On encountering a new problem that they cannot solve, many students immediately give up. It doesn’t have to be this way.
There is a difference between students who welcome and remain engaged in the problem and those who give up easily. The difference is not due to innate factors, but it is mostly the outcome of teaching. With effective teaching all children can acquire attitudes and strategies to become proficient in problem solving—understand the problem, approach the solution process, and stay engaged in the problem using different perspectives.
According to the framers of the Common Core State Standards-Mathematics (CCSS-M) and Standards of Mathematics Practices (SMP), helping students to understand a problem, initiate a solution process, remain with the problem by exploring it from multiple perspectives are important characteristics of teaching. This helps students acquire the ability to enter the solution process and develop mathematical stamina.
Making Sense of Problems
Making sense of the problem means understanding the language, the concept, and the conditions and parameters involved in the problem. Students identify the objectives of the problem. They may not engage in the problem and remain engaged in the problem if they do not understand the problem. To initiate a solution process and to pursue it, students should associate appropriate schemas and procedures with the language, symbols, and concepts involved in the problem.
Mathematically proficient students read a problem carefully, understand the meaning and context of the problem, and explain to themselves the role of particular numbers, expressions, and actions in the problem. They analyze the givens, study the constraints on the quantities in the problem, understand, identify, or determine the unknowns in the problem. They understand intra- and interrelationships amongst knowns and unknowns. They understand the nature of these relationships. They seek entry points to the solution process keeping focus on the goal/s of the problem.
Mathematically proficient students analyze the problem, consider analogous situations, try special cases, and simpler forms of the problem (changing numbers, e.g., changing fractions into whole numbers, relaxing constraints in the problem, or reducing the number of variables) to gain insight into the problem and solution process.
They classify and organize the information into tables, charts, or groups. They search for regularity, patterns, or trends. They make conjectures about these patterns. They observe and explain correspondences between variables (knowns and unknowns) by forming equations, verbal descriptions, inequalities or diagrams of important features, relationships, and representations. Through these conjectures about the form and meaning of the data, they plan solution pathways and enter the solution process, rather than simply jumping into a solution attempt by choosing a formula or procedure.
Students who have acquired a concept, skill, or procedure using diverse language and a multiplicity of strategies have flexibility of thought to explore multiple ways of entering the problem.
Building Mathematics Stamina: Perseverance
Perseverance means having the self-discipline to continue a task in spite of difficulties and dead ends. It is a function of skills and attitudes. Albert Einstein said, “It’s not that I’m so smart, it’s just that I stay with problems longer.” Perseverance is a necessary ingredient for student achievement. One of the reasons students do not persevere in solving problems is lack of flexibility of thought. When they exhaust their ability and options to think about the problem, they do not have stamina for solving problems. Students develop perseverance when they are taught with rigor.
The requirements of rigor—understanding, fluency, and ability to apply, 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, mathematics and real-life problems. Finally, it is demonstrated in the ability to communicate this understanding. To achieve a level of mastery/rigor among students, mathematics educators need to balance expectations, instruction, and assessments.
We can help students continue thinking about a problem by modeling the many different questions they can ask about a difficult problem.
Effective teachers use a variety of language, questions, and methods to derive a concept or procedure. For example, let us consider a problem:
A science book has 251 pages and a mathematics book has 197 pages.
Teacher: What question can we ask so we have a subtraction problem from this information? Students formulate questions. If they do not, she articulates several questions:
- How many more/extra pages are in the science book than the math book?
- How many less/fewer pages are in the math book than the science book?
- What is the difference in the number of pages in the science and math books?
- How many pages should be added to the math book so that it will have the same number of pages as the science book?
- How many fewer pages should be in the science book to have the same number of pages in the math book?
- How many pages are left in the science book if we took away as many pages as the math book?
Through this process of generating questions, over a period of time, students develop flexibility of thought about additive reasoning, in general, and subtraction, in particular. These children, in future, will find several ways to enter the solution process of any subtraction problem involving numbers other than whole (e.g., fractions, decimals, integers, algebraic expressions, etc.). Effective questions build student stamina for problem solving. This should be a regular process in a mathematics class.
