Understanding Problem Solving and the Common Core State Standards for Mathematics
Luzviminda “Luchie” B. Canlas
“All young Americans must learn to think mathematically, and they must think mathematically to learn.”1
The Common Core State Standards for Mathematics require that we have high expectations and provide strong support for all our students. This involves deep conceptual understanding and knowledge of best practices on our part. We must always be committed to providing appropriate and rigorous support to meet our students’ diverse needs of. We must always be looking for multiple and varied opportunities to deliver instruction that is fun, challenging, meaningful, relevant, and motivating for our students. By doing so, we will be able to give all of them access to the mathematics that they need to make them college and career ready.
Aside from the content standards, The Common Core State Standards for Mathematics have eight Standards for Mathematical Practice. These mathematical practices are as follows:
- Make sense of problems and persevere in solving them.
- Reason abstractly and quantitatively.
- Construct viable arguments and critique the reasoning of others.
- Model with mathematics.
- Use appropriate tools strategically.
- Attend to precision.
- Look for and make use of structure.
- Look for and express regularity in repeated reasoning.
This article’s purpose is to focus on the first standard for mathematical practice: make sense of problems and persevere in solving them. We will try to “unpack” this standard and understand what it really means to help our learners make sense of problems and develop perseverance among them. Try to reflect and ask your self these questions: How do students make sense of problems? How do we build perseverance in students?
What does making sense of problems mean?
In other words, what are the characteristics and practices of proficient problem solvers? Good problem solvers know the difference between information that is relevant and information that is irrelevant when solving problem situations. They know a variety of ways to solve problems and they know which strategies or methods are more efficient than others. Good problem solvers know how to use symbols, diagrams, lists, illustrations, tables and charts to help them solve a problem. They know which operations to use to carry out their computations to find the solution. They use estimation to predict possible answers. They ask questions as they solve and they always check answers to verify their work. They exercise precision and accuracy while performing the work. They know how to represent mathematical situations numerically, algebraically, geometrically, graphically, and verbally. They apply logic and reasoning as they construct possible solutions. They recognize patterns and they can provide suitable counterexamples to prove their point. They also know how to explain and justify their work.
How do we build perseverance in our students?
Perseverance in our students is probably responsible for more of their academic success than any other trait. Building perseverance is not easy to do, but it is possible. You need to exercise perseverance, yourself, in order to do this successfully. Remember that our students are in varying degrees of natural persistence. We know from experience that some of them give up easily in the face of challenges and difficulties. Some of them demonstrate “flightiness” (sadly often interpreted as irresponsibility) while others show unwavering willpower, determination and resolve when confronted with problems in spite of repeated failures and mistakes. Knowing who our students are is always key. What can we do to teach perseverance?
- We can teach students perseverance by gradually increasing their level of interest in a task. It is like building their stamina as they read or complete a task.
- Provide genuine encouragement and positive, immediate feedback as they solve a problem.
- Provide support but do not take charge. Build their self-esteem and point out what they do correctly to boost their morale. Lack of persistence is usually the result of self-doubt, lack of self-esteem and repeated failure with no one helping to identify cause of “breakdown”.
- Students who lack perseverance may have experienced many failures; success may seem impossible to attain. Therefore, we need to craft lessons and activities where they will develop self-confidence and success. We need to know their readiness and interest levels when writing our lessons. We need to differentiate our lessons and know how to create tiered tasks.
- Do not blame the student when they make mistakes. Let them know that mistakes are okay because these are opportunities to learn.
- Build community in your classroom so that students support each other when someone needs help. Teach them how to “learn how to think together”. Research says that group IQ is higher than the average of individual IQs.
- Let them take risks and allow them to explore and investigate mathematically. These types of activities encourage trial and error so it’ll teach them that it is alright to fail at certain times. The mistakes will lead to deeper understanding and that they should not be discouraged if their “plans” don’t work out sometimes. These things happen.
- Get them the habit to finish what they’ve started.
- Encourage self-reflection and metacognition. Model this whenever possible.
- Give students grade-level appropriate tasks, problems, or projects of gradually increasing length and difficulty. Their pride in successful completion of one task will eventually lead to developing perseverance for the next challenge. Hopefully this persistence will lead to resilience.
- One of the Principles of Learning states that we need to “recognize effort”. this means being very specific in what students have done well instead of just saying “good job”. Describe to them what they have done successfully and what they have done unsuccessfully (which processes and its consequences) so that they learn to be persistent and strategic.
Indeed, it takes a tremendous amount of time, effort and determination to teach our students to make sense of problems and to persevere. But it is worth the investment. The benefits are incredible. We are faced with this seemingly massive challenge of making the implementation of the rigorous Common Core State Standards happen. We can do this if we take it one step at a time – one standard at a time. Taking sufficient time to “unpack” and understand what each of the mathematical practices mean, highlighting the importance of each statement so that we can adjust, improve, modify, and refine our instruction, is what we need to do. We must continually nurture the mathematical thinking in all our students by encouraging them to give reasons, explain, and ask questions. By doing these practices, we will help them grow mathematically exponentially. Let’s all make it happen now.
1. National Research Council, 2001.Adding It Up: Helping Children Learn Mathematics
. J. Kirkpartick, J. Swafford., and B. Findell, eds., p. 1. Mathematics Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC. National Academy Press.
If you have a question or comment about this article e-mail Luchie.