More, (b) research on master teachers, and (c) research on cognitive supports.
The 10 Principles of Instruction are as follows:
- Principle 1: Begin a lesson with a short review of previous learning: Daily review can strengthen previous learning and can lead to fluent recall.
- Principle 2. Present new material in small steps with student practice after each step. Only present small amounts of new material at any time, and then assist students as they practice this material.
- Principle 3. Ask a large number of questions and check the responses of all students: Questions help students practice new information and connect new material to their prior learning.
- Principle 4. Provide models: Providing students with models and worked examples can help them learn to solve problems faster.
- Principle 5. Guide student practice: Successful teachers spend more time guiding students’ practice of new material.
- Principle 6. Check for student understanding: Checking for student understanding at each point can help students learn the material with fewer errors.
- Principle 7. Obtain a high success rate: It is important for students to achieve a high success rate during classroom instruction.
- Principle 8. Provide scaffolds for difficult tasks: The teacher provides students with
temporary supports and scaffolds to assist them when they learn difficult tasks. - Principle 9. Require and monitor independent practice: Students need extensive, successful, independent practice in order for skills and knowledge to become automatic.
- Principle 10. Engage students in weekly and monthly review: Students need to be involved in extensive practice in order to develop well-connected and automatic knowledge.
On this page, we have gathered a collection of guides for how the principles might be applied to the maths classroom. The guides have been written by Nathan Barker, Maths Teacher, Jersey College for Girls, Jersey.
This content was originally produced as part of the Accelerate programme, a Department for Education-funded early career teacher programme designed and delivered by Education Development Trust with the Chartered College of Teaching. It is used here with kind permission of Education Development Trust.
Principle 1: Begin a lesson with a short review of previous learning: Daily review can strengthen previous learning and can lead to fluent recall.
In the words of Fields Medallist Bill Thurston (1990) “Mathematics is a tall subject”, there is a large proportion of concepts which build on previous concepts. Thurston (1990) likened the structure to a scaffoldingProgressively introducing students to new concepts to suppor... More, with many interconnected supports. Unless the previous layer is in place it would be impossible to build successfully on top.
Regular review can help strengthen the previous concepts so that newer ones can be constructed. As Rosenshine (2012) states, there have been successful experiments in the area of elementary school mathematics where teachers have used review time to check homework, go over problems and practice concepts and skills that are needed for automaticity.
However, in what form and with what focus should each review take?
Retrieval practice
Retrieval practice (see The Learning Scientists for a short introduction) is the process of actively retrieving information from the long term memory and was noted by Rosenshine (2012) that the ‘most effective’ teachers spend the first five to eight minutes of every lesson recalling prior learning.
As Young (2019) states ‘retrieval practice is not all about quizzes’ and asks how we include in our planning the task of getting the ‘learning out of’ the students. Free revision and retrieval sites exist, such as SENECA Learning, but careful consideration as to the construction of the questions must be given. From a practical perspective you could use Sherrington’s (2019) six key principles to help inform the construction and purpose of your retrieval practice questions and the guide accessible at https://www.retrievalpractice.org/.
1 | Involve everyone |
2 | Make checking accurate and easy |
3 | Specify the knowledge |
4 | Keep it generative |
5 | Make it time efficient |
6 | Make it workload efficient |
Table 1: Sherrington (2019) six key principles
The goals of the retrieval practice can be varied over time but each five to eight minute slot may fall into one of the categories below. Sometimes the practice will fall into more than one category, but you should be mindful as to the purpose you had in mind for the retrieval practice and the time constraints you have imposed.
- Automaticity: To become mathematically competent, learners need to develop a rich foundation of factual and procedural knowledge (Hodgen et al., 2018). Regular retrieval practice can aid with fluency of the retrieval of mathematics facts and the application of mathematical methods.
For example, helping students to recall definitions of quadrilaterals you could use a Silent Self Quiz (Sherrington, 2019) where the name of the quadrilateral is omitted but the definition is there. Over time you could mix this up by adding in some names and removing their definitions. As you get to know your students you will become aware of the types of mathematical facts they are misremembering and this will help inform the structure of the retrieval practice for automaticity.
- Feedback and review: Research tends to show that feedback has a large effect on learning (Hodgen et al, 2018). However, Hodgen et al. (2018) notes that feedback should be used sparingly and predominantly reserved for more complex tasks. What is perhaps helpful is that within a homework task there may be certain skills that the whole class may need some additional questions on.
For example, after a homework and feedback session on solving linear equations there may have been issues where the equations have involved fractions. This may lead you to the conclusion that the students need to practice their fraction skills. The homework task could inform the fraction skills required to be reviewed.
- What skills do we need today: As we have already established, new mathematical concepts are built on and use previous mathematical ideas. In planning a lesson it is important to consider what previous mathematical skills the students will need at their fingertips. If retrieval practice questions are designed so that they reduce the difficulty of the task during initial learning of new material this will have a positive effect on the acquisition of new knowledge (Martin, 2016).
For example, when solving quadratic equations via factorising as a goal for a lesson there are two skills that will need to be at the students fingertips: factorising quadratic expressions and solving linear equations of the form ax+b=0. Two sets of review questions that review these skills will help the students reduce the difficulty of the tasks of the new concept.
- Error and misconception correcting: There is a debate about how we classify errors and misconceptions in mathematics. I have found it helpful to separate errors based on careless calculations (a misread question or a typo) from misconceptions, which fall into a huge body of research but can be summed up by a lack of understanding of the concept that has been taught. However, as Rushton (2014) states ‘many of the studies were carried out in the 1970s and 1980s, and the misconceptions that were identified then may not be as relevant today’. As you get to know your students, you will discover the misconceptions that they possess.
A good source of information are examiners reports for qualifications. These reports give you a depth of knowledge for a given question.
During a review, you could construct one of the following in order to evaluate if your students have fallen to a common (or perhaps non-common) misconception:
- Select questions from test papers that have the required misconception identified by the examiner’s report and use this to ‘test’ if your students have developed this misconception since being taught the material;
- Create your own questions that leads the student to the misconception you wish to ‘test’ if your students have developed this misconception since being taught the material;
- Use previous knowledge of how students have answered a question to design a question using the misconception and ask the students to identify where in the solution the issues are.
References
Hodgen J, Foster C, Marks R et al. (2018) Evidence for Review of Mathematics Teaching: Improving Mathematics in Key Stages Two and Three. London: Education Endowment Foundation.
Martin A J (2016) Using Load Reduction Instruction (LRI) to boost motivation and engagement. Leicester: British Psychological Society.
Rosenshine B (2012) Principles of Instruction: Research based principles that all teachers should know. American Educator, Spring 2012.
Rushton N (2014) Common errors in Mathematics. Research Matters: A Cambridge Assessment publication 17: 8–17.
Sherrington T (2019) 10 Techniques for Retrieval Practice. Teacherhead. Available at: https://teacherhead.com/2019/03/03/10-techniques-for-retrieval-practice/ (accessed 20 April 2019).
