Let’s face it... School improvement goals almost invariably have something to do with reading and writing. Keep in mind that school improvement provides school-wide goals. All departments are expected to contribute to improvements in literacy, not just English and social studies. But math? Can reading and writing really be incorporated into math in a meaningful way? I say yes.
As we ready our students for the SBAC tests and for life after high school, we need to be thinking about how to prepare them to be successful in a variety of situations. There are two critical skills that we must focus on in secondary math education: 1) technical writing and 2) reading for information/understanding.
I had an interesting experience with technical writing already this school year. In my geometry classes, I did a mini-unit on proof writing. When students were told that we would be writing paragraph proofs, you should have seen the looks of dismay on their faces! Paragraphs... in math? Is that even possible? When the students saw that writing a paragraph proof simply meant putting their two column proof into complete sentences with some transitions, they were relieved to find that this was not a difficult task. They also learned that formal writing could include mathematical symbols. Students need exposure to this type of writing, especially if they are to enter a technical field.
Additionally, I think we need to remember that writing does not necessarily mean assigning papers or essays. If a student is asked to explain their process in solving a problem or to justify an answer using data, this counts as writing! If they are expressing themselves in written language, we are accomplishing the goal to incorporate writing. However, how does one grade a piece of writing without ELA training? At my present district, we have developed a rubric to score a student’s writing based on “readability.” This means checking that, overall, correct grammar is used. It means checking to see if the writing flows well using necessary transitions. And in addition to a readability grade, students can still be scored based on the mathematical content of their writing!
Reading is also necessary and appropriate in the math curriculum. When students leave high school, they need to be skilled in reading for information. They need to make inferences from a text. They need to make sense of mathematical language. Thus, having students read for understanding is an excellent exercise for them. Look for a section in the textbook or for a quality article and have the students work through it at their own pace. Better yet, perform a close read with the whole class (students read the text multiple times, with a new focus each time).
Reading and writing do not have to be an extra burden or another thing to just check off the list. Think about how you can incorporate reading and writing in a meaningful way in your math classes. You owe it to your students!
The following are my thoughts on the article, Selecting and Creating Mathematical Tasks: From Research to Practice (see below).
Although this article was written fifteen years ago, its ideas are very relevant to math education today. There is a very close correlation between the Levels of Demand described by Smith and Stein and Webb's Depth of Knowledge, which is widely being used in schools today. Moving from low- to high-level in Levels of Demand/Webb's Depth of Knowledge, we have: memorization/recall, procedures without connections/skill concept, procedures with connections/strategic thinking, and doing mathematics/extended thinking. It is interesting to read that teachers had some significant discrepancies in their classification of certain mathematical tasks. Often times teachers feel that tasks are more demanding than they actually are. Developing accurate labels for various tasks is vitally important in the preparation of our students for standardized tests (i.e. SBAC). Additionally, we must ensure that we are challenging students with higher-level tasks, as Smith and Stein point out that through these tasks students have higher engagement resulting in higher levels of learning.
In my classroom, here are a couple examples of different levels of demand:
1) Higher-Demand: Geometric Proof Group Activity (There is not a set procedure as to proving any given statement. Students must have a good understanding of definitions, theorems, etc. and be able to apply them. The unpredictable nature causes anxiety in some students.)
2) Lower-Demand: Practice with Area and Perimeter (Students must practice determining area, perimeter, side length, etc of regular geometric objects with some given information. This is very procedural and students can memorize the formulas in order to solve the problems.)
Source:
Smith, M. S., & Stein, M. K. (1998, February). Selecting and creating mathematical tasks: From research to practice. Mathematics Teaching in the Middle School, 3, 344-350.
Having taught in a couple of different school districts over the past couple of years and after conversations with teachers in other school districts, I have really started to wonder about the appropriate placement of Geometry in the high school math curriculum. I have seen many approaches, but I am not sure any of them is ideal.
My previous school district took on the most traditional approach: Algebra 1 first, Geometry in the middle, and Algebra 2 last. The idea is that students need a solid Algebra foundation (mostly equation solving) for many of the concepts that they see in Geometry. Some see breaking up the two years of Algebra as a pro as well. They feel that students that dislike Algebra will become more interested in math when they see Geometry in year two.
At my present district, though, the stance is taken that students will benefit from taking the two Algebra courses successively. It is also reasoned that students receive enough equation solving exposure in junior high to be successful in a high school Geometry course. In my first two months teaching freshman Geometry courses, though, I have had a couple of concerns. Some students’ algebra skills are weak enough that I do feel it interferes at times. I find myself taking for granted that my students will have a certain algebraic skill only to find that I have to back-track and slow down. The more advanced students pick up on it quickly, while others tend to struggle. It is still early, so I am keeping an open mind to see how this plays out the rest of the year. Another concern of mine is that my freshman are not always mature enough as learners for the challenges that Geometry presents. In this course, there is so much application and visual/spatial reasoning. This seems to be a huge transition from junior high mathematics.
