Problem Solving
Michael Fitchett
Just Think Mathematics
· Rationale
· “Just Right” Problems
· Launch, Explore, Summarize Format
· Designing Problems
· Differentiation
· Extensions
Rationale
Without the application of mathematics to solve real-world problems, mathematics would be a useless endeavor. Traditionally, math curriculums have emphasized “ procedural skills” that key on computation alone. Computational fluency is only one part. Without problem solving, computation would be without a purpose. The most mathematically powerful part of the school day is the Problem Solving session.
Most, if not all, important mathematical concepts and procedures can best be taught through problem solving. That is, tasks or problems can and should be posed that engage students in thinking about and developing the important mathematics they need to know.
Van de Walle (2001, p. 40)
The value of teaching mathematics through problem solving include:
· Problem solving places the focus of the students’ attention on sense making.
· Problem solving develops “mathematical power.” Students solving problems will be engaged in all eight outcomes of the Common Core Math Practices.
· Problem solving provides ongoing assessment data that can be used to make instructional decisions. This also provides an excellent way of communicating to parents about a child’s progress.
· It is a lot of fun! Teachers who experience teaching in this manner tend to never return to a teach-by-telling mode. The excitement of students’ developing understanding through their own reasoning is worth all the effort.
“Just Right” Problems
It is important that the word problems given do not have a prescribed or memorized rule or method for solving them. The level of complexity must begin at a student’s instructional level. Problems should have appropriate ideas to engage and solve and still be challenging and interesting. Word problems given should require justifications and explanations for the answers and methods used to find them. The word problems can be view as the “vehicle” for teaching the math content.
In literacy, children need the “Just Right” book to build their reading power. A book that is not too easy for them and not too difficult. (Vgotsky +1) For this to happen, you will need to know where your students are with regards to comprehension and fluency. Much like a “Just Right” book, having “Just Right” problems are important to meet the students where they are currently. Therefor, it is important to think and plan for the math content level currently and your plan for where you want your students to grow.
Launch, Explore, Summarize Format
If we are to give children an opportunity to think and persevere through tasks, it is important to make sure the majority of problem solving sessions is spent on students solving the problems!
Launch: Launching the problem is simply providing the word problem and the materials necessary to work. This may include sticker labels (1 inch by 2 inches , Avery 1561) to be put in a page of their hard back composition books. Launching the problem may also include reading the problem for your non-readers or having your struggling readers read the problem to you with your guidance. Be careful not to show or explain how to do the problem. You must be careful not to take an engaging, cognitive task and “whittle” it down to the point where the students are not having to think about how to solve at all.
There will be times when you may need to clarify or redirect the learning after you have launched the problem. If you notice that the majority of students have totally missed the mark by misreading the problem or the misunderstood the logic in the problem (adding two numbers in problem verses multiplying them), then stop the lesson and redirect by asking questions regarding the problem-not telling them! Have them go back in and seek for understanding in the problem.
Explore: The majority of time during problem solving should be the students “grappling”, thinking, reasoning, organizing, conversing, and explaining. Since “meaning making” is at the core of problem solving, it will be important for students to solve the problems the way they understand them. It is starting where the students are and gradually “moving/growing” them to solve more problems with larger numbers using more sophisticated and efficient strategies.
During the explore, the teacher “walks the room” watching carefully for common methods of solving at the same time conferring with students who need a little direction. As the teacher assesses the progress, it will be important to be thinking about what will need to be summarized during the summarize portion of the problem solving session. Was the problem too easy, too difficult? Was the logic in the problem too clear, too vague? If the problem was too easy, it may not be necessary to go over it in a whole group setting. If the problem was too difficult, it may be necessary to adjust the level the next day rather than to push for understanding. The explore portion of this session is a great opportunity to use formative assessment to plan for the next day or days word problems
Summarize The actual “teaching” occurs during the summarize portion of the problem solving session. Here, the teacher has had a chance to observe how students solved the problems and now has to make some decisions as to what to teach. This could be reviewing the problems where students had difficulty. This also could be taking the students to the next level of sophistication regarding a particular strategy. Many times the summarize portion may lead to a connection or, re-launch for tomorrow’s problems. There may be times when a teacher need not go over the problems because the majority of students were successful solving the problems independently. Sometimes, going over one of the bonus problems (see extensions) is appropriate because the operation used (addition, subtraction, multiplication, division) is the same as the word problem but number size in the bonus problem is larger and thus the teacher can extend the learning using larger numbers.
A word about Math Strategies Perhaps the biggest teacher decision is deciding which strategy to emphasize when solving the math computation in the problem. These are the “building blocks” to developing mathematical power. Movement through the strategies enables students to solve word problems with meaning. Students become more efficient in the more complex strategies as their confidence and computational skills increase. When using the operation of addition, strategies may include the most simple strategy such as counting all when adding two numbers, to a more complex strategy such as composing and decomposing two-digit numbers mentally to solve.
