When teaching maths… Instead of this, Do this

I’ve worked with a number of schools and teachers over the past few years and I have come across many common challenges and questions. So, I thought it might be useful to put together these infographics that break down what teachers need to stop doing and what to do instead when it comes to teaching maths.

You can check out the infographics here 👇

Download the PDF version here.

Keep reading if you want a brief overview of each section.

Practice

Many schools that are aligning their practice to the science of learning are starting to understand the benefits of providing daily retrieval practice through things like “Daily Reviews”, “Do Nows” and “Warm Ups”. What you call it is less important than how it is actually run. There are some essential ingredients that need to be included to make it have the desired impact and that involves applying the Desirable Difficulties principles (retrieval practice, spaced practice, interleaved practice and generation effects).

When the same activity is done each day (e.g. Number Talks, Maths Groups, Skip Counting), without intervention or extension, students who are able to do it one day, are still able to do it the next and students who can’t do it one day, are still unable to do it the next. So, the desirable difficulties are not achieved.

For building fluency, it needs to be understood that you need to acquire the knowledge before you can build fluency. So, if students are put under time pressure before developing accuracy of facts, then this can lead to feelings of anxiety. Fluency practice needs to be at their instructional level. This can be worked out through Curriculum-Based Assessments (see assessments).

Instead of this	Do this
Thinking that because you’ve taught something, students have learnt it.	Provide daily retrieval practice opportunities with concepts spaced out over time.
Getting students to recall the same concepts everyday.	Apply the principles of desirable difficulties and have students thinking hard when recalling information.
Believing that students don’t need to be able to recall their basic maths facts with automaticity.	Run timed fluency practice sessions every day at their instructional level.
Having whole-class timed fluency practice sessions on the same problem-set.	Ensure students have accuracy with the facts before putting them under time pressure.
Solely focusing on rote memorisation of basic maths facts.	Equip students with mental strategies to use when they cannot automatically recall facts.
Running daily maths groups without any specific learning intentions.	Make sure every practice activity has a clear purpose, like building fluency, maintenance or applying knowledge.
Starting every lesson with number talks.	Get more students thinking by giving the whole class more chances to participate.

New Content

I’ve written previously about the hierarchical nature of mathematics and so this means that if students don’t have the pre-requisite knowledge, then the most effective and efficient way for them to gain it is by the teacher teaching it to them. Applying the gradual release of responsibility enables the teacher to gradually fade away the prompting and scaffolds (see how I put this into action here).

We also want to use formal language instead of informal language because it helps students develop their conceptual understanding, explain their answers and gives them the vocabulary to differentiate between similar concepts.

Instead of this	Do this
Believing that students learn better if they discover things for themselves.	It is more efficient and effective to teach them first because then misconceptions aren’t picked up.
Assuming students know what to do if you’ve shown them.	Go through the gradual release of responsibility and provide multiple opportunities to respond with specific feedback during the guided practice phase.
Feeling that students by a certain age don’t need to be using concrete materials.	Help students understand abstract concepts by using concrete and visual representations and clearly connecting them to the abstract ideas.
Feeling that students who are a certain age need to be using concrete materials.	Fade-away the reliance on manipulatives because it is a slower way to operate. Move through the concrete-pictorial-abstract phases as fast as possible, but as slow as necessary.
Thinking that if students respond with correct answers during guided practice that they understand it.	Ensure that they have independent practice opportunities to demonstrate that they can do it without prompting and scaffolding.
Believing that the academic language in maths is too complex.	Break down the vocabulary using morphology, etymology and examples and non-examples.
Thinking that once a concept has been taught that students can apply it in any setting.	Provide minimally different problems that support students in noticing connections and understanding the concept.

Curriculum

I don’t think there’s much conjecture over what needs to be taught in maths, but there’s more debate around how it’s presented, when different concepts are introduced and what aspects are prioritised. The two key ideas to understand are:

1. Knowledge is domain specific, so each new unit of work needs to be presented in a way that treats the learners as novices.
2. Novices go through the stages of learning described in the Instructional Hierarchy (see here).

Motivation can certainly be increased through connection, but those connections don’t have to be to “real-world” activities. In fact, often we manufacture these tasks to involve real-world things, but end up with totally unrealistic scenarios. e.g. Max had 58 avocadoes that he had to share between 4 friends. How many do they have each? All the students are thinking about is, “Why does Max have 58 avocadoes?”

The “Big Ideas” and “connectionist” approach sounds great, but often it ends up taking the focus away from the main idea e.g. Let’s go look for shapes outside and the students just ends up chasing butterflies.

