Oh Polynomials. My least favourite unit by far in the Foundations of Math and Pre-Calculus 10 course I am teaching. Find the greatest common factor, least common multiple, factor these trinomials, collect and simplify like terms, the swimming pool has a width of 5x + 1 and a length of x + 2… YAWN.
How can I frame a boring, completely algorithmic and skill-based unit into something that's relevant and meaningful for my students? I am borrowing Dan Meyer's definition for relevance here.
It Begins with a Question…
A colleague asked me today, "How much time do you have for homework at the end of class?" This was a surprising question to me, and as I thought back over the 10 day unit, my answer was almost none. The question sparked a great dialogue between us about our approach to teaching the same content in our respective classrooms. It really made me think. I realized that while I still dreaded teaching polynomials, I had found a way to improve the way I taught it from first semester that required less rote work and more thinking.
One thing that has not changed, however, is that I avoid teaching FOIL method like the plague. It only works for expanding binomials and does not apply for polynomials with more than two terms. After I read this article I was convinced I would never need FOIL in my classroom:
For a good laugh:
Some things that came up in our discussion:
1. Algebra Tiles
Definitely a hate-hate relationship. As a math teacher, I am obligated to entertain this idea and I do admit it has its benefits, especially in lower level math classes when students are initially being exposed to distributive property and the like. The problem, however, was that my students were already armed with the skills and knowledge of multiplying and factoring polynomials. Moreover, the limitations of using tiles far exceeded the benefits, in my opinion. Algebra tiles do NOT work for: polynomials higher than degree two, multiplying more than two polynomials, and multiplying polynomials with more than three terms. This meant that it took more effort for students to understand how and why it works.
Nevertheless, we spent a few classes examining algebra tiles and their usefulness. Rather than approach it from the typical standpoint of using algebra tiles as a manipulative, I wanted students to see the link between the algebraic and pictorial representations of polynomials. This took work and was not as straightforward as it seemed. A big takeaway for me was that students gained much more out of the experience when they were able to physically manipulate the tiles and arrange them into their "factored forms." Last semester, I "taught" algebra tiles by merely showing them examples and drawing them on the board. It took a bit more prep, but this semester I printed eight sets of tiles (positive and negative) in my classroom and had students manipulate them instead.
If we were to spend any more time on the unit, or if this was a lower grade level, as an enrichment activity I would have students discuss the limitations of algebra tiles and look for ways to address them.
2. Picture Talks
I like to use Sarah VanDerWerf's Stand and Talks as a format for students to discuss picture prompts. I find that the buy in for engagement is much higher when the prompt is linked to physical movement. My favourite questions for photo prompts are: "What do you notice?" and "What do you wonder?"
What are my photo prompts, you ask?
That's right. Algebra tiles.
Goals for students:
I like this activity because it is easy to differentiate and works well as a "minds on" for any topic. Asking students a general question like what they notice/wonder means that lower ability students can comment on ANY aspect of the photo (e.g. "there are blue and green rectangles") while higher ability students can be pushed towards making observations based on any mathematical patterns or relationships they observe (e.g. "the green tiles represent positive polynomials and red tiles are negative").
3. Factoring Method - Criss Cross or Sum Product?
I've had the great fortune of only having one prep and a spare block this semester (for friends and readers who don't teach, that's teacher jargon for FREE TIME, kinda. The details are not important). Anyway, I've been making drop-in's to my fellow colleagues classrooms with my new-found "free time" and one thing I picked up was the importance of proper SEQUENCING. For instance, a natural progression for factoring trinomials might look as follows:
That, together with a quick exercise on sum/products, helped me push students towards seeing the relationship between the factored form of a trinomial, and the sum/product method.
I prefer this method over the traditional "criss cross" method for a few reasons
Which One Doesn't Belong? (WODB)
Fantastic activity for building up thinking skills and vocabulary. Each student picks one of the expressions and must argue why that one doesn't belong.
"27x^2 doesn't belong because it is the only expression that has a coefficient with a perfect cube"
"45x^2 doesn't belong because it is the only expression that has a coefficient with 5 as one of its prime factors"
More WODB prompts can be found here.
How it works: One student is chosen to stand/sit at the front of class facing the audience, they are in the "hot seat". Behind them, a vocabulary term is shown for the rest of class to see. Students in the audience must help the student in the hot seat guess the vocabulary word by miming, explaining the definition, or giving examples. They may not use any part of the word in their explanation.
