I recently attended a webinar hosted by Christina Tondevold – The Recovering Traditionalist – that synthesized recommendations given for supporting students struggling in mathematics through intervention into three important and distinct pieces that should be present at each intervention opportunity.

IES Practice Guide

The Institute of Education Sciences (IES) published their practice guide "Assisting Students Struggling with Mathematics: Response to Intervention (RTI) for Elementary and Middle Schools" in an effort to bring the best available evidence and expertise to give influence on the systemic challenge of math interventions. Their goal was to "formulate specific and coherent evidence-based recommendations" to address the challenge of identifying and providing students who need more help high-quality interventions. The guide aims to give clear, and practical information for use by educators.

"Students struggling with mathematics may benefit from early interventions aimed at improving their mathematics ability and ultimately preventing subsequent failure."

The guide provides eight recommendations, developed by a panel of researchers and practitioners with expertise in various dimensions of the topic, intended to help identify and address the needs of students who struggle with and require assistance in mathematics through focused interventions. The recommendations are listed in the guide with references to their evidence rating – the strength of the research evidence that has shown the effectiveness of a recommendation. Rather than attending to all eight recommendations individually, they were synthesized by Christina and her colleague. They presented that to condense the strong and moderate evidence recommendations would reveal three parts that would create a high quality, student-centered math intervention:

1) Number Talks – opportunity to see and talk about numbers

2) Story Problems – focus on key concepts in the structure of the problems using "bet lines"

3) Games – used to practice facts in a fun way

These parts can be referred to as "Talk, Think, Practice". I'll go into the details of each part below, but her timeline of implementation is to spend approximately 10-15 minutes on each part of the intervention . Once the content is determined, this 25 to 35 minutes of extra math instruction (not the same as classroom instruction – goal is to find gaps & fill them in intervention) can be broken up into 5-10 minutes on the Number Talk, 10-15 minutes in Story Problem work, and 10 minutes on Games. If time doesn't allow for Talk, Think, AND Practice each intervention session, be sure to include a Number Talk in every session, but then you could alternate the Story Problems and Games. So, we always want to include the Talk, but it could be paired with the Think on one day, then with the Practice the next time. For instance, day 1 would be 10 min. Number Talk then 10 in Story Problems; day 2 would be 10 min. Number Talk then 10 min in Games – but all the problems/prompts/games would be covering the same content – i.e. making ten, or multiplication facts of the same family.

Number Talks are an opportunity to see numbers, and to talk about numbers. The routine gives students the chance to practice their flexibility with numbers, and to talk about their strategies. Number Talks are a documented way to help students make sense with numbers – it could be a single computation problem where students discuss their strategies, but typically using number strings within the Number Talk is the best way to fill gaps where there are misunderstandings. Number strings focus on a certain concept, and this purpose gives students the opportunity to see the same thing in several ways or several times. If the student struggles with the content of the Number Talk, the number strings allow students to try the same concept again, but with different numbers or configurations. So, it is important that you are clear in your purpose for the intervention – what's the concept you are covering? This will assist you in selecting an appropriate set of number strings to use in your Number Talk.

Story Problems give students an opportunity to apply the same concept covered in the Number Talk. The numbers and/or strategies of interest were just talked about, so it primes them for the Story Problem, but the focus with this "Think" part is to get them involved in a Story Problem's context. The way to do this is by taking the problem line by line, with "bet lines" rather than having them circle/underline words in the problem. Mathematical Bet Lines emphasize that students should make sense of a problem and takes the focus away from getting straight to an answer. This strategy is structured as a conversation between the teacher and the student(s). The teacher begins by showing and reading only the opening sentence of a problem, then asking, "What do you bet will happen next?". The students offer a prediction, or a "bet", of what will come next. The goal with this approach is to attend to making sense of the Story Problem, and keeping the mathematical relevancy of the predictions at the forefront. This make Story Problems much more interactive, and engages the student(s) into the context of a problem. As you continue, reveal the next line, then pause to allow student(s) time to make and discuss their bets. Giving the opportunity to grapple with the language of the problem rather than plucking numbers out of it allows for engagement in mathematics for understanding. Finally, once all the information has been shared, it is time to make a prediction about what question they will be asked, which will lead to (For more information, here is an article from Teaching Children Mathematics – Supporting Sense Making with Mathematical Bet Lines)

Games are used to practice facts in a fun way, rather than relying on flash cards or worksheets. This "practice" part of the intervention is an opportunity to engage students with their facts, build fact fluency, while also creating an atmosphere of entertainment in relation to their facts. There are many resources when it comes to games for fact practice, but you want to take care that it is not just "fancy" way to do flashcards. It should be a Game they are interacting with that requires the use of their facts. A version of these types of games are referred to as "evergreen" games. Evergreen games are games that have general rules that never change. Once you teach children the rules you can use the game for a variety of math concepts. For example, the rules of Memory never change….but what "matches" they are looking for can change with each new concept you want to focus on. Some other examples of these types of games include Bump, Capture 4, and "I have, Who has…" A few resources can be found here and here. As the Number Talk and the Story Problem parts were focused on similar content chosen as a needed for closing gaps, the Game part of the intervention should be linked in this same or similar content also. The power here is linking the three parts over similar content – it is giving the student an opportunity to experience the content they struggle with, or have gaps in their understanding of, in multiple ways – they Talk about the numbers/strategies in the Number Talk, then Think about them in the context of a Story Problem, and finally Practice those same skills with the Games.