Instead of this	Do this
Overloading students with big ideas.	Break lessons into smaller, manageable chunks and revisit concepts regularly.
Assuming that the foundational maths concepts are easy to learn.	Assess if the foundational concepts have been mastered, rather than assuming.
Focusing only on getting through the curriculum.	Ensure students truly understand key concepts by providing multiple practice opportunities and that they can do things accurately and independently.
Prioritising conceptual or procedural understanding.	Understand that students need both conceptual and procedural understanding and it is bidirectional.
Thinking that teaching algorithms doesn’t build conceptual understanding.	Know that understanding how to use an algorithm not only builds their conceptual and procedural knowledge, but is generally also the most reliable and efficient way to solve a problem.
Presenting a number of different strategies to tackle a mathematical problem when the concept is new.	Develop fluency in one strategy first, then introduce new strategies. Be intentional with the problems presented, to highlight when particular strategies might be useful.
Treating maths as a series of unrelated topics.	Connect different maths concepts to show how they build on each other.
Believing that every lesson must be entirely grounded in real-world applications.	Let the maths be the fun part and only bring in real-world applications when they are relevant to the students.
Differentiating the curriculum because some students are perceived as being better at maths than others.	Differentiate through the types of practice opportunities and scaffolding that is provided.

Problem-Solving

Again, the debate isn’t about whether or not we want our students to be engaging with problem-solving activities or rich tasks. The issue is when these tasks are presented to students. Some teachers will speak about we need to aim for “productive struggle”, when more often than not, this just leads to destructive struggle.

The main reason is that if we present problems to students when we know they don’t have the required knowledge, then it just leads to them feeling frustrated because they can’t actually visualise what they need to do.

Also, problem-solving doesn’t have to mean complete discovery learning. In fact, it’s extremely helpful to provide students with an Attack Strategy (e.g. UPSCheck) and Schema-Based Instruction. This provides them with a framework to approach word problems with and builds their self-regulations skills.

Instead of this	Do this
Beginning lessons with open-ended tasks before the required knowledge has been gained.	Engage learners by teaching them what they need to know and allowing them opportunities to be successful at it. Use worked examples to reduce cognitive load and accelerate learning.
Only providing problem-solving tasks for fast finishers and as extension activities.	Ensure all students have daily opportunities to tackle word problems rooted in concepts they have already mastered, using a structured problem-solving strategy that has been explicitly taught.
Allowing students just to provide their answer.	Emphasise that mathematics is more than just coming up with the correct answer. Ask why, how and what if questions.
Providing scaffolds and prompts when students have mastered concepts.	Encourage students to apply what they have learnt to novel problem types.
Believing that once fluency has been acquired that students won’t need guidance anymore.	Students might still need to be explicitly taught when to apply certain skills and how to bring isolated skills together.
Linking keywords to specific operations in word problems.	Provide students with an attack strategy or easy-to-remember series of steps they can use to guide their approach to solving word problems. Also, teach them that we can use certain strategies for different types of schemas (problems which have similar types of structures).
Feeling that more learning will happen if problems are solved in small groups.	Only use group work if students have specific roles and responsibilities that will benefit the learning of all students in the group.

Assessments

The main questions to ask are:

• What is the purpose of each assessment?
• Does each assessment provide me with the data that I need in a reliable and valid way?

Instead of this	Do this
Relying on pre and post assessments to tell you if students have learnt something.	Understanding students’ prior knowledge is important, but it’s more effective to use formative assessments to guide your next steps than to test concepts they haven’t learned yet.
Placing students in intervention groups based on standardised assessments.	Use Curriculum-Based Measurements (CBM) as a universal screening tool to quickly assess whether students are performing at grade level.
Relying on national standardised assessments like NAPLAN to tell you whether your school is going well or not.	Understand that often standardised assessments are all word problems. So, they actually assess whether or not students are able to apply their learning.
Relying on externally developed summative assessments.	Design summative assessments that align with what was taught and mirror the problem formats used in class.
Assuming that students will build fluency just through practice.	Use Curriculum-Based Assessments (CBA) to measure fluency with Digits Correct Per Minute (DCPM) and identify specific skill gaps for remediation.

New Online (Live) Maths Course

If you want to learn more about the ideas presented in this post and how to apply them in the classroom, then make sure you sign up to my online (live) course 👇

Just so you know that this isn’t just me deciding what works and doesn’t work, this is my reference list 👇

References

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

Alfieri, L., Brooks, P. J., Aldrich, N. J., & Tenebaum, H. R. (2011). Does discovery-based instruction enhance learning? Journal of Educational Psychology, 103(1), 1-18. https://doi.org/10.1037/a0021017

Archer, A. L., & Hughes, C. A. (2010). Explicit instruction: Effective and efficient teaching. Guilford Publications.

