Emphasize Learning of Mathematical Concepts Through Solving Problems
Classroom Examples of Emphasizing Learning of Mathematical Concepts Through Solving Problems
Many people learned mathematics when the teacher explained a rule or concept, demonstrated the associated procedure, and then had students practice it. However, once a procedure has been memorized, it is difficult to go back and build understanding. Students develop deep mathematical understanding when they construct, examine, and justify methods for solving problems. With these tasks, they become better able to recall skills and concepts as needed and to apply them flexibly in new situations. In the classroom, group work allows students to engage in discussion, make presentations, and take charge of their learning.
A significant example of the effect of first building conceptual understanding is demonstrated in a study of first grade Latino students (Fuson, Smith, & Lo Cicero, 1997). Students used a penny frame, then progressed to a visual representation (ten sticks and dots) drawn by the students. Their performance was higher than that of their U.S. counterparts. Through this means of conceptual development, most first graders were able to accurately carry out a ten-structured solution to two-digit addition and subtraction problems to explain their regrouping. Additionally, Carpenter et al. (1999) found that students who were presented mathematical processes within a story were more able to see the reasonableness of the solution and could better estimate answers than when presented with just numbers. They also were more willing to construct their processes.
Fuson, C.F., Smith, S.T., & Lo Cicero, A M. (1997). Supporting Latino first graders’ ten-structured thinking in urban classrooms. Journal for Research in Mathematics Education. Equity, Mathematics Reform, and Research: Crossing Boundaries in Search of Understanding, 6(28), 738-766. Available to order online from http://www.jstor.org.
Carpenter, T.P., Fennema, E., Franke, M.L., Levi, L., & Empson, S.B. (1999). Children’s mathematics: Cognitively guided instruction. Portsmouth, NH: Heinemann.
1. Real-word problems (multiple grades)
Simple real-world problems can be posed in to emphasize reasoning and multiple problem solving. For example, a “calendar math” problem follows: Given the day of the month, i.e., 24th, students generate multiple operations of producing 24 as the answer (e.g., 99 plus 1 minus 75 minus 1) and can create story problems that would also produce 24 as the answer.
For other math problem-solving activities see:
http://www.stfx.ca/special/mathproblems/
http://www.figurethis.org/challenges/toc.htm
2. Counting (elementary)
In an early primary classroom, teach students to decompose whole numbers up to ten by using objects. For example, for the number six, ask students to arrange six beans in all the combinations that compose six, e.g., 1 and 5; 2 and 4; 3 and 3. This type of activity builds the understanding of the number six before they use abstract symbols and procedures like addition and subtraction. In an intermediate classroom, teach the concept of equivalent fractions by having students fold and cut paper into halves and sixteenths prior to teaching the algorithms for calculations with fractions. Other manipulatives that can be helpful for teaching the foundational concepts of fractions are Cuisenaire rods, fraction pieces, or groups of objects such as beans or M & Ms.
http://illuminations.nctm.org/LessonDetail.aspx?id=U147
3. “Number Talks” (elementary)
“Number Talks” are conversations about numbers, including observations of the arrangement of objects, comparisons, and computation to provide opportunities for children to work with computation in meaningful ways. The teacher posts a problem and asks the children to solve it without pencil and paper. After a period of silent thinking time, the teacher asks students to share their processes. For example, if the problem is 17 plus 18, some responses might be: counting on fingers; 15 plus 15 equals 30 plus 2 (from the 17) equals 32 plus 3 (from the 18) equals 35; 10 (17 minus 7) plus 10 (18 minus 8) equals 20 then add 15 (7 plus 8) and you get 35; or 17 plus 17 equals 34 plus 1 equals 35. Students will be more diverse in their strategies if they conduct this conversation after solving the problem mentally rather than with paper and pencil. Given paper and pencil, they tend to go to the algorithm.
http://www.didax.com/articles/number-talks.cfm
http://content.scholastic.com/browse/article.jsp?id=11537
4. “Constant Dimensions” (middle school)
In the “Constant Dimensions” problem, middle school students measure the length and width of a rectangle using both standard and non-standard units of measure to discover that the ratio of length to width of a given rectangle is constant, in spite of the varied units. Students chart their measurements, create a scatter plot, and graph the ordered pairs on a coordinate grid. Students examine the pattern, which shows a slope with a constant ratio, and learn that a rule relates the length and width for this rectangle: The length is always 1.5 times the width, regardless of the unit of measure.
http://illuminations.nctm.org/LessonDetail.aspx?ID=L572
5. “Problems with a Point” (middle and high school)
This database of mid-level problems is searchable by topic, time required, suggested technology, required mathematical background, and the habits of mind that students develop or use as they work. Problems can be sequenced to build on student knowledge and to create individualized learning sequences.
http://www2.edc.org/mathproblems/default.asp
6. Learning about recursion (high school)
In a high school mathematics classroom teaching problem solving and recursion, you can help students develop a stronger understanding of recursion by introducing a wildlife management scenario and how a population of animal changes over time in order to predict how the population may change in the future. Before symbolic representations (subscript notation and recurrence relations) are introduced, have students develop an understanding of mathematical modeling using NOW- NEXT in order to learn that proper procedures will provide information about how a population changes, but not the count of the population. At this point, introduce the symbolic representations, algorithms, and use of calculators.
Hart, E. (1998). Algorithmic problem solving in discrete mathematics. In L.J. Morrow & M. Kenney (Eds.), Teaching and learning algorithms in school mathematics: 1998 yearbook (pp. 251-267). Reston, VA: National Council of Teachers of Mathematics.
8. Calculating compound interest (high school)
In a high school classroom covering compound interest, include the following line of questions to elicit student reasoning. How are the formulas for interest compounded differently (daily, monthly, annually)? When would one type of compound interest rate be more or less desirable? (Compare saving money in a standard saving account to growth in annuity, and then buying a car to credit card debt.) Students should use the Internet to find and compare how lending institutions describe their methods for calculating compound interest. Students should compare their findings. How are descriptions and methods alike and different?
http://www.criticalthinking.org/resources/k12/TRK12-remodelled-lessons-high-school.cfm#275