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How Do I Know My Students Understand the Concepts?

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How Do I Know My Students Understand the Concepts?

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Assessment Strategies: When, Why, and How to Use Them

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Assessment Strategies: When, Why, and How to Use Them

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🔹 Diagnostic Assessment

What: Done before a unit or at the beginning of the school year

Why: To find out what students already know! This will help me gauge how much time I may need to spend on a topic.

How: Use short, simple tasks that require basic knowledge on concepts

Example: “Write everything you know about angles.”

🔹 Formative Assessment

What: Done during a unit

Why: To check understanding and guide teaching. This can help me see if I need to reteach concepts or give more examples.

How: Use exit tickets, group work, or class discussions

Example: After a lesson on the Pythagorean Theorem, students explain the rule in their own words.

🔹 Summative Assessment

What: Done after a unit

Why: To evaluate overall learning

How: Use quizzes, projects, or performance tasks

Example: Create a poster showing how to solve right triangles using sine and cosine.

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2. What Are Some Advantages of Visible Thinking Routines?

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2. What Are Some Advantages of Visible Thinking Routines?

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🔹 Make Thinking Clear

Simple Explanation: Students show what they are thinking. What steps they used, ideas they had, etc.

Example: “I used to think… Now I think…” helps reveal how their understanding changed.

🔹 Catch Misconceptions

Simple Explanation: Helps teachers see what’s unclear

Example: A student says, “You can use SOH CAH TOA on any triangle” is good opportunity to correct!

🔹 Student Reflection

Simple Explanation: Encourages students to think about their learning and recognize where they may need additional improvement

Example: At the end of a lesson, students write a headline: “Trig ratios are shortcuts to solve triangles!”

Additional Ideas: https://www.nctm.org/Publications/TCM-blog/Blog/Transforming-the-Culture-of-Math_-Routines-for-Making-Thinking-Visible/

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By Jancey Clark, posted October 23, 2017 - Show your thinking
As mathematics classrooms continue to shift from a focus on procedures to more conceptual understanding, students are being asked to "show their thinking." This isn't always easy. Visible Thinking routines can provide students with a structure and scaffold to support making connections, using knowledge effectively, and communicating ideas during problem solving.

In the book Making Thinking Visible: How to Promote Engagement, Understanding, and Independence for All Learners, authors Ritchhart, Church, and Morrison suggest ways to use thinking routines in all subject areas and across all grade levels. As an elementary school instructional coach, I have been excited to see how these routines can be used in the math classroom. The following routines help students see that math is about more than finding the answer.
Develop a deeper understanding of content
The Connect-Extend-Challenge (CEC) routine is perfect for introducing new concepts and skills because it encourages students to draw on prior knowledge and experience. Instead of just asking students to solve a problem, ask them how it connects with other problems they have solved. Then ask them to think about what's new to them-how the task extends their thinking further and deeper than the previous lesson. Finally, asking students to identify what challenges they are facing with the task will provide useful formative data and inform your future instruction.
Promote meaningful collaboration
As mathematicians, our students must be able to communicate a claim, make generalizations, recognize patterns, and provide evidence. The Claim-Support-Question routine guides students in doing so. Start with a claim, such as, "All multiples of nine are also multiples of three." Then ask students to work in pairs or small groups to discuss whether this claim is true or false. They must also provide support, or evidence, using manipulatives and examples. Finally, invite students to share any remaining questions that have been raised and have not yet been answered. This routine will highlight student misconceptions and disagreements, and it will keep the discussions focused on the evidence and how it supports or refutes the claim.
Scaffold students' thinking and learning abilities
To support students in uncovering patterns and developing equations, use the See-Think-Wonder routine with Visual Patterns. First, show students a pattern and ask, "What do you see?" In response, students describe patterns and talk about how they would count the objects. Then, move on to asking, "What do you think?" Instruct students to think about what the next step in the pattern would be. This will lead to the question, "What do you wonder?" Students might wonder about what the 100th step would look like and how they could find out. This discussion will lead students to a conceptual understanding of equations.
Build a culture of engaged thinkers and learners
These are just a few of the Making Thinking Visible routines that can be used in the math classroom. When used explicitly and consistently, Visible Thinking routines will promote active thinking and deep learning. In addition, students will become capable of thinking independently and communicating effectively.
Your turn
How are you transforming the culture of math teaching and learning at your school this year? Use the comment section below to share your ideas and plans for transforming the culture at your school, or join us on Twitter @TCM_at_NCTM using the hashtag #TCMblog.

Jancey Clark is an elementary school teacher and instructional coach who is passionate about personalized learning for both students and teachers. In her current role as elementary school learning coach at the American International School in Riyadh, Saudi Arabia (AIS-R), Clark facilitates professional learning through workshops, coaching cycles, and team planning. As a member of the Near East South Asia (NESA) Council of Overseas Schools Mathematics Collaborative Core, she helps develop and promote professional learning opportunities in mathematics for the NESA community. When she is not learning with colleagues at AIS-R, she enjoys sharing and reflecting on learning through Twitter (@jancey5) and her blog ( EDventures in Coaching).