Ashman, G., Kalyuga, S., & Sweller, J. (2020). Problem-solving or explicit instruction: Which should go first when element interactivity is high?. Educational Psychology Review, 32, 229-247.

Axtell, P. K., McCallum, R. S., Mee Bell, S., & Poncy, B. (2009). Developing math automaticity using a classwide fluency building procedure for middle school students: A preliminary study. Psychology in the Schools, 46(6), 526-538.

Ausubel, D. P. (1968). Facilitating meaningful verbal learning in the classroom. The Arithmetic Teacher, 15(2), 126-132.

Barroso, C., Ganley, C. M., McGraw, A. L., Geer, E. A., Hart, S. A., & Daucourt, M. C. (2021). A meta-analysis of the relation between math anxiety and math achievement. Psychological Bulletin, 147(2), 134.

Beilock, S. L., & Willingham, D. T. (2014). Math anxiety: Can teachers help students reduce it? ask the cognitive scientist. American educator, 38(2), 28.

Benson, I., Marriott, N., & McCandliss, B. D. (2022, July). Equational reasoning: A systematic review of the Cuisenaire–Gattegno approach. In Frontiers in Education (Vol. 7, p. 902899). Frontiers.

Berthold, K., Eysink, T. H., & Renkl, A. (2009) “Assisting self-explanation prompts are more effective than open prompts when learning with multiple representations,” Instructional Science, 37 (4), 345–363.

Birnbaum, M. S., Kornell, N., Bjork, E. L., & Bjork, R. A. (2013). Why interleaving enhances inductive learning: The roles of discrimination and retrieval. Memory & cognition, 41, 392-402.

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).

Booth, J. L., Lange, K. E., Koedinger, K. R., & Newton, K. J. (2013). Using example problems to improve student learning in algebra: Differentiating between correct and incorrect examples. Learning and Instruction, 25, 24-34.

Booth, J. L., Cooper, L. A., Donovan, M. S., Huyghe, A., Koedinger, K. R., & Paré-Blagoev, E. J. (2015a). Design-based research within the constraints of practice: AlgebraByExample. Journal of Education for Students Placed at Risk, 20(1–2), 79–100. 

Booth, J. L., Oyer, M. H., Paré-Blagoev, E. J., Elliot, A. J., Barbieri, C., Augustine, A., & Koedinger, K. R. (2015). Learning algebra by example in real-world classrooms. Journal of Research on Educational Effectiveness, 8(4), 530-551.

Booth, J. L., McGinn, K. M., Barbieri, C., Begolli, K. N., Chang, B., Miller-Cotto, D., … & Davenport, J. L. (2017). Evidence for cognitive science principles that impact learning in mathematics. In Acquisition of complex arithmetic skills and higher-order mathematics concepts (pp. 297-325). Academic Press.

Borko, H., & Livingston, C. (1989). Cognition and improvisation: Differences in mathematics instruction by expert and novice teachers. American educational research journal, 26(4), 473-498.

Braithwaite, D. W., & Goldstone, R. L. (2013). Integrating formal and grounded representations in combinatorics learning. Journal of Educational Psychology, 105(3), 666.

Bransford, J., Brophy, S., & Williams, S. (2000). When computer technologies meet the learning sciences: Issues and opportunities. Journal of Applied Developmental Psychology, 21(1), 59-84.

Brophy, J. E. (1991). Conclusion to advances in research on teaching, Vol. II: Teachers’ knowledge of subject matter as it relates to teaching practice. Advances in research on teaching, 2, 347-362.

Bruner, J. S. (1966). Toward a theory of instruction. Harvard University Press.

Burns, M. K. (2002). Comprehensive system of assessment to intervention using curriculum-based assessments. Intervention in School and Clinic, 38(1), 8-13.

Burns, M. K., Codding, R. S., Boice, C. H., & Lukito, G. (2010). Meta-analysis of acquisition and fluency math interventions with instructional and frustration level skills: Evidence for a skill-by-treatment interaction. School Psychology Review, 39(1), 69.

Burns M., Aguilar L. N., Young H., Preast J. L., Taylor C. N., Walsh A. D. (2019). Comparing the effects of incremental rehearsal and traditional drill on retention of mathematics facts and predicting the effects with memory. School Psychology, 34(5), 521–530. https://doi.org/10.1037/spq0000312

Bursal, M., & Paznokas, L. (2006). Mathematics anxiety and preservice elementary teachers’ confidence to teach mathematics and science. School science and mathematics, 106(4), 173-180.

Byun, S. Y., Irvin, M. J., & Bell, B. A. (2015). Advanced math course taking: Effects on math achievement and college enrollment. The Journal of Experimental Education, 83(4), 439-468.

