The Best Ways to Teach Problem Solving

For decades, elementary and middle school classrooms have relied on step-by-step acronyms to teach word problems. You likely know them: CUBES, RACES, or READ. These checklists instruct students to circle the numbers, underline the question, box key words, and execute a predetermined operation. 

While well-intentioned, these mnemonic frameworks often inadvertently bypass actual mathematical thinking. When a student learns that "altogether" means add or "left over" means subtract, they aren't solving a problem—they are performing keyword substitution. The moment a problem introduces a subtle twist, the checklist breaks down, leaving students feeling helpless and stranded.

Moving beyond checklists requires shifting our goal from getting students to complete a task to helping them develop genuine problem-solving capacity.

Keyword strategies create a false sense of security. Consider this classic example:

"Maya has 12 stickers. She has 4 more stickers than Leo. How many stickers does Leo have?"

A student trained on word-problem checklists immediately spots the word "more" and adds the two numbers to get 16. The checklist worked as designed, but the reasoning failed completely. Real problem-solving requires students to visualize relationships, not scan for linguistic shortcuts.

So let's look at four strategies that build authentic problem solvers.  It means moving past formulaic steps to encourage flexible, adaptable thinking.

1. Launch Tasks with "Notice and Wonder" Before handing students a question to calculate, strip the actual question and numbers away. Present a situation—such as "A store is holding a sale on three types of apples"—and ask students two simple questions: What do you notice? What do you wonder? This forces learners to sense-make the context and construct meaning before jumping into calculations.

2. Standardize Visual Representation Instead of hunting for procedural keywords, train students to map out the structure of a problem visually. Tools like bar models or tape diagrams help students visualize relationships between quantities:

• Are we joining two parts to make a whole?

• Are we comparing two unequal quantities?

• Are we splitting a total into equal groups?

Once a student can represent the relationship visually, choosing the correct operation becomes intuitive rather than guesswork.

3. Shift to Collaborative, Vertical Spaces Inspired by Peter Liljedahl’s Building Thinking Classrooms, shift students away from isolated desks and paper worksheets. Having students work in small, randomly assigned groups at vertical non-permanent surfaces (like dry-erase boards) encourages risk-taking and active dialogue. Erasing a mistake on a whiteboard feels low-stakes, whereas erasing paper creates friction.

4. Ask "Keep-Thinking" Questions When a student gets stuck, the instinct is often to offer a hint that reveals the next step. Instead, offer prompts that keep the thinking on the student:

• "What do you know so far that isn't written in numbers?"

• "Can you draw what is happening in this scenario?"

• "Does your answer make sense in the real world?"

  
  

 Summed up, the primary goal is to find the correct answer by understanding mathematical relationships.  Students need to scan for action words and verbs by visualizing, discussing, and modeling. These two things should lead to flexible reasoning paired with resilience.  Teaching students to solve problems isn't about giving them a blueprint for every scenario—it's about building their confidence to navigate unfamiliar territory when there isn't a blueprint at all.  Let me know what you think, I'd love to hear.  

Mastering Area in Geometry Using Strategic Acceleration.

Geometry is often where math starts to feel tangible again. After months of abstract algebra, students finally get to work with shapes, angles, and visual spaces. But when teachers introduce the unit on area, a familiar hurdle appears: many students stumble, confusing perimeter with area, or freezing up when faced with composite shapes.

In the past, catching students up on area meant falling back on old-school remediation. A student struggling with the area of a triangle or circle might be pulled out of class to do elementary school grid worksheets, drilling formulas they memorized and forgot yesterday. This approach alienates older students, making them feel like they are permanently falling behind.

So, how do we modernize math recovery for the geometry classroom, specifically when tackling area? The answer lies in strategic acceleration.

Traditionally, geometry instruction relies heavily on rote formula memorization: Base times height divided by two. Pi r squared. But when students only memorize formulas without understanding why they work or what area actually represents, they struggle the moment a shape is turned sideways or transformed into a composite figure.

