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:
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).
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?
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).
