My goal for my end-of-course project is to develop a mini-unit plan on Constructions. The geometry teachers in my math department have decided to spend approximately four class periods (one hour each) on constructions, including a summative assessment. I would like to incorporate some hands-on constructions, but also involve some technology in the form of Geometer’s Sketchpad (this is what we have used in the past) or Geogebra. Here are my thoughts for a general outline:
Day One:
I would like the students to do some comparing of the different hands-on ways to do constructions. I found a great resource that asks students to compare the following: miras, patty paper, and compass/straight edge. I really like this idea! However, I might eliminate the patty paper to save a bit of time and because my students are already familiar with using the miras. I’ll have to give this some thought. After doing our comparisons, the students will work on various compass and straight edge constructions, keeping a bulleted list of their steps in sequential order. I will model a couple at the start. Each student will be assigned a partner with a similar ability level. While the concept is still new, I like that students will be able to bounce ideas off of each other but that one person will hopefully not dominate the discussion.
Day Two:
This is going to be the Geometer’s Sketchpad/Geogebra day. I will need to spend some time going over the various functions in the program so that students will be able to take off and do some constructions on their own. I may create a short GS or GGB tutorial that repeats some of the constructions from the previous day as a warm-up. From there, I will have a self-paced activity that contains several new constructions. I’d like to research how to make my activity more creative than just having a list of constructions. One idea I have (that may or may not work) would be to have students take the shapes of their constructions and create some kind of picture out of them. (If students print their sketches, this might be something they can do at home.)
Day Three:
I’m not totally decided on what I would like to during this class period. I want to extend the students’ knowledge in an engaging way. Right now, the only idea I have is from Science vs. Magic. This constructions “game” does not allow the students to move to the next level without completing the construction on the current level (similar to a video game). However, with the range of abilities of the students in my classes, I may have to differentiate instruction and offer a couple of activities. I think some of my students would really struggle with this one without some extra support.
Day Four:
On the last day of the mini-unit, students will be given a summative assessment (probably a large quiz). I plan to spend the first half of the hour reviewing (perhaps going over the activities from Day Three) and the second half of the hour giving the assessment.
So that is a synopsis of what I've been tossing around in my head, but I have yet to start putting anything together. If anyone has some resources/activities to share, I would really appreciate it!
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!
Upon reading Authentic Tasks in a Standards-Based World by Edwards, Harper, and Cox, my inner battle begins to rage once more. My desire to take students to a deeper level in their learning is coupled with my desire to cover all of the standards. But how can I do both? The limiting factor is always this: time. Whether we like to admit it or not, the decisions we make as teachers are all too often dictated by time.
In the Authentic Tasks article, the authors use “The Meeting for Lunch Problem” as an exemplar. Students were given a very open ended question that asked where three people in different cities could meet for lunch such that each person had to travel the same distance to get there. Students were given the freedom to investigate and most of them successfully discovered that finding the circumcenter was indeed the method that would lead to a solution. If I am being honest with myself, this is where I would have likely ended the activity. After all, this lesson provided students with an application to a required standard. Check it off and move on, right? Not for Edwards, Harper, and Cox! They encouraged their students to ask probing questions and allowed them to take the activity in a new direction. When students discovered that in some triangles, the circumcenter was obviously out of the way for the three travelers and a foolish way to decide on a meeting place, they developed a revised goal in the “Meeting for Lunch” goal. Together with their teachers, they decided to find the meeting point that would result in the least total distance traveled for all three people involved. Even though this was uncharted territories (even for the teachers), they pressed on. After a lot of studying and a bit of research, they found that the “Fermat Point” would achieve their goal of finding the least total distance. Now, mind you, the Fermat Point is not mentioned anywhere in the Common Core State Standards (CCSS). Does that make studying it a waste of time? Absolutely not. These students applied their knowledge of the various circle centers in this problem as a lead-in to the critical thinking and problem solving that enabled them to extend their learning. They performed at a higher Depth of Knowledge level than they would have had they stopped after simply finding the preconceived circumcenter solution. The students took ownership of their learning.
