Recently, I wrote a post: A Systematic and Structured Approach to Teaching Primary Mathematics (Part 1). Part 2 is coming! However, I felt it was important to address a few points first as I’ve come across a lot of misconceptions since the initial article.

## Why Is It So Hard To Get It Right In Education?

In education, a lot of effective teaching has been prevented from happening due to deficits in human thinking. Firstly, it’s harder to unlearn something than it is to learn something new. Unfortunately, most teacher training institutions have not set teachers up for success in what they need to know about how learning happens (see: Strong Beginnings: Report of the Teacher Education Expert Panel). We’ve also seen specific reports that support this on reading (Buckingham & Meeks, 2019) and mathematics (Fahey et al, 2021). This has meant that teachers are entering the profession affected by the Dunning-Kruger effect (thinking they know more than they do) and with misconceptions of effective teaching practices.

Secondly, humans are biologically inclined to want to feel a sense of belonging to a group. So, they will look for like-minded people who align with their way of thinking (which isn’t necessarily the right way). This is where we see a sense of tribalism develop (I’ve written about it here) and people then interpret things through the lens of their tribe. For instance, if anything that you read in this article doesn’t align with your current thinking about teaching primary mathematics, there’s a good chance that you will dismiss it!

“The fear of social death is greater than the fear of physical death.”

— Brooke Herrington, Sociologist

## Good Teaching Is Good Teaching

Aligning your beliefs to a certain “side” in the “Maths Wars” doesn’t help. It hinders your ability to look at new ideas with an open-mind. Over the years, I’ve observed arguments and accusations being made over different pedagogical approaches, two things have stood out.

- There are good and bad versions of both “traditional” teaching methods and “inquiry-based” teaching methods. For example, a common accusation made against explicit instruction is that it is dominated by the teacher and the students are passive. I would agree that this would be a poor version of delivering instruction, but I wouldn’t call it explicit instruction. Also, critics of inquiry-based learning will often exaggerate the extent that students are left to discover things on their own.

- Outwardly criticising the other side doesn’t help anyone. It can alienate them and often the accusations are unsubstantiated. Unless you have actually observed a teacher in action and then sat down with them to understand their decision-making, it’s actually very hard to know how someone teaches. As I’ve just explained, there are a lot of terms used in education that can be interpreted differently, so until we have a shared language and understanding, we will continue to have misconceptions.

**5 Key Principles for Teaching Primary Mathematics Effectively**

I have come up with these 5 principles to follow because I thought it was important to address some of the misconceptions that are out there when it comes to teaching primary mathematics.

The Five Principles are:

- 1. Understand That Mathematics Is Highly Hierarchical
- 2. We Need Both Conceptual And Procedural Understanding
- 3. Use Concrete, Pictorial And Abstract Representations Based On The Needs Of The Student(s)
- 4. To Think Like A Mathematician, We Need To Teach Them Maths
- 5. Aim For Productive Learning, Rather Than Productive Struggle

## 1. Understand That Mathematics Is Highly Hierarchical

In the primary classroom, fundamental mathematical concepts play a crucial role in shaping the mathematical journey for our young learners. From basic counting and number recognition to essential operations, these concepts serve as the building blocks for more advanced mathematical skills (Price, Mazzocco & Ansari, 2013). Just as phonics lays the groundwork for reading and spelling, mastering these fundamental maths skills provides a solid foundation and enhances students’ problem-solving abilities.

This figure from von Aster & Shalev, 2007 illustrates a developmental model of cognitive number representation that is hierarchically organised.

David Geary (2011) performed a five-year longitudinal study that followed 177 students from first grade through to fifth grade to find out what measures predict later mathematical achievement. He found competency in the following aspects to be vital:

- understanding of the relation between number words, Arabic numerals (e.g. 1, 3, 8) and the underlying quantities they represent, as well as skill at fluently manipulating these representations
- knowledge of the mathematical number line
- basic skills in arithmetic (i.e., skilled use of counting procedures, decomposition, and fact retrieval in problem-solving)

### Practical implications

By acknowledging the hierarchical nature of mathematics, you can plan your instruction accordingly. Start by assessing your students’ current level of understanding and provide targeted support to fill any gaps. Then, scaffold their learning experiences by systematically introducing new concepts in a logical order. This approach ensures that students have a solid foundation and it fosters their confidence before moving on to more challenging topics.