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Transforming the Culture of Math: Routines for Making Thinking Visible

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By Jancey Clark, posted October 23, 2017 - Show your thinking As mathematics classrooms continue to shift from a focus on procedures to more conceptual understanding, students are being asked to "show their thinking." This isn't always easy. Visible Thinking routines can provide students with a structure and scaffold to support making connections, using knowledge effectively, and communicating ideas during problem solving.

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How Would I Use a Visible Thinking Routine? 

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How Would I Use a Visible Thinking Routine? 

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🔹Think, Sort, Connect, Explain

Simple Explanation: Make a concept map to organize ideas

Example: Students list and group key ideas in trigonometry, like “right triangles,” “sine,” “angle,” and “ratios.”

This will show me how the students connect different topics together, and I can have them explain WHY things connect

🔹 Routine: I used to think… Now I think…

Simple Explanation: Shows learning progress

Example: “I used to think pi was only for circles. Now I think it’s also used in radians.”

Reflections bring more thoughtfulness to students and help them recognize their progress and how their thinking has changed

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Other graphic organizers I Could Use That Highlight Students’ Conceptual Understanding

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Other graphic organizers I Could Use That Highlight Students’ Conceptual Understanding

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🔹 Frayer Model

Simple Explanation: A 4-box chart (definition, example, non-example, picture)

Example: Use it for “rational number” — definition in one box, examples in another, etc.

Extremely helpful for students to see examples and non-examples to know how to recognize different mathematical situations

🔹 Venn Diagrams

Simple Explanation: Show similarities and differences

Example: Compare sine and cosine functions — where are they the same, where do they differ?

🔹 T-Charts

Simple Explanation: Compare two things side by side

Example: Sine Rule vs. Cosine Rule — when to use each one

T-charts are used for graphs, so being familiar with using it for comparisons and functions can grow confidence in using t-charts

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What math topics lend themselves well to performance assessment tasks? 

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What math topics lend themselves well to performance assessment tasks? 

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🔹 Linear Functions

Follows a clear process (identify slope, y-intercept, graph it)

Easy to check each step for errors

Students can apply it to real life (money, time, distance)

Assessment Tip: Look for accurate graphs, correct equations, and interpretation of slope.

Example: Students design and compare cell phone plans using graphs and explain which one is better in different situations.

🔹 Trigonometry

Has step-by-step problem-solving (identify triangle type, choose rule, solve)

Visual and measurable progress (diagrams, real measurements)

Easy for students to get peer or teacher help on solving right and non-right triangles

Assessment Tip: Check correct rule use (SOHCAHTOA, sine rule, cosine rule), diagram labeling, and justification.

Example: Students use angles and distances to find the height of buildings and compare their results.

🔹 Geometry (especially area, volume, and scale drawing)

Students follow formulas and logical steps

Can create real-world models or drawings

Errors are easy to spot (incorrect calculations or scale)

Assessment Tip: Check use of correct formula, accuracy in drawings, and explanation of choices.

Example: Students redesign a classroom or playground to scale, calculating space used and presenting a proposal.

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How do you develop core skills such as communication, self-management, and research skills in the math classroom? What might you do in the future to improve your practice in developing these skills?

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How do you develop core skills such as communication, self-management, and research skills in the math classroom? What might you do in the future to improve your practice in developing these skills?

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🔹 Communication Skills

What it means:
Helping students explain their thinking clearly — in writing, out loud, with visuals, or by showing steps.

How I build it now:

I ask students to explain how they got their answers, not just show the final number.

I use “Think-Pair-Share” — students think on their own, then talk to a partner, then we discuss as a class.

I have students write short reflections after solving a problem: “What did I do? Why did it work?”

Example:
After solving a word problem using the Pythagorean Theorem, students explain each step as if teaching it to someone else.

Future ideas:

Start using math journals where students write regularly about what they’re learning.

Use sentence starters to help students talk more confidently (e.g., “I noticed…”, “I figured out…”, “I was confused about…”).

🔹 Self-Management Skills

What it means:
Helping students stay organized, work independently, manage their time, and keep going when things get tough.

How I build it now:

I give clear checklists for big projects so students know what to do and in what order.

I break work into smaller chunks with short deadlines to help students manage their time.

I model how to use strategies like highlighting important information or crossing out wrong answers.

Example:
During a multi-day math task, students get a progress sheet where they check off each part they’ve completed (like drawing diagrams, showing work, writing explanations).

Future ideas:

Give more time for goal setting at the beginning of a unit (“What do you want to get better at?”)

Include mini-lessons on how to handle frustration or distractions.

🔹 Research Skills

What it means:
Helping students learn how to ask questions, find information, and apply it to real-world situations.

How I build it now:

I do short investigations where students find math in everyday life (e.g., "What’s the safest form of transportation?" using data).

I guide students to reliable sources for math facts and models.

I ask students to create questions, not just answer them.

Example:
Students research different sports statistics and use averages or percentages to compare players.

Future ideas:

Use more open-ended projects where students pick a math topic connected to something they’re interested in (like music, gaming, or fashion).