Unfortunately, in the back of mind is still this issue of time. I wish I could push my students to a deep level of understanding with each and every topic, but I would simply run out minutes in the day and days in the school year. Even though I can’t turn every lesson into a three day mini-project in the computer lab, I must strive for balance. When I see a great learning opportunity to push my students, I must take it. Other times I must use my professional judgement to decide when that extra push isn’t quite as necessary. The most important thing is that my students are learning how to think at a deeper level on a regular basis.
Another highlight of the article for me was its quote from the Common Core State Standards for Mathematics (CCSSI 2010):
These Standards do not dictate curriculum or teaching methods... [A] teacher might prefer to teach a topic of his or her own choosing that leads, as a byproduct, to students reaching the standards for topics A and B. (p. 5)
How refreshing! For some reason, this quote makes me feel better about going deeper with some standards than with others. The very institution that developed said standards is giving me permission to do it my way! I can hardly wait.
Professor Golden: I would appreciate feedback regarding the detail of my writing. Did I give enough detail to make my viewpoint known? Did I provide the right amount of information about the article to facilitate my reader’s understanding? Thanks!
It has been a couple of years since I taught geometry, and I only taught it for one year at that. Needless to say, my geometry skills are a little rusty. As a result, taking Modern Geometry (MTH 641) at Grand Valley State University this semester is proving to be a challenge, but in a way that I feel is helping me to become a better teacher. My professor has allowed the focus of the course to be on the teaching of geometry rather than purely geometry content, as all of the students are currently teaching mathematics at some level. This has given me the opportunity to analyze my approach and get down to the foundational building blocks of this fascinating branch of mathematics.
Over the past couple of weeks, we have been looking in detail at Euclid’s Elements. Reading the propositions in the mathematical language of Euclid’s day has challenged me to think about the true meaning of each statement. I cannot rely on my intuition or tidbits I memorized long ago. Rather, I feel it has placed me on an even playing field with my students. I actually feel as though I am learning it for the first time. One of our assignments this week was to take one of Euclid’s propositions from Book 3 (circles) and “modernize” it. I chose Propositions 22: The sum of opposite angles of quadrilaterals in circles equals two right angles. My version reads as follows: The sum of the measures of any pair of opposite angles in an inscribed quadrilateral equals 180 degrees. When I scrutinized the proof, one of Euclid’s methods of reasoning was easily recognizable to me. His proof said, “in any triangle the sum of the three angles equals two right angles.” Clearly two right angles is the same as 180 degrees, which indicates that he is referring to the modern-day Triangle Sum Theorem. His next justification was more of a puzzler. He claimed that two angles CAB and BDC were congruent because they were “in the same segment BADC.” This took some digging. When I looked back at Proposition 21, I found that this claim was in reference to the fact that a central angle subtended by two points is equal to twice the inscribed angle subtended by the same two points (paraphrased). By comparing two different inscribed angles subtended by the same two points, their measures were inferred to be the same. This, I finally realized, was an application of the Central Angle Theorem. From there I was able to finish my proof with ease. (See below.)
This assignment proved to be a great exercise! It took me out of my comfort zone and required genuine thinking and learning to take place. I have a renewed respect for historical mathematical texts, and I will look for a ways to have my high school students examine a piece or two sometime this year. It will also fit in great with the math department’s goal at my school to incorporate content literacy into the curriculum.
Professor Golden: I would love to hear some feedback from you with regards to some historical texts that might be the appropriate level for high school students, or even an idea for how to incorporate historical reading into an assignment or project. Thanks!
Last week my students saw and practiced their first method of proof in my freshman geometry class: Proof by Indirect Reasoning. I knew this would seem a strange concept to them. "If I think something is false, why would I say it is true?" was a question I knew I would be asked. In reality, I looked out at a sea of blank stares after showing the first example. Teaching how to write proofs takes time, though, and after a bit of practice it is finally starting to click.
