Model with Mathematics: Real World to Mathematics and Back

No problem can withstand the assault of sustained thinking. Voltaire

The Standards for Mathematical Practice (SMP in CCSS-M) describe mathematically productive ways of thinking that support both learning and applications by modeling mathematics in the classroom. Providing these experiences has to be an intentional decision on the part of the teacher. In other words, students learn mathematics concepts and procedures using models, on one hand, and use mathematics to model real problems, on the other. This activity is the basis of scientific and many social science innovations. Students need to experience this aspect of mathematics in the classroom from the very beginning.

Models and practical applications of mathematics have three distinct roles in mathematics learning. The first purpose of modeling and applications of mathematics is to motivate students to learn, engage, and see the relevance of mathematics. Here students learn mathematics by using concrete and representational models. This requires choosing the right manipulative, instrument, model or pedagogical tool to learn a mathematics concept, procedure or mathematical language.

In the second case, students apply mathematics—when they have learned a concept, skill, or procedure, to solving real life problems. The second case brings the appropriate mathematical knowledge and methods to match the demands of the problem. The third aspect deals with generating new mathematics or a model to solve a problem where one direct mathematics idea is not available. Throughout history, this twin process of modeling to learn new mathematics and solving novel problems by developing/ discovering models has solved real problems and generated new mathematics ideas—concepts, procedures. This is the interplay of pure and applied mathematics.

The facility of modeling mathematics is an example of the mathematical way of thinking and demonstration of competence in mathematics.

The modeling process, as application, spans all grade levels and applies mathematics that students know up to that grade level to solve “real” and “meaningful” problems. A simple example of modeling is the application of fractions to solve problems relating to rates of increase and decrease in various situations.

Deep Mathematical Understanding and Flexibility
A great divide often exists between students’ conceptual understanding, their procedural skills, and their ability to apply what they know. An even larger divide is that students may have conceptual and procedural knowledge but they have difficulty in applying mathematics ideas and realizing the power and relevance of mathematics.

The belief that applying mathematics is complex and complicated for many students and is separate from learning the concept and skills often leads many teachers to stop short of this most important step of teaching problem solving as part of each lesson. However, application of mathematics should not be a separate activity. While students who learn mathematics in a traditional fashion perform well on customary, standardized assessments, they tend to do poorly on tasks that require them to apply the math concepts to real problems. Students who learn mathematics through a modeling lens are better able to perform on both traditional and non-routine assessments.

Students too often view what happens in the math classroom as removed from and irrelevant to the real world. When a task can tap into a student’s innate sense of wonder about the world around him or her, that student becomes engaged in the problem-solving process. But when we can pique the interest of students through problems that have a basis in reality, we encourage them to question, investigate, and problem solve. Modeling bridges this gap and allows students to understand that to resolve many of the situations around them involve and require mathematics. When students engage in rich modeling tasks, they develop powerful conceptual tools that increase their depth of understanding of mathematical concepts and improve their abilities and interest in mathematics.

The concept of mathematical modeling, as a mathematics practice, has an important place in implementing the Common Core State Standards for Mathematics (CCSS-M). This practice emphasizes a student’s ability to realize the power of mathematics by applying mathematical tools to solve problems. Mathematical modeling demonstrates the power of mathematics for learners. Throughout their schools, students should use mathematical models to represent and understand mathematical relationships.

Levels of Knowing and Modeling
At each stage of mathematics learning (intuitive, concrete, pictorial/representational, abstract/symbolic, applications, and communication) and in mastering its components (linguistic, conceptual, and procedural) problem solving plays an important role. At the intuitive and concrete levels, a real life problem not only acts as a “hook” for students to see the role of mathematics as an important set of tools but also gets them interested in that concept.

At abstract/symbolic and the applications levels, applying the concepts, procedures, and skills shows how those elements are used and integrated, so students learn the strength and limitation of a particular mathematical tool.

When students have acquired a set of concepts and procedures and face a real life problem, they try to model the problem in mathematical form and solve it. This takes several forms: word problems, problem solving, and modeling. Because these are not isolated activities, problem solving, modeling, and application must be embedded throughout students’ learning of mathematics.  To make sense of developments in the natural, physical, and even social sciences and to solve the related problems involves looking for and developing mathematical models.

By incorporating mathematical modeling in their classrooms, teachers can motivate more students to enter STEM fields and to solve real life problems in social sciences and humanities. Integrating computers and calculation tools with mathematics methods, many of the social science problems are amenable to mathematical modeling.

Problem Solving: Model for Introduction to Mathematics Concept
Real life examples can introduce mathematics concepts and bring the real world into the mathematics classroom.  A real world scenario motivates students to see mathematics as relevant to their lives and increases the desire to learn that mathematics idea. In this situation, a teacher moves students, explicitly, from real-world scenarios to the mathematics in those scenarios.

For example, an elementary school teacher might pose a scenario of candy boxes with an equal number of candies in each box and represents it as repeated addition and then relates and extends “the repeated addition of a number,” “groups of objects,” or the tile pattern in the yard to see the “area of a rectangle” into the concept of multiplication.

An upper elementary grade teacher poses a scenario of candy boxes with a number of candies with different flavors in each box to help students identify ratios and proportions of flavors and ingredients.

A middle school teacher might represent a comparison of different DVD rental plans using a table, asking the students whether or not the table helps directly compare the plans or whether elements of the comparison are omitted.

A high school teacher shows several kinds of receptors (parabolic dishes) and poses a set of questions to instigate a discussion why parabolic receptors are optimal shapes to receive the sound, radio, and micro-waves. This discussion instigates the study of parabolas, in particular, and quadratic equations in general. Similarly, the discussion of waves of different kinds might instigate a discussion of Sinusoidal curves in an algebra, trigonometry or pre-calculus class.