The Learning Scientists (nd.) Retrieval practice. Available at: http://www.learningscientists.org/retrieval-practice (accessed 20 April 2019).
Thurston W P (1990) Mathematical Education. Notices of the AMS 37: 844–850.
Young C (2019) Retrieval Practice. Mathematics, Learning and Technology. Available at: https://colleenyoung.wordpress.com/lesson-planning/retrieval-practice/ (accessed 20 April 2019).
Principle 2. Present new material in small steps with student practice after each step. Only present small amounts of new material at any time, and then assist students as they practice this material.
The second principle relates to how we deal with the working memory of our students when teaching new material. Our working memory is the place where we process information and it can only handle a few bits of information at once (Rosenshine, 2012) or as Barton (2018) puts it ‘working memory is best viewed as the place where thought occurs’.
Cognitive Load TheoryAbbreviated to CLT, the idea that working memory is limited ... More (Sweller et al., 1998) is concerned with the limits of working memory and Load reduction instruction refers to the instructional approaches that seek to reduce or manage cognitive load in order to optimize students’ learning and achievement (Martin, 2016). Learning is believed to occur when information is successfully moved from the working memory and stored in the long-term memory (Kirschner et al., 2006).
For more detail of the cognitive science aspect of this for mathematics, Cambridge Mathematics (Cambridge Mathematics, 2017) has a short overview of the main research in this area which links to the papers for the interested reader. From this ‘Mathematics Espresso’ there are a few key points that are worth highlight here and will form the basis of the classroom techniques below.
Difficulties students face in learning new material
According to Cambridge Mathematics (2017) ‘Students who struggle with mathematics may have difficulties with working memory’.
The consequence for the classroom is that students may attribute issues with a new concept not due to the new concept itself but due to the difficulties they have with their working memory. This may affect their attitude towards learning mathematics and potentially increase their ‘mathematics anxiety’ (Centre for Neuroscience in Education, 2019).
What classroom strategies could we use to support pupils who struggle?
Manipulatives/Number lines: Allowing students to have a visual representation of the object of study can reduce their demand on their working memory (Cambridge Mathematics, 2017).
For example, number lines could be used to help students understand the relative size of fractions. Lesson 10 in Frykholm (2010) uses a number line to explore the relationships between unitary fractions and compare their relative positions. Another representation for this topic is a fraction wall (Figure 1) as it can be used to help remind students of the relative sizes of unitary fractions which can then inform how they combine fractions using addition and subtraction.
Figure 1: Fraction wall
Repetition: Regularly repeating key information or instructions can help to reduce the demands on working memory. This means that students do not need to keep the full set of instructions for a task in their limited working memory and instead can use this ‘space’ for the mathematical task they are working on.
For example, I find it useful to use an interactive whiteboard with a slider to reveal additional instructions over time. The students know where to look for the instructions but do not need to hold all instructions at once.
Breaking down tasks into smaller steps: The majority of new concepts in mathematics will be built upon previously learnt foundations.
For example, when adding fractions there is a journey a student will need to go on first with practice after each stage (see principles five, six and nine) before they can access this material. For instance, one path could be:
- What is a fraction? This would involve students identifying the two components, the type of splitting/part (the denominator) and the amount of that splitting/part (the numerator).
- When are fractions equivalent? This would involve students identifying that we can create equivalent fractions (I find a fraction wall is a useful representation to use in this situation).
- Adding fractions with common denominators. These can be used to strength the concept that the numerator is how many and the denominator is the type of splitting. When adding the fractions with common denominators, we are counting objects of the same ‘type’.
- Adding proper fractions where the sum is less than a whole. This can then build on the idea that if two fractions do not have a common denominator then they are not the same type of object, so we cannot just count how many we have. This strengthens the need for finding equivalent fractions and introduces another application of lowest common multiple (lcm).
- Adding proper fractions. Finally, a conversation can then be had about the form we would leave our answers in if the sum is greater than one.
- The above is just one way to look at the break down, but even this breaking down will have previously learnt skills. For such recall, a review based on the ideas of principle one could help students reduce their cognitive load in that lesson where the recall is required.
Designing instruction using variation theory: It may be helpful to consider variation when designing not only tasks, but instructional strategies (Kullberg et al., 2017) and make that variation explicit to the students (Mason, 2011).
For example, when teaching completing the square as an algebraic technique, by considering the variations that will occur in a question, the teacher can understand how to break down the algebraic technique to students.
A larger list of strategies is given in Martin (2016). Some of these instructional strategies will feature in future principles.
A word of warning
Unfortunately, although popular activities ‘working memory training alone does not lead to better outcomes in mathematics attainment’ (Cambridge Mathematics, 2017). Although there is some evidence that isolated working memory training may improve children’s performance on working memory capacity measures (St Clair-Thompson et al, 2010) there is not reliable evidence that this leads to improvement in mathematics attainment (Melby-Lervag and Hulme, 2013).
Therefore we must be careful to place the emphasis perhaps on the retrieval practice of skills learnt before (see principle one) rather than on generic memory techniques, since fluency and automaticity are ‘vital means of reducing the burden on working memory’ (Rosenshire, 1986, 2009).
References
Barton C (2018) How I wish I’d taught Maths: Lessons learned from research, conversation with experts, and 12 years of mistakes. Woodbridge: John Catt Educational Ltd.
Cambridge Mathematics (2017) Why is working memory important for mathematical learning? Mathematical Espresso. Available at: https://www.cambridgemaths.org/espresso/view/working-memory-for-mathematics-learning/ (accessed 20 April 2019).
Centre for Neuroscience in Education (2019) What is Mathematics Anxiety? University of Cambridge. Available at https://www.cne.psychol.cam.ac.uk/math-memory/what-is-mathematics-anxiety (accessed 8 May 2019).
Frykholm J (2010) Learning to Think Mathematically with the Number Line. The Math Learning Centre.
Kirschner PA, Sweller J and Clark RE (2006) Why minimal guidance during instruction does not work: An analysis of the failure of constructivist, discovery, problem-based, experiential, and inquiry-based teaching. Educational Psychologist 41: 75–86.
Kullberg A, Runesson Kempe U and Marton F (2017) What is made possible to learn when using the variation theory of learning in teaching mathematics? ZDM Mathematics Education 49(4): 559–569.
Mason J (2011) Explicit and Implicit Pedagogy: variation theory as a case study. In: Smith C (ed.) Proceedings of the British Society for Research into Learning Mathematics 31(3): 107–112.
Melby-Lerag M and Hulme C (2013) Is working memory training effective? A meta-analytic review. Developmental Psychology 49(2): 270–291.
Rosenshine B V (1986) Synthesis of research on explicit teaching. Educational Leadership 43: 60–69.
Rosenshine BV (2009) The empirical support for direct instructionA method of instruction in which concepts or skills are taug... More. In Tobias S and Duffy TM (ed.) Constructivist instruction: Success or failure? New York: Routledge.
St Clair-Thompson H, Stevens R, Hunt A et al. (2010) Improving children’s working memory and classroom performance. Educational Psychology 30(2): 203–219.