Carnine, D. W. (1976). Effects of two teacher‐presentation rates on off‐task behavior, answering correctly, and participation 1. Journal of Applied Behavior Analysis, 9(2), 199-206.

Carnine, D., Silbert, J., Kameenui, E. J., & Tarver, S. G. (1997). Direct instruction reading. Columbus, OH: Merrill.

Carroll, W. M. (1994). Using worked examples as an instructional support in the algebra classroom. Journal of Educational Psychology, 86(3), 360–367. 

Caviola, S., Toffalini, E., Giofrè, D., Ruiz, J. M., Szűcs, D., & Mammarella, I. C. (2021). Math performance and academic anxiety forms, from sociodemographic to cognitive aspects: A meta-analysis on 906,311 participants. Educational Psychology Review, 1-37.

Chi, M. T. H., de Leeuw, NK, Chiu, M.-H., & LaVancher, C. (1994). Eliciting self-explanations improves understanding. Cognitive Science, 18, 439-477.

Clarke, B., Doabler, C. T., Turtura, J., Smolkowski, K., Kosty, D. B., Sutherland, M., … & Baker, S. K. (2020). Examining the efficacy of a kindergarten mathematics intervention by group size and initial skill: Implications for practice and policy. The Elementary School Journal, 121(1), 125-153.

Codding, R. S., Volpe, R. J., & Poncy, B. C. (2016). Effective math interventions: A guide to improving whole-number knowledge. Guilford Publications.

Coe, R., Rauch, C. J., Kime, S., & Singleton, D. (2020). Great teaching toolkit: evidence review.

Crowe, A. R. (2010). “What’s math got to do with it?”: Numeracy and social studies education. The Social Studies, 101(3), 105-110.

Davis‐Kean, P. E., Domina, T., Kuhfeld, M., Ellis, A., & Gershoff, E. T. (2022). It matters how you start: Early numeracy mastery predicts high school math course‐taking and college attendance. Infant and Child Development, 31(2), e2281.

DeFouw, E. R., Collier-Meek, M. A., Daniels, B., Codding, R. S., & Veiga, M. (2021). Does intervention session length matter? A comparison of a math intervention across three durations. Journal of Behavioral Education, 1-27.

Dehaene, S. (2011). The number sense: How the mind creates mathematics. OUP USA.

Doabler, C. T., Baker, S. K., Kosty, D. B., Smolkowski, K., Clarke, B., Miller, S. J., & Fien, H. (2015). Examining the association between explicit mathematics instruction and student mathematics achievement. The Elementary School Journal, 115(3), 303-333.

Duhon, G. J., Poncy, B. C., Krawiec, C. F., Davis, R. E., Ellis-Hervey, N., & Skinner, C. H. (2022). Toward a more comprehensive evaluation of interventions: a dose-response curve analysis of an explicit timing intervention. School psychology review, 51(1), 84-94.

Dyson, N. I., Jordan, N. C., & Glutting, J. (2013). A number sense intervention for low-income kindergartners at risk for mathematics difficulties. Journal of learning disabilities, 46(2), 166-181.

Finn, J. D., & Zimmer, K. S. (2012). Student engagement: What is it? Why does it matter?. In Handbook of research on student engagement (pp. 97-131). Boston, MA: Springer US.

Forbringer, L., & Weber, W. (2021). RtI in math: Evidence-based interventions. Routledge.

Frye, D., Baroody, A. J., Burchinal, M., Carver, S. M., Jordan, N. C., & McDowell, J. (2013). Teaching math to young children: A practice guide (NCEE 2014-4005).

Fuchs, L. S., Fuchs, D., Compton, D. L., Powell, S. R., Seethaler, P. M., Capizzi, A. M., … & Fletcher, J. M. (2006). The cognitive correlates of third-grade skill in arithmetic, algorithmic computation, and arithmetic word problems. Journal of Educational Psychology, 98(1), 29.

Fuchs, L. S., Bucka, N., Clarke, B., Dougherty, B., Jordan, N. C., & Karp, K. S. (2021). Assisting students struggling with mathematics: Intervention in the elementary grades. Educator’s practice guide (WWC 2021006). What Works Clearinghouse.

Fyfe, E. R., McNeil, N. M., & Borjas, S. (2015). Benefits of “concreteness fading” for children’s mathematics understanding. Learning and Instruction, 35, 104-120.

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

Gersten, R., Jordan, N. C., & Flojo, J. R. (2005). Early identification and interventions for students with mathematics difficulties. Journal of learning disabilities, 38(4), 293-304.

Gersten, R., Beckmann, S., Clarke, B., Foegen, A., Marsh, L., Star, J. R., & Witzel, B. (2009). Assisting students struggling with mathematics: Response to intervention (RtI) for elementary and middle schools. IES National Center for Education Evaluation Practice Guide.

