Learn accessible standard algorithms that students can understand, carry out easily, and bring to fluency. Students initially make drawings that facilitate understanding and explaining.

A standard algorithm uses single-digit operations and concepts of place value (Fuson & Beckmann, 2012/2013; The Common Core Writing Team, 6 March 2015). We clarify in this paper that there are several standard algorithms for each kind of multidigit computation and summarize criteria for choosing better algorithms to teach. We discuss the better standard algorithms for multidigit and decimal subtraction, multiplication, and division that we have found to be accessible to students and to parents in our years of work across many schools across the United States. By accessible we mean that students can understand, explain, and carry out accurately the written methods. We teach students to make math drawings that facilitate their understanding by relating each step in the drawing to each step in the written accessible standard algorithm.

The power of Number Talks and extensions that can build to an equitable Math Talk Classroom

Litkowski et al. compare preschoolers’ performance on three counting items to various standards. We clarify that the items Litkowski and colleagues found to be too easy for kindergarten were actually goals for 4s/PKs in the National Research Council’s report Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity but that they were included as kindergarten standards to ensure that all children had an opportunity to learn these crucial competencies. The helpful analysis in their article of the variability across present state early childhood standards indicates that the kindergarten Common Core State Standards–Mathematics need to remain unchanged for the same reason. We suggest that research funding in early childhood is better spent on research on high-quality instructional contexts for all children than on survey research. And we address the important question of what more-advanced children should learn in kindergarten by pairing standards those children already know with crucial standards that need a lot of time and attention.

Many different methods of multidigit computation have been used historically and are now used around the world, but the context in which multidigit computation now occurs has changed. The worldwide availability of electronic calculators has decreased the need for complex computations. The emphasis now can be on understanding methods as well as performing them. This paper outlines a research program conducted over thirty years to find and test multidigit computation methods that are mathematically desirable and that many kinds of students and teachers can understand and explain. A nurturing Math Talk classroom environment, in which students made and explained math drawings supported sense-making by students and teachers. Powerful and simple math drawings were also developed and assessed. The methods and math drawings identified by this research for multidigit adding, subtracting, multiplying, and dividing are described. Examples are given of student explanations with the drawings. The criteria for deciding which methods are mathematically desirable are given, and the methods are judged by these criteria. Some methods that are common in various countries but that are difficult and may stimulate errors are described so that they might be replaced by the best methods identified by this research. How these methods fit the math standards of two different countries, the United States and China, is described. Sense-making about and using the identified best methods can reduce errors and engender understanding.

This chapter summarizes current research that is relevant to the learning and teaching of addition and subtraction of whole numbers ranging from simple small number situations through the multidigit algorithms. It summarizes results from these areas of research but, because of the relative amounts of research available, concentrates most heavily on children's thinking. The chapter concerns single-digit sums and differences between 10 and 18, more complex addition and subtraction situations (and word problems), and the several more complex solution procedures children use to solve such situations. Many children in kindergarten and first grade will use the direct modeling solution procedures discussed in the section called "Simple Addition and Subtraction Situations", but many other children will use the more sophisticated procedures discussed in the "Levels of Solution Procedures" section. The chapter differentiates children's conceptual structures by developmental level and by the size of the numbers added and subtracted.

This paper addresses the perceived opacity of fraction computation by summarizing results of classroom design experiments focused on making fraction concepts and computation meaningful to students and teachers. A nurturing Math Talk classroom environment, in which students made and explained math drawings supported sense-making by students and teachers. Students were able to make drawings of length models connected to symbolic computations for all fraction operations, and most students were able to explain their thinking. Correct performance on computations was considerably higher than that of U.S. students using traditional textbooks and was more comparable to performance of East Asian students. The paper shows the length drawings for each fraction operation and summarizes the nature of students' errors and how to overcome these. Length drawings (bar and number line models) connected to fraction symbols and words can focus students on unit fractions in all operations. Important difficulties to overcome are that fraction notation does not differentiate the roles of the numerator and denominator, and the notation shows the number but not the size of unit fractions. Multiplying fractions by multiplying each unit fraction and not the whole fraction magnitude is a general method, and division of fractions can begin by dividing numerators and denominators.

We respond to a call to analyze issues of curriculum standards and to present alternative storylines by addressing criticisms of the Common Core State Standards in early childhood. We describe a storyline from multiple media and evaluate this storyline's criticisms, focusing on the criticism that the standards are developmentally inappropriate. We review research and conclude that the criticism is invalid and may reflect a historical belief in the primacy of development over learning rather than the research record. Misinterpreting or ignoring relevant research has equity consequences because it may particularly harm those children most in need of learning support in learning grade-level mathematics. Fortunately, theory and research illuminate learning trajectories that help all children meet these standards.

This paper briefly overviews my research in supporting children to learn number concepts by relating number words, research-based visual supports, and math symbols. I first outline my approach to helping children build relationships between the use of concrete materials and the building of abstract concepts. I then focus on two crucial early aspects of building meanings for numbers: (1) understanding break-apart partners such as 5=3+2 that support addition and subtraction with small numbers and children's moving on to Level 2 counting on and algebraic problem representations, and (2) the use of visual five-groups in understanding numbers 1-1000 and in drawings to support multi-digit computations. The research-based learning path of visual-spatio supports is shown and discussed for each topic, including examples of children'smath drawings for representing word problems algebraically and for multi-digit computations. I have found math drawings to be a key visual support that helps children transition to working meaningfully with symbols and words alone. I close with a brief discussion of the difficulties children have with the number line. This overview can provide a framework within which future research on number learning by individuals with trisomy 21/Down syndrome can proceed.

