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## Learning Trajectories for Vertical Coherence

Three learning trajectories and their connections show how to promote vertical coherence in PK–12 mathematics education.

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## Using TileFarm to Support Emerging Multiplication

Symmetric dot patterns are a particularly powerful object for investigation, providing opportunities for foundational learning across PK–5. We found that second-grade students naturally used repeated addends to count symmetric dot patterns created using the new software TileFarm.

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## Chivalry, at Least “Math Chivalry,” Is Not Dead!

A cartoon exploring a problem about order of operations is coupled with a full-page activity sheet.

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## Questioning the Order of Operations

Abandon mnemonics and make stronger connections between the operations and properties of arithmetic.

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## Mystery Fractions

This activity engages students in problem solving while exploring key concepts of number theory, such as divisibility and divisibility tests, place value, fractions, and scale factors.

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## Hook and Hold

Set sail to explore powerful ways to use anchor charts in mathematics teaching and learning.

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## Improving Preservice Secondary Mathematics Teachers' Capability With Generic Example Proofs

Preservice mathematics teachers are entrusted with developing their future students' interest in and ability to do mathematics effectively. Various policy documents place an importance on being able to reason about and prove mathematical claims. However, it is not enough for these preservice teachers, and their future students, to have a narrow focus on only one type of proof (demonstration proof), as opposed to other forms of proof, such as generic example proofs or pictorial proofs. This article examines the effectiveness of a course on reasoning and proving on preservice teachers' awareness of and abilities to recognize and construct generic example proofs. The findings support assertions that such a course can and does change preservice teachers' capability with generic example proofs.

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## Using Constructed Knowledge to Multiply Fractions

Observe fourth graders' thinking in action as they connect the multiplication of whole numbers to arrays.

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## Helping Young Children See Math in Play

Capitalize on opportunities for mathematical concepts to emerge in common preschool contexts, such as doll corners and block centers.

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## Early Understanding of Equality

One of the first math symbols introduced=the equals sign=underpins much of the algebraic reasoning a child will use in later years.