1. Using procedures in a flexible manner e.g. knows to switch the addends
2. Can use benchmark numbers – 10, 100
3. Understands the magnitude of numbers e.g which is bigger/ smaller

Math facts make up only a small part of maths. Boaler (2015) emphasises that students need to learn to calculate through number sense and spend more time on problem solving and reasoning. Students need to learn reasoning and not just a set of rules and procedures which are disconnected from meaning making.

This concept of number sense is not spontaneously acquired by all students in the early years of schooling and like phonemic awareness requires explicit teaching. Gersten & Chard (1999) describe number sense as the mathematical equivalent of phonemic awareness to literacy development and like phonemic awareness it requires teaching for many students and cannot be relied upon to be learnt by immersion or osmosis.

Students need to know that 4 is made up of 1+1+1+1+1 or 2+2 or 3+1 or even 2+1+1 (Butterworth, 2019). The research (Gersten & Chard, 1999) shows that by teaching number sense many students with learning disabilities would benefit.

Number sense is critically important to students mathematical development (Boaler, 2015) and is the “foundation for all higher level mathematics” (Feikes & Schwingendorf, 2008) in (Boaler, 2015). The development of number sense is inhibited by the overemphasis on memorisation of math facts, the more we emphasise memorisation the less students think about numbers, their relationships and use number sense. Gersten & Chard, (1999) used three representational systems: conventional mathematical symbols, thermometer (vertical number line) and a horizontal number tracks and their findings suggested that by providing the students with a deep understanding of number sense, they will reduce the procedural deficits in later primary grades as they will have a greater understanding of the procedure.

Developing spatial sense and reasoning is an important factor in developing number sense (Matney, Porcella, & Gladieux, 2020). Shaw (1990) states that “the development of spatial sense is vital for learners of all ages”. The Quick Blocks program (Matney, Porcella, & Gladieux, 2020) develops students spatial sense by briefly revealing images of 3D cubes and then students are given time to construct the images with manipulative blocks. The students look, build, look again and then modify. This is a further development of the skill of subitizing which is also a key component of developing number sense skills.

A key factor in accurate and fluent arithmetic is the memory retrieval of basic number facts. Prior to this, students need to develop counting and counting strategies as a foundation during the preschool years (Lambert & Spinath, 2014). Fuchs, et al., (2008) identified that students with mathematical difficulties have often failed to make shift to memory-based retrieval for basic number facts. In their study, which used drill and practice to secure basic number facts, they taught students if they “just know” to say the answer and if not immediately fall back on counting strategies. We need to move students from using counting skills to using additive strategies and beyond.

Dyscalculia

The DSM V defines Dyscalculia (also known as Developmental Dyscalculia to differentiate it from acquired Dyscalculia) as a specific learning disorder with an impediment in mathematics: number sense, memorisation of arithmetic facts, accurate and fluent calculation plus accurate math reasoning (American Psychiatric Association, 2013).

There is some evidence that number sense is moderately hereditary and is often co morbid with other learning difficulties e.g. ADHD, Dyslexia (Butterworth, 2019). The current research suggests there are 24 males to 1 female with Dyscalculia; however, the research is limited.

Research relating to students with Developmental Dyscalculia show that they have a deficit in the development of their Approximate Number System (ANS) (Piazza, Faocoetti, Trussardi, Berletti, & Conte, 2010) (Budgen & Ansari, 2016).

Typically developing children develop a mental counting line around 6 years of age (Case, et al., 1996) and the mental number line appears to be a critical big ideas in solving addition and subtraction problems in year 1 (Gersten & Chard, 1999).

Developing the ANS is an important component of intervention for these students as Piazza, Faocoetti, Trussardi, Berletti, & Conte (2010) showed “a clear association between dyscalculia and impaired "number sense”. (Butterworth, 2019) also recognises a core deficit in the “number module” in the brain.

Accurate assessment of young learners is challenging, especially to enable a correct diagnosis of a learning difficulty. Sadly, many schools operate a “wait to fail” approach”, whereas there is extensive research backing the use of early intervention (Butterworth, 2019).

The research for teaching literacy, is to teach to the child’s needs and this will not ‘harm’ the child but may reduce or remove a diagnosis of a learning difficulty later (Denton, 2012). The same principle is applied in the Mathtastic program relating to the teaching of number.