Using Effective Concrete Materials
To build stamina, younger students should be exposed to a multiplicity of concrete objects (e.g., Visual Cluster cards, TenFrames, Cuisenaire rods, fraction strips, Base-Ten blocks, Unifix cubes, pattern blocks, Invicta Balance, etc.) and diagrams and pictures (number line, Venn diagram, empty number line, bar model, graphic organizers, tape diagrams, tables, charts, graph paper, etc.) to understand strategies based on decomposition/ recomposition of numbers and facts and solve problems. Counting materials and strategies based on them build neither the flexibility nor the stamina for problem solving.
Middle and high school students may, depending on the context of the problem, transform numbers (fractions, decimals, and percents, algebraic expressions) using concrete models and the properties of numbers, operations (associative, commutative, or distributive properties to simplify numbers and expressions), change the viewing window on their graphing calculator to get the information they need (e.g., to observe the behavior of a polynomial, trigonometric, or rational function near the origin or at a specific point; compare it with the “parent function,” etc.), or use Algebra tiles, Geoboard, geogebra, Invicta Balance, etc., to arrive at relationships and equations involving variables.
Monitoring Progress and Evaluating Success
Mathematically proficient students monitor and evaluate their progress and change course if necessary. They check their answers to problems using a different method, and they continually ask themselves: “Does this make sense?” “Does this answer the questions in the problem?” (e.g., analyze partial and final answers). They can explain their solution approach and try to understand others’ approaches to solving problems, and they identify correspondences between different solution approaches. All of these activities, habits, and attitudes help them to be engaged in the problem resulting in perseverance. Students develop and improve perseverance when they realize that mathematics is thinking and making mistakes. It is also a process, not just finding the answer. It happens when we ask:
- Step One: What is it that we are trying to find out here? This is the question we ask in the real world. And this is the most important part of doing mathematics. People, including our students, need practice and opportunities in asking the right questions. This should be a group activity as group work as a strategy is critical to good mathematics work and student engagement. Group work generates better understanding of problems and then multiple entry points. It is also critical in countering inequities in mathematics achievement by different groups of students in the classroom.
- Step Two: Next is to take that problem and turn it from a real world problem into a mathematics problem—express it as a relationship between the elements (variables and quantities) that define or have created the problem. This translation from real word situation expressed in the native language to mathematics language is an important step in doing mathematics.
- Step Three: Once we have defined a relationship (an expression, an equation/inequality, or a system of equations/inequalities, etc.), we manipulate these relationship(s) and that involves formal mathematics—this is the computation step. Through computation, we transform the relationships into an answer in a mathematical form. This is an important step, but for developing interest in mathematics, we should not begin with this step.
- Step Four: When we have dealt with the computation part of mathematics, we need to then turn it back to the real world. We ask the question: Did it answer the question? And we also verify it—a crucial step.
- Step Five: To create interest and involvement, we need to now engage students in collective reflections by sharing different strategies and their relative efficiencies and elegance.
Completing this loop keeps our students grounded in the reality and power of mathematics. The majority of students will repeat these steps in their real life. And a small percent of students will have the satisfaction of repeating the steps in the context of mathematics and sciences only.
Teachers need certain attitudes, skills, and habits of mind for developing children into effective problem solvers with stamina. They need to practice the following:
- Believe in each child’s ability to improve and achieve higher in mathematics.
- Expect and help them to finish what they start and when they are stuck, providing scaffolding with enabling questions to continue in the task.
- Avoid accepting excuses for unfinished work.
- Give positive feedback when a child puts forth extra effort or takes initiative.
- Help students realize that everyone makes mistakes, but what is important is to keep trying.
- Demonstrate and motivate them to try new things.
- Encourage children to take responsibility for their work and make constructive choices.
Students become mathematically proficient and persevere in solving problems when teachers model these skills and choose meaningful problems to solve. They create conditions for students’ engagement in problems; that in turn develops perseverance. Students are engaged when problems are contextual, moderately challenging yet accessible, have multiple entry points, and are amenable to various solution approaches (intuitive, concrete, pictorial, abstract, on the one hand, and arithmetical, geometrical, and algebraical, on the other). It develops a variety of tools.
For example, using the Empty Number Line (ENL) approach to solving addition and subtraction problems rather than jumping into applying the standard procedure has many more entry points to the solution and can be solved using multiple ENLs (e.g., the problem: the difference 231 – 197 can be arrived at by at least different ENLs with a deeper understanding of numbersense (number concept, arithmetic facts, and place value) and problems solving. Arriving at the answer this way will keep them engaged.