A statistics teacher brings in a big bag of MMs to the class and asks:  “Without counting all of the MMs, how do we determine the number of MMs of different colors, as close as possible to their distribution in the bag?” Students might say: “It is easier to count them, why go through all that?” The teacher responds by posing the problem: “Yes, you can count the MMs, but how do we determine the population of fish in the pond or the number of particular species of animal in the wild as we cannot directly count them?” In this process, she shows the power of sampling method in real life and therefore the reasons to learn it.

The role of these problems is to motivate students to learn mathematics and show the power of learning the tools of mathematics. To achieve this goal, the problems have to be of sufficient interest and diversity. They should show the relevance to the topic being studied. The mathematics in them should be transparent.  Finally, they should be accessible to children.

Problem Solving: Applications of Mathematics
The first application of a mathematics concept, procedure or a skill is in the form of word problems. Word problems, while more demanding than pure computation problems, are typically presented in the context of a specific mathematics content area or skill, and are solved with a particular method or algorithm; therefore, students do not apply much mathematical reasoning. Word problems can serve as one example of problem solving; however, typical word problems in mathematics classrooms are “concocted.”  Often, they have no resemblance to realty, so to call them as applications is stretching the meaning of the word “problem solving.” However, if teachers routinely mix different types of problems, involving several mathematics concepts, they can help solicit mathematical reasoning.

Problem solving, on the other hand, is when students need to decide what mathematics, concept, skill, or procedure is involved for solving the problem. Problem solving is more advanced than word problems because it requires students to (a) translate native language to mathematics expressions, relationships (equations, inequalities, formulas, etc.), (b) interpret what mathematics skills, concepts, and procedures are needed to solve the problem, (c) make assumptions and approximations to simplify a complicated situation, and realize that these may need revision later, (d) determine how to find the answer, and (e) to make sense of the solutions in terms of the conditions of the problem and the solution sought.

Problem solving takes place when students have acquired a certain set of mathematics concepts, procedures and skills and the teacher presents problems that they can solve by using these newly learned skills. The role of the teacher is to identify and present these problems to students. These are focused application problems where the math content and the skills needed to solve problems have a close match between the problem and skill set, but they are not recipe oriented.

There are three kinds of applications of mathematics: (a) intra-mathematical, (b) interdisciplinary, and (c) extra curricular.

In intra-mathematical applications, a student learns a new mathematical skill, concept, and procedure and can apply this to solve problems in other parts of mathematics. To be successful in this context, teachers must be cognizant of the connections that can be made in different parts of the mathematics curriculum at that grade level and even higher grades. The role of modeling in mathematics, in this context, is making connections between different branches of mathematics and discovering new relationships about mathematics concepts. Students who engage in modeling in the math classroom have increased mathematical autonomy and flexibility in the ways they use mathematics.

For example, in early grades, students may learn the property of commutative property by using Cuisenaire rods:
2 + 5 = 5 + 2
models with math 1

In this case, the model is used to learn a mathematics concept. On the other hand, they may use Cuisenaire rods to solve an addition problem at first grade (I spent $9 on Monday and $7 on Tuesday.  How many dollars did I spend?) by constructing and then writing an addition equation to describe the situation.
model with math 2

In the middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. On the other hand, a student might use paper folding to see the division of fractions or visual cluster cards to learn the operation on integers.

In inter-disciplinary applications, a student learns a new concept, mathematical skill, or a procedure and can apply it to another discipline.  For example, a student learns the concept and operations of fractions and now can apply this knowledge in the “shop class.”  A student learned the concept of transformations in geometry and now can create a collage by using tessellations in art class.  The student just learned how to solve linear equations, so now she can use this skill to solve a problem in chemistry class. A school-based project integrating learning from several disciplines is a good example of this type of application. To achieve this objective of inter-disciplinary applications, teachers should be aware of the interconnections of the mathematical concepts and the use of mathematics in other disciplines of students’ curriculum.

In extra-curricular applications, a student learns a mathematics concept, procedure, and a skill and applies these to problems in everyday situations outside of the curriculum.  Here the teacher finds problems from the real world to connect with mathematics skills.

The goal of mathematical modeling, at this stage, is for students to pose their own questions about the world and to use mathematics to answer those questions. Quite naturally, most students want to know there is some utility in what they’re learning, that a lesson is not just an isolated lesson with no future use. In each section, in each module, they should be able to see what they are learning as relevant to their own lives and their own careers.

Discovering Mathematics: Modeling as Content Category
Throughout history, individuals have generated mathematics knowledge to solve practical problems. On the other hand, some mathematicians focus on mathematics for the sake of mathematics. Many others are interested in mathematics for its power, its tools, its approach to problem solving and modeling problems. The mathematical tools available at any given time are the means for innovation, inventions, determining the standards of living at that time. For example, in the twentieth century, most science, engineering and technology problems were tackled by the tools of calculus, but with the advent of calculators and computers, it is possible to extrapolate the data and find solutions using discrete methods. In such problems, there is need to integrate the mathematical tools that are based on continuous models (functions, calculus, etc.) and discrete models (finite difference methods, probability, statistics, etc.). In this scenario, students use their mathematics skills to discover new mathematical tools and skills.

Modeling as conceptual content category means using mathematics models to generate and learn new mathematics concepts. It is more than just using a concrete material, pictorial representation to learn a mathematics concept. For example, the study of transformations (both rigid and dynamic) to geometric and algebraic objects gives rise to the study of geometrical concepts, understanding of curves, functions, and conic sections. Similarly, in statistics and probability, we create, model, or simulate an idea to study it.

