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A Teacher's Guide To Explaining the CONUNDRUM OF MATHEMATICAL WORD PROBLEMS

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When writing my book was in its infancy, I thought I was a genius. I was certain the reason why students struggled with word problems was mostly because we were not assessing their mathematical…

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A Teacher's Guide To Explaining the CONUNDRUM OF MATHEMATICAL WORD PROBLEMS

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real world word problems can often be funny

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Maths dance party | maths symbols dance | #dance #funny #funnyshorts #viralvideos #viral#youtube

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by Diane Kue

Mathematics has existed in every civilization throughout history. Its worldwide presence is why so many people claim it as a universal language. Further perpetuation of the claim is present in the classroom. A classroom full of students who may or may not all speak the same language can successfully and individually solve the same problem. After all, doesn't 1 + 1 = 2 everywhere, no matter the language, country, or culture if the symbolic representation shows that one whole, joined with another one whole, makes two wholes?

However, the idea that math is a universal language because the answer to a problem is always the same no matter what language is spoken can be misleading. This notion focuses on the answer; it does not acknowledge that the to solve for the answer-or even the perspectives on the processes-can vary. When we focus on the processes to solve, we can discover and unveil various perceptions on or ways to look at problem solving. We find the whys behind the hows of problem solving.

For example, the Chinese used negative integers based on a need to calculate owed money in 600 BCE. Their "invention" or "discovery" was based on a cultural, economic need due to open trading. Europeans, on the other hand, did not use negative numbers because of their view that "they darken the very doctrines of the equations" (Maseres, 1758). However, because they could not deny the need for them, they created "laws of arithmetic" to define negative numbers. The Chinese and European approaches differ not just because they use a different process but because their perception of the concept-their perception of the problem to be solved-was different.

Examine the three pictorial examples below of solving the same problem: 12 x 12 = 144 The process used by the solver is distinctly different in each example, yet the answer for all three is the same.

At a glance, none of these pictorial representations share a perspective to solving the same problem. They have distinctively different approaches from a pedagogical perspective as well.The ancient Egyptian method represents an understanding of binary to solve successfully (McGann, 2008). The Chinese method reflects an understanding of place value to solve successfully. The last example represents equal grouping to solve successfully and is a method we use in the United States. Although 12 x 12 = 144 appears to be a simple multiplication problem, each culture had/has a different foundational understanding of multiplication and, consequently, varies in their approach and process to solving.
Thus, stating that mathematics is a universal language deceptively omits how diverse problem-solving can be, even when the final answer is the same.
As math teachers, we can reflect on our practices to ensure that we encourage students' various processes to problem solving and embrace their differing perspectives to those processes-even if we do not immediately recognize or understand them. In understanding that perspective impacts the solving process, sharing perspectives becomes imperative to enriching learners. In my last article for this blog, I referenced the need for the teacher to gain access to student thinking to address student needs. When students gain a mathematical perspective on problem-solving, the teacher must find a balance between guidance and student-led self-exploration (Huang & Barlow, 2013). Language is adamantly bound to this balance to explain concrete, pictorial, or abstract perspectives. Without it, concepts and ideas cannot be explored, connections cannot be made, and application across disciplines does not occur.

Activity: Read-Represent-Explain

Read-Represent-Explain is a classroom activity that balances teacher guidance, student perspective, and sharing of process thinking (Mendoza & Beene, 2022). A brief synopsis of this activity is as follows:

1. The teacher reads aloud a problem.

2. Students represent how they solve the problem concretely, pictorially, and/or abstractly.

3. Students explain their process to a partner/small group.

As teachers walk with intention to observe students solving and explaining, they gain access to thinking and perspective and can provide guidance if needed (Motley, 2022). The balance between guidance and self-exploration occurs when students lead discussions and learning, and teachers shift from explaining to occasionally asking questions that encourage further exploration of ideas. Teachers and students also gain new perspectives from one another. By asking questions, they honor one another's knowledge and further develop content-specific language skills (Ewing, 2021).

Mathematics is universal. However, how a problem is solved can be nuanced and differ based on perspective. With this understanding, we best serve our students when we create learning opportunities in which our students can share and gain problem-solving perspectives.
Diane Kue, author of Solved: A Teacher's Guide to Making Word Problems Comprehensible, will be hosting "Teaching Math to English Learners" on February 23, 2023 with Adrian Mendoza, author of Teaching Math to English Learners, and Dr. Jim Ewing, author of Juan Jose, You are Especial.

References
Ewing, J (2021). Math for ELs: A "PATH" for success: 4 mindsets to ensure success for English learners. TEPSA Leader, 34(4), 1-4. TEPSA. https://cdn.ymaws.com/membership.tepsa.org/resource/collection/E8A406D5-CC66-40FA-A81E-56D02D6E1A19/fall-leader-2021.pdf

Huang, R., Barlow, A.T. (2013). Matches or discrepancies: Student perceptions and teacher intentions in Chinese mathematics classrooms. In Kaur, B, Clarke, D., Anthony, G., Ohtani, M. (Eds.), Student voice in mathematics classrooms around the world (pp. 161-181). Sense Publishers.

McGann, K (Director). (2008). The story of maths: The language of the universe [Film]. Open University; BBC.

Mendoza, A, and Beene, T (2022). Teaching math to English learners. Seidlitz Education.

Motley, N (2022). Small moves, big gains. Seidlitz Education.

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The Role of Perspective in the Math Classroom

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by Diane Kue Mathematics has existed in every civilization throughout history. Its worldwide presence is why so many people claim it as a universal language. Further perpetuation of the claim is present in the classroom. A classroom full of students who may or may not all speak the same language can successfully and individually solve...

