Word Problems

Activate a complete comprehension picture from the language of the word problem

Expository Writing

Activate a complete comprehension picture (idea) of the entire essay to be written

Word Problems

Translate the complete comprehension picture from the language of the word problem into a mathematical comprehension picture

Translate the mathematical comprehension picture from the language of the word problem into a mathematical equation

Expository Writing

Translate (organize) the complete comprehension picture (idea) of the entire essay to be written into a general sequential order of paragraphs: introduction with thesis statement, body with supporting evidence, and conclusion

Translate the complete comprehension picture of the entire essay to be written into a specific sequential order in the form of a detailed outline, paragraph by paragraph

Word Problems

Activate the required mathematical abilities to solve the equation

Expository Writing

Activate the required written expression abilities to transform the detailed outline into complete sentences and paragraphs

Word Problems

Activate and sustain the complete comprehension picture of the word problem throughout problem solving and know the solution makes sense

Expository Writing

Activate and sustain the complete comprehension picture of the entire essay throughout the planning and writing process and know the thesis statement, supporting evidence, and conclusion make sense

Word Problems

Solving word problems encompasses all math content. Any and all math applications can be expressed in any word problem form. The ability to independently solve word problems in general requires students’ abilities to synthesize, generalize, and translate all their math learning from the language of the word problems to the mathematical comprehension picture and apply the mathematics to solve them.

Expository Writing

Solving the challenges of expository writing encompasses all conceivable content/information/ideas that can be expressed in the different types of expository essay styles. The ability to independently write expository essays in general requires students’ abilities to synthesize, generalize and translate their content knowledge/ideas using word finding, sentence construction, grammar, and mechanics into expository forms of written expression.

Of Course You Hated Word Problems:
Current Research in Cognitive Science and Neurobiology
Explains Why

Nancy Fike Knop PhD, ET/P

Abstract. Brain pathways for processing words and processing numbers do not completely overlap. Mathematical thinking has its own set of brain pathways. As a result, solving word problems requires translation between these symbolic languages, making the math more difficult, leading to anxiety and avoidance. Math learning requires systematic, incremental development from concrete to abstract understanding using numbers, not words. Learning to solve word problems is a separate, though important, process.

It’s perfectly natural to hate word problems. You couldn’t help it. Our brains are wired one way for quantity/number symbols and a different way for language/letter-word symbols (Almaric & Dehaene, 2018, Fuchs et al., 2016a). Having to dip in and out of language areas of your brain to use separate quantity areas makes math more difficult. It’s like feeling your way through a room with your eyes closed. You can use hearing and touch to figure out where things are, but sight is easier – different areas of the brain are involved. Why make it more difficult?

Specialists in the brain and learning talk about domain-general and domain-specific brain functions. The hippocampus (memory formation), deep in the middle of your brain, and the frontal lobes (planning and inhibition of irrelevant information) are domain-general: you use them for all kinds of learning and thinking. Areas on the top toward the back, the parietal lobes (quantity), and the temporal lobes, on the left side inside your temple (language) are more domain-specific. In other words, these areas are wired into pathways you developed for specific kinds of thinking and communicating (Battista et al., 2018). Seeing, hearing, and touch all have their own specialized brain areas, too.

You were born with a brain area and pathways already specialized for recognizing, estimating, and comparing quantity (amount, loudness, brightness) and a separate area and pathway for recognizing spoken language. Your senses send information to those specific areas for interpretation. As you learned to count and read, letter and number combinations were stored in your memory and new connections were formed, but they developed in different places, with different pathways. Both domain-specific areas and domain-general pathways are involved (Qin et al., 2014, Vogel & De Smedt 2021, Wu et al., 2017).

For reading, your knowledge grew from hearing and understanding language into decoding written words, then combinations of words, then sentences, then chapter books. You combined that with encoding: writing words, sentences, paragraphs. If you were fortunate, you learned to do creative and expository writing. The left temporal and frontal lobes developed important pathways for this.