Sweller J, van Merrienboer JJG and Paas FGWC (1998) Cognitive Architecture and Instructional Design. Educational Psychology Review 10(2): 251–296.
Principle 3. Ask a large number of questions and check the responses of all students: Questions help students practice new information and connect new material to their prior learning.
Students need to practice new material and this is supported by teacher questioning and student discussion (Rosenshine, 2012). There are different types of questioning that can be seen in the classroom, in this principle I will concentrate on one type for the mathematics classroom: Teacher questioning. In subsequent principles I will look at how written questions support a student’s learning at different stages.
Underground Mathematics (2019a) produced a bundle of resources from their website attempting to address the question of how, as teachers, we can facilitate students asking questions about the mathematics to themselves, their peers and their teachers. Underground Mathematics (2019a) identifies two key questions to consider:
- When do you and your students ask questions in the mathematics classroom?
- What types of questions are asked and why do we ask them?
By considering these two questions when planning a lesson we as teachers can make the space and time in lessons available to questioning and use some of the types of questioning below to check for student understanding.
Teacher questioning: For an overview of question structures that can be used, see Mason and Watson (1998). However, in this section I will only look at the following:
- Are you sure? and How do you know?
It is important to create a positive mathematical culture of questioning within the classroom (Bowland Maths, 2008). By asking students to firstly make a commitment to their solution or answer and then ask them to justify their position, it opens students up to the hypercorrection effect (Hawes, 2010). Hypercorrection deals with the idea that if pupils are more confident about their answer, then find out they are wrong, they are then more likely to remember the correct answer in the long term (Butterfield and Metcalfe, 2001).
- What’s the same and what’s different?
Variation theory looks at what is the same and what is different about a mathematical structure (Watson, 2018). This line of questioning can be used when making comparisons between mathematical objects where either the variation or the lack of variation divulges a deeper structure.
For example, the resource ‘Discriminating’ (Underground Mathematics, 2019b) looks at several statements about the quadratic equation ax2+bx+c=0 and asks about the effect of certain variation on the constants a, b and c. In Figure 1, a condition is given on the coefficient b. This would lead students to think and ask questions about what effect this has on the number of solutions to the quadratic equations but also is this dependent on the value of a and c. You could allow students to generate these questions and note them down on a large whiteboard so that other students can see the type of questions being asked in the room.
Figure 2: Underground Mathematics, Discriminating (2019).
Through this, students’ understanding of the connection between the algebra and geometry can be explored and vocalised to check for understanding.
- Is there another way?
Way (2011) places this in the final discussion questions of their categorisation of questions in the mathematics classroom. This question can draw together the ‘efforts of the class and prompt sharing and comparison of strategies and solutions’ (Way, 2011). Through this the teacher can bring out the mathematical thinking that has occurred in the class and this can act as an opportunity to check the understanding of the students.
- Is it always, sometimes or never true?
For example, in Underground Mathematics (2019b) resource allows students to consolidate their understanding of the discriminant and make connections between algebra and geometry. In this style of questioning students are required to justify their choice of must, can’t or may (see Figure 1). This allows a teacher to pick about what students do understand about this topic but also allows the students to ask the questions of each other.
An additional example can be found in the Swan (2005, p21–23). In the number example (Figure 2) statements are provided that can be used with students to develop their capacity to explain, convince and prove. One of the teacher’s roles here is to encourage students to think more deeply by asking questions such as “is this one still true for decimals or negative numbers?”. Questions like this also allow the teacher to see which students are able to adapt their arguments and apply their previously learnt knowledge.
Figure 3: Evaluating mathematical statements, (Swan, 2005, p22).
It is important to remember that ‘it is necessary to plan the teaching by choosing question items that suit the student population, the teaching goals, the different needs and the teacher’s own teaching style’ (Aizikovitsh-Udi & Star, 2011). Therefore, all the suggestions given above should be considered along with your own teaching environment. Further work on examining your own classroom culture and the place questioning takes in it can be found in Pennant (2013) and Bowland Maths (2008) have produced a module designed to look at the characteristics of questioning.
References
Aizikovitsh-Udi E and Star J (2011) The skill of asking good questions in mathematics teaching. Procedia-Social and Behavioral Sciences, 15: 1354–1358.
Bowland Maths (2008) Questioning and reasoning. Bowland Maths. Available at: https://www.bowlandmaths.org.uk/materials/pd/online/pd_05/pdf/pd_05_handbook_full.pdf (accessed 29 April 2019).
Butterfield B and Metcalfe J (2001) Errors committed with high confidence are hypercorrected. Journal of Experimental Psychology: Learning, Memory, & Cognition 27(6): 1491–1494.
Hawes DR (2010) Confidently wrong? Psychology today. Available at: https://www.psychologytoday.com/intl/blog/quilted-science/201005/confidently-wrong (accessed 29 April 2019).
Pennant J (2013) Developing a classroom culture that supports a problem-solving approach to mathematics. NRICH. Available at: https://nrich.maths.org/10341 (accessed 29 April 2019).
Rosenshine B (2012) Principles of Instruction: Research based principles that all teachers should know. American Educator, Spring 2012.
Swan M (2005) Standards Unit: Improving learning in mathematics: challenges and strategies. Department for EducationThe ministerial department responsible for children’s serv... More and Skills Standards Unit. Available at: https://www.ncetm.org.uk/public/files/224/improving_learning_in_mathematicsi.pdf (accessed 8 May 2019).
Underground Mathematics (2019a) Asking questions in the classroom. University of Cambridge. Available at: https://undergroundmathematics.org/bundles/asking-questions-in-the-classroom (accessed 29 April 2019).
Underground Mathematics (2019b) Discriminating. University of Cambridge. Available at: https://undergroundmathematics.org/quadratics/discriminating (accessed 29 April 2019).
Underground Mathematics (2019c) Discriminating – Teacher support. University of Cambridge. Available at: https://undergroundmathematics.org/quadratics/discriminating-teacher-support (accessed 29 April 2019).
Watson A (2018) Variation in mathematics teaching and learning: A collection of writings from ATM: Mathematics Teaching. Derby: Association of Teachers of Mathematics (ATM).
Watson A and Mason J (1998) Questions and prompts for mathematical thinking. Derby: Association of Teachers of Mathematics (ATM). Available at: www.atm.org.uk (accessed 29 April 2019).
Way J (2011) Using questioning to stimulate mathematical thinking. NRICH. Available at: https://nrich.maths.org/2473 (accessed 29 April 2019).
Principle 4. Provide models: Providing students with models and worked examples can help them learn to solve problems faster.
Rosenshine (2012) states that “teacher modelling and thinking aloud while demonstrating how to solve a problem are examples of effective cognitive support”, so it is important to think about how we present (or prepare) new material and examples to students. Worked examples are also key in mathematics and “allow students to focus on the specific steps thus reduce cognitive load on their working memory” (Rosenshine, 2012) and as we have seen in an earlier principle, the working memory is a limited resource.
