Using your imagination, show or describe 10 ways to connect dots A and B.

1)    What information is needed for you to calculate the area of the square?

2)    How can the two triangle measurements help you figure out the area of the square?

3)    Figure ADE is a right triangle. Where is the right angle? Where is the hypotenuse?

4)    How can you use the Pythagorean Theorem to figure out the length of line segment AD?

5)    What is the length of line segment AD? What is the length of line segment AB?

6)    What is the area of the square?

Figure ABCD is a square. Figure ADE is a right triangle. What is the area of the square?

1) What information is needed to calculate the perimeter of the triangle? What information do you already have?

2) How can you figure out the length of line segment AD? You are given the area of square ABCD; how can that help you discover the length of AD?

3) How is the area of a square or rectangle usually calculated? How can you calculate this area, if you don’t know the length of any of the sides?

4) But the area has already been calculated. Can you think backwards to figure out what the length of the sides must be?  Are the sides all the same length? How do you know for sure? What number times itself equals 4? What is the square root of 4 square meters? What is the length of AD?

5) How can you figure out the length of line segment AE? You know the length of AD and DE, which are two sides of the triangle. What kind of triangle is it? Which side of the right triangle is the hypotenuse? If this is a right triangle, can you use the Pythagorean Theorem to calculate the length of the hypotenuse? What is the Pythagorean Theorem: Is it a + b = c? Is it a² + b² = c²? Does the Pythagorean Theorem tell you the length of the hypotenuse, or the length of the hypotenuse squared?

6) Extra hint for students who already know about 3-4-5 right triangles: double the measurements of AD and DE; what do you notice?

Figure ABCD is a square with an area of 4 square meters. Figure ADE is a right triangle. One side measures 1.5 meters as indicated. What is the perimeter of the triangle?  What prior concepts or procedures do students need in order to solve this problem?

1) What information is needed for you to calculate the circumference of the circle? Is that information presented on the drawing? Can you figure out the needed information from the information that is given to you in the picture? How will you do it?

2) What does “isosceles” mean? What is the length of side ED of the triangle ADE? How can you use the Pythagorean Theorem (a2 + b2 = c2) to figure out the length of side AD? What is the length of it? Is Line AD part of the triangle? Is it also part of the square?

3)  Do we know the exact value of the square root of 2? What do you get if you multiply the square root of 2 times the square root of 2? How do we calculate the area of square ABCD? What is the area?

4) Draw a diagonal from A to C. How can you use the Pythagorean Theorem to figure out the length of Line AC? What is the length of it? Is Line AD part of a triangle? Is it also part of the circle? Is the diagonal Line AC the same as the diameter of the circle? How do you know? If you know the length of the diameter, how  can you calculate the circumference of the circle? What is it?

Figure ABCD is a square. Points A, B, C, and D are all located on the circumference of the circle. Figure ADE is an isosceles right triangle. One side measures 1 meter.

What is the area of the square? What is the area of the circle?  (Use 3.14 for ∏.)

What prior knowledge/skills do students need to bring to this problem?

This week we have a different type of problem. In order for students to solve a problem in a given lesson, they often need an understanding of and fluency with prior concepts, procedures, definitions, or theorems.  If the needed concepts or procedures are missing, the path can be closed to them. Equally important, in order to support their students’ discovery of a successful path, teachers, parents, and tutors also need to know what prior knowledge and skills are required to solve a particular problem.

So for this week, what concepts and procedures does a student need in order to prove that the distance around the inner object is shorter than the distance around the outer object? We will post some suggestions next week. In the meantime, post some yourself by leaving a comment.

1. Put your left index finger on zero

2. With your right index finger, add 8 (Do not touch zero. Touch 1, 2, 3, 4, 5, 6, 7, 8.)

3. Say, “I just added 8.”

4. Slide your left finger from the zero to the 8 and say, “That’s one group of 8.”

5. With your right index finger, add 8 more (Touch 9, 10, 11, 12, 13, 14, 15, 16.)

6. Say, “I added 8 two times.”

7. Slide your left finger from the 8 to the 16 and say, “That’s two groups of 8.”

8. With your right index finger, add 8 more (Touch 17, 18, 19, 20, 21, 22, 23, 24.)

9. Say, “I added 8 three times.”

10. Slide your left finger from the 16 to the 24 and say, “That’s three groups of 8.”

11. How many times did you add 8? How many groups of 8 is that? How many is in each group? At what number did you end? So 3 x 8 = ?”