Gickling, E. E., Shane, R. L., & Croskery, K. M. (1989). Developing mathematics skills in low-achieving high school students through curriculum-based assessment. School Psychology Review, 18(3), 344-355.

Gunderson, E.A., Ramirez, G., Beilock, S.L., & Levine, S.C. (2012). The relation between spatial skill and early number knowledge: The role of the linear number line. Developmental Psychology, 48, 1229–1241.

Haring, N. G., & Eaton, M. D. (1978). Systematic instructional procedures: An instructional hierarchy. The fourth R: Research in the classroom, 23-40.

Hemmings, B., Grootenboer, P., & Kay, R. (2011). Predicting mathematics achievement: The influence of prior achievement and attitudes. International Journal of Science and Mathematics Education, 9, 691-705.

Hickendorff, M., Torbeyns J. & Verschaffel L. (2019), Multi-digit Addition, Subtraction, Multiplication, and Division Strategies. In: Fritz A., Haase V.G., Räsänen P. (Eds.) International Handbook of Mathematical Learning Difficulties. Switzerland: Springer. 543-560. Doi: 10.1007/978-3-319-97148-3_32

Hill, H. C., Rowan, B., & Ball, D. L. (2005). Effects of teachers’ mathematical knowledge for teaching on student achievement. American educational research journal, 42(2), 371-406.

Hintze, J. M., Christ, T. J., & Methe, S. A. (2006). Curriculum‐based assessment. Psychology in the Schools, 43(1), 45-56.

Hughes, C. A., Morris, J. R., Therrien, W. J., & Benson, S. K. (2017). Explicit instruction: Historical and contemporary contexts. Learning Disabilities Research & Practice, 32(3), 140-148.

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.

Hurrell, D. (2021). Conceptual knowledge or procedural knowledge or conceptual knowledge and procedural knowledge: Why the conjunction is important to teachers. Australian Journal of Teacher Education (Online), 46(2), 57-71.

Hwang, J., Riccomini, P. J., Hwang, S. Y., & Morano, S. (2019). A systematic analysis of experimental studies targeting fractions for students with mathematics difficulties. Learning Disabilities Research & Practice, 34(1), 47–61. https://doi.org/10.1111/ldrp.12187

Hwang, S., & Son, T. (2021). Students’ Attitude toward Mathematics and Its Relationship with Mathematics Achievement. Journal of Education and e-Learning Research, 8(3), 272-280.

Jitendra, A. K., Griffin, C. C., Haria, P., Leh, J., Adams, A., & Kaduvettoor, A. (2007). A comparison of single and multiple strategy instruction on third-grade students’ mathematical problem solving. Journal of Educational Psychology, 99(1), 115–127. https://doi.org/10.1037/0022-0663.99.1.115

Jordan, N. C., Hanich, L. B., & Kaplan, D. (2003). A longitudinal study of mathematical competencies in children with specific mathematics difficulties versus children with comorbid mathematics and reading difficulties. Child development, 74(3), 834-850.

Jordan, N. C., Kaplan, D., Ramineni, C., & Locuniak, M. N. (2009). Early math matters: kindergarten number competence and later mathematics outcomes. Developmental psychology, 45(3), 850.

Jordan, N. C., Devlin, B. L., & Botello, M. (2022). Core foundations of early mathematics: refining the number sense framework. Current Opinion in Behavioral Sciences, 46, 101181.

Kabel, M., Hwang, J., & Hwang, J. (2021). Lessons learned from a rural classroom study: Transitioning from concrete to virtual manipulatives to teach math fact fluency to students with learning disabilities. Journal of Curriculum Studies Research, 3(1), 42–68. https://doi.org/10.46303/jcsr.2021.7

Kaminski, J. A., & Sloutsky, V. M. (2013). Extraneous perceptual information interferes with children’s acquisition of mathematical knowledge. Journal of Educational Psychology, 105(2), 351.

Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding it up: Helping children learn mathematics (Vol. 2101). National research council (Ed.). Washington, DC: National Academy Press.

Kilpatrick, J. (2014). History of research in mathematics education. In S. Lerman (Ed.), Encyclopedia of mathematics education. Dordrecht: Springer. https://doi.org/10.1007/978-94-007-4978-8_71

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.

Kiss, A. J., & Christ, T. J. (2019). Screening for Math in Early Grades: Is Reading Enough? Assessment for Effective Intervention, 45(1), 38-50. https://doi.org/10.1177/1534508418766410

Koon, S., & Davis, M. (2019). Math Course Sequences in Grades 6-11 and Math Achievement in Mississippi. REL 2019-007. Regional Educational Laboratory Southeast.

