3 Ways to Assess Math Understanding More Deeply
New Mathematics Framework
A new Mathematics Framework for the State of California has just released for public comments. Jo Boaler was one of 5 writers of this new framework. The framework is worth some consideration by educators beyond California. Some highlights of the recommendations in the new proposed framework are these:
An asset-based, mindset infused approach, with a strong social justice orientation.
An approach to mathematics of “Investigating and Connecting” big ideas. It proposes that students do not spend time doing questions for the sake of doing them but investigate ideas with a purpose of “making sense of the world,” “predicting what could happen” or “impacting the future.” Sounds very PYP to me!
A new definition of fluency - neither number sense nor “fluency” are about speed, they are about flexibility.
K-12 chapters on two important areas of mathematics – number sense and data science and three Standards for Mathematical Practices.
Every grade, TK-10, now has a network map. The maps set out the big ideas and the connections between them. A document sharing the big ideas and network maps is available HERE.
The California Department of Education (CDE) shared ideas for integrating digital tools, with wonderful teacher vignettes and similar standards guidance for ELA/ELD. The full 500 page document is available HERE. You can also find all of these resources by visiting a new page on our website titled California Maths that is linked to a new item on our top menu called "California."
Learn more about this project from Jo Boaler HERE.
The Importance of Data Literacy
In this article - New Data Science Standards Are Needed for a Data-Filled World. Here’s What We Propose - Jo Boaler explains the reason behind creating the Big Ideas website and the importance of teaching children K-12 data literacy.
Jo Boaler and the YouCubed team have created a website - Big Ideas - that focuses on the development of Data Literacy. The need for data literacy has increased in recent years, not only to prepare students for 21st century jobs and support an informed citizenry. This has implications throughout the school years: teachers of younger students have the important task of helping them develop data literacy, and as students move through the grades they can start to learn the exciting and new discipline of data science.
The Big Ideas website offers ideas for the development of data literacy and data science through the grades!
For each grade range, the website links to tasks and data talks that youcubed has curated which are appropriate for addressing the content at that level. Data talks support student development in reading and understanding data visuals – an important form of data literacy. The ones found on the Big Ideas website are among the most popular data science resources and a full collection can be found HERE.
Integrating Effective Teaching Practices
This document covers:
Three specific teacher talk moves that boost student engagement and mathematical thinking
Actionable tips for teachers, including how to phrase class instructions
Common pitfalls to avoid and how to make the most of class time
Four Teacher-Recommended Instructional Strategies for MathClick/Tap top View
Cindy Garcia, Danielle Ngo, Patrick Brown, and Andrea Clark share their favorite math instructional strategies.
Concrete Representational Abstract
Cindy Garcia has been a bilingual educator for 14 years and is currently a district instructional specialist for PK-6 bilingual/ESL mathematics. She is active on Twitter @CindyGarciaTX and on her blog:
The single most effective strategy that I have used to teach mathematics is the Concrete Representational Abstract (CRA) approach.
During the concrete step, students use physical materials (real-life objects or models) to explore a concept. Using physical materials allows the students to see and touch abstract concepts such as place value. Students are able to manipulate these materials and make sense of what works and what does not work. For example, students can represent 102, 120, and 201 with base 10 blocks and count each model to see the difference of the value of the digit 2 in each number.
During the representational step, students use pictures, images, or virtual manipulatives to represent concrete materials and complete math tasks. Students are making connections and gaining a deeper understanding of the concept by creating or drawing representations.
During the abstract step, students are now primarily using numbers and symbols. Students working at the abstract stage have a solid understanding of the concept.
The CRA approach is appropriate and applicable to all grade levels. It is not about the age of the student but rather the concept being taught. In 3rd grade, it is beneficial to students to have them use base 10 blocks to create an open-area model, then draw an open-area model, and finally use the multiplication algorithm. In algebra, it is STILL beneficial to practice using algebra tiles to multiply polynomials using an open-area model.
The CRA approach provides students P-12 to have multiple opportunities to explore concepts and make connections with prior concepts. Some teachers try to start teaching a concept at the abstract level, for example, the standard algorithm for multiplication. However, they soon find out that students have difficulty remembering the steps, don’t regroup, or don’t line up digits correctly. One of the main reasons is that students don’t understand this shortcut and they have not had the concrete & representational experiences to see how the shortcuts in the standard algorithm work.
