When I was growing up in India, I recall having to take regular Math quizzes that tested 'Mental Math' in my elementary school years. Mental Math was essentially a test of how quickly we were able to solve the given arithmetic problems. Thanks to this early exposure, computational Math was never an issue for me even in later years when I took competitive exams. However what did take some getting used to was math that was not pure numbers but tested other aspects such as pattern recognition, spatial reasoning, logic - all of which I will put under the broad header of 'critical thinking.' Here's a look at these two components of math and how you can get your child exposed to both. Read the rest→
Defining Computational Math and Critical Thinking
Curriculum details posted on official websites sound a little complicated due to the terminology used so it is hard to decipher the actual concepts that they are referring to. In layman's terms here is a rough guide to what falls under the banners of computational math and critical reasoning.
Computational Math: This is sometimes referred to as basic math or arithmetic and contains what most of us typically associate with math. In the early years, it may contain concepts around basic arithmetic operations which then progress to more complex topics of decimals, fractions and then further onto algebra, geometry etc. The basic thread is that this math is largely numerical and based on established relationships and procedures which students need to absorb and remember in order to be able to solve problems within a given time.
Critical Thinking & Problem Solving: This can cover a range of concepts and tests skills beyond numerical abilities. Gaining proficiency in this area does not necessarily come with repetition but it requires application and perseverance on the part of the student. Some of the types of problems that fall under this area include:
- Logic: There are 3 students(L,M,N) and 3 ice-creams (vanilla,chocolate,mint). The tallest person likes Vanilla. N is taller than M but shorter than L. The shortest does not like mint. Who likes chocolate? Many of us will have come across these problems at some point in our academic life. While I don't typically find these problems to be part of an actual curriculum, I have seen my son bring these home as part of optional math 'challenges' that he is given in school.
- Spatial Reasoning: Problems dealing with shapes and surface area fall under this. Examples include:
o Showing a shape and asking how it would look when rotated
o Estimating number of faces painted in a 3Dfigure
o Problems of symmetry
- Pattern Recognition: These problems are sometimes the easiest to start with and can include patterns that are numerical or pictorial.
- Complex Word Problems: Similar to logic problems, these problems test less the math abilities but more the ability of the student to comprehend and infer based on what is stated. Students may also need to sift through the information to identify what is relevant to the problem.
· School Curriculum & Approach to Teaching:
Many schools and curricula across the world have a well-established record of providing a good foundation in basic and computational arithmetic. Reasoning and problem solving have only recently started emerging as important aspects of math education.
In the US, the new Common Core curriculum seems to have its focus on fewer but a deeper understanding of mathematical concepts. The Core Standards will focus on "developing the critical-thinking, problem-solving, and analytical skills students will need to be successful." While procedural skills will be emphasized, focus will also be on conceptual understanding, linking of topics across grades and the practical application of math.
Similarly recognizing issues with mathematical achievement in the US based on results such as PISA, the National Council of Teachers of Mathematics (NTCM) which focuses on improving math education for all students in its recent publication stressed (among other points) that there should be a move away from blind learning of procedures. In their words "Effective teaching of mathematics engages students in solving and discussing tasks that promote mathematical reasoning and problem solving and allow multiple entry points and varied solution strategies."
Resources for Computational and Critical Thinking:
Both types of skills are equally important to build overall math capabilities. In fact the building blocks of computational math provide the foundation to add on the critical thinking skills. So while students get exposed to basic math skills through school, parents who are looking to build critical thinking math skills from an early age may need to supplement what’s done in school.
Some after-school Math programs like JEI and Eye Level have specific modules on Critical Thinking. In addition, local and regional Math contests and Olympiads do a great job of testing all round math skills. You can also find some online sites and apps that provide fun math activities. If critical thinking is your specific focus area then check out the following websites:
- Khan Academy: Some interesting problems using basic principles of multiplication and division.
- Logic Puzzles: The website has more than 25,000 puzzles and the student can set the level of difficulty. They also have apps for iPad and Android tablets.
- Math is Fun: The website has a number of interesting starter puzzles on numbers, logic and alphabets.
- Math Kangaroo: The quintessential math competition, it has a range of complex math problems for the math lover.