While there has been a big movement (and with good reason) around the science of reading, there has been less support for the science of math. Yes, there is a science of math. It refers to an evidence-informed approach to teaching and learning mathematics.
I’ve taken a deep dive into the world of Mathematics over the last couple of years and it hasn’t been pretty! There are still many dominant voices who are more aligned with an approach where students should start with “exploring” concepts before they have been taught what the concept is. I don’t want to turn this into a research-dense slugfest as to why we shouldn’t take this approach, but I will just touch on it.
MYTH: It is more effective to discover your own learning
What we know about how we learn:
- We can only learn when we are attending to something (Willingham, 2010)
- We need to be motivated in order to attend to something (McCrea, 2020)
- Meaningfulness can increase motivation by creating a connection (Shimamura, 2018)
- However, we don’t need to discover things for ourselves to create that connection (Kirschner, Sweller & Clark, 2006)
- Productive struggle does not lead to increases in motivation or student learning outcomes (Hughes and Riccomini, 2011)
- Desirable difficulties is NOT the same as productive struggle. Desirable difficulties refers to making the learning more difficult in order to make the learning last through retrieval, spacing and interleaving (Bjork and Bjork, 2011)
- In fact, as the expert, the teacher has a better chance (and it’s part of their job) of helping create those connections for students by sequencing concepts in small steps, providing multiple models and using stories/analogies (Kirschner, Sweller & Clark, 2006; Rosenshine, 2012; Willingham, 2010)
This is how we learn things in English, Mathematics, History, Science and every other subject that you can think of. We DO NOT have a “numeracy brain” and a “literacy brain”. Everything we learn requires us to make connections to existing knowledge within our long term memory. This is more likely to occur if concepts are sequenced in small steps. It doesn’t change whether it is for the subject Mathematics or English.
However, we do learn some things differently based on whether it is biologically primary knowledge or biologically secondary knowledge. Some maths ideas can be categorised as biologically primary, meaning that they can be learnt without someone explicitly teaching it to them. For example, older infants can distinguish three crackers from two crackers, but not from four crackers (Feigenson et al, 2002). The graphic below highlights what needs to be explicitly taught versus what is biologically primary knowledge (Thanks to Glen Fahey from CIS for bringing this study to my attention).
Sarah Powell, Elizabeth M. Hughes and Corey Peltier wrote this brilliant paper for the Centre for Independent Studies – Myths That Undermine Maths Teaching. It debunks a number of myths about Mathematics.
My systematic and structured approach to teaching primary mathematics
I have been working with a team of teachers (shout out to the SoLaR Collective) on developing some professional learning and resources to support teachers in implementing evidence-informed mathematics teaching. To try and develop consistency and a shared understanding of what can be a part of an effective primary mathematics lesson, I decided to put together a lesson structure.
It is by no means the absolute only way of teaching maths and I’m sure that I will continue to edit it in the future. However, I feel it offers a pretty good structure and is relevant for most primary classes.
| Time | Stage | Overview |
| Pre-Lesson Planning | What are the learning intentions and success criteria? What is the prerequisite knowledge? What are the common misconceptions? What is the key vocabulary? What are the hinge questions? How can you sequence the concept in small steps? | |
| 15-30 min | Daily Review | Principles to follow – Retrieval practice – Spaced practice – Interleaving – Prerequisite knowledge Concepts to focus on – Basic maths facts – Number sense – Vocabulary and concepts – Mental Maths – Multiplication tables – Problem-solving |
| New Concept Development | ||
| 10 min maximum | Teacher-led instruction | Modelling – Live worked examples (have multiple examples prepared) – Build conceptual and procedural understanding at the same time – Examples and non-examples – Strategy comparison – use concrete, pictorial, and abstract representations |
| 20 min + | Practice | Guided and independent – Example-problem pair – Provide multiple practice opportunities – Be prepared to reteach it Scaffold tasks for novices – Backwards Fading – Pair-shares Deliberate practice – Sufficient time to work without prompts and scaffolds – Interleaving Reasoning – Self-explanation prompts – Mistake analysis Problem-Solving Bar Method UPAC – Understand – Plan – Answer – Check Games |
| Assessment | Checking for understanding (Mini whiteboards, cold calling, choral responses, turn and talk) Quizzes Short answer Multiple choice |
For this blog post, I’m only going to go over the first two phases. I will look at New Concept Development in Part 2.