Danielle Ngo is a 3rd grade teacher and Lower School math coordinator at The Windward School. She has been a teacher for 10 years and works primarily with students who have language-based learning disabilities:
Growing up, so many of us were taught that there is one right answer to every math problem, and that there is one efficient way to arrive at that conclusion. The impetus to return to this framework when teaching math is a tempting one and one I’ve found myself having to fight actively against during my own classroom instruction. In my experience, the most effective way to counter this impulse is to mindfully increase the discourse present during my math lessons. Encouraging discourse benefits our students in several ways, all of which solidify crucial math concepts and sharpen higher-order thinking and reasoning skills:
Distributes math authority in the classroom: Allowing discourse between students—not just between the students and their teacher—establishes a classroom environment in which all contributions are respected and valued. Not only does this type of environment encourage students to advocate for themselves, to ask clarifying questions, and to assess their understanding of material, it also incentivizes students to actively engage in lessons by giving them agency and ownership over their knowledge. Learning becomes a collaborative effort, one in which each student can and should participate.
Promotes a deeper understanding of mathematical concepts: While the rote memorization of a process allows many students to pass their tests, this superficial grasp of math skills does not build a solid foundation for more complex concepts. Through the requisite explanation and justification of their thought processes, discourse pushes students to move beyond an understanding of math as a set of procedural tasks. Rather, rich classroom discussion gives students the freedom to explore the “why’s and how’s” of math—to engage with the concepts at hand, think critically about them, and connect new topics to previous knowledge. These connections allow students to develop a meaningful understanding of mathematical concepts and to use prior knowledge to solve unfamiliar problems.
Develops mathematical-language skills: Students internalize vocabulary words—both their definitions and correct usage—through repeated exposures to the words in meaningful contexts. Appropriately facilitated classroom discourse provides the perfect opportunity for students to practice using new vocabulary terms, as well as to restate definitions in their own words. Additionally, since many math concepts build on prior knowledge, classroom discussions allow students to revisit vocabulary words; use them in multiple, varied contexts; and thus keep the terms current.
Patrick Brown is the executive director of STEM and CTE for the Fort Zumwalt school district, in Missouri, an experienced educator, and a noted author:
The COVID-19 pandemic is a sobering reminder that we are educating today’s students for a world that is increasingly complex and unpredictable. The sequence that we use in mathematics education can be pivotal in developing students’ understanding and ability to apply ideas to their lives.
An explore-before-explain mindset to mathematics teaching means situating learning in real-life situations and problems and using those circumstances as a context for learning. Explore-before-explain teaching is all about creating conceptual coherence for learners and students’ experiences must occur before explanations and practice-type activities.
Distance learning reaffirmed these ideas when I was faced with the challenge of teaching area and perimeter for the first-time to a 3rd grade learner. I quickly realized that rather than viewing area and perimeter as topics to be explained and then practiced, situating learning in problem-solving scenarios and using household items as manipulatives can illustrate ideas and derive the mathematical formulas and relationships.
Using Lego bricks, we quickly transformed equations and word problems into problem-solving situations that could be built. Student Lego constructions were used as evidence for comparing and contrasting physically how area and perimeter are similar and different as well as mathematical ways to calculate these concepts (e.g., students quickly learned by using Legos that perimeter is the distance around a shape while area is the total shape of an object). Thus, situating learning and having students use data as evidence for mathematical understanding have been critical for motivating and engaging students in distance learning environments.
Using an explore-before-explain sequence of mathematics instruction helps transform traditional mathematics lessons into activities that promote the development of deeper conceptual understanding and transfer learning.
A Whiteboard Wall
Andrea Clark is a grade 5-7 math and language arts teacher in Austin, Texas. She has a master’s in STEM education and has been teaching for over 10 years:
If you want to increase motivation, persistence, and participation in your math classroom, I recommend a whiteboard wall. Or some reusable dry erase flipcharts to hang on the wall. Or some dry erase paint. Anything to get your students standing up and working on math together on a nonpermanent surface.
The idea of using “vertical nonpermanent surfaces” in the math classroom comes from Peter Liljedahl’s work with the best conditions for encouraging and supporting problem-solving in the math classroom. He found that students who worked on whiteboards (nonpermanent surfaces) started writing much sooner than students who worked on paper. He also found that students who worked on whiteboards discussed more, participated more, and persisted for longer than students working on paper. Working on a vertical whiteboard (hung on the wall) increased all of these factors, even compared with working on horizontal whiteboards.
Adding additional whiteboard space for my students to write on the walls has changed my math classroom (I have a few moveable whiteboard walls covered in dry erase paint as well as one wall with large whiteboards from end to end). My students spent less time sitting down, more time collaborating, and more time doing high-quality math. They were more willing to take risks, even willing to erase everything they had done and start over if necessary. They were able to solve problems that were complex and challenging, covering the whiteboards with their thinking and drawing.=
And my students loved it. They were excited to work together on the whiteboards. They were excited to come to math and work through difficult problems together. They moved around the room, talking to other groups and sharing ideas. The fact that the boards were on the wall meant that everyone could see what other groups were doing. I could see where every group was just by looking around the room. I could see who needed help and who needed more time to work through something. But my students could see everything, too. They could get ideas from classmates outside of their group, using others’ ideas to get them through a disagreement or a sticking point. It made formally presenting their ideas easier, too; everyone could just turn and look at the board of the students who were sharing.