We address common criticisms of the Common Core State Standards Mathematics, evaluating them based on comprehensive reviews of existing documentation and research to better ground future debates and to ameliorate negative effects of possible misconceptions or misinterpretations. The four main criticisms follow. (1) No one who helped develop the standards had any expertise in the education of young children. (2.) The CCSSM dictates scripted curricula and didactic instruction rigidly applied to all children at the same pace. (3.) The standards emphasize academic skills and leave no time for play, exploratory approaches, or social-emotional development. (4.) The standards are too early and therefore developmentally inappropriate for children in the early grades. We conclude that these criticisms are not valid, and that, given the importance of mathematics to academic success in all subjects, all children need and deserve to build a robust knowledge of mathematics in their earliest years and can do so if we use the research knowledge and research-based standards and programs presently available. We summarize and exemplify the research-based balanced approach to teaching based on learning trajectories that can provide guidance for engaging and developmentally appropriate mathematical experiences that have been demonstrated to help all children learn to high standards.

Howe (2014, Three pillars of first grade mathematics, and beyond. In: Li Y. & Lappan G. (eds), Mathematics curriculum in school education, Springer, Dordrecht, pp 183-207) identified three pillars of first grade mathematics and beyond that described central mathematical and sense-making aspects of major Common Core State Standards Math (National Governors Association Center for Best Practices, Council of Chief State School Officers. 2010) domains. This chapter builds on each pillar by sharing visual models that have been powerful in helping students learn the aspects identified by Howe. Visual models are central core ideas and practices in the CCSS-M and deserve attention and discussion. The researchbased examples discussed here are simple math drawings that students can make and use in their own ways in problem solving and explaining of thinking. Such drawings support the math talk discussions that are at the heart of the CCSS-M and of the mathematical practices. They enable (Howe's, 2014, Three pillars of first grade mathematics, and beyond. In: Li Y. & Lappan G. (eds), Mathematics curriculum in school education, Springer, Dordrecht, pp 183-207) three pillars to come to life in the classroom. Teachers and students can come to appreciate all of these pillars: Pillar I, the power of robust understanding of the operations of addition and subtraction including situations that givemeaning to the operations and levels of single-digit addition and subtraction; Pillar II, an approach to arithmetic computation that intertwines place value with the addition/subtraction facts; and Pillar III, making connections between counting number and measurement number.

Analyses show that criticisms of CCSSM are incorrect. Research also provides guidelines for appropriate, effective, and joyful teaching and learning.

Preparing children to be successful in mathematics begins with what they learn before they reach 1st grade. Math knowledge in prekindergarten and kindergarten predicts school achievement in math and in other topics, such as reading. Indeed, early math knowledge is one of the strongest predictors of math grades in high school, high school graduation, and college entry. But children enter kindergarten with a huge range of numerical knowledge and skills. So early educators must be attentive to practices that will aid in closing the gaps in learning. They can attend to this by following the guidance outlined in the National Research Council’s report, Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity, which identified math concepts that young children can and should learn. Especially important are children’s competence with quantity and number, as well as geometry and spatial reasoning. The NRC’s research summary and recommendations helped guide the Common Core State Standards in mathematics for kindergarten through 2nd grade.

The results of the Fuson and Li (ZDM Math. Educ. 41:793–808, 2009) analysis of the major early numerical aspects and learning supports for single-digit and multi-digit adding and subtracting in a representative Chinese textbook series and a US textbook series (Math Expressions) are related to the Chinese standards and to the US Common Core State Standards for these topics. Similar learning paths and visual-quantitative supports for mathematical thinking were identified in the textbooks from both countries, the US standards, and the experimental Chinese standards (2001). The new Chinese standards (2011) were less specific about learning paths and supports, though these appeared in examples. Criteria for judging the best variations of the multi-digit adding and subtracting variations were proposed and used. This analysis identified the best variations as the “New Groups Below” for adding and the “Ungroup First” for subtracting. The somewhat different levels in the adding and subtracting learning paths for East Asia and the US are summarized.

Abstract This chapter is an overview of central research-based perspectives that support teaching-learning for understanding and for fluency. We summarize the Class Learning Path Model that integrates two theoretical foci – a Piagetian focus on learning and a Vygotskiian focus on teaching – and specifies phases in learning that reflect Vygotsky’s assertion about the move from spontaneous to scientific concepts. Major aspects of the model were drawn from national research-based reports. This model connects understanding and fluency with a focus on mathematically important but also accessible methods in the middle and on maths drawings and other supports for understanding these methods. Such methods can be generated by students and can bridge from less-advanced student methods to formal methods that are unnecessarily complex. For three maths domains in Grades Kindergarten through Grade 6, we illustrate and discuss methods in the middle and drawings (diagrams) that support these methods: problem solving and especially the full range of word problem situations with each quantity the unknown; multidigit addition, subtraction, multiplication, and division; and ratio and proportion. Central features of the Common Core State Standards Mathematical Practices (CCSSO/NGA 2010) in these domains are identified, and how these can support understanding and fluency are briefly discussed. Further aspects of how the pedagogical supports help students move through the Class Learning Path in their own individual ways, and implications for research and for designing maths programmes are then discussed.