1: Mathematical Thinking, Problem Solving and Math as a Language
- Page ID
- 180982
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Thinking Mathematically to Solve Problems
Elementary school teachers should understand the interplay between thinking mathematically and problem-solving because these skills are foundational to students' success in mathematics and beyond. Mathematical thinking involves understanding why concepts work, making connections, and applying knowledge, while problem-solving is the active process of using these skills in new and challenging situations. Together, they help students move beyond memorization and toward a deeper understanding of math. (Devlin, 2012; NCTM 2020)
Developing problem-solving skills in young children is essential for their long-term success in mathematics and beyond. Early exposure predicts later achievement, as deep mathematical thinking builds a strong foundation for advanced learning. It also fosters positive mathematical identities, helping all children, regardless of background, see themselves as capable problem solvers. Rather than rote memorization, problem-solving promotes conceptual understanding by encouraging reasoning, pattern recognition, and connection-making. Engaging, thought-provoking problems increase student motivation and enjoyment, leading to higher achievement, especially when children collaborate and explore multiple strategies. (Feikes, Schwingendorf, and Gregg, 2018; Seda and Brown, 2021). These skills align with key mathematical practices, such as reasoning, explaining, and making connections, as emphasized in the Common Core Standards (CDE, 2013). Ultimately, mathematical thinking extends beyond numbers, equipping students with critical thinking and perseverance, essential for lifelong success.
Teachers play a vital role in nurturing these abilities. By creating supportive learning environments, designing rich mathematical tasks, and facilitating meaningful discussions, teachers can help young learners become confident, capable problem solvers who see math as a tool to explore and understand the world around them. Chapter 1 will be the jumping off point of our journey to learn to be better mathematical thinkers, problem solvers, and ultimately better teachers of mathematics.
Chapter Learning Outcomes
Essential Questions: How do we approach complex problems in everyday life? What strategies can help us become better problem solvers? How can understanding mathematics as a language help us communicate complex ideas across different disciplines and real-world contexts?
Learning that Transfers: You will engage with problem-solving strategies and mathematical thinking to develop flexible, adaptable skills. These skills extend beyond solving mathematical problems and equip you to tackle complex, real-world situations and be empowered to innovate and think critically in diverse contexts.
- SLO 1.1 Students will apply and evaluate multiple problem-solving strategies, including Polya's 4-step process, mathematical thinking, and algebraic reasoning to mathematical problems and real-world scenarios.
- SLO 1.2 Students will analyze, translate, solve, and interpret mathematical problems and real-world scenarios.
- SLO 1.3 Students will create a mathematical problem-solving scenario inspired by real-world observations and analyze it using strategies such as counting, measuring, identifying patterns, or quantifying spatial relationships.
- SLO 1.4 Students will be able to analyze and use mathematical language, symbols, and structures to communicate mathematical ideas effectively, demonstrating an understanding of precision and logic in problem-solving.
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Puddles and Paths: A Lesson with Grandpa |
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Morgan and Grandpa, a fine pair to see, Were headed to the park, just as happy as could be. Morgan skipped and she twirled, her heart full of glee, And she called out, “Grandpa, oh Grandpa, look at me!” They walked down the lane when they came to a stop. A puddle so giant, it looked like a swamp! “What do we do?” Morgan asked with a frown. “We can’t jump over or walk around town.” “Let’s think,” said Grandpa, his eyes all aglow. “Problem-solving will guide where we go. We’ll start with step one: just look and define. What’s the problem, dear Morgan? What clues can you find?” “The puddle is wide,” Morgan said with a grin. “But a path of small rocks runs right through the middle and in!” “Step two,” Grandpa said, “is to think up a way. Could we hop on the rocks to keep our shoes out of the spray?” “Let’s try it!” said Morgan, as brave as could be. She leapt to the first rock, then cried, “Grandpa, look at me!” Step three was in action as Grandpa came too, And soon they were over, their path good as new. “What did we learn?” Grandpa asked with a cheer. “Step four is to check: was the solution clear?” “It worked!” shouted Morgan. “The rocks kept us dry. Problem-solving is fun when we give it a try!” |
Story written by ChatGPT with thoughtful prompt engineering