The best approaches and strategies in teaching mathematics are not always so straightforward. Some students progress at much faster rates than others. Some teaching strategies for math are more effective with certain students than others, too.

In this webinar from the Consortium on Reaching Excellence in Education (CORE), learn five hallmarks of rigorous math instruction and top teaching strategies for increasing rigor in mathematics to help students successfully:

- Build conceptual understanding in math
- Increase procedural fluency in math
- Improve problem solving in math
- And more!

To help students achieve a rigorous math understanding, math instruction must be rigorous as well. There are five fundamental aspects of math rigor that should be embedded into instruction and taught to students to help them achieve mathematical understanding.

Math lessons should be focused on:

- Recognizing Coherence and Connections in Math

Create lessons that help students recognize and understand math connections within and across grades as well as between math concepts. Students should see how concepts and skills are related, including how they build on prior learning and can be applied to future learning, too. For example, a student should understand how the properties of fractions are closely related to the properties of whole numbers.

- Building Conceptual Understanding in Math

Take lessons beyond teaching students how to simply answer a problem; support studentsâ€™ abilities to access concepts and connect them together, such as how converting fractions with unlike denominators to like denominators connects to the idea of equivalents. Students who have built a strong conceptional knowledge should be able to explain connections clearly, from writing about them to discussing them verbally.

- Mastering Applications and Problem Solving in Math

Teach students to confidently solve non-routine types of problems that they do not automatically understand how to solve or immediately have the solutions for. When students encounter the unfamiliar, they must be able to think it through and apply math concepts that they do know to solve the unknown.

- Building Flexibility and Procedural Fluency in Math

It is also important to help students achieve speed and accuracy with facts and simple calculations related to the particular procedures and skills they are learning. But there is more to fluency than memorization. Keep in mind that students have to be fluent with skills such as mental math, estimation and use of counting strategies.

- Aligning Assessments

Finally, select and administer assessments that measure the four fundamental aspects of rigor listed above: coherence, conceptual understanding, problem-solving, and procedural fluency. Why? What is on assessments can often affect what is taught in the classroom. If teachers do not assess students on these skills, this could limit teaching around them. Assessments must be rigorous, too, and should assess the things that are important for students to learn.

For students to have a rigorous math understanding, math instruction itself must be rigorous. Work the five fundamental elements of math rigor listed above into your own teaching strategies for math to help all students master the concepts and skills they need to be successful and achieve true, rigorous mathematical understanding.

**Video Transcript**

Dean Ballard: So this process, though, is really built on thinking about students gaining a rigorous understanding of mathematics. And, in order to achieve that, I want to talk about what we mean by rigorous. Because we don’t meant that that’s more math idea or more complex math ideas crammed into a shorter amount of time. It’s really about connecting ideas together, being fluent with them, and having them be useful to kids in a way in which they can apply the mathematics. And we think about it in terms of, really, five fundamental ways or aspects that I call rigor. And I’m going to briefly go over those five aspects here. The first one is coherence and connections. We see students doing the math, but we want them to see the connections that are within and across grades. And within, between, concepts and how they build on each other.

Dean Ballard: When we think about, for example, recognizing how fractions, the properties of fractions, are related to properties of whole numbers. For instance, if you’re converting fractions to the same denominator when we add and subtract fractions? How that’s connected, actually, to aligning place value with whole numbers. When we add ones to ones and tens to tens and hundreds to hundreds? That’s the same thing as when we want to add one-half, halves to halves and thirds to thirds and fourths to fourths. It’s really about adding like units. And that leads into the idea, in algebra, of adding and subtracting like terms together. So making these connections is really important, in terms of understanding, and it’s what builds rigor for kids. And we don’t want kids to think about the standards and each particular skill or concept as just a single, discrete thing that’s not connected. But we want them to see them how they’re related together and how they can extend that learning and connect it to both prior and previous learning.

Dean Ballard: And this kind of feeds into the next thing which is conceptual understanding. The teachers are teaching more than just how to get the answer. Instead they’re supporting students’ ability to access concepts from numbers and from different perspectives so that students are able to see the math as more than just a set of, like I said, discrete concepts but they see the connections between them. Again, like I was talking about converting fractions with unlike denominators to having like denominators and seeing how that’s connected to the idea of equivalence, making equivalent fractions. We want students to be able to build concepts together. And when students can demonstrate this conceptual knowledge in their core mathematics classes? That’s really getting to that conceptual understanding and that justification, applications that they make and having them explain those connections. Discussing them, writing about them, all sort of the discourse that we’re looking for in classrooms. That’s what we’re talking about, about building that conceptual understanding.

Dean Ballard: And then this extends to applications and problem solving. We want students to be able to problem solve with mathematics and that means solving problems for which they don’t really know automatically what the answer is, or the way to solve it. They have to think it through. They have to apply the math to model real-world problems and engage in tasks. And this is what we call non-routine types of problems we’re asking them to solve here. These are kinds of problems, like I said, where they don’t automatically know how to get to the solution. It may apply the concepts they’ve just learned, but it isn’t such an automatic application that it becomes just practice. They actually have to think through, “Well, how do I apply it in this situation?”

Dean Ballard: But, of course, we also do want fluency and flexibility. This is part of rigor that students do build speed and accuracy with facts and simple calculations and with the procedures and skills they’re learning. But there’s more to fluency than just memorization. We should keep in mind that students have to be fluent with the skills such as mental math, estimation, and use of counting strategies. Fluency, you know, think about it in terms of just adding seven or knowing what seven means. We want students to know that two plus five and six plus one are different ways to get seven, that three-eighths is the same as three one-eighths put together. And not only to know these things, but to know when to use them in appropriate ways. So students still need to build fluency and flexibility. They do need both immediate and distributed practice. This is what helps them build these routines in so they can retain the knowledge, can perfect it, and it becomes a part of that foundation which they can build on. But let’s keep in mind that this is just one part of rigor, an important part but there’s more to it than just this.

Dean Ballard: It also brings us to the fifth bullet I’d like to mention about rigor. And it doesn’t seem like it might belong in this set, but I really feel it does. And that’s aligned assessments. We want teachers to use assessments that are really rigorous and robust. And that means that we want assessments that include fluency with facts and include conceptual understanding and include problem solving. And if we don’t assess students on these then it’s likely that we’re going to start short-cutting our teaching on them. Because it’s been my experience that, when push comes to shove and we’re up against pacing deadlines, we often start to just look at what’s on the test and just teach to that. You know, what do they need to know to pass the test? And, unfortunately, then the test becomes the guide for our teaching and our curriculum rather than the standards and the curriculum guiding what should be on the assessment. So it becomes very important that the assessments are rigorous and assess the things that we think are important for kids to learn.

Dean Ballard: And I know that there’s more, of course, and I’m not saying that all teachers do this all the time. And there’s certainly more to their instruction than just teaching to tests or worrying about what’s on a test. But I think that most of us now are in the Common Core age, we’re seeing the Smarter Balanced, PARCC tests, and other assessments and worried about those and what’s on those and making sure we’re teaching to those. So the quality of those assessments becomes even more important. And it’s really a disservice if those assessments do not assess the things that we think are important. And I think it’s sort of, it’s irresponsible to pretend that what’s on the assessment does not affect what we teach.