{"id":26442,"date":"2026-04-21T08:16:33","date_gmt":"2026-04-21T15:16:33","guid":{"rendered":"https:\/\/www.nwea.org\/blog\/?p=26442"},"modified":"2026-04-21T08:16:34","modified_gmt":"2026-04-21T15:16:34","slug":"increase-math-engagement-and-retention-with-the-discovery-learning-model","status":"publish","type":"post","link":"https:\/\/www.nwea.org\/blog\/2026\/increase-math-engagement-and-retention-with-the-discovery-learning-model\/","title":{"rendered":"Increase math engagement\u2014and retention\u2014with the discovery learning model"},"content":{"rendered":"<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-26444\" src=\"https:\/\/www.nwea.org\/blog\/uploads\/2026\/04\/increase-math-engagement-and-retention-with-the-discovery-learning-model_850x300_hero.png\" alt=\"\" width=\"850\" height=\"300\" srcset=\"https:\/\/www.nwea.org\/blog\/uploads\/2026\/04\/increase-math-engagement-and-retention-with-the-discovery-learning-model_850x300_hero.png 850w, https:\/\/www.nwea.org\/blog\/uploads\/2026\/04\/increase-math-engagement-and-retention-with-the-discovery-learning-model_850x300_hero-300x106.png 300w, https:\/\/www.nwea.org\/blog\/uploads\/2026\/04\/increase-math-engagement-and-retention-with-the-discovery-learning-model_850x300_hero-768x271.png 768w, https:\/\/www.nwea.org\/blog\/uploads\/2026\/04\/increase-math-engagement-and-retention-with-the-discovery-learning-model_850x300_hero-720x254.png 720w\" sizes=\"(max-width: 850px) 100vw, 850px\" \/>In <a href=\"https:\/\/www.nwea.org\/blog\/2025\/5-ways-to-help-students-develop-a-growth-mindset-in-math-class\/\">\u201c5 ways to help students develop a growth mindset in math class,\u201d<\/a> I wrote about several ways instruction in mathematics could borrow from instructional practices in ELA to bring about better student mindsets, opportunities, and conversations in class. Today, I\u2019d like to focus on another discipline that is prioritizing a teaching strategy that mathematics should absolutely be leveraging: the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Discovery_learning\">discovery learning model<\/a>.<\/p>\n<p>Consider for a moment the kinds of activities that make a science lesson successful. Some of the things that come to mind are collaboration, inquiry-based activities, the scientific method, and hands-on experiences. And what do all of these have in common? An opportunity for <a href=\"https:\/\/link.springer.com\/chapter\/10.1007\/0-306-48671-7_7\">teachers and students to experience critical exploration together in the classroom<\/a>.<\/p>\n<p>In many ways, mathematics instruction has much to learn from the discovery learning model approach. Most of the time, we don\u2019t think of treating math in this way\u2014as a discipline with ideas that should be explored, discovered, and made sense of authentically. All too often, the math subject matter becomes a cycle of stating the mathematical principle up front, modeling how to solve a problem using that principle, and then having students repeat it. In what ways could having students <em>discover<\/em> the mathematical principles themselves bring about more impactful learning experiences for our students?<\/p>\n<h2>What is the discovery learning model?<\/h2>\n<p>As noted in <a href=\"https:\/\/journals.physiology.org\/doi\/epdf\/10.1152\/advances.1998.275.6.S4\">\u201cA theoretical foundation for discovery learning,\u201d<\/a> the main objective of discovery learning is encouraging students to make sense of new information by relying on knowledge they already hold about the world. Marilla Svinicki, the article\u2019s author, explains that \u201cRather than being passive recipients of large amounts of relatively unconnected information, students are being asked to make their own connections between what they are learning and what they have experienced in real life.\u201d<\/p>\n<p>The discovery learning model is rooted in the cognitive model of learning, she goes on to explain, \u201cwhich focuses on what goes on in the mind of the learner as new information is acquired.\u201d It prioritizes the following:<\/p>\n<ul>\n<li><strong>Active learning. <\/strong>Active learning encourages students to focus on a task and key ideas so they can respond to questions related to a new topic and get immediate feedback on their comprehension from their teacher. Active learning can help with retention and motivate students, because they know they\u2019re expected to participate.<\/li>\n<li><strong>Meaningful learning. <\/strong>In the context of the discovery learning model, meaningful learning refers to the power of relying on students\u2019 knowledge and experiences to make meaning of new information. It also points to the power in having students explore a process independently, rather than simply following directions; work with their preconceived notions on a topic; benefit from a concrete approach, rather than an abstract one; explore new information in real-world contexts; understand how the knowledge can be used outside the classroom; and solve problems more independently.