To solve a problem, you must understand how the quantities and values relate to one another in a mathematical setting (quantitative analysis). A quantity is a measurable entity, and the value of a quantity is the number, or the number and the unit, that represents the measured or counted quantity (Clement and Bernhard, 2005). But if you have students that have gaps in foundations or language, or are intimidated by math, how can you approach teaching quantitative analysis within a word problem?

I once searched ted.com for inspiring math teacher speakers, and the teacher I found awed me and transformed my perspective. Dan Meyer (2010), in his presentation, Math class needs a makeover, argued that when we give problems to students, they are not involved in the formulation, the building, the process thinking of the math in the world around them. In his video, he gives a few examples of how to facilitate student-led thinking around math, and in his examples, he takes away the math “to level the playing field of intuition” (Meyer, 2010).  It is a video that I encourage everyone to watch. Thinking of of your students who have gaps in mathematical foundations, an activity in which the math is removed may seem like a backward approach. After all, a traditional classroom environment is “filled with students and adults who think of mathematics as rules and procedures to memorize without understanding the numerical relationships that provide the foundation for these rules” (Parrish, 2014, 4). If I described your classroom, one in which students must memorize a set of procedures and use mnemonic devices to problem-solve, please don’t think I am pointing fingers. I just described my own classroom for many years. However, after trying Meyer’s paradigm-shifting approach, the purpose of this student-led backward design to building and discovering the mathematics in a problem worked for a multitude of reasons. The approach fosters learning in response to inquiry and challenging situations within the learners’ threshold of understanding. This is a characteristic of constructivism, in which each person brings his or her own collection of knowledge to apply to the situation. It is here that Meyers argues that removing the math allows those with less mathematical knowledge an equitable opportunity for engagement. Leveling the playing field of intuition happens when a challenge occurs that prior knowledge alone cannot master. A “disequilibrium” arises that promotes learning, or the development of new or modified concepts as an adaptation (Schifter, 1996, 39). During this development, students build their conceptual knowledge of how mathematics fit in their understanding.

One major component to success is that students intimidated by mathematical conversations feel secure enough to contribute to the process because they do not feel a deficiency in mathematical discourse or concepts will prevent them from contributing to solving the problem. Students being able to build on one another’s understanding empowers the group and perpetuates a safe environment for learning. Applying this idea of taking away the math to promote the math is the first task I will introduce. 

Structuring taking away the math
When I first tried this activity, it was not easy generating questions without adding quantities, relationships, and/or values, but with practice, I succeeded. For a balanced understanding of the concept, non-examples are just as important as examples. Here are some non-examples of questions that remove the math and why:
 

When asking a question, it is important to phrase it in a way that only presents a problem. Students need to build their understanding. They need to sort through what quantities and relationships are important, and assign the values for each quantity. Here is an example list of questions I tried with students in my attempt to level the playing field:

• About how many books are in our library?
• How many pizzas do I need to buy for our pizza party?
• How many minutes have we spent reading The Lightning Thief by Rick Riordan?

Of the three examples, only one question was met with resistance and frustration. Can you guess which one and why?
I had failed the first time I tried to remove the math. If you guessed the library books question was the one that failed, you are correct. I took my third-grade students to the school library, told them to look around, and asked them “About how many books do you think are in our library?” I had gotten the librarian’s blessing to use our 30-minute library time working through this math problem. I am glad I did not quit when my students could not (collectively or individually) problem-solve the estimated number of books in our library, although that half-hour was a huge mess. Some students immediately started counting each book, one-by-one. Others complained that there were too many books to count and there was not enough time. Some came close and talked about multiplying the number of bookshelves times another number, but they couldn’t agree on what the other number would be. It was good that they understood the relationship between the bookshelves and total number of books, but they struggled with finding the value of their second quantity and had no academic vocabulary to express it. I still remember the librarian’s pity-smile when she told us the exact number of books before we left, defeated. Upon reflection, I think asking my students to estimate the number of books in our library was a little overwhelming for them. How could I have done a better job setting them up for success? At the time, I didn’t know. Now, after some experience, I have gotten over the embarrassment of that moment. The students were not as strong in building upon one another’s knowledge because they were not used to communicating and problem-solving in the environment I had placed them. They needed scaffolded support with language—expressing what they needed to know—and most importantly, I needed to set them up for success with a much smaller and more invested task before overwhelming them with calculating an estimated 15,000 books.