Koscinski S. T., Gast D. L. (1993). Use of constant time delay in teaching multiplication facts to students with learning disabilities. Journal of Learning Disabilities, 26, 533–544. https://doi.org/10.1177/002221949302600807

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

Kyriakides, L., & Creemers, B. P. (2008). Using a multidimensional approach to measure the impact of classroom-level factors upon student achievement: A study testing the validity of the dynamic model. School effectiveness and school improvement, 19(2), 183-205.

Lahme, B., McLeman, C., Nakamaye, M., & Umland, K. (2019). The number line: Unifying the evolving definition of number in K–12 mathematics. Notices of the American Mathematical Society, 66(9), 1465-1470.

Lee, J. S., & Ginsburg, H. P. (2009). Early childhood teachers’ misconceptions about mathematics education for young children in the United States. Australasian Journal of Early Childhood, 34(4), 37-45.

LeFevre, J. A., DeStefano, D., Coleman, B., & Shanahan, T. (2005). Mathematical cognition and working memory. Handbook of mathematical cognition, 361-377.

Marton, F., & Pang, M. F. (2013). Meanings Are Acquired from Experiencing Differences against a Background of Sameness, Rather than from Experiencing Sameness against a Background of Difference: Putting a Conjecture to the Test by Embedding It in a Pedagogical Tool. Frontline learning research, 1(1), 24-41.

Mason, E. N., Benz, S. A., Lembke, E. S., Burns, M. K., & Powell, S. R. (2019). From professional development to implementation: A district’s experience implementing mathematics tiered systems of support. Learning Disabilities Research & Practice, 34(4), 207-214.

Mayer, R. E. (2004). Should there be a three-strikes rule against pure discovery learning?. American psychologist, 59(1), 14.

Merlo, S. (2024). The Science of Maths and How to Apply It. CIS Analysis Paper 71 https://www.cis.org.au/publication/the-science-of-mathematics-and-how-to-apply-it/

Morano, S., Randolph, K., Markelz, A. M., & Church, N. (2020). Combining Explicit Strategy Instruction and Mastery Practice to Build Arithmetic Fact Fluency. TEACHING Exceptional Children, 53(1), 60-69. https://doi.org/10.1177/0040059920906455

Muijs, D., Kyriakides, L., Van der Werf, G., Creemers, B., Timperley, H., & Earl, L. (2014). State of the art–teacher effectiveness and professional learning. School effectiveness and school improvement, 25(2), 231-256.

National Mathematics Advisory Panel (NMAP). Foundations for Success: The Final Report of the National Mathematics Advisory Panel, U.S. Department of Education: Washington, DC, 2008

Nelson, G., & McMaster, K. L. (2019). The effects of early numeracy interventions for students in preschool and early elementary: A meta-analysis. Journal of Educational Psychology, 111(6), 1001.

Niemivirta, M., Tapola, A., Tuominen, H., & Viljaranta, J. (2022). Developmental trajectories of school‐beginners’ ability self‐concept, intrinsic value and performance in mathematics. British Journal of Educational Psychology.

Norton, S. (2012). The use of alternative algorithms in whole number computation. Interventional Journal for Mathematics Teaching and Learning, 2-16.

Nguyen, T., Watts, T. W., Duncan, G. J., Clements, D. H., Sarama, J. S., Wolfe, C., & Spitler, M. E. (2016). Which preschool mathematics competencies are most predictive of fifth grade achievement?. Early childhood research quarterly, 36, 550-560.

Núñez, R. E. (2017). Is there really an evolved capacity for number?. Trends in cognitive sciences, 21(6), 409-424.

Ok M. W., Bryant D. P. (2016). Effects of a strategic intervention with iPad practice on the multiplication fact performance of fifth-grade students with learning disabilities. Learning Disability Quarterly, 39(3), 146–158. https://doi.org/10.1177/0731948715598285

Peltier, C., & Vannest, K. J. (2017). A meta-analysis of schema instruction on the problem-solving performance of elementary school students. Review of Educational Research, 87(5), 899-920.

Peltier, C., Sinclair, T. E., Pulos, J. M., & Suk, A. (2020). Effects of schema-based instruction on immediate, generalized, and combined structured word problems. The Journal of Special Education, 54(2), 101-112.

Pesek, D. D., & Kirshner, D. (2000). Interference of instrumental instruction in subsequent relational learning. Journal for Research in Mathematics Education, 31(5), 524-540.

Polya, G. (1945). How to solve it; a new aspect of mathematical method. Princeton University Press.

Powell, S. R., Stevens, E. A., & Hughes, E. M. (2019). Math language in middle school: Be more specific. TEACHING Exceptional Children, 51(4), 286-295.