Modeling with Mathematics
When students themselves find or encounter real world problems and want to solve them, they are modeling mathematics at the highest level. One distinct difference between typical problem solving and mathematical modeling is that modeling frequently involves interpretation or analysis of an essentially nonmathematical scenario. This content conceptual category reflects a modeling cycle involving a series of operations.  Students must:

  1. identify a problem,
  2. study the scenario that gave rise to the problem to determine what the important factors or variables are, interpret these mathematically, identify variables in the situation and selecting those that represent essential features,
  3. observe the nature of the data—looking for regularities (e.g., if the data is increasing at a constant rate, it may be modeled by a linear system; if the change is constant at the second level of iteration, it can be modeled by a quadratic function; if each entry in the data is a constant multiple of the previous entry, it is modeled by a geometric/exponential function, etc.),
  4. develop and formulate a tentative mathematical model by selecting arithmetical, geometric, graphical, tabular, algebraic, or statistical representations that describe relationships between the variables,
  5. use the model to analyze the problem situation mathematically, draw conclusions, and assess them for reasonableness of the solution,
  6. analyze and perform operations on these relationships obtained by the modeling process to draw conclusions,
  7. test the solution to determine whether it makes sense in the context of the problem situation,
  8. interpret the results of the mathematics in terms of the original situation, and
  9. if the solution makes sense and they have a mathematical model for this type of problem, validate the conclusions by comparing them with the situation, then either improve the model or express the model formally in mathematical terms – if it is acceptable, and report the conclusions and the reasoning behind them.

However, if the problem is not adequately solved, the learning from this trial is incorporated to improve the model.

The iterative process and interpretation of the solution are hallmarks of the modeling process. A vital part of modeling is interpreting the solution and comparing it to reality.

By high school, students might use pictorial, numerical, algebraic, geometrical, trigonometric, functional, probabilistic, statistical, and computational methods to solve real life—social and physical science problems.  Choices of functions, diagrams, assumptions, range of constraints, and approximations are present throughout the modeling cycle.

Why Modeling?
There are common misconceptions about mathematical modeling. Many teachers view mathematical modeling as a process of showing the students how to approach or solve a problem. The first two types of applications described above are not true examples of mathematical modeling. They are uses of models to learn mathematics. The key feature of mathematical modeling, as defined by the NCTM and the CCSSM, is that students seek or encounter a problem, to solve. The teacher is the facilitator and guide in the process, but the modeling is done primarily by the students under her guidance: students select a problem, select the mathematics, integrate the skills and concepts, and then explain what they have done.

Mathematical modeling is not just a type of word problem or problem solving—it is mathematics being practiced; it is applications of mathematical ways of thinking. Modeling represents a shift from learning math to doing math. Modeling can be differentiated from word problems as it does not usually call for the use of one method or algorithm in order to solve the problem. Standard word problems or even problem solving in school mathematics curricula do not model realistic problem situations for problem solving, whereas modeling presents students with realistic problem-solving experiences requiring strategizing, using prior knowledge, and testing and revising solutions in real contexts.

The inclusion of modeling in the math classroom increases student engagement, depth of understanding, and provides opportunities for investigation, contribution, and success for all learners. Students involved in problem solving and inquiry-based activities such as modeling develop a positive disposition toward mathematics.

There is a place in the classroom for each type of problem, but it is possible to take a typical word problem and adapt it in such a way that it increases the depth of knowledge required to solve it. Open-ended modeling problems allow students to use mathematical tools and prior knowledge including measurement, proportions, map reading, scale drawings, and geography to make decisions and justify those decisions in a real-life problem. Students may use multiple methods to come to a conclusion and multiple representations to demonstrate their understanding. They must explain their solutions and use writing in mathematics to explain their reasoning.

Mathematical modeling is an effective practice for all students even for those who have a history of poor performance on traditional mathematics tasks. By incorporating modeling tasks into the classroom, we recognize all students as important contributors to the decision- making and investigation of the problem at hand. Because a good modeling task will be based in real-life experience, all students have the ability to make contributions based on their prior knowledge. Such tasks also emphasize and require a broader range of mathematical abilities than algorithmic exercises, and therefore allow a broader range of students to emerge as being capable. Students who may have a history of poor performance in math when their abilities and understanding are assessed solely on narrowly defined tasks and assessments can demonstrate significant ability and potential when given the opportunity to problem solve in a real-life modeling context. It is therefore important for teachers to emphasize to students that any (mathematically valid) solution for which they can make a strong argument is “correct.” This flexibility in thinking, and departure from the idea of only one correct solution, encourages students.

When students understand that they each have a unique contribution and valid voice in problem solving, they are more likely to become involved. Even students who have struggled in the past will contribute and share their thinking, and will be less likely to rely on the work of more successful students. Because modeling tasks can utilize a broad range of mathematical abilities, a broader range of students can emerge as capable mathematical learners.

Not all modeling problems have several feasible solutions, but the process always presents opportunities for different approaches and diversity of thinking, resulting in greater chances for success for students with diverse backgrounds and experiences. As these students gain confidence in their ability to contribute to the problem solving, they begin to develop a sense of mathematical autonomy.

Mathematical Empowerment
Our job as teachers is to present the tools, show students when and how they are used, and then provide a context in which they can choose the appropriate tools for a given problem. When we provide a larger toolkit, students begin to approach problems in a variety of (perfectly valid) ways and gain confidence in their abilities. The more opportunities are given to students to make mathematical decisions, the more they are encouraged to use their tools to explore and reason about mathematical problems. They grow increasingly confident in offering their ideas and methods and are better able to take the initiative when presented with a novel task.

Students should also see the teacher as a problem solver. When a teacher participates in problem solving as a senior learner and persists in solving a problem with students and explore different ways of solving the problem, students learn from that cognitively and affectively. When they see the teacher excited about learning, interested in students’ ideas and thinking, and willing to explore new ways of teaching and learning, even unmotivated learners convert into students who are excited about learning and proud to consider themselves mathematical learners. When teachers are open to finding different ways of solving a problem, they are more likely to create mathematically confident students.

Success and empowerment are the key factors to student motivation. The student who has tasted success is more persistent in solving problems and demonstrates meta-cognition in learning. Successful learners are curious and desire greater challenge. As they build confidence in their own thinking, their intrinsic motivation increases. They are viewed, and view themselves, as developing math experts. They have the confidence to make mathematical decisions, to approach a problem from one direction and, if necessary, change direction and try another. They will use the mathematical tools in their toolkit and decide when and how they might be helpful in a given problem. Therefore, a vital part of our work as math teachers is to encourage our students to become originators of ideas rather than merely recipients of content.