Additionally, the task did not have a personal value to them that calculating the number of pizzas I needed to buy for them did, or the prideful investment that only finishing an 87,223-word book could muster. (The latter question came towards the end of the year.) When I asked the students how many pizzas I needed to buy for our party, some of the quantities included: the number of slices each person would eat, the number of students in the class, how many different pizzas they would want, and if they included their teacher. Of all these quantities, they collectively sifted through to determine which ones were needed to answer my question and assigned their values. During this process, they discarded quantities that were not important, such as the number of the types of pizza to buy. After looking at values, they realized that there was a relationship between quantities missing “as a result of making sense of the situation’s quantitative structure” (Clement and Bernhard, 2005). They had not thought about the number of slices in one pizza. On the inside I was jumping for joy. It was a beautiful teaching moment. I wish I could travel back in time and record their dialogue. Had I given up after the embarrassing library incident, I would have robbed my students of the many opportunities they had to generate their own approach to mathematical problem-solving.

I hope that reading about my failure helps you so that if you fail as well, you know that success is also possible. Our students are not always exposed to guiding their own learning within a community so it takes time as they learn how to learn from one another and you will need to guide and model language as they communicate. I hope that if you try this method, you try it often. There is a lot of talk about culturally responsive teaching. CRT, among other things, is about student voice--acknowledging, respecting and applying it in a safe environment that encourages different perspectives. This is an example. Your students will learn an appreciation for the knowledge and perspective their peers' bring to their own understanding as they collectively build their mathematical portfolios.

This post is very long. Thank you for your investment. In my next post, I will continue this concept and structure as it applies to word problems. Please comment and add your perspective; I would love to learn from you.

Clement, L and Bernhard, J, “A Problem-Solving Alternative to Using Key Words,” Mathematics Teaching in the Middle School Vol. 10 Issue 7

Meyer, Dan. (2010 Mar 6). Math class needs a makeover [video]. YouTube. https://www.ted.com/talks/dan_meyer_math_class_needs_a_makeover/transcript?language=en 

Parrish, S. Number Talks: Whole Number Computation (Sausalito, CA: Math Solutions, 2014).

Schifter, D. What’s Happening in MATH CLASS? Envisioning new practices through teacher narratives (New York, NY: Teachers College Press, 1996). 

An archetype is a typical example of a certain person. In literacy, there are several character archetypes such as the hero and villain. When solving word problems, there are a few student archetypes I had each year that I had named for my own baseline pedagogical understanding. Because this is based on my personal observations, you may have additional input. Please feel free to share in the comments.

Student archetypes when solving word problems:

Lost Souls: These students struggle with just reading the word problem, let alone computations. For a variety of reasons often beyond their control, they have gaps in literacy and mathematics foundations and without scaffolded instruction, will end up directionless lost souls. Teachers must intervene immediately and strategically to provide Lost Souls with the right support so they can gain confidence and skills.

Tunnel Visionaries: These students forego reading and immediately apply whatever operation they are most comfortable with to the word problem despite whether or not it is correct. (In the younger grades, they often add every number.) Tunnel Visionaries do not see a value to reading comprehension, only to mathematical operations. Because they are eager to solve the problems using the shortest route, they often get wrong answers. 