Powell, S. R., Bos, S. E, & Lin, X. (2021). The assessment of mathematics vocabulary in the elementary and middle school grades. In A. Fritz-Stratmann, E., Gürsoy, & M. Herzog (Eds.), Diversity dimensions in mathematics and language learning (pp. 313–330). De Grutyer. https://library.oapen.org/handle/20.500.12657/50212

Powell, S. R., Namkung, J. M., & Lin, X. (2022). An investigation of using keywords to solve word problems. The Elementary School Journal, 122(3), 452–473. https://doi.org/10.1086/717888

Powell, S., Hughes, E. & Peltier, C. (2022) Myths That Undermine Maths Teaching. Analysis Paper 38, Centre for Independent Studies

Powell, S., King, S. & Benz, S. (2024)  Maths Practices You Can Count On: Five Research-Validated Practices in Mathematics. Analysis Paper 62, Centre for Independent Studies

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.

Rakoczy, K., Pinger, P., Hochweber, J., Klieme, E., Schütze, B., & Besser, M. (2019). Formative assessment in mathematics: Mediated by feedback’s perceived usefulness and students’ self-efficacy. Learning and Instruction, 60, 154-165.

Rea, C. P., & Modigliani, V. (1985). The effect of expanded versus massed practice on the retention of multiplication facts and spelling lists. Human Learning: Journal of Practical Research & Applications.

Renkl, A., & Eitel, A. (2019). Self-explaining: Learning about principles and their application.

Riccomini P. J., Stocker J. D., Morano S. (2017). Implementing an effective mathematics fact fluency practice activity. TEACHING Exceptional Children, 49(5), 318–327. https://doi.org/10.1177/0040059916685053

Ritchie, S. J., & Bates, T. C. (2013). Enduring links from childhood mathematics and reading achievement to adult socioeconomic status. Psychological science, 24(7), 1301-1308.

Rittle-Johnson, B., & Star, J. R. (2009). Compared with what? The effects of different comparisons on conceptual knowledge and procedural flexibility for equation solving. Journal of Educational Psychology, 101(3), 529.

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.

Robinson, C. S., Menchetti, B. M., & Torgesen, J. K. (2002). Toward a two‐factor theory of one type of mathematics disabilities. Learning Disabilities Research & Practice, 17(2), 81-89.

Rohrer, D. (2012). Interleaving helps students distinguish among similar concepts. Educational Psychology Review, 24, 355-367.

Rohrer, D., Dedrick, R. F., & Stershic, S. (2015). Interleaved practice improves mathematics learning. Journal of Educational Psychology, 107(3), 900.

Rolf, K. R. (2022). Examining the Effectiveness of Explicit, Systematic Mathematics Vocabulary Instruction for Students with Learning Difficulties and Disabilities in a Specialized Setting.

Root, J., Saunders, A., Jimenez, B., & Gilley, D. (2023). Essential components for math instruction Considerations for students with extensive support needs. TEACHING Exceptional Children, 56(1), 34-43.

Sarnecka, B. W., & Carey, S. (2008). How counting represents number: What children must learn and when they learn it. Cognition, 108(3), 662-674.

Schmidt, W. H., & Houang, R. T. (2007). Lack of focus in the mathematics curriculum: Symptom or cause. Lessons learned: What international assessments tell us about math achievement, 65-84.

Schoenfeld, A. H. (2002). Making mathematics work for all children: Issues of standards, testing, and equity. Educational researcher, 31(1), 13-25.

Schutte, G. M., Duhon, G. J., Solomon, B. G., Poncy, B. C., Moore, K., & Story, B. (2015). A comparative analysis of massed vs. distributed practice on basic math fact fluency growth rates. Journal of School Psychology, 53(2), 149-159.

Schwonke, R., Renkl, A., Salden, R., & Aleven, V. (2011). Effects of different ratios of worked solution steps and problem solving opportunities on cognitive load and learning outcomes. Computers in Human Behavior, 27(1), 58-62.

Shrager, J., & Siegler, R. S. (1998). SCADS: A model of children’s strategy choices and strategy discoveries. Psychological science, 9(5), 405-410.

Siegler, R., Carpenter, T., Fennell, F., Geary, D., Lewis, J., Okamoto, Y., … & Wray, J. (2010). Developing Effective Fractions Instruction for Kindergarten through 8th Grade. IES Practice Guide. NCEE 2010-4039. What Works Clearinghouse.

Siegler, R. S., Duncan, G. J., Davis-Kean, P. E., Duckworth, K., Claessens, A., Engel, M., … & Chen, M. (2012). Early predictors of high school mathematics achievement. Psychological science, 23(7), 691-697.

Simonsen, B., Fairbanks, S., Briesch, A., Myers, D., & Sugai, G. (2008). Evidence-based practices in classroom management: Considerations for research to practice. Education and treatment of children, 351-380.