If a student lacks foundational number sense or fraction fluency, forcing them to memorize formulas doesn’t fix the gap; it just masks it. Traditional remediation treats this by slowing down the entire class or pulling kids out of grade-level content. Strategic acceleration, however, keeps students anchored in high-rigor geometry while providing just-in-time support right when they need it.

Imagine your geometry class is diving into a real-world project: calculating the cost of materials to lay sod and build a patio for a backyard landscape design (involving composite shapes made of rectangles, triangles, and semicircles).

Instead of spending two weeks re-teaching basic multiplication or decimals to the whole class beforehand, a modernized recovery approach uses a quick, targeted diagnostic:

• If a student struggles with multiplying decimals while calculating square footage, the teacher provides a quick, five-minute targeted scaffold during the design phase.

• If another student forgets how to decompose a complex shape into simpler triangles, the teacher uses visual manipulatives or a digital applet right then and there to rebuild that spatial intuition.

Students stay engaged because they are working on a compelling, grade-level project, but they receive the precise building blocks required to succeed.

To make area recovery stick, classrooms are shifting away from flat worksheets toward tactile learning. Students struggling with area formulas benefit immensely from physically cutting paper shapes apart to prove that a parallelogram is just a rearranged rectangle, or using grid overlays to see why a triangle's area is half of a rectangle's.

By blending grade-level geometric challenges with just-in-time arithmetic support, we turn an area unit from a source of frustration into an opportunity for visual discovery.  Let me know what you think, I'd love to hear.

Happy Labor Day

 

How Middle Schools Are Modernizing Math Recovery

Remember the traditional approach to helping students who fell behind in math? It usually looked like a punitive cycle of endless remediation: pulling struggling students out of class, forcing them to sit through isolated worksheets from lower grade levels, and drilling them on basic skills while their peers moved forward.

Not only did this bore students to tears, but it also widened the very achievement gap it was supposed to close. By keeping kids stuck in the past, we unintentionally locked them out of the rigorous grade-level content they deserved.

Fortunately, post-pandemic academic recovery has evolved. Middle schools across the country are shifting away from outdated remediation models and embracing a powerful alternative: strategic acceleration.

The logic behind old-school remediation seemed harmless enough: If a student doesn't understand sixth-grade fractions, let's stop teaching them seventh-grade math and make them practice fifth-grade fractions until they get it.

In reality, this approach is a recipe for disaster. When students are perpetually held back to review foundational concepts, they miss out on critical grade-level instruction. Over time, the gap between where they are and where they need to be becomes an unbridgeable chasm. More importantly, it signals to students that they aren't "math people," fueling the very engagement cliff we try so hard to avoid.

Strategic acceleration takes a fundamentally different approach. Instead of looking backward and slowing down, it looks forward and prepares students for what is coming right now.

Acceleration is built on just-in-time support rather than just-in-case review. When a middle school math class is about to dive into a new unit on solving multi-step algebraic equations, the teacher doesn't spend three weeks reteaching negative numbers from scratch. Instead, they embed a brief, targeted review of negative integers right at the exact moment students need that specific skill to tackle the grade-level problem.

This model relies on a few core practices:

• Prerequisite Diagnostics: Identifying the exact, micro-skills required for an upcoming unit rather than assuming a student needs an entire year of reteaching.

• Scaffolded Grade-Level Access: Ensuring every student—regardless of where they started the year—opens the same grade-level textbook and engages with the same core standards.

• Embedded Support: Providing targeted coaching during class time so that intervention happens fluidly rather than segregating students.

By modernizing how we approach math recovery, we send a clear, empowering message to our students: You belong in this grade-level classroom, and we will give you the tools you need to succeed here.

Strategic acceleration bridges foundational gaps without sacrificing rigor. It transforms middle school math from a source of anxiety and remediation into a launchpad for future success, proving that every student can rise to the challenge when given the right support at the right time.  Let me know what you think, I'd love to hear. 

Algebra I Lesson Plan: Analyzing Local Housing Market Trends

Today, I'm sharing an Algebra I lesson plan that can easily be used to expose students to Quantitative Literacy. This gives you an idea of how to integrate the topic into the classroom.