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by Diane Kue When I was young, I watched my uncle bend his neck to walk through the doorway only to then hit his head on my parents’ light fixture. I had never seen someone so tall in my life, and …

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by Diane Kue When I was an elementary classroom teacher, I was often told that my students needed to read in every subject because “there is reading in every content area.” I could see how reading …

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Two Ways To Integrate Problem-Based Learning In A Unit (And Another To Avoid) - Robert Kaplinsky

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Here are two ways to integrate a problem-based lesson into your unit as well as a third that should be avoided. I use an 8th grade standard on the volume of spheres, cones, and cylinders as an example.

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Today’s math curriculum is teaching students to expect -- and excel at -- paint-by-numbers classwork, robbing kids of a skill more important than solving problems: formulating them. Dan Meyer shows classroom-tested math exercises that prompt students to stop and think.

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Using Prompts and Sentence Stems to Expose Mathematical Thinking When Problem-Solving

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by Diane Kue How students solve a problem—whether it is a standalone equation, expression, or a word problem—can be quite perplexing, especially when answers are unexplained. Students’ approaches a…

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Several of the most important and relevant lessons in Skew The Script's AP Statistics curriculum.

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A Teacher's Guide To Explaining the CONUNDRUM OF MATHEMATICAL WORD PROBLEMS

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In the final chapter of my book , my brother made the above statement. Although he is an apt mathematician, how he learned about word problems at school was disconnected to their application to the…

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Steve Wyborney's Blog: I'm on a Learning Mission.

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Welcome to Splat! You are only moments away from a VERY POWERFUL, highly interactive number sense strategy that can be used at any grade level!

This post includes 50 (fifty!) free, downloadable PowerPoint math lessons!

Watch the video, download some (or all) of the lessons, and experience what happens when you unleash this resource in your classroom!

I've been excited to click the Publish button on this post for several weeks! It's time!
UPDATE: The Fraction Splat! Series (with 20 more downloadable lessons) is now available!
After reading this blog post, take a look at THE 12 MOST POPULAR MATH STRATEGIES AND DOWNLOADS ON THIS BLOG.

Splat Through 10
How many dots are under the Splat?

Splat Through 20
How many dots are under the Splat?

Multiple Splats
(Note: Splats that are the same color must cover the same number.)
Splat 3.1 - multiple splats Splat 3.2 - multiple splats Splat 3.3 - multiple splats Splat 3.4 - multiple splats Splat 3.5 - multiple splats
Instant Multiple Splat
(Note: An additional feature of this level is the opportunity to ask, "What could the total be?")

Splat 4.1 - Instant Multiple Splats Splat 4.2 - Instant Multiple Splats Splat 4.3 - Instant Multiple Splats Splat 4.4 - Instant Multiple Splats Splat 4.5 - Instant Multiple Splats
2-Color Splat
(Splats of different colors must have different values.)

Splat 5.1 2-color Splat Splat 5.2 2-color Splat Splat 5.3 2-color Splat Splat 5.4 2-color Splat Splat 5.5 2-color Splat Splat 6.2 Instant 2-color Splat Splat 6.3 Instant 2-color Splat Splat 6.4 Instant 2-color Splat Splat 6.5 Instant 2-color Splat
Instant 2-Variable Splat
(Splats of different colors must have different values. Splats of the same color must have the same value.)
Splat 7.1 Instant 2-variable Splat Splat 7.2 Instant 2-variable Splat Splat 7.3 Instant 2-variable Splat Splat 7.4 Instant 2-variable Splat Splat 7.5 Instant 2-variable Splat
Number Splat
(Note: Only the numbers circled on the screen may be used.)
Splat 8.1 - Number Splat Splat 8.2 - Number Splat Splat 8.3 - Number Splat Splat 8.4 - Number Splat Splat 8.5 - Number Splat Splat 9.1 - Instant Number Splat Splat 9.2 - Instant Number Splat Splat 9.3 - Instant Number Splat Splat 9.4 - Instant Number Splat Splat 9.5 - Instant Number Splat
Instant Number Splat with 2 Variables
(Note: Only the numbers circled on the screen may be used, and Splats of different colors must have different values.)
Splat 10.1 - Instant 2-Variable Number Splat Splat 10.2 - Instant 2-Variable Number Splat Splat 10.3 - Instant 2-Variable Number Splat Splat 10.4 - Instant 2-Variable Number Splat Splat 10.5 - Instant 2-Variable Number Splat
NOTE: I wrote each of these lessons using PowerPoint, so I recommend playing them in PowerPoint. If you do not have PowerPoint, you may want to download the PowerPoint viewer so that you can play the lessons.

UPDATE: The Fraction Splat! Series (with 20 more downloadable lessons) is now available!
Other Posts and Resources Which May Be of Interest Tiled Area Questions The Maze Hundreds Chart
Introducing Cube Connectors
Provide Massive Space to Notice
Primary Tile Questions
3 Powerful Tile Strategies (and 40 new downloadable pages)
Please take a moment to subscribe to the blog so you'll be notified each time new resources are posted!

I would treasure hearing about your experiences using Splat! to promote number sense in your classroom!

All my best,
Steve

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Splat! - Steve Wyborney's Blog: I'm on a Learning Mission.

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Welcome to Splat! You are only moments away from a VERY POWERFUL, highly interactive number sense strategy that can be used at any grade level!

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New Gallup Poll: Many American Adults Wish They Had More Math Skills

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A vast majority of more than 5,000 respondents said the subject is important in both their personal and work lives; managers note a lack of skills.