Adventures in Corequisite STEM Calculus: Warm-Up Routines that Build Curiosity, Confidence, Conceptual Connections, and Community

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classroom where students are gathered around cards at a desks

Hello again to all fellow adventurers!

This is the second post in my blogging journey, Adventures in Corequisite STEM Calculus. My goal in this blog series is to share pivotal course design decisions and instructional strategies that can open the world of calculus to all students, irrespective of their prior math experiences.

In my first blog, I shared the success of San Diego Mesa College’s inaugural open access, supported calculus class. Thirty-one of thirty-nine students earned a passing grade in my class, and twenty-seven had plans to take Calc 2. But these are not just numbers on a spreadsheet. These numbers represent real, impactful changes in students’ academic paths and their relationship with mathematics. This was evident to me in the first two weeks of this semester, when nine students from this inaugural class made it a point to visit—often with their classmates from last semester, to share the next step in their STEM journey or just say hi. (Usually, only one or two students out of 100 will circle back for a chat, but I have never had nearly a quarter of a former class come round!) It was heartwarming to see this direct evidence of the STEM community we built together last semester.

As we embark on a new semester, filled with new faces and challenges, I find myself reflecting on the elements that fostered the supportive learning environment my students and I created in the fall. A couple of weeks into the new term, the positive vibes and potential for success are palpable in my class, and I aim to communicate in this blog series the specific strategies that contribute to making this happen.

In this blog post, I will focus on how we begin each class meeting—a crucial 15-30 minutes that promote curiosity and deep engagement, paving the way for a productive class session. We’ll explore three of my favorite types of warm-up activities: Card Sorts, Which One Doesn’t Belong?, and Notice and Wonder. As you continue reading, you will find logistical details for using these warm-ups, examples, and most importantly an explanation of how these activities build curiosity, confidence, conceptual connections, and community.

My Class Meeting Schedule: My supported calculus class meets Mondays and Wednesdays 8:35 – 11:00am and Friday 8:35 – 11:45am. The structure for each class meeting follows a similar trajectory, with the exception that on Fridays we use the extra 45 minutes for our weekly quizzes and reassessments. We will explore what happens after the Warm-up activity in future posts.

A warm-up is a simple and versatile routine that sets the stage for a productive learning experience. For me, a good warm-up activity motivates students to explore and question – it generates curiosity. It invites students to engage in a task that is intellectually safe and fun while also fostering critical and creative thinking. It is an easy way to support deeper learning and develop a community of learners that values everyone’s knowledge and growth. A supported calculus class affords more time to use warm-up activities, though I have successfully integrated them into all of my classes.

Tip: Warm-up activities are best when they are collaborative. In my class, students quickly get into the routine of teaming up (usually groups of three, sometimes pairs) and getting to work on a task that is projected on the overhead or grabbing their handout – one per group. Latecomers are integrated into existing groups or form new ones, depending on how much time is left and if they’re able to catch up quickly.

Tip: While students are working, our embedded tutor fields questions and provides some guidance. I mainly observe and listen to students’ conversations, gathering insights that will inform the conversation when we come back together as a class. This usually takes about 5-15 minutes. When it’s time to wrap-up and reflect (another 10-15 minutes), there is lively discussion. I typically facilitate this by highlighting the ideas of quieter students or asking students to contribute ideas I overheard in their group – elevating the voices of students who may be less confident in their ideas.

Card Sorts

The Card Sort is one of my favorite strategies to kickstart a calculus adventure. In a Card Sort activity, each group of students is given one set of cards. Each card has a different piece of information and students work together to sort the cards based on the task at hand.

The beauty of Card Sorts lies in their simplicity and versatility. They can be tailored to any topic and the tasks can vary to accomplish different types of learning goals. The three examples below illustrate this versatility.

Example: A Card Sort with an Open-ended Task

Below is an example of a Card Sort with an open-ended task: Sort the cards based on what you notice. An open-ended Card Sort is a good option when students have different levels of familiarity with a topic because there are many points of entry and many good answers. I have used this particular Card Sort as a warm-up exercise early in the semester before a short lesson on polynomial functions that precedes a discussion of limits. Without prior knowledge, students can group the polynomials by different features that mathematicians recognize as end behavior, the existence of extrema, whether the function has zeros. Students with more advanced knowledge about polynomials will be able to contribute observations about the effect of degree or the leading coefficient on the shape. If the warm-up is laying the groundwork for introducing limits, in the wrap-up you could ask questions similar to those in slides 10-22 in this Desmos activity on polynomial functions that I created based on the work of Amy McNabb. (I don’t use Desmos activities in class because I want students to be interacting with each other instead of their devices, but this gives you a sense of how to transition from the warm-up activity and lay the groundwork for the day’s lesson where polynomial functions will be used.)