Algebra I Topic Classification

This lesson falls under Linear Functions and Modeling (specifically: Writing Linear Equations in Slope-Intercept Form and Interpreting Slope and $y$-Intercept in Context). It can also be paired with units on Scatter Plots and Lines of Best Fit.

Lesson Overview

• Grade Level:  Algebra I

• Objective: Students will write, interpret, and use linear equations to analyze housing market price changes over time and make predictions about future affordability.

• Duration: 1-2 class periods (approx. 45–60 minutes)

Materials Needed

• Printed or digital student handouts with local housing market data sets.

• Graphing calculators (or online graphing tools like Desmos).

• Rulers and colored pencils.

1. Warm-Up: The Real Estate Check (10 Minutes)

Begin with a hook that connects to quantitative literacy:

• The Hook: Project a recent headline or local real estate listing showing the median home price in your city.

• The Question: Ask students: "If a house cost $200,000 ten years ago and costs $320,000 today, did it increase by the same dollar amount every single year? How could we model that mathematically?"

• Briefly review slope-intercept form ($y=mx+b$) and remind students what m (rate of change/slope) and b (initial value/y-intercept) represent.

2. Direct Instruction: Setting Up the Model (10 Minutes)

Walk students through how to convert raw housing data into a linear model.

Provide a simplified data set for a fictional or generalized local neighborhood:

• Year 2016 ($x=0$): Median Home Price = $240,000

• Year 2026 ($x=10$): Median Home Price = $340,000

Guide the class through finding the slope (m):

• $$m=\; (340,000-240,000)/(10-0)=100,000=10,000$$

Explain the real-world interpretation:

• Slope ($m=10,000$): The median home price increases by an average of $10,000 per year.

• $y$-Intercept ($b=240,000$): The starting median home price in the baseline year (2016) was $240,000.

• The Equation: $y=10,000x+240,000$ (where x represents years since 2016, and y represents the median price).

3. Guided Practice: Housing Market Investigation (20 Minutes)

Divide students into pairs and hand out a localized activity worksheet using real or realistic regional data.

• Task 1: Plot two data points representing median housing prices from past years on a coordinate plane. Draw a line connecting them.

• Task 2: Write the linear equation for your local market trend.

• Task 3: Use the equation to interpolate and extrapolate:

  • Interpolation: What does the model predict the median price was in 2021 ($x=5$)?

  • Extrapolation: If this trend continues linearly, what will the median home price be in 2036 ($x=20$)?

• Task 4 (Critical Thinking / QL): Ask students: "Is a linear model realistic for housing prices over a 20-year span? Why might housing markets behave more like exponential functions or experience sudden market crashes?"(This bridges Algebra I linear concepts smoothly into future Algebra II exponential units).

4. Independent Practice & Wrap-Up (10 Minutes)

• Have students write a brief 3-sentence "Real Estate Analyst Summary" answering:

  1. What is the annual rate of change in our market model?

  2. Based on your calculation, will a young adult graduating college today be able to buy a home in this area in 10 years if starting salaries stay flat?

• Closure: Review answers as a class, highlighting how abstract algebra concepts directly help us understand personal finance and community economics.

  Here is a lesson that you can easily use in your classroom.  In fact, the data is easy to get from your local real estate agents or the news and that makes it an easy lesson to implement. Let me know what you think, I'd love to hear.  Have a great day.

Why Quantitative Literacy Belongs in Every Math Classroom

For generations, the traditional high school math track has followed a rigid path: push through algebra, conquer geometry, survive trigonometry, and ultimately reach the holy grail of AP Calculus. But as our world becomes increasingly data-driven, a critical question arises: How often does the average adult actually use calculus to solve daily problems?

Enter Quantitative Literacy (QL)—often called numeracy. It is time we brought this vital skill front and center in our middle and high school math classrooms.

Quantitative literacy is the ability to understand, interpret, and critically evaluate numerical information in everyday contexts. While traditional math focuses heavily on abstract symbolic manipulation and theoretical proofs, QL focuses on practical sense-making.

Think about the quantitative challenges adults face every single week:

• Decoding a complex healthcare plan or retirement portfolio.