The introduction of modeling into the math classroom across all grade levels will increase student understanding, interest, and appreciation for the power of mathematics tools. Whether engaged in shorter, more focused modeling problems or more extensive, multi-day projects, students gain confidence in their proficiency as mathematical learners and make connections between mathematics concepts and real-life applications. All students, regardless of background or history with mathematics, have the opportunity to contribute and learn through experiences with mathematical modeling.

Characteristics of Effective Problem Solvers
First, to be a problem solver, one has to be mathematically proficient. Mathematically proficient problem solvers can and do apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. Students who regularly experience modeling with mathematics as a problem solving tool acquire unique characteristics and think of themselves as mathematicians.

They do not shy away from selecting complex problems and they:

  • simplify a complex problem and identify important quantities to look at relationships and they can represent this problem mathematically,
  • ask what mathematics do I know to describe this situation either with an equation or a diagram and interpret the results of a mathematical situation,
  • look for the mathematics learned to apply to another problem and try to solve the problem by changing the parameters of in the problem.

They regularly ask:

  • What model (quantitative, geometrical, algebraic, statistical, probabilistic, or mixed) could be constructed to represent the problem?
  • What are the ways to represent the information in the problem (e.g., create a diagram, graph, table, equations, etc.)?
  • What tools and approaches are appropriate to the problem at hand?
  • How to select and decide which argument makes sense and is reasonable in the context of the problem?
  • How to justify the appropriateness of the solution, explain why it makes sense, and how to convince the group of the reasonableness of the solution?
  • How to make sure that the results make sense?
  • How to improve/revise the model?
  • How do I incorporate the comments and concerns of others in the approach?
  • What is the best way of presenting the solution to others?
  • What further extensions, generalizations, investigations might be interesting or necessary?

In order to support such learners, the classroom must be a place that encourages choice and provides positive feedback regarding competence.  Teachers in such classrooms:

  • Assure all students that they are capable and competent, and their ideas are worth sharing with others and encourage student collaboration,
  • Presents problems that encourage student initiative and provide the opportunity for a variety of approaches and representation,
  • Make available appropriate manipulatives and instructional materials for exploration,
  • Practice and integrates the three roles: didactic, Socratic, and coaching,
  • Spend less time talking and more time listening to student questions and reasoning,
  • Ask more questions, give measured and focused feedback without curtailing creativity and initiatives, seek suggestions for improving solutions, encourage alternative solution approaches.

In such classrooms, students are more likely to dive into a problem and less likely to ask a question like “What do we do now?” As students succeed, they will be more likely to identify a problem, discuss methods for approaching the problem, and begin investigating and discussing different possible methods of solving the problem. When students are motivated from within—when they are excited about participating in their own learning for learning’s sake rather than because of pressure or external rewards—they become empowered learners.

Example 1  In the early grades, students have concrete, pictorial and arithmetic models available. This might be as simple as writing an addition equation to describe a situation. For example:
My team scored 91 points on Monday and 37 points on Tuesday.
How many more points did they make on Monday than Tuesday?
What is the difference in points on the two days?

This problem could also ask:
How many fewer points did my team score on Tuesday than Monday?
How many more points should my team have made on Tuesday to have the same score as on Monday?

A second grade class worked out the problem as:
One child said: ‘I subtracted 1 from 91 to make it 90 and then subtracted 30 from 90 to get 60 and then subtracted the remaining 6 from 60 to get 54. I know 6 and 4 are pairs to make 10. I took 37 away in all. My answer is 54.’

David was all excited. He said: ‘I have a better method. I subtracted 40 from 91 to get 51 and then added 3 to 51 to make sure that I actually subtracted 37. I subtracted and also added.’

Another child said: ‘I first added 50 to 37 and then 3 and then 1 to get the same answer 54.’ He said: ‘I know 7 and 3 are pairs to make 10.’ He showed the work on the empty number line as follows:

Another child said: ‘I added 50 to 37 to get to 87, but then I just added 4 directly to 87 to 91 as I know that 7 plus 4 is 11, so 87 plus 4 is 91.

Another child said: ‘I did it a little differently. I added 3 to 37 to make it 40. And then I added 1 to 40 to make it 41. And then I added 50 to 41 to get 91. I also got 54.’

Another said: ‘I added 4 to make 41 and then added 50 to get 91.’

Another said: ‘I added 3 to get 40 and then 50 to get 90 and then added 1 to get 91. And the answer is the same 54.’

Then students wrote their equations:  91 − 37 = 54 or 37 + 54 = 91.  They concluded that the team made 54 more points on Monday than Tuesday.

All of these examples show that the students are able to apply their understanding of (a) number concept (as demonstrated in the decomposition/ recomposition of numbers), (b) addition and subtraction facts (as seen in number sense), and (c) place value. The discussion and the recording of different number relationships demonstrate all the standards of mathematics practice.

Example 2  In the middle grades, students have pictorial, arithmetical, algebraic and geometrical methods available. For example, a student might apply proportional reasoning to plan a school event or analyze a problem in the community:
Our school survey revealed that 7 out of 8 students have access to iPads for their homework. If there are 128 students in our grade, I wonder how many students have access to iPads. Is that more than the number of students with iPads in the eighth grade where, according to our English teacher, 91 students have access to iPads?

Students in a seventh grade class demonstrated modeling by constructing the following table and then corresponding equation.

Some students in the class approached the problem as:
# of Students with iPads: ___ ___ ___ ___ ___ ___ ___ (7 equal line segments)
Total # of Students:  ___ ___ ___ ___ ___ ___ ___ ___ (8 equal line segments)
# of Students with iPads in my grade: ___ ___ ___ ___ ___ ___ ___
# of Students in my Grade: ___ ___ ___ ___ ___ ___ ___ ___ = 128
The goal now is to find: “What does one line segment represent?”
Now, 8 (___) = 128,
# of students represented by ___ = 128 ÷ 8 = 16
# of Students in my grade with iPads = 7 × 16 = 112.
Thus, the number of students with iPads in my class = 112. Then they extended it to the 8th grade to compare the corresponding numbers.

