25th Anniversary Special: The Problem with Word Problems –
A Research Basis Part I
Word problems, word problems, word problems – the most common requests I receive and the highest anxiety-causing mathematics – everybody gets so worked up about word problems!
Throughout all the courses and over all the 46 years I have been an educator, many of you have heard me discuss the problem with word problems. I talk about how and why word problems are an entirely different category of math development than all the K-12 Making Math Real Simultaneous Multisensory Structured Methods; and consequently, why the teaching of word problems must be entirely different than the teaching of numerically-based mathematics.
The Strong Parallels Between Solving Word Problems
and Writing Expository Essays
In my discussions, I note the strong parallels between the extremely high-order cognitive demands of solving word problems and expository writing. Both are:
- High(est) order cognitive demands on the ability to generalize current knowledge to new connections and on multidimensional executive functions in support of extensive working memory applications that must encompass the open-ended problem solving and critical thinking required for a wide diversity of word problems or for expressing a wide diversity of essay styles and topics:
Activate (Synthesize) Complete Comprehension Picture
Activate a complete comprehension picture from the language of the word problem
Activate a complete comprehension picture (idea) of the entire essay to be written
Translate Complete Comprehension Picture
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
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
Solve the Indicated Task
Activate the required mathematical abilities to solve the equation
Activate the required written expression abilities to transform the detailed outline into complete sentences and paragraphs
Demands on Working Memory
Activate and sustain the complete comprehension picture of the word problem throughout problem solving and know the solution makes sense
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
Demands on Open-Ended Problem Solving and Critical Thinking:
The Highest Order of Generalizability
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.
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.
- Multi-year development (typically 10) starting in kindergarten is indicated for both with respective year-by-year incremental instruction leading to the independent ability to solve an extensive variety of word problems or to write the classic American 5-paragraph expository essay. The significant duration of the multi-year development is directly related to the ongoing developments of front to back connections in the brain, which support the high-order cognitive demands of critical thinking and open-ended problem solving. Front to back connections typically start developing later in life, often starting in early adolescence.
- Neither word problem development nor expository writing development can be taught by assignment. Both require specific and explicit sequential development and there are no shortcuts for either. Representative examples of “shortcuts” for word problems include “Key Words”, “the 5-Step Plan”, “Draw a Picture/Make a Diagram”, “Guess and Check”, “Work Backwards”, etc.; and for expository essays, “The Hamburger Model” and graphic organizers such as “Webs, Mind Maps, and Concept Maps”. Interventions for word problem development and for expository writing must first determine students’ current developmental level within each respective multi-year progression and begin the intervention at that point, Chronological age and current grade level of the student do not factor into where the developmental intervention begins.
- Solving word problems and writing expository essays are extremely valuable developments
- Solving word problems and writing expository essays have been and continue to be extremely challenging for students of all ages
Request For a Research Basis to Explain
Why Word Problems Can Be So Challenging
I requested my colleague, Nancy Knop, PhD, a classroom science teacher, an educational therapist (currently retired) and a research biologist, to write an article based on the most current research on math and language to provide a research basis from cognitive science and neurobiology to explain why word problems are such a problem for so many of us. Dr. Knop and I have collaborated in multiple capacities over the last 20 years including co-presenting at Learning & the Brain Conferences in San Francisco, Cambridge, and Washington D.C., and she has provided me with numerous and valuable research-based resources throughout this 20-year period.
The Problem with Word Problems
One of the most frequent requests I continue to receive from educators and parents throughout the courses and in my private practice is how to deal with word problems. Perhaps since word problems can be so challenging for so many of us, word problems continue to be inappropriately included as part of the domain of numerically-based mathematics and therefore considered as the highest level of mathematical achievement, and consequently are given prominent emphasis on standardized tests, general math assessments, grade level expectations across all grades, and IEP goals and objectives. This is the problem and this is the principal purpose of this article: word problem development with its significant requirement on translating language to math makes it a separate educational development from the teaching of numerically-based mathematics; and therefore, cannot be used either as a way to teach numerically-based mathematics or as a way to assess numerically-based mathematics achievement. Word problems and numerically-based mathematics require separate and explicit instructional practices and separate assessments to serve as respective progress monitors and measures of achievement levels.
Providing Some Relief: When Indicated, Remove the Inappropriate
Inclusion of Word Problems from IEP Goals and Objectives and
Student Study Plans
In addition to wanting to help people understand the unique nature of word problem development, another of the principal reasons I asked Dr. Knop to provide this research-based article is when indicated, to help families provide the necessary basis to schools and IEP teams to have the inclusion of word problems in IEP goals and objectives and student study plans removed. Schools and IEP teams who are unaware of the unique nature of word problem development and inappropriately include word problems, especially for students with reading and language-based learning disabilities, as the means to address the development of numerically-based mathematics have caused profound distress and damage to numerous students and their families for decades. It is my sincerest hope that Dr. Knop’s article, “Of Course You Hated Word Problems: Current Research in Cognitive Science and Neurobiology Explains Why”, can provide much needed relief for students who have been subjected to the contraindicated inclusion of word problems in their IEP goals and objectives and/or student study plans.
Of Course You Hated Word Problems:
Current Research in Cognitive Science and Neurobiology
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.
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.
Please stay tuned for “The Problem with Word Problems – A Research
Basis Part 2: The Problem with Teaching Word Problems”
Be well and stay healthy.
David Berg, E.T.
Founder & Director of the Making Math Real Institute
Creator of the Making Math Real Simultaneous Multisensory Structured Methods