I loved ending the math class with whiteboards covered in writing. It reminded me of all of the thinking and talking and collaborating that had just happened. And that was a good feeling at the end of the day. Use nonpermanent vertical surfaces and watch your math class come alive.
The 3-Act problem-solving activity structure is specifically designed to engage students in mathematical modelling. Three-Act tasks were originally designed by Dan Meyer (2011) (see video below) for use in secondary classrooms and adapted by Graham Fletcher and other educators for elementary school classrooms.
Summary of a Three-Act Task Lesson
The teacher shares a compelling multimedia depiction of a situation through a video or photographs.
Students discuss what they notice and wonder about the video, including mathematical features of the situations.
Students decide on a mathematical question to answer about the situation.
The teacher provides information or resources that students think they need to work on the focal question.
Students work to answer the question.
Students discuss their strategies and solutions.
The teacher may compare and connect students’ ideas or “reveal the answer.”
If relevant to the focal problem, students consider why their modelling was different from the real-world resolution.
Source: National Council of Teachers of Mathematics
Dan Meyer: Real Context Problems - Video
3-Act Math Lesson Examples
Graham Fletcher (K-7) Robert Kaplinsky (K-12) Mike Wiernicki (K-8) Dane Ehlert (6-12) Kyle Pearce (3-12) Catherine Castillo (K-5) Kendra Lomax (K-3) Kristen Acosta (K-5). I See Maths (K-5). Tap Into Their Minds (K-12) Dan Meyer’s (6-12) Jon Orr (6-12)
Student Recording Sheets
3 Reads Problem Solving Strategy
The 3 Reads instructional strategy is designed to develop students’ ability to make sense of problems by deconstructing the process of reading mathematical situations. It is applicable for all grade levels K-12.
Over time, students will internalize this process, thereby creating a heuristic for reading and making sense of mathematical story problems.
The info-graphic captures the flow of the routine, and resources below will support your implementation of the routine.
More Math Resources
PATH: Math Problem Solving Strategy
- P: PLAN & ANALYZE - What is the problem asking me to do/solve?
- A: APPLY & SOLVE - What strategies can I use to solve the problem?
- T: TOOLS FOR APPLICATION -What is the most efficient tool I can use to solve the problem?
- H: HOW? JUSTIFY - How did I arrive at my solution, and how do I know my solution pathway is correct?
Math With Pictures Blog
Graham Fletcher Website(Click/Tap to View)
Graham Fletcher has served in education as a classroom teacher, math instructional lead, and currently as a math specialist. His work with the math progressions and problem-based lessons has led him to present throughout North America and beyond.
Graham is continually advocating for best practices in elementary mathematics by seeking new and innovative ways to support students and teachers in their development of conceptual understanding. He is the author of Building Fact Fluency: A Toolkit for Addition and Subtraction.
NRICH(Click/Tap to View)
Math At Home
Feature packed math activities with activities and games suitable for learners to work on at home without a teacher. LINK
The NRICH Maths Fair is a set of free, hands-on resources. They are designed to be set out simultaneously around a room for children to engage with and move around freely. However, they can be used in a variety of ways and settings, including at home with family members.
Math is Visual(Click/Tap to View)
This website was created to assist in building a better conceptual understanding of mathematics through the use of visuals. The images, videos and resources shared here are intended to help all teachers, parents and students understand that Math Is Visual and we should take every opportunity to teach it that way.
Youcubed(Click/Tap to View)
Youcubed provided a variety Math Problems/Tasks for all grades. Curated by Jo Boaler and Stanford Graduate School of Education | 100% free
The site also has a free student online math course.
Data Talks: Data talks are short 5-10 minute classroom discussions to help students develop data literacy.
Some Non-Screen Activities
Neighbourhood Numbers (K-5): As you are walking with your child there are many conversations you can have about the numbers on the houses and apartments around you. You can ask: What is that number? How do you say it? Are the numbers getting bigger or smaller? By how much? What do you think the next number will be? What do you notice about the numbers on the other side of the street? Are there any patterns? Where are the even numbers? Odd? How do they increase? Is there a starting point?
Nim Games (3-12): Mathematical games are often simple to play, but hard to master and Nim Games are no exception. You only need a handful of beans or coins to play but with so many variations, Nim Games can bring entertainment for hours.
Finger Painting (K-8): Is there really mathematics in finger painting? Yes, of course there is! From finger discrimination and counting to systems thinking about mixing colours this art activity includes mathematics at every stage.