For novices, we can reduce their extraneous load by sequencing concepts in small steps, starting with the language we use. For example, saying, “Three twos is six” is more accessible for students to understand, rather than “Three times two equals six” because it can be confusing for them to hear the word “times” being used when they would naturally link it to telling the time. We can add the mathematical vocabulary after the conceptual understanding is developed. This also links in with the next point of interweaving conceptual and procedural knowledge.

However, “there are over 105 novel mathematics vocabulary terms that children are expected to understand and apply” by the end of Year 1 (Hughes, Powell & Stevens, 2016). So, this means that we need to be highly intentional with what we teach, how we teach and when. Curriculum designers need to think about every aspect of this sequencing from similarities and differences between concepts and how complex they are. The complexity of the concept will affect how fast you move through it e.g. division and fractions are concepts that are difficult for students to develop strong understandings of.

Just developing an understanding of what numbers are is complex. Heck, just hearing the numbers can be tricky for many children. In fact, one of the biggest predictors of mathematical success is actually phonological awareness! As they can’t hear the “een” in “fourteen” and mistake it for “forty”.

This slide from Daniel Ansari’s presentation for Think Forward Educators provides an overview of the complexity involved with understanding whole number.

In, “How Children Learn Math”, Krasa et al (2022) describe five other important aspects of number that need to be understood:

**One-to-one principle:**There must be one unique tag per item counted.**Stable order principle:**The tags used must be ordered the same way consistently over time**Cardinality principle:**The last tag used in the stable order list uniquely represents the cardinal value of the set**Abstraction principle:**It does not matter what you are counting as long as you think of them as entities, real or imagined.**Order-irrelevance principle:**As long as the first two principles are adhered to, it does not matter in what order you count the items

**Other activities**

**Prerequisite knowledge:**know what the prerequisite knowledge is that students need to understand before engaging with the new content. Assess this knowledge before starting the new content.**Daily Reviews:**Ensure that every lesson includes retrieval, spaced and interleaved practice to ensure students build on and maintain their understanding of previously learnt content.**Sequence concepts in small steps:**We don’t just want to build incrementally, but also focus on the smallest aspects first.**Stem sentences:**can also be used to decrease the size of the step that students have to take to complete their answers.

## 2. We Need Both Conceptual And Procedural Understanding

A comprehensive understanding of mathematics requires both conceptual understanding and procedural fluency. Conceptual understanding involves grasping the underlying principles and logical connections in math, while procedural understanding focuses on executing specific mathematical procedures accurately.

While both conceptual and procedural understanding are vital, attempting to focus on them simultaneously can be overwhelming for students. It can lead to confusion and hinder students’ overall learning experience. However, it is not about one or the other or one before the other, but rather they are bidirectional and can be intertwined (Rittle-Johnson, et al, 2015).

This graphic from Kilpatrick et al, (2001) shows how the five strands (adaptive reasoning, strategic competence, conceptual understanding, productive disposition and procedural fluency) are interwoven and interdependent. Similar to Scarborough’s Reading Rope, what you focus on will be dependent on the stage of the student’s mathematical journey e.g. in the early years, more time needs to be spent on developing their number sense (understanding numbers) and then basic maths facts, before a shift is made onto more complex concepts like fractions and Algebra.

### Fluency in basic maths facts opens doors

As noted in Effective Math Interventions (Codding, et al., 2017), “Students without basic fact fluency seem to be less able to grasp underlying math concepts, perform procedural computation tasks, solve word problems, or access higher-level math curricula (Fuchs, et al., 2006; Gersten et al., 1999; Jordan et al., 2003).”

Being fluent DOES NOT mean simply memorising the facts, but knowing the concepts and procedures so well that you are able to recall them quickly and flexibly. So, yes they need to do “drill-like” activities, but not at the expense of not understanding what the numbers represent. It also doesn’t mean that it can’t be fun. Check out Times Tables Rockstars for a great example of this.