Explicit instruction
I do not believe that one single approach to the teaching of mathematics is useful to students all the time and Coe et al. (2014) support this with the idea that quality teaching is multidimensional. There is no ‘recipe’ to great teaching and it is even still not clear what particular isolated elements are necessary for teaching to be effective (Coe et al., 2014).
However, Barton (2018) reviewed the research evidence on what makes great teaching and in the early stages of learning new material for the first time (the early knowledge acquisition phase of learning) he found that an explicit instruction approach was most effective.
What this means for mathematics teaching is that perhaps it is best to explicitly model the new content first, so that students see the kind of responses that are required (Coe et al., 2014).
Barton (2018) also addresses the call for a balance between explicit instruction and inquiry-based learning, with the conclusion that the type of instruction should be determined by the knowledge base of the students. For a mathematics lesson, I interpret this as using open-ended tasks when students possess the content knowledge for the skills involved, having gained this content knowledge from teacher-led instruction.
Worked Examples
Worked examples are hugely important in mathematics (Barton, 2018) as they allow students to see modelled solutions, and give teachers a chance to emphasise important concepts/approaches to questions. This is particularly valuable for novices as demonstrated by Sweller et al. (1998) who found that pupils who learned a subject/skill through solving problems did not perform as well on tests as those who learned by studying equivalent worked examples.
However, how we choose, present and what we then do with those worked examples with the students are all equally important.
For example, when working through an example with a class you could use a pre-recorded video of the example. This will allow you as the teacher not to distract the students as they watch the example (perhaps silently first). Then perhaps with a later worked example the teacher could pause the video at certain points and ask why certain choices were made in the solution. This will have the additional benefit of allowing the teacher to ask questions and check for understanding.
Example-Problem pairs
One of the interesting findings from Trafton and Reiser (1993) was that subjects who solved problems interleaved with worked examples took less time to solve the problems and submitted more accurate solutions than those doing a block of worked examples followed by a block of problems. For the mathematics classroom, this would mean breaking down the worked examples into ‘Example-Problem Pairs’.
For example, using a worked example video followed by a “Your Turn” question can be used to in the early stages of teaching a new skill.
It is important to note that once the new skill has been acquired students should then get a chance to practice independently and with a carefully selected set of questions (Atkinson et al., 2000).
Rosenshine (2012) states that “teacher modelling and thinking aloud while demonstrating how to solve a problem are examples of effective cognitive support”, so it is important to think about how we present (or prepare) new material and examples to students. Worked examples are also key in mathematics and “allow students to focus on the specific steps thus reduce cognitive load on their working memory” (Rosenshine, 2012) and as we have seen in an earlier principle, the working memory is a limited resource.
Explicit instruction
I do not believe that one single approach to the teaching of mathematics is useful to students all the time and Coe et al. (2014) support this with the idea that quality teaching is multidimensional. There is no ‘recipe’ to great teaching and it is even still not clear what particular isolated elements are necessary for teaching to be effective (Coe et al., 2014).
However, Barton (2018) reviewed the research evidence on what makes great teaching and in the early stages of learning new material for the first time (the early knowledge acquisition phase of learning) he found that an explicit instruction approach was most effective.
What this means for mathematics teaching is that perhaps it is best to explicitly model the new content first, so that students see the kind of responses that are required (Coe et al., 2014).
Barton (2018) also addresses the call for a balance between explicit instruction and inquiry-based learning, with the conclusion that the type of instruction should be determined by the knowledge base of the students. For a mathematics lesson, I interpret this as using open-ended tasks when students possess the content knowledge for the skills involved, having gained this content knowledge from teacher-led instruction.
Worked Examples
Worked examples are hugely important in mathematics (Barton, 2018) as they allow students to see modelled solutions, and give teachers a chance to emphasise important concepts/approaches to questions. This is particularly valuable for novices as demonstrated by Sweller et al. (1998) who found that pupils who learned a subject/skill through solving problems did not perform as well on tests as those who learned by studying equivalent worked examples.
However, how we choose, present and what we then do with those worked examples with the students are all equally important.
For example, when working through an example with a class you could use a pre-recorded video of the example. This will allow you as the teacher not to distract the students as they watch the example (perhaps silently first). Then perhaps with a later worked example the teacher could pause the video at certain points and ask why certain choices were made in the solution. This will have the additional benefit of allowing the teacher to ask questions and check for understanding.
Example-Problem pairs
One of the interesting findings from Trafton and Reiser (1993) was that subjects who solved problems interleaved with worked examples took less time to solve the problems and submitted more accurate solutions than those doing a block of worked examples followed by a block of problems. For the mathematics classroom, this would mean breaking down the worked examples into ‘Example-Problem Pairs’.
For example, using a worked example video followed by a “Your Turn” question can be used to in the early stages of teaching a new skill.
It is important to note that once the new skill has been acquired students should then get a chance to practice independently and with a carefully selected set of questions (Atkinson et al., 2000).
Modelling the mathematical thought process live
In mathematics there is also a place for worked examples for problems where there may be more than one approach to take. The modelling here is not of the skills or newly learnt concept but of the skills of a problem solver. Gould (2018) points out the possible pros and cons of solving a problem in real time, but what is certainly true is that planning when to use this approach is important and perhaps should not be used in the early learning of a topic/skill, but when learners are already confident in the material being taught.
References
Atkinson RK, Derry SJ, Renkl A et al. (2000) Learning from examples: Instructional principles from the worked examples research. Review of Educational Research 70(2): 181–214.
Barton C (2018) How I wish I’d taught Maths: Lessons learned from research, conversations with experts, and 12 years of mistakes. Woodbridge: John Catt Educational Ltd.
Coe R, Aloisi C, Higgins S et al. (2014) What makes great teaching? Review of the underpinning research. CEM. Available at: https://www.suttontrust.com/wp-content/uploads/2014/10/What-Makes-Great-Teaching-REPORT/pdf (accessed 14 May 2019).
Gould T (2018) Only smart teachers have the answer? Cambridge Mathematics, Mathematical Salad. Available at: https://www.cambridgemaths.org/blogs/only-smart-teachers-have-the-answer/ (accessed 14 May 2019).
Rosenshine B (2012) Principles of Instruction: Research based principles that all teachers should know. American Educator, Spring 2012.
Sweller J, Van Merrienboer JJG and Pass FGWC (1998) Cognitive architecture and instructional design. Educational Psychology Review 10(3): 251–296.
Barton C (2018) How I wish I’d taught Maths: Lessons learned from research, conversations with experts, and 12 years of mistakes. Woodbridge: John Catt Educational Ltd.
Coe R, Aloisi C, Higgins S et al. (2014) What makes great teaching? Review of the underpinning research. CEM. Available at: https://www.suttontrust.com/wp-content/uploads/2014/10/What-Makes-Great-Teaching-REPORT/pdf (accessed 14 May 2019).
Gould T (2018) Only smart teachers have the answer? Cambridge Mathematics, Mathematical Salad. Available at: https://www.cambridgemaths.org/blogs/only-smart-teachers-have-the-answer/ (accessed 14 May 2019).