Example 3 Observing social and political world events, a group of researchers wanted to know how things go viral on the Internet. In other words, the team wanted to understand how political and social movements, ideas, or products could catch on or fail to do so.

Since two phenomena—disease and social movements use the same word “spread,” the team borrowed from mathematical models used in epidemiology. In biology and medicine, scientists have studied the spread of viruses, disease, and epidemics by using mathematical models. This suggests a possible model for the spread of social and political phenomena.

In medicine, there are susceptibles, infected, and impacted (dead, cured, quarantined); similarly, the researchers reasoned that they could construct a new model to examine the spread of ideas. The team showed that while an individual’s resistance to the spread of a “contagion” might be high, when bombarded by that contagion from many directions, such as happens through Facebook or Twitter, transmission occurs, i.e. you view the activity or participate in it as well. That synergy leads to explosive transmission and we say that something has gone “viral.” This is not only a wonderful example of the use of mathematical modeling to explain a real-world phenomenon but also an example of the generalizability of mathematics and mathematical models. The same mathematics and the same types of mathematical models that can be used to study, for example, the spread of Ebola can be used to study the spread of ideas.

Example 4 Modeling with mathematics means that the students not only understand the concept and procedures but also see that particular method(s) may have limitations and that the context of the problem calls for the applicability and efficiency of model and the method. For example, after students have learned to recognize and use the linear system, they explore real life situations and learn that some situations are modeled by linear systems only under certain conditions. For students to become proficient in modeling, they should have experiences that relate to modeling. Here is an example of a problem (from a textbook commonly used in high schools) a teacher presented to his students to understand modeling:

When a cake is first removed from the oven, its temperature is 370°F. After 3 hours, its temperature is approximately 70° F, the temperature of the kitchen.

  • Does this situation represent a linear system? Why do you think so? If so, represent it as a linear system. If not, why not?
  • Use the information above to write two ordered pairs (x, y), where x represents the time (in hours) since the cake was removed from the oven and y represents the temperature (in degrees Fahrenheit) of the cake at that time.
  • If it is a linear system, write the linear relationship between x and y, in any of the following forms, with general values (two point form; a point and slope form; slope-intercept form; standard form)
  • Find the slope of the line through the two points identified in step 2.
  • Write the linear equation in slope-intercept form or point-slope form.
  • Use the equation from step 5 to estimate the temperature of the cake after 1 hour, after 2-hours, and after 4 hours.

The problem, as given in the book, was straightforward; however, the questions above are reformulated to make sure that the students not only understand the problem but also have a deeper understanding and make connections between different concepts and relate the problem to a realistic situation. After this, the teacher made this problem even more rigorous by asking a series of further questions:

  • Why do you think the information given to you in the problem satisfies the conditions of a linear relationship?
  • Under what conditions can this be modeled by a linear relationship?
  • You know from geometry that two points determine a line, is that condition satisfied here?
  • What does a linear relationship look like in general?
  • What minimum conditions do you need to be able to find the linear relationship in this situation?
  • What is unknown in the linear function you just gave?
  • What is unknown in this equation?
  • When you look at your ordered pairs, will the slope be positive or negative?
  • What will be the orientation of the line?
  • How will you find the slope of this line?
  • What is the formula for slope?
  • Can you find the slope geometrically?
  • You said: “The formula for slope is .”
  • What do m, (y2−y1) and (x2−x1) represent in the formula?
  • What do y2, y1, x2, x1 represent?
  • Will the formula…give the same slope for your line?
  • Why do you think so?
  • Can you prove that the two formulas represent the same slope?
  • Please draw a rough sketch of the line.
  • Based on this sketch, what can you predict about the temperature in the future?

Then the teacher asked his students to solve the problem. Students calculated the slope by considering two points (two ordered pairs): (0, 370) and (3, 70). As the teacher was walking around in the room looking at their work, he asked students:

  • What does -100 mean here?
  • What will be the temperature in five hours? 10 hours?

At this point there was a great deal of discussion amongst students and they began to question whether it was really a linear model. Students came to the conclusion that it was a linear model only till the temperature of the cake reached room temperature, and after that it was not a linear model. The teacher introduced several examples of non-linear and mixed models. Students even brought the idea of a step function.

This is an example of teaching with rigor, making connections, and how mathematics is used to model real-world prolems. The teacher focused only on one problem during the lesson, but students understood the concept at a deeper level rather than solving several problems just applying a procedure.

The requirements of rigor—understanding, fluency, and ability to apply, are parallel to our expectations in reading. A child is a good reader when he or she (a) has acquired fluency in reading (displays speed in decoding, chunking, blending of sounds using efficient strategies indicating phonemic awareness, and word attack), (b) shows comprehension (understands the context, intent, and nuances of meaning in the material read), and (c) is able to use it in real life with confidence (pragmatics—able to read a diversity of materials from different genres and reads for interest and purpose). Mastery in any of these elements alone is not enough because reading is the integration of these skills.

Similarly, rigor in mathematics means a student demonstrates intra- and inter-conceptual understanding, fluency in performing computational procedures and their interrelationships, knowledge of the appropriateness of a particular mathematical conceptual and procedural tool, and ability to apply mathematics concepts and procedures in solving meaningful, real-life problems. Finally, it is demonstrated in their ability to communicate this understanding. To achieve the same level of mastery as in reading, mathematics educators need to balance these elements in expectations, instruction, and assessments.

The writers of the CCSS-M were careful to balance conceptual understanding, procedural skill and fluency, and application at each grade level.

 

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Model with Mathematics: Real World to Mathematics and Back

Construct Viable Arguments and Critique the Reasoning of Others: How Do You Know? Prove it!

The third of the Standards of Mathematics Practice (SMP) is mathematicians’ key occupation: construct viable arguments and critique the reasoning of others in a mathematical discourse. They discover, invent, and develop mathematics knowledge by constantly engaging in this process.