### Practical implications

This means that sometimes the focus will be on teaching the step-by-step procedure of how to solve an addition problem or it could be explaining what fractions actually are. Other times, we might need to get number bonds to ten into their long-term memory to free up space in their working memory for the next concept.

Also, it doesn’t mean that we don’t do problem-solving until they get to high school, but rather the level of complexity needs to be appropriate. e.g. John had 5 apples and he gave 3 to Sam. How many does he have now? This is a word problem that could be understood both conceptually and procedurally by 7-year olds (as long as it’s been taught to them).

Looking at how it can be interwoven. These are the steps that I took in order to get to teaching a three-digit addition word problem. You can see how I moved between conceptual and procedural understanding. The other thing to take into account when teaching a new number is to use accessible numbers. That’s why I taught the algorithm separately to regrouping.

Other useful techniques are:

- Make the connections for students through language e.g. saying 13 is one ten and three ones. Also, this is closer to how number words in Mandarin are structured.
- Strategy comparisons
- Comparing correct vs incorrect procedures
- Reduce the extraneous load by using accessible numbers when teaching a new concept e.g. in the example that I have provided, I taught regrouping separately to the steps of the algorithm
- Self-explanations
- Fluency activities

## 3. Use Concrete, Pictorial And Abstract Representations Based On The Needs Of The Student(s)

**Concrete: **involves physical objects or manipulatives that students can touch and interact with, helping them develop a solid foundation of understanding. These manipulatives make math concepts more tangible and accessible.

**Pictorial**/**Representational**: such as diagrams and charts, visually depict mathematical ideas and serve as a bridge between the concrete and abstract. They help students visualise relationships and patterns, fostering a deeper conceptual understanding.

**Abstract**: includes symbols and mathematical notation, are concise and efficient ways to represent math ideas. They are essential for developing fluency in mathematical language, calculations, and problem-solving. However, it’s important to recognise that relying solely on abstract representations may be challenging for some students who need more exposure to concrete or pictorial representations before they can visualise it themselves.

While the end goal is for all students to be able to use abstract representations, it doesn’t mean that by a certain age we stop using concrete and pictorial representations. If students haven’t got a strong grasp of a concept, without a concrete or pictorial representation, they are required to hold both the old concept and new concept in their working memory at once.

The other misconception is that CPA should be used like a series of steps where you go from one to the next. In actual fact, using all three at once can strengthen the links between each stage.

### Practical Implications

The teacher needs to model multiple worked examples, use example problem-pairs and scaffolded support. You then need to be responsive to the information that your students are giving you. This might mean continuing to use the concrete and pictorial representations for longer than you had intended to. The other consideration to make is to be consistent with the types of manipulatives that you use, rather than having too many. Again, this is to ensure that students maintain their attention on the main thing, rather than the new cute, colourful bear counters!

**Examples**

- You might use Cuisenaire rods to develop our students’ understanding of things like number bonds, equivalence and fractions. The pictorial version of this are bar models (drawings of rectangles that represent both known and unknown quantities).

- Use a number line to demonstrate the relationship between numbers. This allows students to develop their number sense.
- Make the connections explicit! We need to explain the connections between concepts because their knowledge is still developing e.g. make the link between skip counting and multiplication.
- David Morkunas has spoken about how bundling (paddle pop/popisicle) sticks can be a great way to show regrouping because you can bundle and unbundle them, rather than MAB blocks where they are stuck in place.

## 4. To Think Like A Mathematician, We Need To Teach Them Maths

Yes, mathematicians might relish in challenging situations, they can be curious and can monitor their own thinking. However, a novice is not a little expert (Chi et al, 1981; Willingham, 2021). While understanding how a mathematician thinks can provide insights into their problem-solving strategies and approaches, it does not guarantee that one will automatically think better or become a better mathematician. Becoming a proficient mathematician requires practice, a deep understanding of mathematical concepts, and the ability to apply logical and analytical thinking.

