What Does It Mean To Understand Mathematics?
You have probably had a time when you asked your students to complete a math problem, clearly expected them to do well, but as you walked around checking their progress you realized that something strange was going on. Robert Kaplinsky explains in his blog that it is critical that we give students opportunities to develop rigorous mathematical understandings. Procedural skill is still an essential piece but it is just as important as developing their conceptual understanding and the ability to apply mathematics. Often times we teach students how to do mathematics with the belief that they will be able to apply it when the moment comes. Clearly that is not always the case.
Ronbert Kaplinski believes that, "It is critical that we give students opportunities to develop rigorous mathematical understandings. Procedural skill is still an essential piece but it is just as important as developing their conceptual understanding and the ability to apply mathematics. Often times we teach students how to do mathematics with the belief that they will be able to apply it when the moment comes. Clearly that is not always the case."
Boosting Confidence With Student-Driven Math
When teachers noticed a lack of autonomy in the math classroom, a switch to a student-centered curriculum bumped up ownership of learning—and the kids’ ability to talk about their problem-solving approach.
Mathematical Mindset Teaching Guide, Teaching Video and Additional Resources
YouCubed have designed a Mathematical Mindset Guide to help teachers create or strengthen a growth mindset culture. The guide contains five Mathematical Mindset Practices along with links to teaching videos. The videos all show Jo and Cathy teaching middle school students (can be adapted to Primary students). There are different stages described in each practice to help capture the journey of a mathematical mindset classroom and the evidence teachers may collect along the way for their own reflection or for discussion with colleagues. The guide has been designed for teachers to use in the process of self-reflection, or for coaches or administrators to use to encourage a mindset teaching culture. In the interactive version of the guide on the web page, you can click on the arrow buttons in the Expanding descriptors to see a short extract of Jo/Cathy teaching in the ways described.
The goal for the guide is to support a mathematical mindset journey of learning and growth. Teachers can work with the guide individually or in collaboration with others. The guide is intended to be non-judgmental, non-evaluative, and iterative in nature. When using the guide consider the classroom community as a whole rather than the teacher alone. It is also important to note that while the goal of the guide is to communicate all aspects of a mathematical mindset classroom, it is not always possible to find evidence of all practices in one lesson. We encourage teachers, coaches, and administrators to use this guide, and our reflection suggestions iteratively over multiple lessons.
The Three-Act Problem-Solving Activity Structure
The Three-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
Three-Act Problem-Based Lesson Search Engine
Robert Kaplinski has created this highly recommended resource to make it easy to search for Three-Act Problems.
Three-Act Problem Solving Resources & Activities
Three-Act Problem Solving Resources
Three-Act Problem Solving Activities
Two math videos staring TobyBear to be used as a provocation for a Grade 1/2 class inquiry into concepts of Division and fair trading. By Cathie Matthews
Three-Act Problem-Solving Activity Websites
Three-Act Problem Solving Websites
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 M. Acosta (K-5).
I See Maths (K-5).
Dan Meyer (6-12)
Jon Orr (6-12)
Three-Reads Problem Solving Strategy
The Three-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 infographic captures the flow of the routine, and resources below will support your implementation of the routine.
PATH Problem Solving Strategy
PATH: Problem Solving Strategy
Source: The Daily Cafe Website
What is PATH math 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?
More Math Strategies
The NRICH ProjectClick/Tap to View
The NRICH Project aims to enrich the mathematical experiences of all learners. The NRICH website hosts a large collection of math problems and solutions for teachers and students. An interesting section of the website is the Features section where students are presented with a problem which they are encouraged to solve using a variety of strategies and words. Live web-based sessions are scheduled where problem solutions for each feature are discussed. This is wonderful PD and could be used for feedback to students.
Primary Math Strategies & Resources
Proof Through Words (FEATURES): Creating convincing arguments or "proofs" is a key mathematical skill which even the youngest learners can explore with NRICH
For Students: Shape Your Proof (maths.org)
See Events for Teachers and Students where the live problem solutions are discussed. Here you will find how to register for free for the online events for a chance to work on problems from their latest features. The hosts will consider the valuable mathematical thinking that the problems might prompt in your classroom and how they can be embedded into your everyday practice.
Early Years Math Strategies & Resources
Proof Through Words (FEATURES): Creating convincing arguments or "proofs" is a key mathematical skill which even the youngest learners can explore with NRICH
Student Games and Activities
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.
Four Teacher-Recommended Instructional Math Strategies
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.
A List of Math Teaching Ideas/Practices
Math With Pictures Blog |Edutopia |A popular strategy, using pictures in math class gives students another way to access math concepts and can spark great conversations. Other Websites: Number Talk Images; Kristen Acosta Website
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
How Math Journals Help Students Process Their Learning | Edutopia | By reflecting on their math work in writing and drawings, students see what they’ve learned and find areas where they need to improve.
3 Ways to Assess Math Understanding More Deeply [Video] | Edutopia | Getting the answer right is just one part of math assessment—these strategies can provide deeper insight into students' mathematical thinking.