Modifications: Differentiate by giving students the option of bringing a "cheat sheet" of vocabulary terms with them. Prepare students for the activity by giving them cross word or fill in the blank exercise reviewing the vocabulary words for the unit. An "expert round" can include vocabulary not on the cheat sheet. "Challenge round" can be facing a peer or the teacher. Can play in teams or as a class.
his activity was shared by a good colleague of mine. To get BINGO, students must find one "expert" in the classroom to answer each question on the bingo card until all the questions have been answered. The student who answers the question must sign their name. A student may not be asked more than once to answer the same Bingo card.
Wow, I think Fawn Nguyen is absolutely spot on when she says that classroom management is completely dependent on the who the teacher is and the types of students you have. (Fawn is my heroine BTW - just needed to throw that out there). This seems obvious, but it's taken me a while to put this tidbit of knowledge into good practice.
Books like First Days of School and Teach Like a Champion have been invaluable reads, providing tons of practical advice teachers can implement right away. The issue is learning how to filter that knowledge so that it's true to your own teaching style and well-suited to who your students are. Teaching math in an academic classroom is way different than in a college or applied-level classroom, for instance, and not because the material is different per say, but because the students' attitudes towards math differ tremendously. I found that students who are in applied or college-level math courses generally have lower confidence in their math abilities. Subsequently, each wrong answer means another failure added to the list and just reinforces what they already knew, "I'm not good at math." Here, priority #1 is to build a safe and welcoming classroom where a culture of error is the norm, and is celebrated as a vehicle for learning. Likewise, North American students and Asian students also differ in their attitudes towards math. Comparatively speaking, math anxiety seems to be a bigger issue in North America. On the other hand, students in Asia tend to be really good at math, they respect the subject, and they will work hard at it, even when things get tough. In Asia, the norm is repeat and rehearse everything the teacher's taught, but the challenge is to get kids thinking independently and creatively.
Different mindsets on math, as told in memes:
An interesting article here on the influence of culture on achievement in math.
April's Tips on Classroom Management
Okay, so still fairly new at this teaching thing, but here are some things that worked for me:
0. Plan a good lesson. I'm echoing Fawn on this one when I say that having an engaging lesson solves soooooo many potential discipline issues in the classroom. Kids will act out when they are bored. I know this because I WAS this. I mean, I was an A student throughout high school and a MODEL student at that. One summer I took a physics and had a teacher who literally read the textbook to us. I can do that myself, thank you very much. So, rather than sit in silence and boredom, I discovered that the reflective properties of light were pretty fun to play around with. I was particularly intrigued at how various angles of light rays from the window would bounced off the shiny surface of my watch right into - yup, the teacher's eyes.
1. Learn names. I always make it a point to know the names of all my students and connect with them in some way. To me, there's nothing worse than being called "you in the red shirt" or "hey you." Teach students, not the subject.
2. Don't repeat student answers. I first noticed this during my observations of a veteran teacher while I was student teaching, and it completely changes the way discussions flow in the classroom. If a student answers a question, and the teacher repeats the answer (usually in a louder voice or with elaborations), in the students minds this translates to, "Information is not important, unless it comes out of the teacher's mouth." If a student says something really insightful, ask them to repeat it instead - you'll have reinforced two important messages to a) the student: "Your contributions are valuable!", and b) the class, "We have a lot to learn from our peers!" It is so vital for teachers to give students opportunities to be responsible for their own learning.
3. No Opt-Out. I got this one from Teach Like a Champion. The premise is simple, if a student does not know the answer to a question, they cannot get away with "I don't know." You might ask another student for their thoughts, you might provide a hint, you might just say the answer outright, but you will always go back to the student who said, "I don't know." "I don't know" is not an acceptable answer in my classroom. We want students to get from:
"I don't know" therefore "I don't have to try", to
"I don't know, YET" so "I'm going to keep trying."
4. Keep it simple. If you're like me, you probably have a list of 20 different procedures, routines, and policies you'd like your students to do and maintain throughout the year, but this is not realistic! I ended up flopping on most of them. My first two years of teaching have been chaotic, and I'm slowly coming to accept that it will be this way for a while. Focus on the five most important guidelines and procedures that your classroom cannot do without, then build from there.
5. Document everything. The biggest lifesaver for me last year was getting students to fill out "Action Plans" for whenever they made a bad choice. There are many variants of this on Pintrest. Student Responsibility Cards for homework were also cool, but didn't work out that well for my classroom because I didn't follow through on consequences. So I'll keep the first and toss out the latter. Links to some documents I used below.
6. Give logical consequences. When I was little, my punishment for making bad choices was always the same; mum would make me stand facing the wall with my arms up, and fingers pinching my ears. I guess it was supposed to make me feel ashamed of my actions, but it doesn't make sense. Let's remedy the behaviour, and not punish the student. Examples of logical consequences below.