### How Learning Happens

**Red circle:**the intended new concept that is being presented by the teacher**Light grey circles:**minimal schema strength (understanding)**Black circles:**strong schema strength**Black lines:**connections to prior learning

This graphic showcases the differences in how a novice and expert acquire new knowledge. On their own, a novice will struggle to make connections to their prior learning because the new concept is still abstract to them. They need a strong foundation of basic concepts and principles in order to connect to new ideas.

Meanwhile, an expert is able to use their extensive knowledge base to approach new information or problems in a more contextualised and interconnected manner. They can quickly and effortlessly perform routine or familiar tasks, allowing them to focus their cognitive resources on more complex aspects of a problem. Experts have developed mental frameworks or schemas that help them recognise the underlying structure and patterns in a problem. They can quickly identify the relevant information and apply appropriate problem-solving strategies.

We have separate mental models for every bit of information because knowledge expertise is domain specific. Also, or each of these concepts we are at different stages of learning (see below).

This detailed graphic (below) from Dr. Corey Peltier shows how the level of teacher support changes based on where the student is at in their stage of learning.

### Would you attempt to build a house without being taught how to?

Teaching mathematics by starting with a problem without giving our students the knowledge and skills to complete it, can be likened to expecting someone to build a house without providing them with the necessary tools, materials, and instructions. While the problem may serve as the end goal or the blueprint of the house, it is essential to provide explicit instruction and guidance to ensure learners have the foundational knowledge and skills required to solve the problem effectively.

Without demonstrating and explaining, the individuals may struggle to understand the various components of the house, the construction process and the specific techniques required. They may lack knowledge of structural integrity, building codes and appropriate materials. As a result, their attempts to solve the problem might be inefficient, error-prone or even hazardous.

Even if you teach a novice the strategies that a ‘mathematician’ uses, without the schemas to make connections to their prior learning, they aren’t able to solve problems. If you haven’t taught them how to do it, the only people who will be able to build the house without you showing them how to do it will be people who could already do it. When we rely on knowledge that students bring to the classroom from their life experiences, then we are actually creating an equity issue. Also, any success that is achieved is not actually due to effective teaching.

Agodini and Harris (2014) compared four different maths curriculums that used different pedagogical approaches for first and second grade using 789 teachers. It found that students taught with explicit instruction made the greatest gains and the most constructivist maths program had significantly lower scores in all the participating groups.

* If the majority (>80%) of students have the prerequisite knowledge and the task complexity is simple, THEN we can launch into a problem-solving task. I would still be reluctant to turn this into a group task because that adds in an extra layer of complexity – how to work in a group so that everyone is thinking.

### Practical implications

We need to ensure that learners develop a strong foundation in mathematics, allowing them to understand the problem in context and apply appropriate strategies to solve it. Once learners have acquired the necessary knowledge and skills, they are better equipped to engage in problem-solving activities and tackle more open-ended challenges.

This means that if our students haven’t got the prerequisite knowledge required to complete a task, they’re not going to be able to do it. It doesn’t mean we never give our students problems to solve, but rather we should only give our pupils a problem if they actually have a chance to complete it.

### Word Problem Attack Strategies

We need to explicitly teach students an attack strategy. This is an easy-to-remember set of steps (Powell and Fuchs, 2018 – This is a really good read). It should involve:

**Interpreting the word problem’s meaning:**read the problem, identify the question, and determine the central idea of the problem**Finding the missing quantity:**- set up a number sentence or use a graphic organiser
- perform calculations
- label the number answer
- Check whether the answer makes sense

Sample attack strategies (Powell and Fuchs, 2018)

## 5. Aim For Productive Learning, Rather Than Productive Struggle

Anxiety can affect anyone and be about anything. Maths anxiety isn’t so much a whole separate thing, but rather just something people can get anxious about. Often, it can be connected to being in uncomfortable situations, such as not being able to complete a problem. Researchers have found that having to understand mathematical concepts intuitively and the difficulty in conjuring up the spatial imagery is actually what causes the feelings of anxiety (Krasa et al, 2022) This is where “productive struggle” can actually lead to destructive struggle. Can it be prevented? Yes!