Early Intervention

A google search of “evidence based literacy interventions programs primary” gave a result of 11,700,000 (15/12/2021) whereas the same search, when you replace literacy with numeracy, only shows 7,410,000 results (15/12/2021).

This is the reality in a Primary School. There are many Literacy interventions in place but many fewer, if any, numeracy programs.

A few years ago I was at a local network meeting of Learning Support Teachers and we were discussing the intervention programs we each had in place for our primary school students. I asked the question “what about maths?” and no one had anything in place. Gersten & Chard (1999) describe that “mathematics has always been an afterthought” after reading intervention if at all. Mathtastic aims to provide a solution to this gap in early number intervention.

The research is very clear that early intervention regardless of the student’s difficulty is vital to their long-term success and this is shown by the resources dedicated to early intervention by the education and health services. (Hanson & Bruder, 2001) (Ramey & Ramey, 1998). Dowker (2009) pleasingly found that “arithmetic differences are highly susceptible to intervention”.

Math Anxiety

The incidence of maths anxiety is rising as students are expected to memorise math facts through timed testing (Boaler, 2015). This is a limited to way to learn number facts and potentially damaging to students success and engagement in maths.

Students who can demonstrate number sense and can use numbers flexibly show a lot more understanding of maths than those who can just recite rote learnt facts. For example a student may know 7x8=56 by rote but can they use that knowledge to solve 7x7 or 7x9?.

The use of speed pressure, timed testing and blind memorisation is damaging to students and increases anxiety (Boaler, 2015) and is not a component of the Mathtastic program.

Studies have shown that timed responses force students to not rely on counting, however, without this aid the students just didn’t answer the questions (Gersten & Chard, 1999) which again shows the lack of effectiveness of this method to improve student fact fluency. Butterworth (2019) found that students with Dyscalculia were twice as likely to have math anxiety than their neurotypical peers.

Theory behind Mathtastic

Mathtastic has been developed from a range of different research components brought together to ensure students gain the conceptual understanding along with computational. One cannot truly happen without the other (Griffin S. 2004).

The program has been designed to be taught by a teacher in a 1:1 or small group setting for students who have a specific difficulty in number.

During the early years of schooling the focus for students with a learning disability in Literacy is to remediate and teach the skills. Later, the student can be introduced to compensatory measures such as speech to text, immersive reader etc. I believe the same is true for learning maths. We need to intervene at two levels: tools of intervention and tools of access e.g. use a calculator (Dowker, 2009). Some of these access tools need to happen after the student has developed number sense as far as they are able and to achieve a balance between curriculum demands and mathematical skills.

Moving beyond counting in ones is critical, students need to develop additive and multiplicative reasoning.
The National Research Council (National Research Council, 2001) identified 5 key intertwined components of mathematical proficiency. This is the Mathematical Equivalent of Scarboroughs’ Rope (Scarborough, 2001).

For many people they see success in maths as procedural fluency but this is only one component of understanding in maths.

Diagnostic Assessment

The Number Knowledge Test and Placement Tests Overview, (Griffin & Case, 1970) identified a central conceptual structure theory for number.

This is essential as it enables children to make sense of a range of qualitative problems across contexts and also supports the learning of more complex number systems.

The Running Records for maths by Nikki Newton are also a great place to start.

Maths Running Records - Dr Nikki Newton

Progression Model

Mathtastic provides sustained and increasingly spaced opportunities for learning, revision and consolidation of concepts and procedures (Fuson, Kalchman, & Bransford, 2005).

Butterworth (2019) recommends an intensive and cyclical approach to any intervention program.

The gradual release of responsibility model (Pearson & Gallagher, 1983) also advises a spiral review process (Fisher & Frey, 2013).

Learning trajectories

The Learning Trajectories framework is a long established evidence based progression of mathematical skills which is supported by their website Learning Trajectories (Clements & Sarama, 2022).

Mathtastic also draws from this developmental framework.

The development of Mathematical Reasoning

Mathtastic focuses on developing counting strategies and additive thinking. Later modules will develop multiplicative reasoning.

For many students they benefit from additional “wait time” and then after 10-15 seconds for the teacher to ask if they need help or more time. Often, they just need more time. This is a strategy advocated by (Kline, 2008) and the teacher needs to be patient and provide the thinking time.