Nature of Mathematics Knowing
Mathematics is learned and generated by observing concrete situations and models, identifying and extending patterns, using analogous situations, and applying formal logic and reasoning to new and old situations. Developing formal reasoning provides a stronger base for learning and the development of mathematical ideas. In two previous Standards of Mathematics Practice (SMP), the emphasis was on understanding the problem—the language and concepts involved in the problem and then taking the specific concept to a general situation.

Developing reasoning, supporting one’s argument, critiquing another’s approach should not be reserved for high school geometry or advanced calculus; they should be part of all mathematics learning from Kindergarten on. Kindergarteners and first graders should be as familiar as high achieving high school students with the appropriate language (vocabulary, syntax, and mathematics sentence structure) and the development and practice of reasoning and logic (deductive and inductive; direct and indirect) such as: “prove it” “how did you know?” “how did you find out?” “defend your answer” “how can you be sure?” They should know answers to these questions and many others such as: “What definition or result did you use in this approach?” “What is wrong with this answer?” “Do you agree with …?” “why do you agree with … reasoning?” “What conclusions can you make from this?” “Is this a correct inference?” “Do you agree with that person’s reasoning?” “Why?” “Why not?” Development of and insistence on providing reasoning for their statements is not to make mathematics difficult; it is to understand mathematics better, deeper, and with understanding. Such mathematical thinking offers students the choice whether they want to be generators of mathematics knowledge or its users.

The origin of reasoning is intuition. When children’s intuitive answers are encouraged, they feel confident and are ready for formal reasoning. Mathematics is about removing obstacles to intuition and keeping simple things simple. Doing good mathematics is the interplay between intuition and reasoning—making things simple.

Viable Arguments and Critique of Others’ Reasoning
Mathematically proficient students understand and use stated assumptions, definitions, derived formulas, proven theorems, and established results in constructing arguments in the process of forming equations, relationships, and representations.

They can give examples for terms and definitions. They make conjectures and build a logical progression of statements to explore the truth of their conjectures and ideas.

They are able to analyze situations by breaking them into cases and recognize and use counter examples. They justify their conclusions, communicate them to others using mathematical language, and respond to questions and the arguments of others using appropriate reasoning.

Mnemonic Devices and Mathematical Reasoning
One hallmark of mathematical understanding is the ability to justify, in ways appropriate to the student’s mathematical maturity, why a particular mathematical statement is true, where a mathematical rule comes from, and how and when that can be applied.

There is real difference between students who can give the sum 8 + 6 as 14 by counting up or by rote memorization and the difference 17 – 9 by counting down or rote memorization and those who can find the sum by using strategies: decomposition/ recomposition of numbers, making ten, and knowing teens numbers. They see sum 8 + 6 as the outcome of strategies: 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14, or 4 + 4 + 6 = 4 + 10 = 14, or 2 + 6 + 6 = 2 + 12 = 14, or 8 + 8 – 2 = 16 – 2 = 14, or 7 + 1 + 6 = 7 + 7 = 14. They develop mastery (understanding, fluency and applicability) and develop efficient procedures.

There is a world of difference between a student who can summon a mnemonic device (DMSB = Does My Sister Bite or Dead Mice Smell Bad or Does McDonald Sell Burgers) to conduct the long division procedure: divide, multiply, subtract, and bring down and the student who knows why particular digits in the quotient are in a certain place or what will be the probable size (estimate in the correct order of magnitude) of the quotient before and when he completes the division procedure. Learning and applying procedures by just memorizing mnemonic is not mathematics.

Similarly, using the mnemonic device PEMDAS (= Please Excuse My Dear Aunt Sally to implement the order of operations: Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction) is purely procedural and shows a lack of understanding. It is important to know the reasons behind this order of operations (for details see the post on Order of Operations).

  • Addition and subtraction are one-dimensional operations (linear—for example joining two Cuisenaire rods or skip counting on a number line); they are at the same and the lowest level of operations. If both operations appear in the same expression, they are executed in order of appearance, first come first serve (Capture4);
  • Multiplication and division are two-dimensional operations (as represented by an array or the area of a rectangle), therefore, are at a higher level than addition and subtraction, they must be performed before addition and subtraction and if they both appear in an expression should be treated as first come first serve (Capture5);
  • If all the four operations: addition, subtraction, multiplication, and division appear in a mathematical expression, the order should be: Two-dimensional operations first and then the one-dimensional operation in order of their appearance (Capture7).
  • Exponential expressions are multi-dimensional (depending on the size of the exponent, e.g., a 10-cube = 103 is a 3-dimensional expression with an exponent of 3 and a base of 10; therefore, exponentiation operation is more important than multiplication (and division) and definitely higher than addition and subtraction, therefore, must be performed before all of them. Therefore, the order of operations so far is: (Capture9);
  • Grouping operations are expressions included in groups such as brackets, braces, parentheses either transparent and/or hidden (compound expressions in the numerator and denominator of a fraction, function and radical operations are hidden operations. They may involve some or all of the above operations in multiple forms, therefore, are or higher preference than all of the above operations. In transparent grouping operations, the order is parentheses, braces, and brackets. The hidden grouping operations are performed in the context. Inside a grouping operation, the same order as in the above operations is kept.

Hidden operations such as: fraction and radical operations need to be brought to students’ attention. For example, the fraction expression has hidden groupings as the numerator and denominator involve extra operations, even though there is no transparent grouping operation. In order to simplify the fractionCapture16 read it as: (3 + 5) ÷ (3 -1). Therefore, before we simplify the fraction (performing the division operation), we simplify the hidden operations in the numerator and the denominator. Similarly, function and radical operations are hidden: e.g., f(a) = 3a where a = Capture1and x=2).

Therefore, the grouping operations (parentheses, braces, and brackets in this order) are performed first. The hidden grouping is contextual. Then exponential operations need to be performed. After that multiplication and division are in order of their appearance. The last operations to be performed are addition and subtraction in order of their appearance. Therefore, the grouping operations should be performed before all of the other operations. The order of operations, therefore, should be: (Capture2). Here, G represents grouping operations—transparent, hidden, and both.