Through these three parts of student-centered math intervention, the goal is to build student's numeracy. Progress monitoring only measures the right answer and the time it takes. Our students deserve investment in their number sense and mathematical understanding.

A Remarkable Read: Uncovering Student Thinking in Mathematics

We know students learning more when we begin with them – finding out what they know and what they need to learn and then using assessment for learning to help them learn more. Formative assessment informs instruction. It may take many forms, but the Uncovering Student Thinking in Mathematics books focus in on using diagnostic questions –

Mathematics Assessment Probes

  • The purpose of the these probes is to elicit prior understanding and commonly held misconceptions, then allow the educator to make sound instructional choices based on the specific needs of a particular group of students.
  • The structure of a math assessment probe is two tiered, one for elicitation of common understandings and misunderstandings (can come as selected response, multiple choice, or opposing views formats) and the other for elaboration of individual student thinking.
  • The types of understandings and misunderstandings the probes uncover include conceptual understanding and procedural knowledge, as well as common errors and overgeneralizations.

Also included with each probe are teachers' notes, designed around the QUEST cycle of action research:

Questioning student understanding of a particular topic

Uncovering understanding and misunderstandings using a probe

Examining student work

Seeking links to cognitive research to drive next steps in instruction

Teaching implications based on findings and determining impact on learning by asking an additional question.

The probes cover a range of content domains, including Number & Operations, Geometry & Measurement, and Algebra, Data Analysis & Probability.

A Remarkable Read: Mathematics Formative Assessment, Vol 1 & 2

Research shows that formative assessments have the power to improve learning, increase student engagement, and give insight into student thinking.

Written by Page Keeley and Cheryl Rose Tobey, these volumes help inform how to best link assessment, instruction, and learning through the use of Formative Assessment Classroom Techniques (FACTs).

Not only do these strategies inform instructional planning and better meet the needs of all students,

the authors provide important guidance with each technique, including tips and caveats, examples of implementation, and possible modifications.

The best part is that these strategies are relevant for all grades and topic areas – and are specific to MATH!

Mathematics Formative Assessment is a collaboration between the National Council of Teachers of Mathematics and Corwin Press. Volume 1, published in 2011, contains 75 practical strategies and Volume 2, just published in 2017, adds in 50 more strategies.

The first book walks teachers through what a Formative Assessment-Centered Classroom looks like and how to make the shift. It gives guidance on what aspects need to be present in a classroom community to support the use of these techniques.

Also introduced is the "MAIL" cycle – "Mathematics Assessment, Instruction, and Learning" cycle, which includes these stages:

  • Engagement and readiness
  • Eliciting prior knowledge
  • Exploration and discovery
  • Concept and skill development
  • Concept and skill transfer
  • Self-assessment and reflection

Each stage has a single focus question to help direct teachers through the MAIL cycle. Several helpful tables makes connections to assessment and instruction for the teacher of each stage of the cycle, and the connection to learning for the student, as well as what type of assessment should be used during each stage.

The next section of the first volume gives consideration of how to select a formative assessment classroom technique (FACT), and thoughts on planning and using FACTs – starting small, then maintaining implementation, and finally extending implementation. Ultimately, there is no single best FACT or system of FACTs; what works best in one classroom may not work well in another.

Each Formative Assessment Technique (FACT) was selected based on its ability to meet a set of considerations that included,

  • its flexibility in use
  • the level of inquiry it promotes
  • how engaging the technique is
  • the ease of use by teachers
  • how valid is the result
  • multiple benefits to the classroom
  • promoting an opportunity to learn

For each FACT, an overview is provided along with how the technique promotes student learning and informs instruction. Other helpful information with each formative assessment classroom technique in this book includes implementation attributes, tips for design and administration, examples, and any modifications that are possible.

In Volume 1, there is a very helpful chart listing all the FACTs, and aligning them to their primary purpose for promoting learning and informing teaching.

Some of the 75 mathematics formative assessment classroom techniques from Volume 1 include Agreement Circles, Frayer Model, Human Scatter Graph, No-Hands Questioning, Thinking Log, and Whiteboarding.

In Volume 2, two tables are provided that link the examples of a specific FACT in the book with the math content standards they align with as well as the mathematical practices.

Of the 50 more in Volume 2, here is a sampling – Conjecture cards, Feedback Check-ins, Learning Intentions, Partner Strategy Rounds, Ranking Tasks, Slide Sort, and Thermometer Feedback.

The authors provide a rich repertoire of formative assessment techniques that allow learners to interact with assessment in a variety of ways – writing, drawing, speaking, listening, questioning, investigating, and modeling. Also, the math specific examples that these books provide is invaluable, considering math is often lacking in general formative assessment resources!