The Pre-Lesson Phase
| Time | Stage | Overview |
| Pre-Lesson Planning | What are the learning intentions and success criteria? What is the prerequisite knowledge? What are the common misconceptions? What is the key vocabulary? What are the hinge questions? How can you sequence the concept in small steps? |
I don’t think that there is anything groundbreaking in what I am suggesting during this phase, but I just wanted to highlight its importance because this is where we can get in front of any likely challenges. For novice teachers, pre-preparing these also allows them to focus their working memory on being responsive to their class.
Resources to support this phase
- Complete Maths: “Everything you need to teach, learn and assess mathematics.” Complete Maths comes from Mark McCourts team and takes a mastery approach. You can sign up for a free account and you get access to so much greatness!
- NCETM – Primary Mastery Professional Development
- National Numeracy Learning Progression
- Learning & Teaching with Learning Trajectories: Learn about the steps children typically take to learn maths by exploring LTs. Each topic (or trajectory) is broken down into the levels, or steps, children take on their path to being strong mathematicians.
The Daily Review Phase
| Time | Stage | Overview |
| 20-30 min | Daily Review | Fast paced, use routines and learning and engagement icons to minimise the need for explanations. Principles to follow – Retrieval practice: Pulling information out, not putting new information in. Building fluency/automaticity of facts and procedures – Spaced practice: pull out content from last lesson, last week, last month, last term – Interleaving: rather than just having a chunk of questions on the one concept, break it up with some questions from a similar topic e.g. 4 x 5, 4 x 8, 4 x 6, 5 x 8, 4 x 2 – Prerequisite knowledge: review concepts that will be required for your upcoming lesson Concepts to focus on – Basic maths facts: Reviewing basic maths facts such as addition, subtraction, multiplication and division facts can help students develop fluency and speed in mental calculation. – Number sense: Including activities that promote number sense such as number patterns, place value and estimation, can help students develop a deeper understanding of numbers and their relationships. – Vocabulary and concepts: Reviewing key maths vocabulary and concepts can help students solidify their understanding and build a strong foundation for future learning. – Mental Maths: Refers to any mathematical activity completed without the use of any sort of aid – pen and paper, calculator, abacus. This can help students deepen their conceptual understanding of number and mathematical relationships – Multiplication tables: Reviewing times tables on a regular basis can help students develop fluency and automaticity in multiplication, which is an essential component of maths proficiency. – Problem-solving: Including a daily word problem task can help students develop their maths problem-solving skills, as well as promote reasoning. These problems should only be based on concepts that have already been mastered. |
The first time we learn something it is like drawing a shape in the sand on a beach. At the time it is crystal clear, but it soon fades when the tide comes in and washes over it, leaving only a faint imprint. Yet the more we return over time to redraw this shape, the more resilient it will become to the blurring effects of the tide.
Emma McCrea, Making every maths lesson count
The main focus of the Daily Review should be retrieval practice focused on building fluency/automaticity of key facts and procedures. Fluency in mathematics means having a quick and efficient grasp of basic maths facts and procedures, which enables students to solve more complex problems with ease and accuracy. It is not rote learning. Conceptual and procedural understanding should be taught side by side, not one before the other.
In order to achieve fluency in maths, students need to be able to:
- Count with understanding: Students need to be able to count accurately and understand the meaning of numbers, including their magnitude (how far away from zero) and relationship to each other.
- Add and subtract within 20: It is essential that they can add and subtract single-digit numbers within 20 quickly and accurately, using a variety of strategies such as counting on, making ten and decomposing numbers (breaking numbers into smaller parts).
- Multiply and divide within 100: This involves recalling multiplication and division facts up to 100 quickly and accurately and applying these facts to solve problems.