The mnemonic devices are important for remembering the sequence of activities in a multiple step procedure or operation; however, the use of acronyms and memory reminders should be only after students have understood the concepts and procedures and the reason for a particular order of operations. They do not take the place of conceptual understanding and derivation of procedures.

Similarly, there is a difference between a high school student who uses the mnemonic (FOIL) to expand a product such as (a + b)(x + y)= ab +ay +bx +by and a student who can explain where the mnemonic comes from (application of the distributive property of multiplication over addition, applied twice: (a + b)(x + y)= (a + b)x +(a + b)y = ax + bx + ay +by. The student who can explain the rule understands the mathematics and can use the mnemonic device productively as he may succeed at a less familiar task such as expanding (a + b + c)(x + y +z) or (2x + 3y)(-2x2+6xy -5y2).

Another practice that does not develop mathematical reasoning in students is the emphasis and introduction of procedures before the appropriate conceptual schemas are developed. It is important to develop the language containers and the conceptual understanding before a procedure is introduced. Fluency of a procedure or skill without conceptual strategies robs students of applying mathematics with understanding and reasoning. It is, therefore, important to assess them both for understanding before students are asked to apply them. Both conceptual and procedural understanding can be assessed by teachers by using mathematical tasks of sufficient richness and constantly asking the question: how do you know it?

The student should first have the conceptual understanding and then use it to acquire the procedure and only then should mnemonic devices be introduced to remember and automatize the steps.

When mnemonic devices and algorithms/procedures are introduced before conceptual understanding and the development of language containers (vocabulary, terminology, language expressions), students do not show interest in conceptual understanding and apply these without knowing the reasons behind them.

When students are given mnemonic devices before they understand the concept and procedure and the related reasoning, it may be difficult for them to apply the concept, defend their work and reasoning, and communicate their results and understanding. The classrooms where use of these mnemonic devices as a proxy for mathematics is paramount, real interest and passion for mathematics are absent and difficult to achieve.

Deductive and Inductive Reasoning
Many in the general public and non-mathematicians and even some teachers have the misconception that mathematics is a collection of sequential procedures, and the only justification for their actions is the sequence of steps and best case the use of deductive logic. It is true that the foundations of mathematics including arithmetic are established by formal deductive logic. However, in learning school mathematics and even in some higher mathematics, there is an interplay of deductive and inductive logic. In inductive logic, one moves from many specific examples to a pattern, that helps develop conjectures and then we arrive at a general principle—theorem, formula, and procedure. It is a right hemispheric activity—looking for patterns.

Deductive logic, on the other hand, starts from the general principle—formula, theorem, definition, etc. and proceeds to its application to specific situation. It is a left-hemispheric activity—engaging in sequential reasoning. Mathematics reasoning is the interaction of these opposite but complementary activities; it is similar to the corpus callosum integrating the flow of these activities from one side of the brain to the other. In that sense mathematics reasoning is a whole brain activity—integration of thinking originating from kinesthetic to linguistic to spatial orientation/spatial organization to inter- and intra-personal to logico-mathematical intelligence. The integration of inductive and deductive reasoning spans from seeing a concept geometrically/spatially to following step-by-step procedures using sequential procedural logic.

Mathematically proficient students are able to reason deductively and inductively about data, concept, and procedures, making plausible arguments that take into account the context from which the data arose and understanding the nature and quality of the concepts and procedures.

Applying Mathematical Reasoning as Communication
Just like any communication, mathematics communication has two parts: expressing one’s ideas succinctly with reason and understanding others’ ideas correctly. It is defending one’s ideas but also understanding others’ ideas and identifying strengths and finding fallacies in arguments from both sides. It is not enough to defend one’s argument, it is equally important to:
(a) understand and identify others’ reasoning and its validity and effectiveness
(b) recognize the fallacies in one’s own and other’s reasoning and arguments, and
(c) correct the fallacies in one’s own and others’ arguments and approaches in a mathematics context, e.g., problem solving.

This means students are able to compare the soundness, effectiveness, and efficiency of the two (or many) plausible arguments and approaches, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is and how to fix it.

Mathematical reasoning is developmental and contextual. Children are capable of developing reasoning according to their age and mathematics concepts. For example, elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. The concrete materials they use to show their reasoning should be (a) effective, (b) efficient, and (c) elegant. Such concrete arguments should make sense and be correct even though they may not be generalizable or made formal until later grades. For example, even Kindergarten students can easily see the commutative property of addition using Cuisenaire rods and then generalized to numbers and even variables.

Capture3

Later, students learn to determine domains to which an argument/reasoning based on language, diagrams or formal logic applies. At the same time, students, in all grades, can observe, listen or read the arguments of others, decide whether they make sense, and ask questions and add to clarify or improve the arguments.

Developing Mathematics Reasoning
Mathematical reasoning develops when we provide students experiences that help them acquire the component skills of such reasoning. Teachers’ questions aid the development of mathematical reasoning:
(a) What strategy can be used to find the sum 9 + 7?

  • I can count 7 after 9.
  • Can you give more efficient strategy?
  • I can use blue and black Cuisenaire rods.
  • Can you give a strategy without concrete materials?
  • I can use Empty Number Line.
  • Can you give any of the addition strategies?
  • Making ten: (9 + 1 + 6 = 10 + 6 = 16)
  • Making ten: (6 + 3 + 7 = 6 + 10 = 16)
  • Using doubles: (2 + 7 + 7 = 2 + 14 = 16)
  • Using doubles: (9 + 9 – 2 = 18 – 2 = 16)
  • Using missing double: ( 8 + 1 + 7 = 8 + 8 = 16)

(b) What strategy can be used to find the difference 17 – 9?