Similarly, the Bar Model (BM) is an effective problem solving tool involving fractions, decimals, percents, and deriving algebraic equations easily. The area model of multiplication and division is effective for whole numbers, fractions, decimals, and algebraic numbers and for deriving properties of operations (e.g., commutative, associative, and distributive properties of multiplication and subtraction, etc.).
Tools are not enough, however, unless teachers scaffold student work. Questioning, based on formative assessment, is the key to the scaffolding process. Scaffolding is a function of a teacher’s ability in
- doing task analysis—know and establish the trajectory of the development of a concept, skill or procedure, and help students to know the goal of the task,
- being aware of the student’s capabilities, as well as their limits;
- doing continuous formative assessments of students’ assets—cognitive and content (conceptual and skill sets),
- asking enabling questions to move students toward the goal, and gradually fade and remove the support structures, and
- knowing models and approaches best suited for connecting concepts with students.
The response to “good” questions develops conceptual understanding, stick-to-it-ness, and helps them refine the tools—make them effective, efficient, and elegant. The better a teacher gets at asking “why” questions, the better her students are at understanding concepts, staying on, applying tools, and solving problems.
Effective questioning is more than giving students a solution approach, steps for solving a problem, or identifying the typology of the problem. Effective questions invite students to enter the solution process and stay with it. They may include:
- What question(s) are you trying to answer in the problem? What are you trying to find? Can you state that in your own words?
- What information do you have that can help you answer the question in the problem? Do you have enough information to answer the questions raised in the problem?
- Do you know any relationships among the information you have and what you do not have?
- Can you write this information using mathematical symbols?
- Can you write a fact, equation, inequality, formula or a relationship between symbols in the problem?
- How would you show the information in the problem in a different way?
- What other information do you need to answer the question?
- Where might you get that information?
- What other questions do you need to answer before you can answer the question in the problem?
- Have you solved another problem like this before?
- How is this problem like that problem? What is different about this problem?
- Could you solve the problem if the numbers were simpler?
When students have solved the problem, the teacher reengages them by asking:
- Do you have the answer to the problem/question?
- Have you answered the question raised in the problem?
- Which question in the problem does this answer?
- What does your answer mean?
- Does this answer make sense?
- Have you expressed the answer in the appropriate units of measurement or order of magnitude?
- What did you learn from this problem?
- Is there any information in the problem that was not necessary for answering the problem?
- Can we relax the conditions of the problem and still answer the problem?
- Can you write another problem similar to the given problem?
- Can you formulate a more difficult problem?
To demonstrate some of the questions, let us consider a problem:
In a village, 20% of voting age people did not vote during the last election. If only 4,280 people voted, what was the voting age population of the village?
Teacher: What are we looking for?
Students: The total voting age population of the village.
T: What information do we have?
S: The number of people voted? 4,280
S: The percentage of people did not vote? 20%
T: What else do we have? What information can we derive from the given information?
S: The percentage of people who did vote: 80%
S: The percentage of voting age population: 100%
T: What are we trying to find?
S: The voting age population of the village.
T: Can you represent the information by diagram, table, equation, or relationship? Make a start and try to solve it. We will discuss all of the methods used by the class. I will visit all of you and keep an eye on your progress. You can ask me questions when you need help.
At the end she asks children to share all of their methods and their relative merits are discussed. The approaches are shown here.
Method One: Visual Representation Method (line segment, Bar Model or Pie Chart)
The following bar represents the total voting age population.
# of people of voting age = ______________________________ =100% = ?
# of people who voted = _______________________ = 80% = 4,280
# of people did not vote = ______ = 20% =
As we do not know the total population, we represent it by a “?” mark, which is made up of those who voted (longer line or a bar) and those who did not vote (shorter line or shorter bar) (see the bar model below).
Because of 80% and 20% distribution, the line/bar is divided in two sections: the larger section is 4 equal parts and the smaller section is 1 part. The number 4,280 is equal to 4 equal parts and the missing part is one part. Therefore, one part is equal to 4,280 ÷ 4 = 1,070. Then the total number of people of voting age is 5 parts (4 parts + 1 part): 1,070 × 5 = 5,350.