A screenshot of a Desmos activity with the instructions "create groups based on what you notice about the graph shape" and 9 different polynomial functions with their graph and equation below.

Example: A Card Sort as a Matching Exercise

More often, my Card Sorts have a correct answer, such as this one created by Bryan Passwater. In this Card Sort, students match an equation to a limit statement and a graph. Even though there are correct answers, there is not a procedure to follow here. Students can work from any starting point and in any direction. I might use this Card Sort as a warm-up to give students the opportunity to solidify their understanding of a previous lesson on limits, to practice with function and limit notation, and to build toward a more formal lesson on continuity. After the sorting activity, as a class we discuss some of the matches with students explaining their reasoning, focusing on which cards presented the largest challenge and why that was the case.

Tip: Occasionally, a group of students will finish early. I always have a few extension questions in my back pocket to keep the faster groups working. For example, in the previous matching exercise, if students used a process of elimination to finish up their matches, I ask them to focus on the matches that were the most challenging and explain why their matches make sense mathematically. I don’t hesitate to wrap up a Card Sort before all of the groups have finished the sorting. Frequently, only half of the groups have finished when we begin the class discussion. In these situations, I start the debrief by asking the groups that did not finish to pick a card or cards that were the most challenging to them, and we start the conversation there.

Example: A Card Sort with a Sequencing Task

I’ve also used Card Sorts to demystify even the most daunting proofs and derivations, such as using the limit definition to find the derivative of sin(x). As a Card Sort, students order the steps in the derivation of the derivative. This focuses the conversation on the kind of thinking that goes into reading a mathematical proof–noticing the what and articulating the why of each step. This particular Card Sort provides opportunities for students to recognize the use of a trig identity, the use of sophisticated factoring, and the use of the properties of fractions. I used this warm-up activity after a lesson on the limit definition of the derivative for polynomial and rational functions. As a wrap-up, we reconstructed the order with different groups providing the sequential steps in the derivation and the mathematical justifications – if you’re curious, I wrote a quick reflection of how this activity went as part of my evaluation last semester. 

card sort showing the steps of the limit definition of a derivative for f(x)=  sin(x)

Tip: Card Sorts with a sequencing task help students’ build skills in reading and understanding worked examples from a math textbook. This might be particularly valuable in a supported calculus course where students may be at different places with this skill.

How does a Card Sort build curiosity, confidence, conceptual connections, and community?

  • Students engage directly with the material, using their hands as well as their brains. The act of moving slips of paper around creates a ‘rough draft’ space where they don’t have to commit to an idea immediately. This lessens the fear of making a mistake, encourages risk-taking, and builds confidence.
  • The Card Sort prompts curiosity and discussion. Students must argue, reason, and persuade their peers about where each card belongs. This dialogue deepens their understanding of the interconnectedness of calculus concepts, and every student can contribute, even if it is just observing differences in the cards.
  • The Card Sort can be a form of retrieval practice that feels less like a test and more like a puzzle, making the learning process enjoyable and the concepts memorable.
  • The Card Sort can reveal students’ pre-existing knowledge and misconceptions without the pressure of right or wrong answers, providing a safe space for exploration and correction.

Tip: The physical act of moving the cards around is essential, so you need to create cards. This requires some upfront investment in time – and yes, even a few paper cutter battle scars from those last-minute creations, but the deeper learning, community and fun that results is worth the time.

“Which One Doesn’t Belong?”

“Which One Doesn’t Belong?” (WODB) warm-ups are as simple as they are profound, presenting students with four seemingly disparate elements— usually graphs or functions—and challenging them to identify the outlier. But the twist is that, in a well-designed WODB, every choice can be defended, making the ‘right answer’ a matter of perspective and reasoning.

Example

I used this WODB as the first activity in my supported calculus class as the icebreaker. Students were in groups of three. I projected the graphs of four rational functions (though I did not use this vocabulary yet) and asked them, “Which One Doesn’t Belong?”

screenshot of slide with "Which one doesn't belong" and four rational functions pictured.

After letting the groups come up with an argument for a single graph, I asked them to find a reason that each of the graphs pictured might not belong. For the wrap-up, different groups described their reasons for the graph that did not belong. I added to the conversation by providing the mathematical vocabulary (e.g., continuous, asymptote) and notation (e.g., function notation, limit notation) associated with rational functions that formalized their ideas.