Estimating (K-12): Estimating might sound like an activity from young children, and although it is fantastic for them to estimate how many pieces of candy are in a jar or how many crackers come in a box, older students can also benefit from these and take them to the next level.
Emoji Graph (K-12): Lessons
How Tall Is That Tree? (2-12): How many times have you come across a tree and thought: that tree is humongous? What do you measure with? Have a discussion with family and friends when outside about different strategies to determine the height of trees you’re curious about.
Fraction Hunt (2-6): Fractions are all around us! Walk around the house, yard, and neighbourhood with your child. Where do you see fractions? What would you call one part of the whole? What if the objects are different sizes? Can they still be represented as a fraction?
Apple Orchard (4-12): This layout based on how apple trees are planted in orchards leads to some interesting explorations of area, patterns, and growth rates. Younger students can model the situation with beads or beans, while older students can graph the growth rates they find.
Pixel Art On Windows (K-12): We've been inspired by the beautiful art many people are making on their windows with sticky notes or squared pieces of paper to cheer up their neighbours during this time. In this task, we explore some of the math involved and pose a challenge for students to tackle.
Would You Rather Math(Click/Tap to View)
Would You Rather Math presents students with two scenarios of a problem and asks them to choose one of them and then defend their answer. There is no right or wrong answer. It provides opportunities for students to strengthen their critical thinking skills by taking a stance and defending
Math Resources/Articles(Click/Tap to View)
2 Ways to Encourage Reflection on Math Concepts | Edutopia | 2 July 2021 | Open-ended questions guide students to participate and to think mathematically, which cements their learning.
XtraMath - a free online program that helps students improve their math fact fluency
Facilitating Rich Math Tasks Remotely Podcast: Graham Fletcher talks about remote ways to facilitate 3-Act Math Tasks. Learn ways that allow you to connect with your students, create spaces for them to share their ideas, and engage students in productive math thinking from home.
Delta Learn Online Primary Math for Fun: A curated list of sites that parents can use to support their child in deepening their understanding of numbers through online and offline games and activities at home.
Mathology Little Books: Simple online tool with rich math activities
Math At Home - Daily Math Activities and other printable resources. The Daily Math Activities introduces math routines and exploring math in the home. Great examples and ideas provided for easy to implement learning.
SD38 Math Resources: A starter collection of K-7 weekly plans, instructional routines, math games, resources to share with parents and a link to additional vetted online resources.
Pirate Math Equation Quest: (K-3) an extension of the Pirate Math program developed by Vanderbilt University. Includes: single-digit and double-digit additive and multiplicative word problems that include four schemas: Total, Difference, Change, and Equal Groups. It is now available online for free, including the manuals and training videos. Download now while it’s available.
Delta Learn Online Math for Fun: (Gr 4-7) A curated list of sites that parents can use to support their child in practicing math and problem-solving at home through online and offline games and activities.
Nelson Canada: (Gr 4-7) Access to all of Nelson’s Math & Science resources.
SD38 Math Resources: (Gr 4-7) A starter collection of K-7 weekly plans, instructional routines, math games, resources to share with parents and a link to additional vetted online resources.
Math Reference Books
Teaching Student-Centered Mathematics: Developmentally Appropriate Instruction for Pre K-2 (Vol. 1) & Gr. 3-5 (Vol. 2) with Enhanced Pearson eText - Access Card Package (Vol. 1) & (Vol. 2) - 3rd Ed.
By: John Van de Walle, Karen Karp, LouAnn Lovin, Jennifer Bay-Williams
Note: These packages includes the Enhanced Pearson eText and the bound book version.
These books & eText help students make connections between mathematics and their worlds―and helping them feel empowered to use math in their lives. Designed for classroom teachers, the book focuses on specific grade bands and includes information on creating an effective classroom environment, aligning teaching to various standards and practices, such as the Common Core State Standards and NCTM’s teaching practices, and engaging families.
The first portion of the book addresses how to build a student-centered environment in which children can become mathematically proficient, while the second portion focuses on practical ways to teach important concepts in a student-centered fashion. The new edition features a corresponding Enhanced Pearson eText version with links to embedded videos, blackline masters, downloadable teacher resource and activity pages, lesson plans, activities correlated to the CCSS, and tables of common errors and misconceptions.
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Improve mastery and retention with the Enhanced Pearson eText
The Enhanced Pearson eText provides a rich, interactive learning environment designed to improve student mastery of content. The Enhanced Pearson eText is:
Engaging. The new interactive, multimedia learning features were developed by the authors and other subject-matter experts to deepen and enrich the learning experience.
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*The Enhanced eText features are only available in the Pearson eText format. They are not available in third-party eTexts or downloads.
*The Pearson eText App is available on Google Play and in the App Store. It requires Android OS 3.1-4, a 7” or 10” tablet, or iPad iOS 5.0 or later.