Rosenshine B (2012) Principles of Instruction: Research based principles that all teachers should know. American Educator, Spring 2012.
Sweller J, Van Merrienboer JJG and Pass FGWC (1998) Cognitive architecture and instructional design. Educational Psychology Review 10(3): 251–296.
Principle 5. Guide student practice: Successful teachers spend more time guiding students’ practice of new material.
Students should be given sufficient time to rephrase, elaborate and summarise new material (Rosenshine, 2012). When students have been given enough rehearsal time they are able to retrieve the newly acquired material easily and use it to solve problems, a key component of any mathematics lesson. In mathematics, guiding practice can be achieved by balancing the models and worked examples shown to students with additional questions to practice along the way, checking for student understanding at each point (principle six).
Rephrasing in the Mathematics classroom – making connections: Mathematics is an interconnected subject and quite often you can rephrase something new in the form of old language.
For example, studying de Moivre’s theorem in Complex Numbers, you could ask students to rephrase this algebraic statement in terms of geometry. This would give an insight as to which students are ready to apply these ideas in context and thus start their independent practice. The same idea could be applied to the concept of division. There are many ways to say 24 divided by 4 is 6 and asking students to rephrase this allows for stronger links to be made.
Elaborating: One of the most powerful questions a teacher can ask is “why?” This opens up potentially closed questions, as it requires students to justify and elaborate on their initial response. This allows the teacher to “see” if the students have understood the new skill or concept.
Summarising: Summarising in mathematics needs to extend beyond the sharing of strategies. It needs to give students a chance to discuss the mathematical ideas developed in each task and make connections between strategies (Kosiak et al., 2016).
For example a gallery walk in which students circulate throughout the classroom to observe the work of others (Kosiak et al., 2016) and then describe what they have seen, allows the teacher to check if the students have taken on board the skills and concepts of the lesson.
Tools to help guide student practice:
- Choice of examples: When selecting example questions for students to try, it is important that they are relevant to the new content or skills they have just been taught. This will allow you as the teacher to work on any initial misconceptions and deal with them straight away.
For example, after formally teaching completing the square as an algebraic skill, present a selection of example-problem pairs (Barton, 2018) that cover the variation of the initial style of question they will see.
Example One: x2+6x and a your turn of x2+8x.
Example Two: x2+6x+5 and a your turn of x2+8x+1.
Example Three: x2-6x+5 and a your turn of : x2-8x+1.
Example Four: x2-4x-4 and a your turn of x2-6x-9.
This choice of examples, followed by a problem and teacher questioning can probe if the students are ready for independent practice. If not, the teacher can then consider if a repeat explanation is required (Rosenshine and Stevens, 1986).
- Questioning: The questions you ask need to be directly relevant to the new content or skill. It is also important not to extrapolate from some students providing good answers that all students understand. As Rosenshine and (1986) state, another error (particularly with older children) is to assume that it is not necessary to check for understanding, and that simply repeating the points will be sufficient.
Underground Mathematics (2019) provide some useful guidance on questioning, and we can use this to ensure that we generate questions that are specific to the new content or skill that we are teaching.
The following example considers questioning in the context of the Underground Mathematics (2019) resource ‘choose your families’:
Resource | Context | Purpose |
Underground Mathematics, Choose your families. | The students have just finished learning about graphical features of functions. | To review the graphical features of a function. |
When using this resource the teacher is trying to find out what the students “see” in the graphs provided and how this connects to the formal teaching of the graphical features of a function. The table below demonstrates how moving from generic questions to more specific questions makes this questioning more purposeful and better supports guided practice:
Generic question | Specific question |
What do you see? | Can you identify any graphical features that you have seen before?
Do we have a specific name for the features you have identified? |
How could you sort these? | From the features you have listed, which graphs have those features in common?
Can some graphs not have certain combinations of graphical features? |
Can you give another example of that? | Could you suggest a function you have learnt about that fits with the graphical features you have identified here? |
Through each of the areas above it is important that the teacher receives feedback from all of the students.
A concrete tool: Mini-whiteboards
Mini-Whiteboards are an excellent tool for gaining quick feedback on the understanding of the whole class. This will allow you to see who in the room is confident in the new skill, and thus ready for independent practice (Rosenshine and Stevens, 1986) and those that may require additional support in the form of additional explanation, repeated explanation or scaffolding (principle eight) so that they can progress to independent practice. We will explore this idea more in principle six.
References
Barton C (2018) How I wish I’d taught Maths: Lessons learned from research, conversation with experts, and 12 years of mistakes. Woodbridge: John Catt Educational Ltd.
Kosiak J, McCool J and Markworth K (2016) Problem Solving in All Seasons, Grades 3-5. NCTM.
Rosenshine B (2012) Principles of Instruction: Research based principles that all teachers should know. American Educator, Spring 2012.
Rosenshine B and Stevens R (1986) Teaching functions. In: Wittrock M (ed.) Handbook of research on teaching (3rd ed.) New York: Macmillan.
Underground Mathematics (2019) Asking questions in the classroom. University of Cambridge. Available at: https://undergroundmathematics.org/bundles/asking-questions-in-the-classroom (accessed 29 April 2019).
Principle 6. Check for student understanding: Checking for student understanding at each point can help students learn the material with fewer errors.
How can we as teachers know if the students are learning what we think they are learning? Will students be able to answer questions but hide their misunderstanding of a new topic? These are challenging questions, but there are tools available to the teacher to check the students’ understanding during lessons.
Exposing errors and misconceptions
Learners make mistakes (errors and misconceptions) for many reasons, but it is important to identify those mistakes that are symptoms of more profound difficulties (Swan, 2005). The work of Higgins et al (2002) and Swan (2002) shows that mistakes are often the result of consistent, alternative interpretations of mathematical ideas and therefore should not be dismissed as wrong thinking (Swan, 2005).
For example, learners are generalising from their earlier experiences and this helps to explain why students can think that “division always makes numbers smaller” and “shapes with bigger areas have bigger perimeters”.
It is important that teachers are aware of what the previous knowledge of the students and how the new skill or content interacts with it. This will allow the teacher to provoke the mistakes and use them as a learning opportunity (Swan, 2005).
Diagnostic questions
One way to provoke students into making mistakes is through diagnostic questions. Diagnostic questions are designed to help identify, and crucially understand students’ mistakes in an efficient and accurate manner (Barton, 2018).
For example, the website diagnosticquestions.com has a bank of questions that have been developed to specifically help identify, and crucially understand students’ mistakes.
Figure 4: diagnosticquestions.com/Questions/Go#/3531
In the above figure, the answers have been specifically constructed in order to probe student mistakes. For instance, a student answering A may be ignoring place value and simply seeing them as 503, 051, 50 and 51.
As Barton (2018) states, a good diagnostic question is one in which there is one correct answer, three incorrect answers and each incorrect answer reveals a specific mistake for that skill or topic.
“Each time, ask yourself what you would learn about your students from their choice of answers without them needing to utter another word.” (Barton, 2018).