For novices, using the problem-first approach also has ramifications on their self-efficacy, motivation and potential to pick up misconceptions. Another negative of relying on students to discover things for themselves is that they can actually “learn” how to do things the wrong way and embed these misconceptions into their long-term memory. This then becomes another barrier for the teacher to overcome (Roy et al, 2019).

Motivation and connection aren’t dependent on the task, it’s dependent on the students’ ability to make connections to their schema, feelings of previous success and an understanding of the task.

### Are we assessing reading comprehension or mathematical ability?

In order for a child to complete a word problem they need:

- Reading Comprehension
- Mathematical Vocabulary
- Problem-Solving Strategies
- Fluency in Mathematical Operations
- Reasoning

On top of this, often when we try to increase the difficulty of word problems we actually ramp up the literacy difficulty of the problem, rather than the maths aspect of it e.g. including redundant information to try to confuse the learner.

### You need to be engaged to learn, but engagement doesn’t mean learning is happening

A lot of the literature on mathematics focuses on engagement. While engagement is needed to attend to something, it doesn’t mean we’re thinking about the right things. For example, a popular activity to do with students is to cook pizzas when learning about fractions. While the students might be having fun and engaged in what they’re doing, it doesn’t mean that they will actually learn anything about fractions during the lesson.

### Practical implications

Our pupils need their teacher to help them create the connections through sequencing concepts in small steps, telling them what the direct links are and using stories and analogies.

**Speak positively about mathematics:**Our own thoughts and opinions of the subject and our students’ ability (Pygmalion effect) can impact our pupils.**Let the maths be the fun part:**There seems to be an overemphasis on trying to make maths “engaging” by creating ”exciting” activities or making links to the “real world”. Firstly, we never question the need to do imaginative writing with our students. Yet, how many of you have published a fiction book? However, we seem to have to justify the fact that sometimes what we learn in maths will only be relevant for us, when we’re learning maths. However, this doesn’t mean that we take it out of the curriculum because education is about providing opportunities and exposing our students to biologically secondary knowledge. Many of us may never use calculus or trigonometry outside of school. But for others, it may bring a lifelong love, that leads to them becoming an engineer!**Success breeds success:**Teach the students how to get the answers to the questions that we ask them. This means that they can be successful in mathematics. Early success increases the chances of students paying attention in the future (I’ve written about motivation here). Showing them how to get the answer isn’t cheating, it’s teaching.**Decrease the stakes:**Given the nature of mathematics, where there are often definitive right and wrong answers, continual mistakes can be disheartening. However, this highlights the importance of effective teaching. It is crucial to cultivate a classroom culture that embraces learning from mistakes. When a student provides an incorrect answer, it serves as valuable feedback for the teacher, indicating the necessity for potential re-teaching or additional support. Timed testing can also play a role in this process, as teachers can use it to gather data to inform future instruction. Open communication between teachers and students about this purpose is essential to foster a supportive learning environment.**Revoicing: “**involves repeating, rephrasing, or expanding on student talk” (Anthony & Walshaw, 2009).**Use examples and non-examples:**Variation Theory tells us how important it is to show students what a concept is and isn’t. They need to see multiple examples in order to strengthen their mental model of the concept.

## Final thoughts

I know that I’ve written half a book in this blog post, but I thought it was important to try and make each point as clear as possible. Australia has one of the sharpest declines in mathematics among PISA-participating countries. The reality is that we’re not developing the level of fluency that is required in facts and procedures and this is impacting on their problem-solving and reasoning skills.

The most effective and efficient method for giving our students the neccessary mathematical skills and knowledge is through teaching it to them first, providing the required level of scaffolding, checking for understanding and giving them practice opportunities both with and without teacher support. Inquiry-based activities are great for students when they have the prerequisite skills, otherwise we are just setting them up for failure.

In education, our currency is time and you ask any teacher and they’ll tell you, we don’t have enough of it! So, as schools, systems and educators we have a moral imperative to stop leaving our students mathematical success up for chance and teach using the most effective and efficient methods.