Show students how to evaluate sources and explain where their information came from.

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Traditional math assessments tend to provide a narrow gauge of student learning-here's how some teachers are going deeper.

Jacob Lund / Alamy Stock Photo

Math teachers are rethinking student assessments in creative ways that allow them to get a broader picture of kids' conceptual math understanding, writes Madeline Will for Education Week. And while this creative approach to assessing student knowledge is, at least in part, due to remote learning, the strategies are powerful and make sense during a normal school year.

"I think this is good for a lot of us math teachers because it's forced us to rethink what assessments are supposed to accomplish," math teacher Matthew Rector told Will. "In the past, most of us have thought about assessments as ranking tools-give a kid a grade and move on. Assessments should be about moving mathematical knowledge forward."

While teachers have been rethinking assessments for some time, the shift to remote learning "has helped continue the momentum," Trena Wilkerson, the president of the National Council of Teachers of Mathematics told Will. "Teachers are thinking creatively and out-of-the-box in how to assess student understanding and student thinking, and then how to use that to support instructional decisions."

Use Familiar Tech Tools to Get at the Thinking Behind the Math

Rather than routinely asking students to solve a series of equations, Will spoke with teachers who are now asking students to explain a math concept, or "break down a problem and explain how they reach its solution." Students can choose how to record their work: in a Google doc, via video, or by snapping a photo of their work on paper. "It allows them to express their thoughts better," high school math teacher Bobson Wong told Will, and "it's very hard to plagiarize."

High school math teacher Emma Chiappetta likes to ask her advanced linear algebra students to create videos teaching their classmates applications and concepts. To check how effective her students are at explaining algebraic concepts, they each attempt to solve a few problems associated with a peer's video lesson.

Theresa Williams, a middle school math teacher at the University of Wyoming Lab School in Laramie, Wyoming, does five-minute interview assessments to gauge her students' progress and inform her teaching. It's helpful to create a checklist of what you expect a proficient student to be able to say, notes Williams. "It works great for kids who know a lot more than they showed on a [traditional] assessment," and gives students "multiple opportunities to show that they are proficient."

Try Math Magazines or Reflective Journaling

Writing can be a powerful mode of learning "because it engages both hemispheres of the brain," writes middle school math coordinator Alessandra King. "Effective writing also clarifies and organizes a student's thoughts." Some teachers are asking students to do "reflective journaling about math concepts" to assess students' grasp of the material, writes Will, providing a rich snapshot of where kids are at in a math unit.

King likes to get her math students reflecting and writing about math by asking them to create a math magazine where they focus on how math concepts are applied in the real world. "This has been one of my most popular projects-students are amazed to discover some of the myriad applications of math."

She starts by curating a list of math-related articles from newspapers, journals, podcasts, and videos for students to choose from and then summarize for an online magazine. "For assessment, I've created a simple rubric that looks at content understanding, clarity of communication, editing, critical thinking, initiative, and creativity," King says. Especially for students who enjoy reading and writing more than the "computational side of math," the project gives them an opportunity to showcase their understanding of math concepts-while gaining a "stronger appreciation of the usefulness and effectiveness of math."

Assign Projects With Real World Implications

Using the Massachusetts census, secondary math teacher Joey Grabowski's Algebra 1 students select categorical groups like gender or race, and quantitative variables like income or age, then "compare the distributions of two or more groups of people." Then they write a report about their statistical analysis. Using projects rather than unit tests to assess his students gives Grabowski a unique lens into their thinking, writes Will. "[With a statistical report], they are analyzing and critiquing things," he says. "Computers can do a lot of these calculations for us, but they can't interpret data."

When the pandemic hit, high school math teacher Chiappetta moved her statistics students' project proposals online. She traded gallery walks for virtual ones on Flipgrid and asked students to leave feedback for classmates' projects there. "After a certain point, it's not meaningful for me to quiz my students on their execution of calculations," said Chiappetta. "The projects that my students do allow me to assess their ability to apply those calculations in context."

Actively Embrace Mistakes

Creating a mistake-friendly classroom is valuable for all students across academic subjects-but especially in math class, which can be an anxiety provoking environment for students. Carol Dweck, a professor of psychology at Stanford University and author of Mindset: The New Psychology of Success, famously said: "Every time a student makes a mistake ... they grow a synapse."

Will notes that teachers are tapping into this idea in assessments too, normalizing errors and asking students to grapple with problems that are purposely solved incorrectly, requiring students to identify the mistakes and then figure out how to solve them.

Algebra 1 teacher Robert McAusland told Will he likes to give students opportunities for redos, giving students time to work through problems they struggled with in prior assessments. With students learning from home, he found that "initial assessment scores were unnaturally high, possibly because students were looking up answers at home." But as his students became more confident in their abilities to tackle tough problems, "scores are normalizing" and he's been able to convey to students that "learning to understand mathematics is not about right or wrong. ... There are no bad mistakes."

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Creative Ways to Assess Math Understanding

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Traditional math assessments tend to provide a narrow gauge of student learning-here's how some teachers are going deeper.