The information from a diagnostic quiz can help a teacher gain a valuable insight into the students’ understanding of a concept before they embark on independent practice.
Using answers – an alternative approach to diagnostic questions
An alternative way to use a diagnostic question is to provide a set of reasons for each possible answer. Students can then work in small groups and discuss the reasons given for each of the possible answers. The teacher can then circulate the room listening to the discussion and through this gain a better insight into the students ‘understanding of the skill or concept.
When using small group discussions, it might be helpful to read Small group discussion: the teachers role (Swan, p. 39, 2005), as sometimes teachers can appear unsure as to their own role during this time and this will help you to maximise this tool for drawing out what students have understood. A summary of the areas to consider is given in Table 1.
Do make the purpose of the task clear. |
Do keep reinforcing the ‘ground rules’. |
Do listen before intervening. |
Join in, but don’t judge. |
Do ask learners to describe, explain and interpret. |
Do not do the thinking for learners. |
Don’t be afraid of leaving discussions unresloved. |
Table 1: List of dos and don’ts for teachers during small group discussions (Swan, p. 39, 2005).
Mini-whiteboards
As Swan (2005) states, mini-whiteboards are an indispensable resource in the classroom as they allow teachers to see at a glance what every learner thinks and they encourage learners to use private, rough working that can be quickly erased.
For example, if you use mini-whiteboards with open style questioning, such as “Two fractions that add to 1 are … Now show me a different pair”, with a follow up question of “how different can you make your pair from someone else?”. This allows the teacher to “see” the limits of the current understanding of the students in the room. It might be the case that if a student is only providing fractions with common denominators (even after a lesson where this was not a restriction) then they may struggle with independent practice on adding fractions with any denominators.
References
Barton C (2018) How I wish I’d taught Maths: Lessons learned from research, conversation with experts, and 12 years of mistakes. Woodbridge: John Catt Educational Ltd.
Higgins S, Ryan J, Swan M et al. (2002) Learning from mistakes, misunderstandings and misconceptions in mathematics. In: Thompson I (ed.) National numeracy and key stage 3 strategies, London: DfES.
Swan M (2002) Dealing with misconceptions in mathematics. In: Gates P (ed.) Issues in mathematics teaching, Routledge Falmer: 147–165.
Swan M (2005) Standards Unit, Improving learning in mathematics: challenges and strategies. Success for all. Department for Education and Skills Standards Unit, August 2005.
Rosenshine B (2012) Principles of Instruction: Research based principles that all teachers should know. American Educator, Spring 2012.
Principle 7. Obtain a high success rate: It is important for students to achieve a high success rate during classroom instruction.
Rosenshine states that of the two major studies they found that students in classrooms with more effective teachers had a higher success rate, as judged by the quality of their oral responses.
In a study of fourth-grade mathematics, it was found that 82 per cent of students’ answers were correct in the classrooms of the most successful teachers, but the least successful teachers had a success rate of only 73 per cent. What is probably important here is not the percentages, but the fact that the students were feeling success from the effort they have put into learning (Barton, 2018) and thus feel motivated to continue.
Mastery
Mastery is Pink’s (2011) third element of motivation and when students adopt mastery goals rather than performance goals, the American Psychological Association (2015) explains, students persist in the face of challenging tasks and process information more deeply. Students with mastery goals are interested in learning new skills and improving their understanding and competence, whereas students with performance goals are more concerned with proving their ability or avoiding negative judgements (Furner and Gonzalez-DeHass, 2011).
For example, using the NCETM (2019) mastery resources, lessons can be organised into short units and where all students are required to master one set of the lessons’ core concepts before they proceed to the next.
However, as Cambridge Mathematics (2019) warns, ‘mastery’ has multiple meanings and is not used consistently in policy, practice and research.
Success, failure and struggle
A key element of motivation is the feeling of success (Barton, 2018) or the belief that success is in reach. As Middleton and Spanias (1999) argue, students’ perceptions of success are highly influential in forming their motivational attitudes.
Barton (2018) believes that too much experience of past struggle and failure will only dampen the students’ belief that their effort will pay off. He argues that the choice of activities we give our students is crucial and that we should give them a taste of immediate success (within 20 seconds).
For example, the Mathematical Etudes (Foster, 2019) project ‘aims to find creative, imaginative and thought-provoking ways to help learners of mathematics develop their fluency in important mathematical procedures’. These activities allow for an immediate feeling of success but with a deeper, richer structure to explore. This helps students gain the success that they need to continue but also allows the teacher to assess the students’ understanding. The activity Almost One https://nrich.maths.org/13205 is one such activity. The initial success comes from students being able to choose their own fractions and add them together to get an answer. The deeper part comes from trying to solve the larger problem.
Load reduction instruction (LRI)
Martin (2016) states that LRI is based on five principles at key points in the learning process:
- Reducing the difficulty of a task during initial learning
- Instructional support and scaffolding (explored in principle eight)
- Ample structured practice (explored in principle five)
- Appropriate provision of instructional feedback
- Independent practice, supported autonomy, and guided discovery learningAllowing learners to discover key ideas or concepts for them... More (explored in principle nine).
When reducing the difficulty of the task, segmenting (Martin, 2016) can be used to break down the task into component parts and the teacher can encourage the students to see the completion of each component as a success.
For example, when teaching students about ordering and comparing the value of numbers expressed in different ways it would be helpful to break down the stages of this task into understanding the different representations (fractions, decimals and percentages). In each stage, students can first gain success (maybe with the help of manipulatives or pictorial representatives) mastering each representation before attempting to combine them.
Following each stage, the students can receive instructional feedback (Martin, 2016) on more complex tasks (Hodgen et al., 2018) in order to help increase the students perseverance.
For example, detailed feedback could be given on answers to the following questions after studying the fraction stage:
Find a number between 17 and 18 ? Can you find another and another? What can you say about the numbers you can make that are between 17 and 18 ?
These questions are not trivial. Students who have developed their skills of working with fractions will gain a greater sense of success through answering these questions, building perseverance and intrinsic motivation to find possible answers to the questions.
References
American Psychological Association (2015) Top 20 principles from Psychology for PreK-12 Teaching and Learning. Available at: https://www.apa.org/ed/schools/teaching-learning/top-twenty-principles.pdf (accessed 23 July 2019).
Barton C (2018) How I wish I’d taught Maths: Lessons learned from research, conversations with experts, and 12 years of mistakes. Woodbridge: John Catt Educational Ltd.
Cambridge Mathematics (2019) Mastery in mathematics. Mathematical Espresso. Available at: https://www.cambridgemaths.org/espresso/view/mastery-in-mathematics/(accessed 20 July 2019).
Foster C (2019) Mathematical etudes. Available at: http://www.mathematicaletudes.com/ (accessed 20 July 2019).
Furner JM and Gonzalez-DeHass A (2011) How do Students’ Mastery and Performance Goals Relate to Math Anxiety. Eurasia Journal of Mathematics, Science & Technology 7(4): 227–242.