Example: 28 + 57
“20 and 50 is 70. I know 8 plus 7 is 15 because I can make the 8 a 10 by giving it 2 from the 7. That would then be 10 plus 5 which is 15. So, 70 plus 15 is 95.”)
It is important to understand that math strategies are not grade-level specific. Instead, they are progressions that move through the grade levels depending on the needs of the learners. The Common Core Standards do call out some of these strategies to guide grade-level goals. However, it will be necessary to be responsive to all levels of learners in your class.
Designing Problems
Traditionally, word problems were always found at the bottom of the math book page. Many times these problems were skipped, or given to those students who finished early. Often, they were problems that students had no real world connection-just another generic problem to complete. It is ironic that the one “thinking” component of math work was put last.
Looking at the Common Core Mathematical Practices, it is interesting to notice that in order to: make sense of problems and persevere in solving them, reason abstractly and quantitatively, and, construct viable arguments and critique the reasonableness of others, students must have the kinds of tasks to make this possible!
Creating word problems that “connect” to your class may seem arduous at first but when you begin to look at the sequence of math standards and the “steps” that children go through to understand number, it becomes clearer as the year goes along.
When designing word problems, the following factors need to be considered:
· Mathematical Operations
· Number Size
· Logic In the Problem (Is the action clear or more complex?)
· Problem Types (Where is the unknown part of the problem located?)
· Level of “Real-World” Connection and Novelty
Mathematical Operation (addition, subtraction, division, multiplication)
Number Size One of the best way to differentiate for the learners in your classroom is choosing the size of numbers you will use in your word problems. Looking at the word problem above, the different colored numbers can serve as “entry points” to the word problem. Your struggling learners will do the problems that are in their comfort zone. They also may attempt the higher numbers but may use a more basic method to solve such as “counting on” verses a using a doubles strategy. Thus, number size may determine which strategy to use when solving.
Logic and Action Logic in a word problem is perhaps the biggest area of need for young learners. It seems that understanding the logic in word problems has been reduced by merely looking at the two numbers in a problem and reading the last sentence looking for a clue as to the operation one might use. Developing a student’s logical “sense” is more than looking at the “catch phrases” at the end of a word problem like, “How many altogether?” Or, “How many left over.?”
Students need to move from problems that are concrete to problems that are more abstract. Most problems that we, as adults solve, are not necessarily problems that involve straight forward, one calculation. Teachers need to start with problems where the “action” (what is being asked) is very simple and clear. Over the course of the school, year move to non-routine problems. Example:
Routine Problem Non-Routine Problem
Problem Types In order for students to know what is being asked in a word problem, it is important to give students a variety of problems so that they are constantly looking for meaning. Many times, word problems are taught in a unit, such as subtraction where all the problems of course are subtraction problems. Instead, students need to be exposed to not only all of the operations (addition, subtraction, multiplication, division), but also problems where the “unknown” (the answer) is located in different parts of the problem. For example, look at the following problem:
Result Unknown Change Unknown Start Unknown
In each one of these problems, the unknown part is in different parts of the problem. This forces students to read and perhaps model to make sense of what is going on in the problem. Having the unknown in different parts of the problem also may allow students to understand the inverse relationship between addition and subtraction.
(Notice that in the second problem, students may see that to find the difference between 14 and 5, one might “add up” starting at 5 to reach 14 instead of subtract.)
Novelty/ Real World Connection Most of us remember a version of this word problem from our math books: problem:
If the train left the station at 12:00 travelling at a speed of 45 miles an hour, what time will the train…
Problems like these, although the math content embedded in them was important, probably had left many of us with a dislike of word problems. They were mundane and most of us looked at problem solving as a “must do.” Many of the word problems in textbooks do not connect to our lives in any way.
Students are motivated by solving problems that are connected directly to their lives at that particular point in their lives. Providing a direct “real world” connection in a word problem will motivate more students to want to solve the problem.
Here is an example of an actual true story that was turned into a word problem.
I started a 3rd grade year where I taught in a portable. Skunks had somehow gotten under foundation of our classroom.
Other examples of making your problems real world:
· Putting the names of your students in the problem.
· Connect to something that has occurred in class or at school
· Connect from a read-a-loud that was done
· Connect to science or social studies content
· Connect to holidays
· Put spelling words or vocabulary words in the word problems
· Connect to students grade level/age (Example: losing teeth in 1st grade)
Differentiation
Even children in primary grades are smart enough to know how they “stack up” against the other children in the class. This becomes obvious many times because the teacher gives a different set of problems to struggling students and a different task for higher students. With problem solving with the sticker labels, all children can have access to the same problem. This type of task differentiates for students whose number sense may not be as high. It also differentiates for those students who need extensions into problems with larger numbers. Number size in the problem gradually changes to meet the needs of all learners in the classroom. (see sticker label below) Experience has shown that as number sense grows, most students will eventually solve all the problems though the strategy may change as the numbers get higher.