| Mathematics presents students with a mode of | | | | "higher level activities" and "higher level mathematical |
| thinking/reasoning. It includes observation, attention | | | | reasoning." When higher level activities are taught |
| to detail, analysis, synthesis, relevant question asking, | | | | through mere memorization or repetitive activities |
| and problem solving. It involves some valuable traits | | | | devoid of real understanding, they do not involve any |
| like the ability to handle sweat, frustration, dead | | | | reasoning at all. When lower level activities are taught |
| ends, perseverance, and the discovery, hopefully, | | | | in ways that make students really think, then those |
| that there is wonder, joy, and even some exhilaration | | | | students are involved with higher level mathematical |
| at the end. We invite students deeper into or higher | | | | reasoning. And math teaching need not hang its head |
| up this mode of reasoning year-by-year, | | | | and feel inferior to other academic disciplines while |
| subject-by-subject. So what is higher level | | | | focusing on these lower level activities. |
| mathematical reasoning? | | | | Another unfortunate answer to what is higher level |
| A look at some of the approved and adopted texts | | | | mathematical reasoning can be seen in the rush to |
| suggests that a typical answer is, "algebra." Algebra is | | | | complicate problem sets in textbooks. The geometry |
| generally considered to be higher level math thinking | | | | book that the student I tutor is using in school, |
| for today's school students. Constants, variables, | | | | published by a major publisher and state adopted, has |
| coefficients, expressions, equations, quadratic | | | | outstanding higher level math reasoning problems to |
| equations, real, rational, and irrational numbers, and | | | | solve. I'm having as much fun with some of them as |
| combining like terms... If we can just get middle | | | | I'm sure that authors and state committee members |
| school, upper and even lower elementary students to | | | | had. But my student and many in her class are not. |
| start thinking about all of this, we believe that higher | | | | There are precious few problems in any section of |
| level math reasoning is taking place! | | | | this book that allow students to develop a confident |
| But what about geometry students who have | | | | understanding of the basic concepts and procedures |
| already passed Algebra I, but still have not mastered | | | | before "higher level math reasoning" is introduced in |
| basic number sense concepts involving fractions? For | | | | the form of clever and complicated levels of |
| example, I tutored a high school geometry student | | | | application. |
| recently who did not realize that if amount A is half | | | | Rather than leaping to higher level activities that |
| as much as amount B, then amount B must be twice | | | | require fluent reasoning that has not yet been |
| as much as amount A. This student had memorized | | | | developed, the interests of students would be better |
| the formula for determining the measure of an | | | | served if this book (and others like it) presented |
| inscribed angle (it is 1/2 the measure of its | | | | step-by-step contexts of problems of graduated |
| intercepted arc), and had solved many problems | | | | difficulty-each problem based on the reasoning |
| correctly. But when asked to find the measure of | | | | developed in the previous problem, and preparing |
| the arc when given the measure of the angle, the | | | | students for the next step of reasoning represented |
| student was stumped. It seems that for this student, | | | | in the following problem. The proper function of a |
| thinking about basic fractional relationships was | | | | math book is to develop mathematical reasoning, not |
| actually higher level mathematical reasoning-higher | | | | merely to create problems that require its use. By |
| than the current level of understanding. | | | | rushing to over-complicate the problems, textbooks |
| Higher level math reasoning for students is simply | | | | unwittingly exclude many students from success, |
| whatever the next step is from where they are | | | | actually thwarting the development of their reasoning |
| now. The relationship between 1/2 and twice, or that | | | | and forcing them to rely on mere memorization to |
| a group can be both one and many, or that a "1" | | | | cope with their work. |
| sitting in the tens column has a different value than a | | | | Yes we need to keep earlier concepts and |
| "1" in the ones column are all good higher level math | | | | procedures alive by integrating them into the |
| thinking skills for students who do not yet | | | | problems in subsequent chapters, and yes students |
| understand those concepts. People generally consider | | | | need to explore multiple uses and applications, and |
| algebra more abstract than arithmetic, because it | | | | yes they need to use all of this to solve |
| appears to be less concrete-and therefore it must be | | | | mathematical problems and not merely perform |
| the flagship of "higher level mathematical reasoning." | | | | arithmetic calculations. I am not arguing against any of |
| But any concept is "abstract" to the student who | | | | this. But enrichment is enriching and higher level |
| does not understand it yet! | | | | mathematical reasoning is only reasoning when |
| The critical element is not the level of difficulty of | | | | students have access to it. We should take as much |
| the work, but whether or not the work is being | | | | pride in opening up and developing that next level of |
| addressed through reasoning. Students who can | | | | higher mathematical reasoning, whatever it may be, |
| factor quadratic equations because they have | | | | as we do in the creative, clever, complicated, and fun |
| memorized a bunch of rules cannot be said to be | | | | problems our mathematical minds conceive. We |
| applying higher level mathematical reasoning, unless | | | | should remember what it's like for those who are |
| they actually understand why they are doing what | | | | looking to us for guidance. What is higher level |
| they are doing. There is a big difference between | | | | mathematical reasoning for them? |