For math, you learned to use ideas about quantity and the symbols that represented quantity to carry out arithmetic operations: addition, subtraction, multiplication, and division. You used your visual-spatial memory and your hands to move things around that represented quantities. You actually understood concepts. For instance, you laid colored counters on a number line, different colors for each addend, and built a picture of addition in your mind that easily translated to number symbol addition. You saw that multiplication is just a shortcut for repeated addition. You valued all the math facts and operation ideas stored in your memory because it made solving arithmetic problems fluent and successful (Price et al. 2013). Transferring your direct understanding from concrete manipulatives to abstract symbols in a specific, systematic progression allowed you to be able to use the symbols fluently and accurately. You built an automatic recall of the information you needed. Or you could do all this if you had a chance to learn these things in a direct and explicit way, incrementally, using all your senses, with creative practice in clearly recording or expressing your computations and solutions. You developed the ability to do proportional reasoning (fractions, decimals and percents). You developed working memory and executive function (domain-general attributes) in algebra. You activated big picture understanding and deductive conclusions from partially given information in geometry. Your frontal and parietal lobes developed important wiring for this.

What happened when word problems entered the picture? At first, maybe not so much. Perhaps you could translate adding apples and oranges to figure out the total quantity of fruit in the basket. But if that problem was a written problem, it was more difficult than one spoken to you, because you had to decode the letter/word symbols and translate into number and operation symbols: different languages. Any language-based learning issue just added to the difficulty (Fuchs & Fuchs 2002, Geary, 2011, Kennedy, 2020). As the information included got more and more complicated, it could become too difficult. Too many brain areas were involved in sorting it out and translating it. If you didn’t have automatic understanding of the math, understanding literally in your body, involving, seeing, touching, and hearing, or enough practice doing problems like this one, the word problem was a serious challenge. Written words put you back at square one, trying to translate.

Increasingly, students are asked to solve math problems and then to “explain your thinking” using words or written language. Solutions to math problems that show all the steps using numbers DO completely show thinking, in mathematical language. Restating solutions in words requires another translation step. Solutions to word problems should be expressed numerically, with terms labeled as necessary, e.g. ‘15 pieces of fruit total.” That is enough. Once solutions are shown mathematically, it is not necessary, and you should not be required, to write sentences explaining again everything you just explained mathematically.

The demands of solving word problems are very similar to the demands of expository writing. Both require major translation tasks. For word problems: 1) translate the language of the word problem into a big picture of understanding, 2) translate that big picture into its equivalent as an equation, 3) solve the equation. For expository writing: 1) translate the big picture of an idea into an organized sequence (outline), 2) translate the organized sequence into expository language (expository language is different then spoken language), 3) write the essay. It takes many years of laying groundwork and developing the concepts and skills involved (Berg, n.d.). You are fortunate if you are comfortable with either one.

Most of us are not comfortable with word problems and we were not comfortable with them in school. Now, research in cognitive science and neurobiology is yielding an explanation for our discomfort. Our math brain was not designed for words. We developed strategies for coping, but we didn’t develop fluent understanding. Or we hated word problems, developed math anxiety, and avoided any further math (Ashcroft & Moore, 2009, Choe et al., 2019). Alas, real life presents us with “word problems” every day: knowing what 20% off means, balancing checkbooks, understanding simple and compound interest, income and tax rates, or analyzing statistics for accurate interpretation.

Confidence and competence in word-problem solving depends on knowledge of arithmetic (Fuchs et al., 2016b). Although this is basic and essential, it is not enough. For spoken story problems or word problems presented in text, language and reading comprehension is also necessary (Fuchs et al., 2020). For anyone with a learning issue that involves reading, executive function, working memory, or number sense, no matter how gifted in other areas, word problems become even more challenging (Knop & Chou, 2020). This is why arithmetic problem solving and translation of word problems into mathematics must be taught separately and explicitly.

Development of the ability to address and solve word problems is important. It takes many carefully structured years of instruction to develop the ability to analyze, translate, solve and translate again to communicate solutions effectively in both numbers and words. Our brains do not process word problems the same way as mathematical problems presented numerically. Learning to solve word problems is an important adjunct, and it is necessary. But word problems should not be taught or assessed instead of or as the entryway to numerically based mathematics. There must be separate well-designed approaches and progress monitors to both; otherwise it is like trying to teach children to see with their ears.