Amnesiacs: These students often confuse teachers because they seem to "get it" during the lesson, and can do the math without needed support. However, when left alone or when spiraling a lesson, they often act as though they were never taught the content. Amnesiacs have learned basic survival skills. They apply short term tricks to getting through the lesson but never grasped the concept. (If that days' lesson had been over multiplication, they multiply to solve every word problem. However, because they never learned what made the word problem a multiplication word problem, when they have to solve another multiplication problem two weeks later, they are clueless of where to begin.) Amnesiacs may get As and Bs on daily grades, but they struggle on tests.

Computation Onlys: These students surprise teachers because they have strong computational skills but as soon as they are confronted with a word problem they are at a loss of how to solve it. A Computation Only archetype either never learned how math applies to the real world despite their apt ability to solve equations or they struggle with reading comprehension.

Fishes Out of Water: These students have strong literacy skills and most of the time love reading. However, when it comes to math they lose their confidence, feel overwhelmed, and/or struggle with connecting mathematical concepts. They will have no issue with comprehending the word problem, but they will have no clue with comprehending the math in the word problem.

Mathmagicians: These students solve the problems in their heads, or show very little work. When asked how they solved a problem, they often state, "I just know it" leaving the teacher to wonder to what extent they understand their problem and how much might have been a lucky guess. They're often annoyed with classmates that don't just "get it." Without proper scaffolding of how to verbalize their thinking, Mathmagicians will not get to properly showcase their logistical assets.

Junior Teachers: These students might have been teachers in another life. They not only understand word problems, but they also have a way of explaining it to others and seem to enjoy it. They are the most compatible partners. Teachers need to be wary of using them as a second teacher without ensuring that Junior Teachers have enriching activities for their own learning advancement as well.

If you are reading this, then most likely you have one or more of the following challenges:

• My student can perform the math isolated from the word problem. It is understanding what operations and numbers to use in a word problem that is the struggle.
• My student can solve problems with a partner, but not alone.
• My student reads below grade level and cannot comprehend word problems.
• My student reads below grade level and struggles with math foundations.

Word problems are tricky. Instead of numbers and operations dictating how to be solved, words get involved and make mathematicians decide the numbers and operations, determine values, relationships, and decipher what exactly needs to be answered based on what they read. Most people envision numbers and equations when they hear the word mathematics. Not too many people picture words. 

Because words are a key component to word problems, the role in how words are used to unveil how quantities and values relate to one another must be acknowledged, not ignored. However, this is still a novel perspective for many mathematics teachers. We aren't exactly taught in our mathematical pedagogical classes that words matter. (If you were, kudos to you for being ahead of the rest of us.)  This is why there are so many "quick fix" solutions to teaching word problem strategies. A quick google search demonstrates just how popular problem solving strategies are. For copyright purposes, I have made my own poster, but an internet search will show its likeness to what is out there.

Notice that in all the verbs mentioned--circle, underline, box, evaluate, draw, solve and check-- the word comprehend is not present. When reading, comprehension is a mandatory factor to understanding how each word relates to the others. This the the perspective that I am addressing in this post: teaching how to solve word problems can be a conundrum for most mathematics teachers because it requires breaking away from our mathematical viewpoint to teaching and involving a literacy skillset which we had not anticipated. Before looking for the numbers to circle, we must first read how the numbers relate to one another. Before underlining the question, we must grasp how this question is applicable to the problem. Boxing key words is negated when the mathematician-turned-reader no longer uses key words as a replacement for comprehension. Once comprehension is established, the mathematician can evaluate the relationships between the quantities and apply their mathematics skillset to solving the problem.

Solving word problems is not a strategy. From a pedagogical perspective, applying literacy skills to first comprehending the words before addressing the math is a strategic approach to teaching our students lifelong skills.

As a classroom teacher, I was often confused about why quick fix strategies didn't really work. I tried a few different approaches to teaching how to solve word problems. Some of them worked, some didn't. After I left the classroom to write curriculum, I had more time to research and understand what I was doing wrong and some of the things I did right. In later blogs, I will share with you my journey and my research. I hope you share as well so that we can all become stronger educators and impact our students positively.