<\/li>\n<li><strong>Changes in beliefs and attitudes.<\/strong> As Svinicki notes, this characteristic of the discovery learning model refers to \u201ca change in the way students think about advanced knowledge and their ability to conquer it.\u201d Specifically, it\u2019s about supporting learners in understanding that knowledge can be uncovered by them, rather than only coming directly from an authority; processes are more valuable than just facts; and they can take responsibility for and an active part in their learning.<\/li>\n<\/ul>\n<h2>A sample activity<\/h2>\n<p>In my classroom, students were no strangers to visual models. In each lesson, students were either creating, analyzing, or leveraging information from a model. Some of the most powerful conversations ever to be had in our room were based on finding patterns and making conjectures about what was going on in a particular visualization, especially one they\u2019d never seen before.<\/p>\n<p>I started by displaying a model (for instance, Pascal\u2019s triangle shown below) and telling students to take a moment to observe the model independently, looking specifically for number patterns and trying to figure out how the model worked.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-26446 aligncenter\" src=\"https:\/\/www.nwea.org\/blog\/uploads\/2026\/04\/increase-math-engagement-and-retention-with-the-discovery-learning-model_233x203_PascalTriangle.png\" alt=\"A Pascal\u2019s triangle shows numbers grouped in a triangle shape. The number one appears at the apex of the triangle and along the left and right sides. Each number inside the triangle is the sum of the two numbers directly above it.\" width=\"502\" height=\"421\" srcset=\"https:\/\/www.nwea.org\/blog\/uploads\/2026\/04\/increase-math-engagement-and-retention-with-the-discovery-learning-model_233x203_PascalTriangle.png 502w, https:\/\/www.nwea.org\/blog\/uploads\/2026\/04\/increase-math-engagement-and-retention-with-the-discovery-learning-model_233x203_PascalTriangle-300x252.png 300w\" sizes=\"(max-width: 502px) 100vw, 502px\" \/><\/p>\n<p>Then, within our classroom community, members would take turns stating a theory about how a particular aspect of the model functioned. Students would be encouraged to go up to the board and justify their theory so it could be understood or acknowledged by all members of the community. Other members were encouraged to ask questions, offer a counter example, or agree with the conjecture. Once an idea had been proven or disproven by the group, the sharer would return to their seat, and another student and their idea would take the floor.<\/p>\n<p>The goal of the activity was for all students to engage. Engagement could take many forms; students could be engaged by asking clarifying questions, agreeing with an idea, providing an example, proposing a theory, supporting another\u2019s theory, or thinking critically about an idea. In every single instance, my classroom saw high levels of engagement and deep connections made among the many concepts of the discipline. Students were hungry to discover something for themselves or build on a novel idea from a classmate.<\/p>\n<p>The beauty of such a practice is that there are no limits to how far you can dig into a robust model that holds numerous secrets. Although the community may start with simple ideas like \u201cThe first and last number in each row is always 1\u201d or \u201cEach number is the sum of the two numbers adjacent to it in the previous row\u201d when looking at Pascal\u2019s triangle, I\u2019ve had students as young as second grade who dug deep to notice things like \u201cThe sum of each row is double the previous row\u201d or even \u201cShading only the odd numbers reveals an interesting triangular pattern.\u201d<\/p>\n<p>To push students even further, I\u2019d often extend the activity by issuing a challenge, like \u201cCan you find the Fibonacci sequence hiding anywhere in this model?\u201d<\/p>\n<h2>Behind the scenes of discovery learning<\/h2>\n<p>Now that I\u2019ve showcased a sample activity, let\u2019s look at what was happening behind the scenes.<\/p>\n<p>While the age-old \u201cI do, we do, you do\u201d strategy has its place in teaching, authentic discovery remains a vital tool. By allowing students to make conjectures and discover patterns within a model, they can express higher levels of engagement and a more solid ownership over their learning.