Selected References
Amalric, M., & Dehaene, S. (2018). Cortical circuits for mathematical knowledge: Evidence for a major subdivision within the brain’s semantic networks. Philosophical Transactions of the Royal Society B, 373(1740). https://doi.org/10.1098/rstb.2016.0515

Ashcroft, M. H. & Moore, A. M. (2009). Mathematics anxiety and the affective drop in performance. Journal of Psychoeducational Assessment, 27(3), 197-205. https://doi.org/10.1177/0734282908330580

Battista, C., Evans, T. M., Ngoon, T. J., Chen, T., Chen, L., Kochalka, J., & Menon, V. (2018). Mechanisms of interactive specialization and emergence of functional brain circuits supporting cognitive development in children. npj Science of Learning, 3(1). https://doi.org/10.1038/s41539-017-0017-2

Berg, D. (n.d.). Learn how to close the gap in achievement and reach the full diversity of learners. Making Math Real Institute. https://www.makingmathreal.org/about

Choe, K. W., Jenifer, J. G., Rozek, C. S., Berman M. G., & Beilock, S. (2019). Calculated avoidance: Math anxiety predicts math avoidance in effort-based decision-making. Sci. Adv. (5): eeay 1062 doi: 10.1126/sciadv.aay1062

Fuchs L. S., Fuchs D. (2002). Mathematical problem-solving profiles of students with mathematics disabilities with and without comorbid reading disabilities. J. Learning Disabilities 35: 563-573. DOI: 10.1177/00222194020350060701

Fuchs, L. S., Geary, D. C., Fuchs, D., Compton, D. L., & Hamlett, C. L. (2016a). Pathways to third‐grade calculation versus word‐reading competence: Are they more alike or different? Child Development, 87(2), 558–567. https://doi.org/10.1111/cdev.12474

Fuchs, L. S., Gilbert, J. K., Powell, S .R., Cirino, P. T., Fuchs, D., Hamlett, C. L., Seethaler, P. M. & Tolar, T. M.,(2016b). The role of cognitive processes, foundational math skill, and calculation accuracy and fluency in word-problem solving versus pre-algebraic knowledge. Developmental Psychology 52: 2085-2098. doi:10:1037/dev0000227

Fuchs, L. S., Fuchs D., Seethaler, P. M. & Craddock C., (2020). Improving language comprehension to enhance word-problem solving. Read Writ Q 36(2): 142-156 https://doi.org/10.1080/10573569.2019.1666760

Geary, D. C. (2011). Consequences, characteristics, and causes of mathematical learning disabilities and persistent low achievement in mathematics. Journal of Developmental and Behavioral Pediatrics, 32(3), 250–263.

Kennedy, D. (2020) What’s math got to do with it? Math learning disabilities, dyslexia, and ADHD: Understanding the connections, remediating effectively. The Educational Therapist 41(1): 4-8.

Knop, N. F. & Chou, S. H. (2020). Giftedness and Math Difficulty. In C. M. Fugate, W. A. Behrens, & C. Boswell, (Eds.), Understanding Twice-Exceptional Learners, Connecting Research to Pratice (pp 183-216). Prufrock Academic Press.

Price, G. R., Mazzocco, M. M. M., & Ansari, D. (2013). Why mental arithmetic counts: Brain activation during single digit arithmetic predicts high school math scores. Journal of Neuroscience, 33
(1), 156–163. https://doi.org/10.1523/JNEUROSCI.2936-12.2013

Qin, S., Cho, S., Chen, T., Rosenberg-Lee, M., Geary, D. C., & Menon, V. (2014).Hippocampal-neocortical functional reorganization underlies children’s cognitive development. Nature Neuroscience, 17(9) 1263–1269. https://doi.org/10.1038/nn.3788

Vogel, S. E. & B. De Smedt (2021). Developmental brain dynamics of numerical and arithmetic abilities. npj Sci. Learn. 6(22): 1-11 https://doi.org/10.1038/s41539-021-00099-3

Wu, S. S., Chen, L., Battista, C., Smith Watts, A, K., Willcutt, E. G., & Menon, V. (2017, September). Distinct influences of affective and cognitive factors on children’s non-verbal and verbal mathematical abilities. Cognition, 166, 118–129. https://doi.org/10.1016/j.cognition.2017.05.016

About Nancy Fike Knop, Ph.D., ET/P
Nancy Fike Knop, Ph.D., ET/P is a recently retired educational therapist. She worked with children and adolescents as an ET for 20 years, providing prescriptive remediation for challenges and background gaps in math and other subjects. Dr. Knop taught science to grades 7-12 at Head Royce School in Oakland and served as Department Chair for five years. She holds a B.S. degree from the University of Connecticut, a Ph.D from UC Berkeley, and a graduate-level certificate in Educational Therapy from UC Berkeley Extension. Dr. Knop writes and speaks locally and nationally, interpreting primary research about the biology of learning including brain development, the neurobiology of math and language learning, the role of vision, the importance of gesture, and the essential nature of sleep and other environmental influences that relate to learning and learning differences.