Tip: Design your own WODB warm-ups to address specific misconceptions or difficulties you have observed in your class. For example, the four functions y = x^5, y = \sqrt{x}, y = 5^x , y = \frac{1}{3x+2} provide an opportunity to recognize functions that can, and cannot, be differentiated using the power rule. The added challenge of identifying a reason that each function would not belong broadens the conversation to a wider review of points of discontinuity, points of inflection, asymptotes, domain, and range. 

How does a WODB problem build curiosity, confidence, conceptual connections, and community?

  • WODBs engage students at all levels of understanding by providing multiple entry points to the task. A student might initially choose based on a simple observation but deepen their understanding through the discussion of other perspectives. 
  • WODBs are all about comparing and contrasting, a fundamental critical thinking skill that helps students make conceptual connections. 
  • WODBs are like puzzles. They are fun and encourage creative thinking that builds curiosity and confidence.

Tip: By simply putting a WODB on the projector, some students may wait to engage until the discussion. Be sure to have students first work in pairs or groups of three. This provides a safer space for students to test their ideas and paves the way for a richer class discussion.

Notice and Wonder

A Notice and Wonder warm-up is perhaps the easiest to implement. All you need is a slide with something from the day’s material on it – a graph, problem, worked example, real-world scenario.

Example

I recently used this Notice and Wonder in my supported calculus class after previous work with polynomials and continuity. One of the calculus learning goals for the day was the Intermediate Value Theorem and Bolzano’s corollary about the existence of roots. I projected the image below on the overhead. 

I started with the basic prompts: What do you notice? What do you wonder? Students were given a few minutes to discuss their responses with a partner before the class discussion. Working from their observations, we had a great conversation about the likely degree of this polynomial (and why) and about the leading coefficient (and why). We also discussed the end behavior and used limit notation to represent it.

And then, since the goal of the day was to think more about continuity and intermediate function values over an interval, I added new questions: Where must there be roots? Where might there be roots? My students were quick to say there must be a root behind B, but it took a bit of prompting to get them to articulate why they could be sure of that. This simple warm-up helped my students generate the idea underlying the Intermediate Value Theorem and focus on its essential connection to a function’s continuity. This warm-up was inspired by the Desmos’ Calculus Collection: Intermediate Value Theorem, which contains additional prompts that could be used to structure the class discussion.

How does a Notice and Wonder problem build curiosity, confidence, conceptual connections, and community?

  • A Notice and Wonder warm-up intentionally asks students to be curious – and curiosity is a great way to start a learning journey!
  • A Notice and Wonder warm-up provides the opportunity for all students to bring their unique perspectives and questions to the forefront. It is a safe way to ask a question–there is no stigma with wondering. 
  • A Notice and Wonder also cultivates in students a mathematician’s eye– a way of seeing and noting key features of a mathematical object that are important to understanding and problem-solving.

Tip: I recommend the excellent book Choosing to See: A Framework for Equity in the Math Classroom, by Pamela Seda and Kendall Brown. I loved their spin on the typical “what do you notice?” and “what do you wonder?” prompts. They ask students to write down (1) five factual statements about the image and (2) two questions they have about the image. They also give teachers helpful guidance on a routine that accepts each factual statement without evaluating it and, instead, revisits the statements at the end of class to see if the class wants to make adjustments.

As we wrap up this exploration of warm-up activities and continue our journey through the world of supported calculus, let’s carry with us the understanding that how we start each class is important. If we use those first 15 minutes of class to generate curiosity, build confidence, make conceptual connections, and foster community, we have set the stage for a productive learning experience. Whether you plan to lecture or incorporate active learning later in the lesson, a warm-up invites students to think deeply, engage fully, and embrace the intellectual joy to be found in the study of calculus.

If you are on this adventure into supported calculus, planning to start the journey soon, or just want to chat about calculus or teaching math in general, I’d love to connect with you. I am hosting community hours on Thursdays from 12-1pm in a Coreq Calc Discussions Zoom Room every week until May 23rd. Please join me. I am also happy to chat or answer any questions via email (kspoon@sdccd.edu).

Until our next journey together,

Kelly Spoon

Kelly Spoon is a Professor of Mathematics at San Diego Mesa College in the San Diego Community College District. She has been at the forefront of her institution’s response to AB705 and AB1705, playing a key role in coordinating Mesa’s corequisite classes for Intermediate Algebra, Statistics, and, most recently, Calculus.

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