Method Two: Applications of Fractions
The fraction of people who did not vote = 20% (= ⅕) of total number of people of voting age. The fraction of people who voted = 80% (= ⅘) of total number of people of voting age = ⅘ of total = 4280 (the 4 parts out of the 5 equal parts). So 1 part is 4,280 ÷ 4 = 1,070. Therefore, the total = 5 parts =1,070 × 5 = 5,350.
Method Three: Ratio and Proportion Method
Here part = number of people voted, whole = number of people of voting age, percent of people voted is 80% as percent of people did not vote is 20%. We can compare the number of people who voted in two forms: 80 percent vs actual number (4,380) and similarly compare the total # of people of voting age as 100 percent vs. actual number that we do not know and we consider as “?”. We have
4×? = 5(4280) (multiply both sides by 5 and ?; or cross-multiply);
? = 5(4280) ÷ 4 (isolate the “?,” divide both sides by 4),
? = 5(1070)
? = 5,350 (# of people of voting age).
Or, the total number of people of voting age = Number of people who voted + number of people who did not vote = 4280 + 1070 = 5350.
Method Four: Algebraic Method
Let us assume the number of people of voting age is x.
The number of people who did not vote is 20% of x. The number of people who did vote is 80% of x.
Thus, 80% of x = 4,280
.80 × x = 4,280 or ⅘ of x = 4,280 or x = 4,280÷⅘
x = 4280 ÷ .8 = 5,350.
Therefore, the total number of people of voting age = 5,350.
Method Five: Shortcut
To solve the problem, many teachers will just give the formula:
They will say to solve this problem is easy:
If 80% of a number is 4,280, then what is that number?
First, underline is and of in the problem. Then, the number just before is is the number to be placed in place of is and the number in place of of is to be placed in place of of, in the formula. Therefore, we have . Then, they will ask students to solve next ten to twenty problems on a sheet of paper.
This is purely a procedural method and does not emphasize much mathematics. The consequence is that students are unable to apply it if the problem is slightly different or the numbers are placed in a different form or different language. In this method, there is no involvement with language or concepts of mathematics. There are no connections made with other procedures or concepts. Students get the impression that mathematics is just a collection of procedures, and if they can recall the formula but can’t apply it, they give up.
Shortcut methods do not develop perseverance. Perseverance is reached when teachers apply methods that have mathematics and thinking behind them rather than methods that appear like tricks. Students who are familiar with the above four methods will be able to see where this formula comes from and then use it effectively.
Exposure to multiple approaches helps students understand concepts and acquire “stamina” for problem solving. As in any exercise, the stamina is a function of optimal (conceptually efficient) methods, regular and intentional practice, guided reinforcement (coaching and well-designed exercises and homework) and discussions of mathematics processes.
When teachers encourage students to share with the class, their
- understanding of the problem—language and concepts involved in the problem,
- entry points to the problem,
- approaches and strategies to and nature of the solution, and
- the mathematics concepts and procedures involved
students work hard and their mathematics stamina is strengthened. The crucial point is that students need to understand, know and experience that mathematics is not equal to computation. That is what develops perseverance.
Well-crafted mathematics classroom tasks, exercises, and assignments (including homework) hold the potential to make learning and teaching of mathematics focused and relevant and making all students achieve. In planning lessons, effective teachers make decisions about context, mathematics language, content, and rigor. Since homework is generally for reinforcement and practice, they assign homework that achieves those goals and needs to ensure that large chunks of class time are devoted to “why” and “how” questions to develop and reinforce mathematics concepts.
If mathematics is taught using deep learning—emphasis on concepts, language, and multiple models, instead of a performance subject—applying just “closed end” standard procedures, students will see it as important knowledge. Mathematics will become a collection of powerful tools that empower them to think quantitatively to solve problems in their work and lives.
We need to give all students the opportunity to taste real mathematics. Once students have acquired and mastered numeracy and algebraic skills with understanding, fluency and have the ability to apply, then we should use more efficient methods for computations. For example, computers and calculators can do a better job than any human as long as we know what we are doing and when such tools should be used. When relevant and efficient, we ought to use calculators and computers to do computation and engage students to spend more effort on conceptualizing and solving problems.