• Critically evaluating misleading graphs or statistics shared on social media news feeds.

• Calculating the true cost of a loan, mortgage, or credit card interest rate.

• Analyzing environmental or public health data to make informed civic choices.

In short, QL isn't just about getting the right mathematical answer; it’s about understanding what numbers actually mean and using them to make sound, rational decisions.

Quantitative literacy can be integrated into normal math classes.  Too often, schools treat quantitative literacy as a separate elective reserved for students not pursuing traditional STEM pathways. However, QL shouldn't be boxed into an isolated corner. It is a mindset that can—and should—be woven directly into standard algebra, geometry, and statistics courses. Integrating QL doesn't mean throwing out your core curriculum; it simply means transforming how students interact with it.

If you want to blend quantitative literacy into your existing math lessons, you don't need a complete curriculum overhaul. You can start with these practical strategies:

• Swap Out Artificial Word Problems: Instead of asking about "two trains leaving a station at the same time," use real data. Have students analyze local housing market trends, consumer price index changes, or current economic datasets.

• Debunk Media Statistics: Bring in infographics or news articles. Ask students to spot flaws in graph scaling, identify omitted variables, or question whether a reported correlation actually implies causation.

• Incorporate Financial Math: Use linear and exponential functions to explore real-world scenarios like compound interest, student loans, inflation, or the depreciation of a car.

• Prioritize Estimation (Fermi Problems): Give students open-ended questions like, "How much plastic waste does our school generate in a year?" Let them make reasonable assumptions, estimate variables, and defend their calculations.

By integrating quantitative literacy into normal math classes, we bridge the gap between abstract school mathematics and real-world competence. We stop teaching math as a series of dry hoops to jump through and start teaching it as an essential life skill.  Let me know what you think, I'd love to hear.  Have a great day.

How to Rescue Student Engagement in Middle School Math

Remember when math class meant colorful fraction tiles, fun geometry puzzles, and enthusiastic hands shooting up into the air? For many students, that magic starts to fade the moment they cross the threshold into middle school.

Educational researchers and national polling data have tracked a troubling trend known as the student engagement cliff. As children transition from elementary to middle school, their motivation and enthusiasm for learning drop precipitously, hitting an all-time low by the time they reach high school.

In mathematics, this cliff feels more like a sheer vertical drop. Why does it happen, and more importantly, how can educators climb back up?

The root of the engagement cliff in middle school math is largely conceptual. Elementary math is heavily concrete—students count blocks, measure physical objects, and work with tangible quantities.

Suddenly, middle school introduces pre-algebra, negative integers, abstract ratios, and unknown variables like x and y. Math stops being about things you can hold or see and transforms into a language of invisible rules and abstract symbols.

For young adolescents undergoing massive developmental, social, and neurological changes, sitting through rows of repetitive worksheets to memorize procedures they don't understand feels entirely disconnected from reality. The common refrain—"When am I ever going to use this?"—becomes a cry for help.

Fortunately, educators and curriculum designers are fighting back against the cliff by fundamentally reshaping how math is taught in grades 6 through 8. The strategy relies on three powerful pillars:

• Project-Based Learning (PBL): Instead of isolated word problems, students tackle open-ended projects. They might design a tiny home to understand scaling and ratios, or analyze local sports statistics to study probability. Math becomes a tool to build something real.

• Gamified Inquiry: Middle schoolers love competition, strategy, and play. Educators are leveraging interactive platforms and collaborative math escape rooms to turn complex problem-solving into an engaging team challenge rather than a solitary chore.

• Identity-Driven Context: Culturally responsive teaching connects mathematical concepts directly to students' lived experiences and future aspirations, proving that math belongs to everyone.

Reversing the engagement cliff requires us to stop treating middle school math as a dry "gatekeeper" subject meant to weed students out. Instead, we must treat it as a gateway to creative thinking and problem-solving. By trading rote memorization for dynamic, hands-on inquiry, we can help students not just survive the middle school transition, but actually rediscover the joy of math. Let me know what you think, I'd love to hear.  Have a great weekend.