Why do I, as a middle grades math teacher, need to know about calculus and analytic geometry?

To prepare my students for the future.

“Effective mathematics teaching requires an understanding of what students know and need to learn and then challenging and supporting them to learn it well.” NCTM 2000, p. 2

Prepare them for what……… 1.) Connections to other math concepts and subject areas 2.) AP Calculus Exam 3.)College Readiness

1.) Connections to other math concepts and subject areas

In US we teach math using a foundation mentality. We teach something one year, then build on that the next year, and build on to that the next year, etc…. We have to connect where they have been and be able to connect it to where they are going.

According to the NCTM Process Standard IX, Connection, students should be able to: recognize and use connections between mathematical ideas; understand how mathematical ideas interconnect and build on one another to produce a coherent whole; recognize and display mathematics in contexts outside of mathematics Students should “see and experience the rich interplay among mathematical topics, …” Quote taken from our reading NCTM Principles and Standards of School Mathematics

Connections!

Real World Connections Blog Comment I - “Since we are teaching the 21st century student, we as teachers need to make sure we are making those real world connections for our students.” – Kelly Blog Comment II – “Get with you math [science] teacher and work that out for some integration then go to Carowinds and study the velocity of the roller coasters!” – Kara One of the main ideas of calculus is velocity !!!!!!!!

Imagine a roller coaster…

Secant line = Average Rate of Change [average velocity] = Slope Tangent Line = Instantaneous Rate of Change [instantaneous velocity] = Derivative

Wiki Postings: (I will include visuals) Electric companies need to know how much cable is needed to 1) go diagonally across a park (regular math) or 2) hanging in a catenary pattern from two towers (calculus). Architects need to calculate the area of a roof to know the cost and figure the weight on it with and without snow when it is 1) a peaked roof, but essentially flat (regular math) and 2) a complicated, non-spherical shape like the Houston Astrodome (calculus). Calculating the lead needed to hit a moving object for 1) a quarterback throwing to his receiver (regular math) and 2) NASA finding the lead for the Viking 1 aimed at Mars where both Earth and Mars have different elliptical orbits and the relative speed of both planets is constantly changing, not to mention the constantly changing gravitational pull of the Earth, Moon, Mars, and the sun during it’s trip (calculus). In the real world, relationships are rarely as simple as a straight line graph. This is what makes calculus so useful.

Another wiki posting: “Integrals appear in many practical situations. Consider a swimming pool. If it is rectangular, then from its length, width, and depth we can easily determine the volume of water it can contain (to fill it), the area of its surface (to cover it), and the length of its edge (to rope it). But if it is oval with a rounded bottom, all of these quantities call for integrals. Practical approximations may suffice at first, but eventually we demand exact and rigorous answers to such problems.”

2.) AP Calculus Exam-

Accumulation is a major concept on the AP Exam. We read an article about accumulation and the Fundamental Theorem of Calculus. During our activity on Tuesday where we created the sine and cosine curve based on the unit circle, we discussed finding the area under one of the curves (integral) by using rectangles. Adding all of the areas of the rectangles together is the accumulation or area under the curve. The widths of the rectangles can be different. There is varying rates of change; therefore, is not constant. ( See link below.)

Connections!

Middle school- accumulation…finding total amount, total surface area

3.)College Readiness Communicate mathematically

Multiple Representations As we progressed through this class we solved problems using multiple representations. We communicated mathematically in all forms: N- numerical A- algebraic G- graphical S- symbolic Example- CBR Activity High school calculus and college professors are looking for this.

Questioning Techniques Questioning techniques- “teacher found that the focus of their questions shifted from answers and procedures to observations and uses of patterns, comparison’s of different strategies and representations, and connections among mathematical ideas” (Designing Questions to Encourage Children’s Mathematical Thinking p.402)

RIGOR Rigor- teaching through exploration as demonstrated by teachers/writers in articles read in class; must change students thinking from the low level of Bloom’s taxonomy to the higher levels. Students have to become “out of the box” thinkers.

“The practical side [of his paper] is motivated by our general lack of insight into poor quality of calculus learning and teaching in the United States” Thompson- Imagery of Rate and Operational Understanding of the Fundamental Theorem of Calculus As a result of taking this class you should now be prepared, more insightful, and determined to positively impact the calculus learning in our country. Some final words to take with you:

Providing Support Yields Connections Hopefully Involving Calculus