For my initial lesson, I came up with an analogy to present the concept of indirect proof that I felt might help the students relate with this method of reasoning, however backwards it may seem. My advice to my students was this: Think like a lawyer.
Here is how our analogy went:
Me: Alright, who in the class looks especially guilty of a crime today?
After many-a-suggestion, I decided to pick a student that would really play up the part so we could have some fun with it.
Me: What crime did Johnny* commit?
*Name changed for student's privacy.
Johnny: I robbed a bank.
Said a little too quickly, I thought. ;)
Me: Okay. I'm a prosecutor in the courtroom. I think that Johnny robbed the bank, but he is pleading not guilty. This is how I will begin my proof: Let's assume that Johnny is innocent. Johnny, what is your alibi?
Johnny: I was on vacation.
Me: Were you on vacation the entire week of the crime?
Johnny: Yes.
Me: Okay, Johnny. But here's the thing. I have a security tape that shows you shopping here in town the night before the crime took place. You say you were on vacation. Unless you can be in two places at once, I'd say this is a contradiction!
We than had a small sidebar conversation about the definition of a contradiction, which they had already written in their notes.
Me: This means my original assumption - that you were innocent - must be false. Therefore, you are guilty!
This was a fun way to introduce the topic of indirect proof so that my students could make a connection. Everyone has seen this method of reasoning used by a lawyer on TV, right? While it didn't make my students immediately become expert proof-writers, I did feel that it helped them to gain some insight and understanding into how and why this method of proof is used. It also supports the stance of Eric Knuth in his article, Proof as a Tool for Learning Mathematics. He asserts that proofs should not only show that something is true, but also why it is true. This will enable them to better comprehend the underlying mathematics. After all, if students don't understand the why, have we really done our job?
If anyone else has some analogies or strategies for teaching a particular method of proof, I'd love to hear about it in the comments section!
Learning about the Van Hiele levels in my grad class right at the start of the school year once again got me thinking about how I can assist my high school math students to perform at a higher level. With the implementation of Common Core and the looming SBAC assessments, math teachers are feeling the pressure now more than ever to ensure that our students are able to think critically, reason logically, and problem solve effectively. Basic comprehension and limited mathematical skills will in no way be enough when it comes time for mandated testing (nor should it!). I am realizing quickly that not only must we work harder, we also have to work smarter. This means understanding how our students learn.
The concept of Van Hiele levels follows the common educational theme of progressive levels of thinking. What makes it especially useful is that it is specific to geometry! Bloom’s Taxonomy, which seemed to be a hot topic during my undergraduate career, includes the following levels: Remembering, Understanding, Applying, Analyzing, Evaluating, and Creating. The more recent focus since our transition to the Common Core has been on Webb’s Depth of Knowledge (DOK), which includes Recall (level one), Skill/Concept (level two), Strategic Thinking (level three), and Extended Thinking (level four). Both models can be applied to all subject areas, and both mean basically the same thing. Now consider the Van Hiele levels: Visualization, Analysis, Abstraction, Deduction, and Rigor (as outlined in Marguerite Mason’s article). These levels can easily and directly be related to the Geometry Common Core State Standards.
Having Van Hiele in mind will help me to apply Webb’s DOK in my Geometry classes this year and to identify the levels at which my students are performing. I have only taught Geometry once prior to this year (as a sophomore-level course), and in my experience most students came in with Visualization and Analysis skills. Bright students picked up Abstraction, but Deduction was a struggle for almost all of my students. This year, my Geometry students will primarily be freshman. They will not have taken Algebra 1 yet, and I am interested to see if that impacts their capabilities or if my observations remain largely the same. I am hoping to develop strategies and activities that will push my students to use Abstraction and Deduction on a regular basis, so that eventually it comes naturally to them.
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