Monthly Archives: January 2014

Five Observations Regarding Teaching Similarity According to Common Core

For the last two months I taught geometry according to Common Core, at least as it is interpreted by EngageNY in its Module 3 (Similarity and the Pythagorean Theorem). I am “off the reservation” in the sense that I am the only geometry teacher who is not following the book and is “going common core”.

Based on my experience, limited as it is, there are a number of observations that I would like to share. These observations pertain only to my experience in my school and with the students that I have. These students are largely freshmen and sophomores who have gone through Algebra I and, if they are sophomores, through Algebra II.

First, there is dearth of common core curricular material. As far as I can tell, EngageNY is the only entity that publishes a teacher edition on how to teach Similarity and the Pythagorean Theorem in the SPIRIT of common core standards.  There are other places on the Internet where one can find problems and projects geared to common core, but to come as close to a text as we are used to, EngageNY seems to be it.  I should add that the EngageNY module I taught from is for 8th graders – there is nothing for High School Geometry. (Professor Wu’s notes regarding High School Geometry are geared to future math teachers and they are extremely useful as to the common core approach, but they do not constitute a text).  However, given that we are in a transition period to a full blown common core curriculum, and given the mathematical preparedness of most of my students, I saw no problems using this Module.

The second thing that stands out is the unity and the internal logic of the common core approach. Similarity, the Pythagorean Theorem and even trigonometric functions are all based on geometric transformations, specifically dilations. While as a teacher I appreciate the consistency of the approach, I wonder if it truly enhances significantly student understanding. The approach that we choose in teaching math goes directly to the fundamental question of what do we want our students to get out of high school math. Do we just want them to be able to solve problems? In that case a drill and kill could work (more or less). If we want them to think mathematically, then yes – a math course with logical consistency will probably be superior to the current texts. However, students, especially at this age, rarely take the long view and can stand back and appreciate the beauty of a consistent approach to math teaching. In the absence of a randomized, control study, I don’t think that we have yet a clear cut answer. Personally, I am pleased that when I teach, I feel that there is a logical continuation and unity to my teaching.

Third, I have observed that, at least the EngageNY modules are not problem rich.  At the end of it all, math students should be able to solve math problems and being successful at that comes from practice (as well as teacher guidance). My hope is that future texts will have a significant number of problems in them and that those problems include some challenging ones.  I have had difficulty finding challenging problems in similarity. (An example of what I would consider challenging High School problems in similarity can be found here. These are a very far, far cry from the problems we find in our textbooks).

Fourth, common core puts emphasis on students being able to explain their reasoning. I have asked questions that require explaining a result and the outcomes are sad, if often funny. Students (at least mine) have difficulty in answering in complete sentences. Responses are terse as if the students are afraid that a longer paragraph will show that they are not sure of their answers. Because of this, I found that asking “why” and “explain” questions are a good diagnostic tool for assessing students understanding, perhaps more so than checking the results of their problem sets. However, I keep in mind the fact that students have never before been asked to explain their answers, and that gradually, by modeling complete answers in complete sentences, students will improve. I hope.

My last observation has to do with modeling. In common core there is also emphasis on students being able to apply their math knowledge to more complex, real life situations – to do mathematical modeling. Once more, this is not something my students have been asked to do in the past. All previous student experience in math has been structured and structured rather rigidly. The “real life” problems in textbooks are so contrived, they are laughable. They are also duplicates of “regular” math problems. An unstructured project is a challenge for my students. I gave my students their first project – the rolling cups project from the Shell Center. The project is part of their exam in similarity and is the take home part of the exam. We will see what happens.

All in all, teaching similarity according to common core has been interesting and challenging. Part of the challenge is that I am the only one doing this and that the school-wide assessments are written by the other teacher who are teaching the traditional way. They have more time to practice “standard” problems than I  do, because of the time we spent in class going over dilations and how they form the basis of similarity. Also, their assessments do not require explanations of the results – they are simply “find x” problems. The other big part of the challenge is that both I and my students are new to this common core business. we still finding our way, but I think it is a worthwhile endeavor.