Singer-Dudek, J., & Greer, R. D. (2005). A long-term analysis of the relationship between fluency and the training and maintenance of complex math skills. The Psychological Record, 55, 361-376.

Skinner, M. G., & Cuevas, J. A. (2023). The effects of schema-based instruction on wordproblems in a third-grade mathematics classroom. International Journal of Instruction, 16(1), 855880. https://doi.org/10.29333/iji.2023.16148a 

Stannard, L., Wolfgang, C.H., Jones, I., & Phelps, P. (2001). A longitudinal study of the predicted relations among construction play and mathematical achievement. Early Childhood Development and Care, 167, 115–125.

Stevens, E. A., Leroux, A. J., Mowbray, M. H., & Lee, G. S. (2022). Evaluating the effects of adding explicit vocabulary instruction to a word-problem schema intervention. Exceptional Children, 89(3), 275–293. https://doi.org/10.1177/00144029221112290

Stickney, E. M., Sharp, L. B., & Kenyon, A. S. (2012). Technology-enhanced assessment of math fact automaticity: Patterns of performance for low-and typically achieving students. Assessment for Effective Intervention, 37(2), 84-94.

Sutherland, K. S., Alder, N., & Gunter, P. L. (2003). The effect of varying rates of opportunities to respond to academic requests on the classroom behavior of students with EBD. Journal of Emotional and behavioral Disorders, 11(4), 239-248.

Sweller, J. (1994). Cognitive load theory, learning difficulty, and instructional design. Learning and instruction, 4(4), 295-312.

Sweller, J., & Cooper, G. A. (1985). The use of worked examples as a substitute for problem solving in learning algebra. Cognition and instruction, 2(1), 59-89.

Taylor, K., & Rohrer, D. (2010). The effects of interleaved practice. Applied cognitive psychology, 24(6), 837-848.

Thompson, C. A. (2013). Poor quantitative skills of newly insured may affect ability to manage medications. American Journal of Health SystemPharmacy, 70, 1464-1475

Tricot, A., & Sweller, J. (2014). Domain-specific knowledge and why teaching generic skills does not work. Educational Psychology Review, 26(2), 265–283. https://doi.org/10.1007/s10648-013-9243-1

Uttal, D. H., Amaya, M., del Rosario Maita, M., Hand, L. L., Cohen, C. A., O’Doherty, K., & DeLoache, J. S. (2013). It works both ways: Transfer difficulties between manipulatives and written subtraction solutions. Child Development Research, 2013.

VanDerHeyden, A. M., & Burns, M. K. (2005). Using curriculum-based assessment and curriculum-based measurement to guide elementary mathematics instruction: Effect on individual and group accountability scores. Assessment for Effective Intervention, 30(3), 15-31.

VanDerHeyden, A. M., & Codding, R. S. (2020). Belief-Based versus Evidence-Based Math Assessment and Instruction. Communique, 48(5).

VanDerHeyden, A., McLaughlin, T., Algina, J., & Snyder, P. (2012). Randomized evaluation of a supplemental grade-wide mathematics intervention. American Educational Research Journal, 49(6), 1251-1284.

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

Van Gog, T., Kester, L., & Paas, F. (2011). Effects of worked examples, example-problem, and problem-example pairs on novices’ learning. Contemporary Educational Psychology, 36(3), 212-218.

Wai, J., Lubinski, D., & Benbow, C.P. (2009). Spatial ability for STEM domains: Aligning over 50 years of cumulative psychological knowledge solidifes its importance. Journal of Educational Psychology, 101, 817–835.

Watts, T. W., Duncan, G. J., Siegler, R. S., & Davis-Kean, P. E. (2014). What’s past is prologue: Relations between early mathematics knowledge and high school achievement. Educational Researcher, 43(7), 352-360.

Wilkins, J. L. (2008). The relationship among elementary teachers’ content knowledge, attitudes, beliefs, and practices. Journal of Mathematics Teacher Education, 11, 139-164.

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.

Witzel, B., & Clarke, B. (2015). Focus on Inclusive Education: Benefits of Using a Multi-tiered System of Supports to Improve Inclusive Practices: Bradley Witzel, Editor. Childhood Education, 91(3), 215-219.

Woodward, J., Beckmann, S., Driscoll, M., Franke, M., Herzig, P., Jitendra, A., … & Ogbuehi, P. (2012). Improving Mathematical Problem Solving in Grades 4 through 8. IES Practice Guide. NCEE 2012-4055. What Works Clearinghouse.

You, S., & Nguyen, J. (2012). Multilevel analysis of student pathways to higher education. Educational psychology, 32(7), 860-882.