- Use place value and number sense: Students should have a strong understanding of place value and number sense, including the relationships between digits in multi-digit numbers and the ability to use this knowledge to perform operations and solve problems.
Fluency in these areas is important not only for success in mathematics, but also for success in other academic and real-life contexts that require mathematical reasoning and problem-solving skills.
Caveats
For this to work effectively, an enacted, low variance curriculum is needed. Teachers need to know what students have learnt in the past, in order to plan for the present. Recently, I encountered how difficult this can be when trying to plan some lessons that I was demonstrating for other teachers. I could only go off the syllabus and guess as to what sorts of things the students may have encountered in the past and then I tried to match up with the recommended focus areas (mentioned above).
While it wasn’t a massive issue, it meant that I had to spend a lot more time checking for understanding and (re)teaching concepts. You want it to be fast paced to maintain students’ attention.
Resources to support this phase
- Ochre – Review Materials: This is such an amazing resource! It currently features 100’s of Daily Review lessons. It has been built around the Shaping Minds Maths Curriculum
- David Morkunas – Spaced Interleaved and Retrieval Practice: The Key to Long term Knowledge Retention. David’s presented a number of times on this topic and has provided some excellent examples.
- Clare Sealy: How Retrieval Practice Helps Long-Term Maths Skills: How I Wish I’d Taught Maths (6)
- David Costello: Making Math Stick: Tactics for Retrieval Practice
Part 2
In Part 2, I will go over new concept development and what we can do during the teacher led instruction phase and then into the practice phase. I would love to know your thoughts and if you have any questions or comments.
Recommended Mathematics Books
As I mentioned, I’ve gone down a massive rabbit hole when it comes to educating myself on how to teach primary mathematics effectively. This is a list of books that I’ve read/reading and that have influenced my thinking around this:
- Barton, Craig: How I Wish I’d Taught Maths
- Booker, George: Building Numeracy
- Booker, George, Bond, Denise and Seah, Rebecca: Teaching Primary Mathematics
- Cockburn, Anne and Littler, Paul: Mathematical Misconceptions
- Krasa, Nancy, Tzanetopoulos, Karen and Maas, Colleen: How Children Learn Math (I will have Karen on the Knowledge for Teachers Podcast soon)
- Mackle, Kieran: Thinking Deeply About Primary Mathematics
- McCourt, Mark: Teaching for Mastery
- McCrea, Emma: Making Every Maths Lesson Count
- McGrane, Chris and McCourt, Mark: Mathematical Tasks
- Norton, Stephen: Teaching and Learning Fundamental Mathematics
- Pershan, Michael: Teaching Math With Examples
- Stein, Marcy, Kinder, Diane, Rolf, Kristen, Silbert, Jerry and Carnine Douglas: Direct Instruction Mathematics
References
Bjork, E. L., & Bjork, R. A. (2011). Making things hard on yourself, but in a good way: Creating desirable difficulties to enhance learning. Psychology and the real world: Essays illustrating fundamental contributions to society, 2(59-68).
Feigenson, L., Carey, S., & Hauser, M. (2002). The representations underlying infants’ choice of more: Object files versus analog magnitudes. Psychological science, 13(2), 150-156.
Hughes, E. M., & Riccomini, P. J. (2011). Mathematics motivation and self-efficacy of middle school students. Focus on Middle School, 24(1), 1-6.
Kirschner, P. A., Sweller, J., & Clark, R. E. (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(2), 75-86.
Mccrea, Peps (2020). Motivated Teaching: Harnessing the science of motivation to boost attention and effort in the classroom. CreateSpace Independent Publishing Platform
Rosenshine, B. (2012). Principles of instruction: Research-based strategies that all teachers should know. American educator, 36(1), 12.
Shimamura, A. (2018). MARGE: A whole-brain learning approach for students and teachers. PDF available from http://www. bit. ly/2UEi1IB.
Willingham, D. T. (2021). Why don’t students like school?: A cognitive scientist answers questions about how the mind works and what it means for the classroom. John Wiley & Sons.