Yet another sequence I have heard about in a nearby district is to “sandwich” Geometry in the middle of Algebra 1. Their students take the first term of of Algebra 1, then take both terms of Geometry, followed by the second term of Algebra 1. Algebra 2 comes last. The rationale is that in the first half of Algebra 1, the focus is generally on equation solving and linear functions, which is primarily what is needed for Geometry. Then, the more difficult concepts in Algebra 1 would come after students complete the full Geometry course. While this would seem to address my concern about students’ lack of Algebra skills, I still wonder if they would be lacking some cognitive maturity.
So this brings me to the question that has been in the back of my mind for a while... would it be appropriate for students to take Geometry last? The more I think about it, the more I like it! I like that students would take Algebra 1 and Algebra 2 successively. I like that they would have very solid algebra skills at the onset. I like that they would be more mature learners and would be seeing the most intense proof content later in their career (and closer to college). So then, why is this not a common practice???
I would love to get some feedback from fellow math teachers. What is happening at your school? What are the pros and cons? What do you think about the “Geometry last” idea? Do you know of any research done on this topic?
As my unit on Transformations with my Geometry students is quickly approaching, I decided to take a little time to review Jen Silverman's Transformation Unit. Jen does curriculum work for the state of Connecticut and was kind enough to share this unit for feedback purposes and for individual teachers’ use.
The first component of the unit that I took note of was that the content is presented in a way that guides students to discover the key ideas, rather than requiring the teacher to deliver the information. A little history is even worked in! I love that students need to apply their prior knowledge and can work at their own pace! I also liked the “Amazing Vector,” “Translation Puzzle,” and “Flip Flop” (to name a few), where it ties in a little play while students practice their newly acquired knowledge about transformations. The ambrigrams activity is also a great way to spark the interest of more artistic students - very cool! (While the video clip was interesting, I was hoping it would have more mathematical content.)
The GeoGebra lessons were also very interesting to look through. I like how Jen includes screenshots and specific instructions to guide students. I think with the very first GeoGebra activity, though, I would need to do it as a whole class activity rather than cut the students loose. Another option would be to do a short GeoGebra tutorial. In my school, teachers have used Sketchpad in the past, so I’ll have to decide if I want to follow their lead or venture out and use GeoGebra. Either way this hands-on way of allowing students to explore will undoubtedly deepen their interest and understanding.
Another neat idea in Jen’s unit is the idea of creating a quilt using square patterns and 90/180/270 degree rotations. I’m always looking for ways to display student work in a fun, colorful way! As a quick review, I would probably also ask students to plot a point of their choice in their original square, rotate that point, and write the appropriate coordinates in the other three squares.
Another strength of the unit are the opportunities for spiral review - asking students to apply knowledge from earlier in the unit to current topics. Often times students are asked to explain their reasoning, which is a great practice for SBA tests! For example, “Is rotation an isometry? Explain.”
Overall, I think this is a fantastic unit on Transformations. When I have the time, I plan to compare it with my current Transformation unit to see where I can incorporate some of Jen’s ideas. I know that both the students and I will appreciate the explorative nature of these student-led activities. Thanks, Jen!
Conjecture. Sure, my students know what it is. They are even asked to make a conjecture of their own in a homework problem here or there. But should that be the extent at which my students develop their own ideas in a geometry class? Should I really be devoting little to no class time for students to explore mathematical concepts? My graduate professor, John Golden, is challenging me this semester to answer these difficult questions.
I remember one of my least favorite undergraduate math professors, and I remember him well. What was so frustrating about my professor was that he would never just answer my questions! Rather he would follow it up with another question, or give me a “hint” that didn’t seem to help at all. Many times I sat in his office hours only for him to tell me that I just needed to “think about it some more.” Wasn’t it his job to help me?! That’s what all my math teachers did previously. As much as I hate to admit it even today, I have probably never worked harder or learned more in another math class. It was a humbling experience, but also quite satisfying to know what I had accomplished. I left that class feeling much more confident in my abilities.
The reason I bring up my undergraduate math professor in this post about conjecture is that I think I need to follow his lead a bit more with my geometry students. Rather than simply provide definitions and theorems (as is often the case), I need to get my students thinking. After all, spoon feeding leads to helplessness. Just as I learned more and developed a deeper understanding in my undergraduate math course, my students will have a similar experience if I allow them to make conjectures and explore whether or not they are correct. It is the struggle that will cause them to develop into more mature math students. As I continue to teach my current unit on triangles, I will strive to find opportunities in which students can discuss, experiment, and play. I will encourage them to make conjectures before I tell them the desired result. My hope is that they will feel a sense of accomplishment, in addition to the cognitive benefits that will surely surface.
One unit in particular that I am excited about is our construction unit, which will take place during second semester. In the math department, we are revamping this unit to provide more of a challenge for our students. In the past, students were simply given a list of steps for various constructions. After performing those steps, they moved on to the next construction. This was a rote process that required very low-level thinking. By giving our students time to experiment by hand and then on Geometer’s Sketchpad, they will get a taste of what famous mathematicians went through as they created those constructions for the very first time.
Professor Golden: The balance activity we did in class would be great to do with high school students! I would appreciate feedback for any other areas that you think a hands-on activity would fit in well. Thanks!
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