Cognitively Guided Instruction

There are recognised stages of developing number skills which develop as students are exposed to math problems.

1. Count all
2. Count on
3. Use known facts
4. Use derived facts
5. Use automatic facts

Mathastic provides opportunities to use these skills plus to encourage students to move past these and use higher order number skills.

Lesson Components

Outline

Mathtastic seeks to teach students number sense through concrete, pictorial and abstract representations and encourage students to work flexibly with numbers while providing them with the explicit teaching required.

Boaler (2015) emphasises that students need to learning calculating through number sense and spend more time on problem solving and reasoning.

Mathtastic will engage students in activities which promote the understanding of number sense and mathematical reasoning (Gersten & Chard, 1999).

Teachers will be able to choose a level of number sense depending on zone of proximal development (Vygotsky 1978).

Flexibility is also built in to extend or revisit concepts.
The program also uses the research on retrieval practice to revisit previously taught skills and concepts in a systematic way (Agarwal & Bain, 2019)

Program Components

Subitizing

Gelman & Gallistel (1978) identified principles that are required to be able to count
• One to one principle - using one to one correspondence,
• Cardinal principle - students know the last number is how many there are,
• Abstractness – students can count anything,
• Order irrelevance – students know what is and isn’t included in the count
• Stable order principle – students know the number order remains the same.

In the early stages of number development, a number track should be used as this maps developmentally to counting objects, concrete and cardinal counting skills. The number line should be used later once students have established the above skills.

Counting

8 number sense strategies

There are 8 recognised number sense strategies, and these are all addressed in Mathtastic on a spiral curriculum, each time revisiting the same number sense strategy and applying to increasingly bigger numbers.

This is the basis of maths like phonemic awareness is the basis of literacy (Gersten & Chard, 1999).

1. Add 0,1,2, 3, Subtract 0,1,2, 3
2. Add from largest number by counting on, Subtract by counting back
3. Rainbow facts
4. Adding tens, subtract tens
5. Doubles/ halving
6. Near doubles
7. Partitioning numbers by place value
8. Adding and subtracting by compensating, Bridge 10 with a 9, Bridge 10 with 7 or 8, Round and adjust

Concrete, Pictorial, Abstract (CPA)

When learning maths is important for students to be taught the concepts from the concrete, pictorial and abstract levels of understanding.

For many students the concrete is removed too early and students are moved onto the abstract too soon reference.

The CPA model should be used to support all levels of teaching maths and should not be seen as a progression but more as a wheel where we can go between the different representations to achieve deep understanding.

Students work through activities to develop the 8 number sense strategies, each time with an increasing number range. Students are encouraged to represent their Maths in the 5 ways shown on the Conceptual Understanding Pentagon:

• Model
• Words
• Pictures
• Equations
• Contexts

Number Talks

Number talks were developed by Ruth Parker and Kathy Richardson in the 1990’s to develop number sense and maths facts simultaneously. The strategy involves posing an abstract math problem and asking he students to solve mentally.

In a classroom, the teacher then collects all the different ways the problem has been solved and shares these with the class, discussing the different methods.

In Mathtastic this method is drawn upon by asking students to solve a problem in more than 1 way.

In this routine, students are engaged in using language to explain their thinking and are encouraged to self-explain (Rittle-Johnson & Schneider, 2015 ).

However, for some students in the early stages of this will benefit from “teacher think alouds” (Fisher & Frey, Gradual Release oif Responsibilty Instructional Framework, 2013) where the teacher models their mathematical thinking. This technique develops the students metacognition in relation to maths.

Problem Types

There are 11 different problem types for addition and subtraction.

My own research showed that in the average student problem solving tasks only join – result unknown and separate – result unknown were commonly used in tasks.

The 11 problem types ensure students truly understand the problem and how to solve it thinking mathematically.

Games

Games are used throughout to practice skills and concepts, have fun and develop mathematical reasoning while reducing any anxiety associated with math tasks.

Conclusion

Mathtastic provides an evidence based intervention for students whose mathematical knowledge is in the early years of Primary school and develops key skills and understanding which will create the building blocks for more complex mathematical tasks.

References

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Mathtastic Research

Introduction

What is maths all about?

Number Sense