## Recommended Resources

- PaTTAN Mathematics: The 2023 Math Conference that featured people such as Dr. Paul Kirschner, Sarah Powell, Dr. Elizabeth Hughes, Dr Brian Poncy, Dr Marcy Stein, Dr. Bradley Witzel, Dr. Paul Riccomini, Dr Amanda Vanderheyden and many more is freely available
- Think Forward Educators Maths Network: I am part of the TFE Maths Committee and there are a number of freely available webinars on offer. Also, we have the amazing Sarah Powell scheduled (7 August) to deliver on What’s important for maths intervention?
- What Works Clearinghouse: Assisting Students Struggling with Mathematics: Intervention in the Elementary Grades
- Neil Almond: Quality First Teaching Checklist: The 10 Most Effective Strategies For Primary Schools

## More info

Currently, there is a lack of professional learning on offer for primary mathematics that is closely aligned with the science of learning. If you would like support in implementing this at your school, feel free to contact me at: brendan@learnwithlee.net

## References

Agodini, R., & Harris, B. (2014). How four elementary math curricula perform among different types of teachers and classrooms. Technical Report.

Anthony, G., & Walshaw, M. (2009). Effective pedagogy in mathematics (Vol. 19). Belley, France: International Academy of Education.

Buckingham, J., & Meeks, L. (2019). Short-changed: Preparation to teach reading in initial teacher education. MultiLit Pty Ltd.: Macquarie Park, Australia.

Chi, M. T., Feltovich, P. J., & Glaser, R. (1981). Categorization and representation of physics problems by experts and novices. Cognitive science, 5(2), 121-152.

Fahey, G., O’Sullivan, J., & Bussell, J. (2021). Failing to teach the teacher: an analysis of mathematics Initial Teacher Education. The Center for Independent Studies. Repéré à: https://www. cis. org. au/wp-content/uploads/2021/11/ap29. Pdf.

Fuchs, L. S., Bucka, N., Clarke, B., Dougherty, B., Jordan, N. C., Karp, K. S., … & Morgan, S. (2021). Assisting Students Struggling with Mathematics: Intervention in the Elementary Grades. Educator’s Practice Guide. WWC 2021006. What Works Clearinghouse.

Geary, D. C. (2011). Cognitive predictors of achievement growth in mathematics: a 5-year longitudinal study. Developmental psychology, 47(6), 1539.

Hughes, E. M., Powell, S. R., & Stevens, E. A. (2016). Supporting clear and concise mathematics language: Instead of that, say this. Teaching Exceptional Children, 49(1), 7-17.

Krasa, N., Tzanetopoulos, K., & Maas, C. (2022). *How Children Learn Math: The Science of Math Learning in Research and Practice*. Routledge.

McCourt, M. (2019). Teaching for mastery. John Catt.

National Research Council, & Mathematics Learning Study Committee. (2001). Adding it up: Helping children learn mathematics. National Academies Press.

Powell, S. R., & Fuchs, L. S. (2018). Effective Word-Problem Instruction: Using Schemas to Facilitate Mathematical Reasoning. TEACHING Exceptional Children, 51(1), 31–42. https://doi.org/10.1177/0040059918777250

Price, G. R., Mazzocco, M. M., & Ansari, D. (2013). Why mental arithmetic counts: brain activation during single digit arithmetic predicts high school math scores. *Journal of Neuroscience*, *33*(1), 156-163.

Rittle-Johnson, B., Schneider, M., & Star, J. R. (2015). Not a one-way street: Bidirectional relations between procedural and conceptual knowledge of mathematics. Educational Psychology Review, 27, 587-597.

Von Aster, M. G., & Shalev, R. S. (2007). Number development and developmental dyscalculia. Developmental medicine & child neurology, 49(11), 868-873.

An interesting read. I work in Maths Education too and advocate for inquiry, engagement and productive struggle but not in the ways described in your blog. I agree with the points made here. Your version of explicit instruction doesn’t match other descriptions of it I have seen just as my approach doesn’t match how you describe inquiry. My observation is that good pedagogy is a mix of strategies and the labels and assumptions that try to describe it are not necessarily helpful.

The key in my opinion is depth of teacher knowledge about maths so the can lead students to understand maths and not just get answers. Judy Hartnett EdD.