Hodgen J, Foster C, Marks R et al. (2018) Evidence for Review of Mathematics Teaching: Improving Mathematics in Key Stages Two and Three. London: Education Endowment Foundation.
Martin AJ (2016) Using Load Reduction Instruction (LRI) to boost motivation and engagement. Leicester: British Psychological Society.
Middleton JA and Spanias PA (1999) Motivation for achievement in mathematics: Findings, generalisations and criticisms of the research. Journal for Research in Mathematics Education 30(1): 65–88.
NCETM (2019) Mastery. National Centre for Excellence in the Teaching of Mathematics. Available at: https://www.ncetm.org.uk/resources/47230 (accessed 22 July 2019).
Pink DH (2011) Drive: the surprising truth about what motivates us. London: Penguin.
Rosenshine B (2012) Principles of Instruction: Research based principles that all teachers should know. American Educator, Spring 2012.
Principle 8. Provide scaffolds for difficult tasks: The teacher provides students with temporary supports and scaffolds to assist them when they learn difficult tasks.
Maybin, Mercer and Stierer (1992) offer this definition of scaffolding:
It is help which will enable a learner to accomplish a task which they would not have been quite able to manage on their own, and it is help which is intended to bring the learner closer to a state of competence which will enable them eventually to complete such a task on their own.
Rosenshine (2012) states that scaffolding includes modelling the steps or thinking aloud by the teacher as he or she solves the problem, but also includes tools such as cue cards, check list and partially completed model answers.
For a more comprehensive review of the state of scaffolding in mathematics education see Bakker et al. (2015). Below are a couple of approaches to scaffolding that I have used in my own mathematics teaching.
Questioning as a means of scaffolding
I have already written about questioning in various contexts in the Mathematics classroom (principles three, five and six). NRICH (2019) have been a successful provider for rich mathematical problems and all NRICH activities come with Teacher Resources that can assist a teacher in scaffolding the various types of activities they have. I have always found that having an understanding about how a task may be approached by students will help me determine where additional support may be needed. This support could be with getting started, making a connection within a problem, or understanding a bigger picture.
For example, in Almost One https://nrich.maths.org/13205/, six fractions are listed and the task is for students to choose to add some together to get as close to 1 as possible. The support that is on offer to help the teacher aid the student is in the form of a “Getting Started” which asks the students to identify a large (bigger than 12) and a small fraction to see how close they can get to one. This identifies something that a student can do straight away and provides them with an early success.
In the teacher resources, a possible approach to the problem is given (thus scaffolding the approach to the problem for a teacher) and highlights that NRICH’s own Fraction Wall https://nrich.maths.org/4519 is a possible support (and a manipulative) that could help the students tackle this problem.
Manipulatives and representations as a scaffold
As Hodgen et al. (2018) summarise, manipulatives offer powerful support to learners but teachers must help the students link the materials they are using to the mathematics of the situation, to appreciate the limitations of concrete materials and to develop related mathematical images, representations and symbols.
For example, when working on the topic of symmetry, having a paper version of the shapes that the students can fold, flip and rotate can help them build up an internal mental image that they can then draw on in future tasks. Once students have had sufficient experience with the physical objects, they will then become less reliant on them in the future.
When scaffolding may not be so helpful
If mathematical tasks contain too much scaffolding they could provide too much of a crutch impeding the development of students perseverance and their ability to actively make connections with exciting knowledge or existing mental models in the long term memory (Barre and Simon, 2017, van de Pol, 2015).
References
Bakker A, Smit J, and Wegerif R (2015) Scaffolding and dialogic teachingThe effective use of talk for teaching and learning, involvi... More in mathematics education: introduction and review. ZDM Mathematics Education 47(7): 1047–1065.
Bare C and Simon L (2017) When Scaffolding is Not Helpful. The Ohio Journal of School Mathematics. Ohio Council of Teachers of Mathematics. Available at: https://library.osu.edu/ojs/index.php/OJSM/article/view/5838/4802 (accessed 18 November 2020).
Hodgen J, Foster C, Marks R et al. (2018) Evidence for Review of Mathematics Teaching: Improving Mathematics in Key Stages Two and Three. London: Education Endowment Foundation.
Maybin J, Mercer N and Stierer B (1992) Scaffolding in the classroom. In: Norman K (ed.) Thinking Voices: The Work of the National Oracy Project. London: Hodder & Stoughton, pp. 186–195.
NRICH (2019) Millennium Mathematics Project. Available at: https://nrich.maths.org/(accessed 25 July 2019).
Rosenshine B (2012) Principles of Instruction: Research based principles that all teachers should know. American Educator, Spring 2012.
van de Pol J, Volman M, Oort F et al. (2015) The effects of scaffolding in the classroom: support contingency and student independent working time in relation to student achievement, task effort and appreciation of support. Instructional Science 43(5): 615–641.
Principle 9. Require and monitor independent practice: Students need extensive, successful, independent practice in order for skills and knowledge to become automatic.
Independent practice is necessary because a good deal of practice is needed in order to become fluent and automatic in a skill (Rosenshine, 2012). Building on the previous principles (the introduction to new material, questioning, modelling, guided practice, checking understanding, provide a high success rate) independent practice allows the embedding of the newly learnt material so that the students can access this later.
For example in https://nrich.maths.org/2670, without sufficient practice at solving linear equation students will find it challenging to apply the ideas to more complex problems.
However, if students are not suitably prepared for their independent practice this increases the demand on the teachers time by individual students and runs the risk of the student running out of time to address errors as the teacher tries to fix gaps in the students prior knowledge.
In addition to this, if students are given a diet of questions that are all too similar or have no purpose teachers run the risk of wasting time and giving students a false impression of the problems they can solve.
Intelligent practice – making the most of the time you have in the classroom (Barton, 2018)
As Barton (2018) points out, well-designed practice questions can help students develop procedural fluency, whilst also allowing the students to make connections. During independent practice, students have time to self-explain, go back over previous work, make and correct mistakes.
For example, Barton (p. 256, 2018) combines variation theory with the development of his intelligent practice questions to produce a selection of questions on calculating the gradient between two points:
1. (0,0) and (3,6) | 6. (0,0) and (7,-2) |
2. (3,6) and (0,0) | 7. (0,0) and (-2,7) |
3. (6,3) and (0,0) | 8. (0,0) and (-2,0.5) |
4. (-6,3) and (0,0) | 9. (0,0) and (-2,2) |
5. (-6,-3) and (0,0) | 10. (0,0) and (-2/5 ,2/7 ) |
Table 2: A sequence of questions on calculating the gradient between two points (Barton, p. 256, 2018).
In his selection of questions, he keeps one of the coordinates (the origin) the same and varies the other. This allows the students to practice the skill of finding the gradient between two points but with the added benefit of making full use of the time available in lessons.
More examples of intelligent practice questions (using variation to make the most out of independent practice) can be found at https://variationtheory.com/.