<\/p>\n<p><a href=\"https:\/\/files.eric.ed.gov\/fulltext\/EJ753970.pdf\">The authenticity of a task also plays a vital role<\/a> in student learning dispositions and motivation. A shared problem is brought to the attention of the learning community, and together we wrestle with the mathematical secrets that are there for the taking. Once we home in on something, we can all practice the art of listening to a theory and weighing the validity of the presented idea. There\u2019s a reason critiquing the reasoning of others is a key facet of mathematical practice in most academic standard frameworks; this mathematical practice provides a meaningful path for <a href=\"https:\/\/scholarworks.umt.edu\/cgi\/viewcontent.cgi?article=1505&amp;context=tme\">deeper content knowledge and application beyond the discipline<\/a>.<\/p>\n<p>Additionally, the discovery learning model sheds light on an aspect of mathematics all too often ignored, oversimplified, or simply relegated to only grades K\u20132 by sets of content standards: pattern knowledge. Research suggests that <a href=\"https:\/\/repository.nie.edu.sg\/server\/api\/core\/bitstreams\/f6f9cfc7-1da7-4f25-a265-89d691da7163\/content\">pattern knowledge is predictive of proficiency in algebra<\/a>, and there is also a wide array of research to support the idea that success in algebra (especially eighth-grade algebra) is <a href=\"https:\/\/files.eric.ed.gov\/fulltext\/EJ753970.pdf\">correlated with higher college attendance and sustained interest in mathematics-related fields<\/a>.<\/p>\n<p>Another great benefit of the discovery learning model is the ability to meet students where they are. A student who tests below grade level may focus on the additive relationships or counting patterns in Pascal\u2019s triangle while other students may be able to make connections to a wider range of concepts, for example. Planning student experiences with <a href=\"https:\/\/www.nwea.org\/blog\/2024\/opening-up-students-mathematical-thinking-using-open-middle-math-problems\/\">Open Middle math problems<\/a> is a great way to get more bang for your buck in a single lesson; it keeps your learners with special needs\u2014including both the students below and above grade level\u2014engaged in the thinking.<\/p>\n<p>And thanks to the interconnectedness of mathematics, you can go further with some students who are ready to chew on something a bit more complex. A subset of the community may be able to recontextualize \u201cEach row is double the previous row\u201d as \u201cEach row\u2019s sum shows a consecutive power of 2\u201d or even \u201cShading the odd numbers reveals a pattern\u201d as \u201cShading the odd number reveals a fractal known as Sierpinski\u2019s triangle,\u201d which in and of itself contains a lot of new patterns to discover. (For more on meeting the needs of students ready for a greater challenge in particular, I encourage you to read <a href=\"https:\/\/www.nwea.org\/blog\/2026\/ways-to-provide-differentiated-instruction-for-advanced-learners\/\">\u201c3 ways to provide differentiated instruction for advanced learners.\u201d<\/a>)<\/p>\n<h2>Why the discovery learning model is so valuable in math instruction<\/h2>\n<p>Research supports that, with appropriate teacher guidance, <a href=\"https:\/\/pure.rug.nl\/ws\/portalfiles\/portal\/81069469\/Meta_analysis_of_inquiry_based_learning_Effects_of_guidance.pdf\">inquiry-based learning can be more effective than many other instructional methods<\/a>. An approach like this represents two major shifts from the traditional mathematics instruction model.<\/p>\n<p>First, the <em>what<\/em> of the learning task is not immediately obvious. Students are not given an opportunity to quickly see what a lesson intends and then disengage from the conversation. A task centered in discovery\u2014especially shared discovery\u2014can pull at the curiosity of students, making them sit forward and take note of the wealth of possibilities able to be discovered.<\/p>\n<p>Second, the <em>how<\/em> of the inner workings in the mathematics is not explicitly stated either. Students have every opportunity to find their own way through the terrain, which is ripe with ideas, methodologies, and connections. They have the time and space to observe the theories of others and try out their own hunches in a safe space that welcomes unique viewpoints and savors disagreement in strategy.<\/p>\n<h2>The two components of creating a community of observation<\/h2>\n<p>The process of creating a classroom community that is not only comfortable with but also maximizes learning possibilities with a discovery learning model is no small feat.<\/p>\n<p>It\u2019s no surprise that <a href=\"https:\/\/static.nsta.org\/connections\/middleschool\/201107harris.pdf\">inquiry-based learning is best supported by solid classroom management<\/a>. Once routines for your community are firmly in place, your classroom will be ready to intentionally dive into the unknown to explore and more concretely build foundations in the mathematical ideas your core curriculum is already hoping to foster.