  • Using teens number: 17 – 9 = 7 + 10 – 9 = 7 + 1 = 8
  • Using making ten: 17 – 9 =10 + 7 – 7 – 2 = 10 – 2 = 8
  • Using doubles’ strategy 17 – 9 =18 – 1 – 9 = 18 – 10 = 8
  • What to add to 9 to get to 17: 9 + 1 + 7 = 17 = 9 + 8 = 17, so 17 – 9 = 8.

(c) What is the nature of the figure formed by joining the consecutive mid points of a quadrilateral?

  • To get a sense of the outcome of this construction, I will first consider a special case of quadrilateral: a square or a rectangle.
  • What does the constructed figure look like in such a special case?
  • What if the quadrilateral is concave? Is this assumption correct? What is your answer in this case? Why?
  • What if it is convex quadrilateral? What is your answer in this case?
  • Is it true in both cases?
  • Is it true for any quadrilateral?
  • Can you prove it by geometrical approach?
  • Can you prove it by algebraic approach?

(d) How many prime numbers are even?
What is the definition of prime numbers?
How many factors does an even number have?

  • 2 has 2 factors, namely, 1 and 2
  • 4 has 3 factors, namely, 1, 2, and 4.
  • 6 has 4 factors, namely, 1, 2, 3, and 6.
  • All even numbers, except 2 have more than 3 factors.

What conjecture can you form?

For upper grades:
Can you predict the nature of any even number?
Can you prove that ___ is the only even prime number?
Is a square number a prime number?
Why is a square number not a prime number?
If n is a prime number, what can you say about n + 1?

(e) Is the product of two irrational numbers always an irrational number?

  • What is the definition of an irrational number?
  • Is every number an irrational number?
  • Why? Can you prove it?
  • If not, why?

Can you give a counter example to justify your answer?

(f) Will the range of the data change if every piece of data is increased by 5 points?
David says: It will increase by 5. Is he right?
Why? Can you prove it?
Melanie says: It will not change. Is she right?
If not, why? Can you prove it?
Can you give a counter example to justify your answer?

(g) What other central tendencies are affected by such a change? Why? Explain.
One of your classmates just stated: Such a change will not change the median of the data, is this true? Why?

When students are given opportunities to make conjectures and build a logical progression of statements to explore the truth of their conjectures, they learn the role of reasoning and constructing arguments. The teacher should constantly ask questions such as: “How did you get it?” “What did you do to get this?” “Can you explain your work?” When teachers ask children to explain their approach to finding solutions and the reasons for selecting the particular approach, children develop the ability to communicate their understanding of concepts and procedures and the ability to trust their thinking. Some questions are applicable to all grade levels:

  • What mathematical evidence would support your assumption/ approach/strategy/solution?
  • How can we be sure of that ….?
  • How could you prove that …?
  • Will it work if …?

However, some questions should be at grade level. For example, at the high school level the questions can be more content specific.

Question: Your classmate claims that the quadratic equation: 2x2 + x + 5 = 0, has no real solutions. This can be followed by questions such as:

  • What is a solution to an equation?
  • What is a real solution?
  • Do you agree with this claim?
  • Why? Why not?
  • What information in the equation assures you that it does not have any real solutions?
  • How did you determine that this does not have a real solution?
  • Can you change the constants in this equation so that it will have two real solutions?
  • Only one real solution.

Teachers should analyze general situations by breaking them into special cases and ask students to recognize, use and supply examples, counter examples, and non-examples. This can be exemplified by questions such as:

  • What were you considering when …?
  • Why isn’t every fraction a rational number?
  • Is every rational number a fraction?
  • Is every fraction a ratio?
  • Is every ratio a fraction?

To help children how to learn to justify their conclusions, communicate them to others, and respond to the arguments of others, teachers can ask questions such as:

  • How did you decide to try that strategy?
  • Do you agree with David’s statement? “Between two rational numbers, there is always a rational number.”
  • Why do you agree?
  • How will you find it?
  • Why don’t you agree?
  • Do you have a counter example?
  • Is this statement true for all real numbers?
  • Why?
  • Is every square a rectangle?
  • Why?

Analogies and Metaphors as Aids to Mathematical Reasoning
Students’ reading comprehension is improved when their thinking involves the understanding of analogy, metaphor, and simile. Similarly, the use of analogies and metaphors is an example of reasoning in mathematics, particularly in the initial stages of learning a concept. Students need to learn to reason by using analogies and reason inductively about data, making plausible arguments that take into account the context from which the data arose. For that teachers need to follow with questions such as:

  • What is different and what is same about this problem and the other you solved before?
  • Did you try the method of the previous problem?
  • Did it work?
  • If it did not work, how did you know it did not work?
  • Why did it not work?
  • Could it work with some changes in your approach? Why or why not? What changes would you make?
  • How did you decide to test whether your approach worked?

One of the important aspects of thinking children need to develop is to know the conditions under which a particular definition, formula, or procedure applies and the parameters of its limitations. To develop this ability, teachers could ask questions about the content under discussion. For example:

  • Is 3,468 divisible by 4?
  • Yes or no?
  • Why? Justify your answer without actually dividing the number by 4.
  • Is this number divisible by 12? Why? Justify your answer without actually dividing the number by 4.
  • In a fraction Capture blog 6.02, if a = b, and b ≠ 0, then the fraction is equal to 1.

Do you agree wit this statement? If so, can you prove it? If not, can you give or construct a counter example to this situation?

Students need focused training and support in comparing the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in an argument, explain what it is. For this a teacher may articulate questions that focus on:

  • How to differentiate between inefficient and efficient lines of reasoning?
  • How to focus and listen to the arguments of others and ask questions to determine if the reasoning and the direction of the argument make sense?

Finally, teachers should ask clarifying questions or suggest ideas to improve/revise student arguments.

All skills, from cognitive to affective to psychomotoric, can be improved by efficient and constant practice. In classrooms where expectations of high levels of rigor are standard, students develop proper mathematics reasoning and are keen to identify others’ reasoning and critique it.

 

Construct Viable Arguments and Critique the Reasoning of Others: How Do You Know? Prove it!