How many are in each group? How many groups are there?  How many are there altogether?

12. Try an experiment: Start at zero and add three, eight times. What do you notice?

Mathematics presents students with a mode of thinking/reasoning. It includes observation, attention to detail, analysis, synthesis, relevant question asking, and problem solving.  It involves some valuable traits like the ability to handle sweat, frustration, dead ends, perseverance, and the discovery that there is wonder, joy, and even some exhilaration at the end.  We invite students deeper into or higher up this mode of reasoning year-by-year, subject-by-subject. So what is higher level mathematical reasoning?

A look at some of the approved and adopted texts suggests that a typical answer is, “algebra.” Algebra is generally considered to be higher level math thinking for today’s school students. Constants, variables, coefficients, expressions, equations, quadratic equations, real, rational, and irrational numbers, and combining like terms…  If we can just get upper and even lower elementary students to start thinking about some of this, we believe that there is more of an opportunity for higher level math reasoning.

But what about geometry students who have already passed Algebra I, but still have not mastered basic number sense concepts involving fractions? For example, I tutored a high school geometry student recently who did not realize that if amount A is half as much as amount B, then amount B must be twice as much as amount A. This student had memorized the formula for determining the measure of an inscribed angle (it is 1/2 the measure of its intercepted arc), and had solved many problems correctly. But when asked to find the measure of the arc when given the measure of the angle, the student was stumped. It seems that for this student, thinking about basic fractional relationships was actually higher level mathematical reasoning—higher than the current level of understanding.

Higher level math reasoning for students is simply whatever the next step is from where they are now. The relationship between 1/2 and twice, or that a group can be both one and many, or that a “1” sitting in the tens column has a different value than a “1” in the ones column are all higher level math thinking for students who do not yet understand those concepts. People generally consider algebra more abstract than arithmetic, because it appears to be less concrete—and therefore it must be the flagship of “higher level mathematical reasoning.” But any concept is “abstract” to the student who does not understand it yet!

The critical element is not the level of difficulty of the work, but whether or not the work is being addressed through reasoning. Students who can factor quadratic equations because they have memorized a bunch of rules cannot be said to be applying higher level mathematical reasoning, unless they actually understand why they are doing what they are doing. There is a big difference between “higher level activities” and “higher level mathematical reasoning.” When higher level activities are taught through mere memorization or repetitive activities devoid of real understanding, they do not involve any reasoning at all. When lower level activities are taught in ways that make students really think, then those students are involved with higher level mathematical reasoning. And math teaching need not hang its head and feel inferior to other academic disciplines while focusing on these lower level activities.

Algebra is not the problem in itself. Thinking that it accomplishes the need for higher level reasoning and application is.

Another unfortunate answer to what is higher level mathematical reasoning can be seen in the rush to complicate problem sets in textbooks. The geometry book that the student I tutor is using in school, published by a major publisher and state adopted, has outstanding higher level math reasoning problems to solve. I’m having as much fun with some of them as I’m sure that authors and state committee members had. But my student and many in her class are not. There are precious few problems in any section of this book that allow students to develop a confident understanding of the basic concepts and procedures before “higher level math reasoning” is introduced in the form of clever and complicated levels of application.

Rather than leaping to higher level activities that require fluent reasoning that has not yet been developed, the interests of students would be better served if this book (and others like it) presented step-by-step contexts of problems of graduated difficulty—each problem based on the reasoning developed in the previous problem, and preparing students for the next step of reasoning represented in the following problem. The proper function of a math book is to develop mathematical reasoning, not merely to create problems that require its use. By rushing to over-complicate the problems, textbooks unwittingly exclude many students from success, actually thwarting the development of their reasoning and forcing them to rely on mere memorization to cope with their work.

Yes we need to keep earlier concepts and procedures alive by integrating them into problems in subsequent chapters, and yes students need to explore multiple uses and applications, and yes they need to use all of this to solve mathematical problems and not merely perform arithmetic calculations. I am not arguing against any of this. But enrichment is enriching and higher level mathematical reasoning is only reasoning when students have access to it. We should take as much pride in opening up and developing that next level of higher mathematical reasoning, whatever it may be, as we do in the creative, clever, complicated, and fun problems our mathematical minds conceive. We should remember what it’s like for those who are new to all of this. What is higher level mathematical reasoning for them?