Purposeful practice – questions that allow for the novice and expert to gain something from completing them (Barton, 2018)
In principle seven we considered the principles of Load Reduction Instruction (Martin, 2016) and we can return to the final principle here: independent practice, supported autonomy and guided discovery learning.
This happens after there has been modelling by the teacher and ample guided practice with checking the students’ understanding after key concepts. Without the previous principles, novice learners may not learn from solving problems without a structure to reduce the cognitive load (Sweller, 1988, Barton, 2018).
To address this, Barton (2018) identifies a category of problems that fit into the title Purposeful Practice. These problems ‘enable students to develop that all-important inflexible knowledge and procedural fluency that is key in the transition from novice to expert’ (Barton, 2018). Unlike the Intelligent Practice questions, these questions provide for a mix of student type (expert/novice) in the classroom.
For example, the Mathematical Etudes (Foster, 2019) are one way to perform purposeful practice. Figure 5 (Foster, 2015) gives an expression polygon etude.
Figure 5: Taken from Foster (2015).
The students are required to solve equations that they form between two expressions if they are linked by a line. Once the students have found the solutions, as Foster (2018) points out, the solutions produce a pattern that lead students to comment on it and offers further challenge in creating their own version.
Barton (2018) lists the following principles that purposeful practice is based around:
- Students need to experience early success (principle seven).
- There must be plenty of opportunities to practice the key procedure.
- The practice should feel different.
- Opportunities must exist for students to make connections, solve problems and think deeper.
- The focus is always on the practice.
Using these principles enables us to support all students to access tasks and achieve success.
References
Barton C (2018) How I wish I’d taught Maths: Lessons learned from research, conversation with experts, and 12 years of mistakes. Woodbridge: John Catt Educational Ltd.
Foster C (2015) Expression polygons. Mathematics Teacher 109(1): 62–65.
Foster C (2018) Developing mathematical fluency: Comparing exercises and rich tasks. Educational Studies in Mathematics 97(2): 121–141.
Foster C (2019) Mathematical etudes. Available at: http://www.mathematicaletudes.com/ (accessed 20 July 2019).
Martin A J (2016) Using Load Reduction Instruction (LRI) to boost motivation and engagement. Leicester: British Psychological Society.
Rosenshine B (2012) Principles of Instruction: Research based principles that all teachers should know. American Educator, Spring 2012.
Sweller J (1988) Cognitive load during problem solving: Effects on learning. Cognitive Science 12: 257–285.
Principle 10. Engage students in weekly and monthly review: Students need to be involved in extensive practice in order to develop well-connected and automatic knowledge.
It is not until students are required to retrieve previously acquired skills or content and apply it to new situations that you can truly determine if they have learnt that material. Rosenshine (2012) states the research on cognitive processing suggest that extensive readings of a variety of materials, frequent review and discussion and application activities help students increase the number of pieces of information in their long-term memory and organise this information into patterns and chucks.
In the mathematics classroom, there has been recent insight into how you could create these classroom activities.
Barton (2018) devotes a whole chapter to ‘Long-term Memory and Desirable Difficulties’, in which he brings together recent research on how long-term memory works and research from cognitive psychological scientists. Of particular interest is Bjork’s (2011) theory of disuse. Instead of memories decaying and disappearing after long periods of disuse, they actually remain in the long-term memory but are less accessible.
Spaced practice and interleaving
The advantages provided by the spacing (distributed) practice of problems that tackle previously taught material is enhanced when there are connections to be made between the problems over time (Benjamin and Tullis, 2011).
For example, when I am reviewing a previously taught concept it could be in one of the following ways:
- The class has finished a body of work on a new topic and has completed their independent practice on this area. I will then introduce some questions that have the new concept present but previously learnt material is required to provide a full solution to the question.
- The class had tackled a previous review question and found a concept in the question challenging. I will then introduce some questions that have the concept present (some in obvious and some in non-obvious ways).
In both situations, the discussions about the questions allow for connections to be made to previously completed work.
For more information on spaced practice and interleavingAn approach to learning where, rather than focusing on one p... More see the Learning Scientists (2019).
Rohrer et al (2014) note that interleaving improves mathematics learning by strengthening the association between each kind of problem and its corresponding strategy. Rohrer et al. (2014) examines the need for extending the practice from superficially similar problems, the so called SSDD (same surface, different depth) (Barton, 2018).
For example, start with a shape, image or context and keep the surface structure of the questions as similar as possible. Then construct either four questions on different topics or four variations of the same deep structure. Figure 1 gives an example of one I use with A Level Mathematics students.
Figure 6: Image created using Desmos.
A website has been created to help teachers source more of this problems, https://ssddproblems.com/.
Low stakes quizzes – as an addition (not an alternative) to same surface, different depth problems
Barton (2018) identified that quizzes as a regular part of the learning process; where the performance of the students is not recorded in a formal way, it would aid students in the retrieval process of mathematical skills and facts. He proposed a structure for these quizzes, one of which was to use the quizzes for things the students have encountered before and make them not topic specific. This was in order to reap the benefits of Spacing and Interleaving Effects.
For example, with my Year 10 and 11 classes, we use low stakes quizzes to review previously taught material from Years 7 to 9. I prefer to have eight quick questions that have simple solutions but focusing on skills (i.e. factorising quadratics) and facts (i.e. geometric facts for quadrilaterals).
I use them at natural breaks in my weekly lessons, so the quiz could be at the start, part way through (I use this particularly when I have a lesson timed for the end of the day) or end of a lesson. The questions are always printed, the solutions added to a review book so that the students can keep their own personal log of the skills and facts that they need to review.
We have now reached the end of the principles and, as Rosenshine and Stevens (1986) state:
It would be a mistake to claim that the teaching procedures which have emerged from this research apply to all subjects, and all learners, all the time. Rather, these procedures are most applicable for the “well-structured” … parts of any content area, and are least applicable to the “ill-structured” parts of any content area.
As always when teaching we should think carefully about how we construct our lessons for our learners over time and not just apply a check box approach.
References
Barton C (2018) How I wish I’d taught Maths: Lessons learned from research, conversations with experts, and 12 years of mistakes. Woodbridge: John Catt Educational Ltd.
Benjamin AS and Tullis J (2010) What makes distributed practice effective? Cognitive Psychology 61: 228–247.
Bjork R A (2011) On the symbiosis of remembering, forgetting, and learning. Successful remembering and successful forgetting: a festschrift in honour of Robert Bjork. New York: Psychology Press.
The Learning Scientists (2019) Six Strategies for Effective Learning. Available at: http://www.learningscientists.org/downloadable-materials (accessed 28 July 2019).
Rohrer D, Dedrick RF and Burgess K (2014) The benefit of interleaved mathematics practice is not limited to superficially similar kinds of problems. Psychonomic Bulletin Review 21(5): 1323–1330.
Rosenshine B and Stevens R (1986) Teaching functions. In Wittrock M (ed.) Handbook of research on teaching (3rd ed.) New York: Macmillan.
Rosenshine B (2012) Principles of Instruction: Research based principles that all teachers should know. American Educator, Spring 2012.
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