<\/p>\n<p>Another valuable teaching strategy that paves the way toward implementing learning like this is the think aloud. In fact, the success of this method hinges on students understanding how to share their thinking, which requires many of them hearing how you, the teacher, think as well. Metacognitive teaching strategies <a href=\"https:\/\/pure.rug.nl\/ws\/portalfiles\/portal\/95440471\/Effectiveness_of_self_regulated_learning_strategies_on_academic_performance.pdf.pdf\">substantially impact student performance in writing, science, math, and reading<\/a> and have <a href=\"https:\/\/pure.rug.nl\/ws\/portalfiles\/portal\/56352857\/Long_term_effects_of_metacognitive_strategy_instruction_on_student_academic_performance.pdf\">long-term positive effects<\/a> on student academic performance, too. Starting early and often with talking out your stream of consciousness when attacking a novel math problem can help your students feel more confident in the hazy space between the complete unknown and knowing exactly what to do. And that\u2019s where authentic learning happens.<\/p>\n<p>With these precursors in place, it just becomes a question of what to provide your learning community to explore together. Ideally, the models or visuals you use should be saturated with ideas that connect to the learning your class has already undertaken or topics they are about to define more clearly. But just because students haven\u2019t learned a formal definition or procedure for something does not mean they don\u2019t already have the <a href=\"https:\/\/faculty.washington.edu\/ekazemi\/theory%20into%20practice.pdf\">cognitive predisposition to explore that idea and find their own way through it<\/a>.<\/p>\n<h2>Selecting the right model for your classroom<\/h2>\n<p>Selecting great mathematical models to build student engagement, natural curiosity, and confidence in the discipline is an opportunistic endeavor. While mathematics already has its own set of effective models (like the hundreds chart, multiplication table, or <a href=\"https:\/\/mathforlove.com\/games\/prime-climb\/teach-with-prime-climb\/\">Prime Climb hundred\u2019s chart<\/a>) and interesting pattern displays from history\u2019s mathematicians (like Pascal\u2019s Triangle, Fibonacci\u2019s Spiral, and Sierpinski\u2019s Triangle), there are many other possible\u2014and even simpler\u2014applications of this approach to whole group discourse and inquiry.<\/p>\n<p>Data literacy continues to be an aspect of mathematics instruction that deserves more airtime, for example. Compiling data tables, graphs, or even infographics into a session of discovery for students could yield some profound conclusions about the specific data, how the data visualizations work, or what the data tells us about the real-world context it represents.<\/p>\n<p>Geometry also naturally lends itself toward visualization. Using angle relationships and definitions of polygons from a fourth-grade lesson, students could have the potential to analyze and make sense of many aspects of high-school level diagrams given the proper time and space to co-construct meaning, allowing you to get the most out of the discovery learning model.<\/p>\n<p>Finally, consider where a strong visualization inundated with information and points of entry could replace the beginning of a math lesson you are planning, an intro to an upcoming unit, or a day of instruction entirely. With proper planning and adequate support during the task, students will rise to the occasion and just might surprise you with how much they can figure out on their own.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>In \u201c5 ways to help students develop a growth mindset in math class,\u201d I wrote about several ways instruction in mathematics could borrow from instructional practices in [&hellip;]<\/p>\n","protected":false},"author":345,"featured_media":26447,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"_yoast_wpseo_metadesc":"Using a discovery learning model in your math classroom can help increase students\u2019 engagement with the content, and aid in retention, too.","footnotes":""},"categories":[649],"tags":[619,627],"grade_level":[830,831,832,833],"product":[],"theme":[],"coauthors":[{"id":345,"name":"Aaron Kugler, NWEA","link":"https:\/\/www.nwea.org\/blog\/author\/akugler\/","avatar_urls":{"24":"https:\/\/www.nwea.org\/blog\/uploads\/2026\/01\/Aaron-Kugler-bio-pic-2-24x24.png","48":"https:\/\/www.nwea.org\/blog\/uploads\/2026\/01\/Aaron-Kugler-bio-pic-2-48x48.png","96":"https:\/\/www.nwea.org\/blog\/uploads\/2026\/01\/Aaron-Kugler-bio-pic-2-96x96.png"}}],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v19.14 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Increase math engagement\u2014and retention\u2014with the discovery learning model - Teach. 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