Yay! So excited to actually start contributing to the #MTBoS (Math Twitter Blogosphere) community, and to start blogging more in general! This Sunday Funday blogging initiative is the perfect excuse to set aside some me time each week and reflect on my teaching.
I'm now going into my third year of teaching, and so far, each year has been in a different country, which has made each "first day" even more special.
My First First Day
In my very first day of my very first full time teaching job in Kazakhstan (blog post here), I spent the first day getting to know my students, telling them a bit about myself, talking to them about my expectations for the class, taking selfies of all the students, and giving them some general advice about how to succeed in math class. I found that it was important and effective to start building those relationships with my students from day one, and by learning all their names as quickly as possible, I let them know that I notice them and care about them.
Prior to preparing my first day lesson plans, I soaked up as much information as I could with all the resources that were available to me. I had read First Days of School by Harry and Rosemary Wong and a few other teaching books, browsed the internet for countless hours looking for ideas and inspiration, watched this entire video by Agape Management, and looked for elements of each that I thought would be suitable for my teaching style. What didn't work, however, was the fact that I did not start the year knowing where I wanted my students to be by the end of the year. This was difficult because I didn't know much about the culture, the style of teaching that students were accustomed to, and I had never taught a class full of ELL students before (hence why nobody laughed at my jokes). Moreover, I did not have full autonomy over the classroom (it was supposed to be a co-teaching type environment but ended up feeling more like I was "guest teaching" a few times a week); my co-teachers were not fluent in English, and had different visions of how they wanted to run their classrooms, which made it difficult to have consistency when it came to expectations and rules.
What ended up happening was that the first day allowed me to start building relationships with my students, but it did nothing to help me manage my classroom (because nothing was consistently enforced). If I could re-do my first day, I would spend more time getting to know my co-teachers, and specifically, these are the questions I would ask:
1. What are your classroom rules and expectations?
2. What are your beliefs about learning in math? (i.e. How do students learn best?)
3. What are your beliefs about teaching in math? (i.e. How can teachers best reach their students?)
4. Describe a typical day in the math classroom for you.
I learned that it is important not to go in assuming that your teaching partner will have the same views about teaching and learning as you do, and not only that but that I needed to take the time to get to know and understand their views! Had I done so much earlier I would have discovered that hands-on activities, student investigations, or differentiated teaching and learning weren't a common tools in their teaching toolbox. The general style of teaching I observed included very fast-paced progression through the units, with lots repetition and mental computations, but very little time spent developing the concepts or looking at their applications. Knowing this, I would have modified my first day presentation to include some math activities that integrated both styles of teaching.
The Second First Day
Country: South Korea
Grades: 8 - 10
In my second year, I taught in South Korea and had full control over my own classroom, which made it significantly easier to plan and organize everything the way I wanted to. My first interaction with my students, however, was not on the first day of class. We had an "orientation day" in which both students and parents attended brief 10 minute presentations by all their teachers.
I began by greeting every student and parent at the door with a handshake. I called the students by name as they walked into the classroom, which took a lot of time for me to learn beforehand, but was so worth the reactions! Prior to meeting all of them, I borrowed the previous years' yearbook and memorized the faces and names of all the students I would be teaching in my classes. Some of them looked stunned that I knew their names already, when none of them had a clue who I was yet!
At the front of the room, I had copies of letter to parents and the course syllabus which I asked each student to pick up as they walked in. In my presentation, I talked briefly about who I was, my educational background, and what students can be expecting to learn this year. My primary goal was to let them know that I care about them and their learning, and that while this year would be challenging, they would also be supported by me.
Then, on the actual first day of class, I had students fill out a "Get to Know Me" form, we played an icebreaker game (two truths and a lie - my favourite to this day), I talked about the rules and expectations, and I ended the day by teaching them my class dismissal routine. What I DIDN'T do (but wish I did), however, was any science, and that's about to change for this year.
This Upcoming School Year
Subject: Chemistry (?), TBD
Grades: 9 - 12 (?) TBD
As in the past, my main goals for the first day of school are:
1) get to know my students, and
2) set the tone for the rest of the year,
but how I plan to achieve them will change somewhat.
1 - Getting to know my students.
Ideally, I would like to learn student names as quickly as I can, before the first day, if possible. But regardless, I would still like to use name tents with feedback, an idea that Sara Vanderwerf talks about in her blog. I think this is a great way to connect with students individually and on a more personal level. I would also like to take pictures of the students with their name tents so I have a visual record as well. A modification I will make to Sara's version of the name tents is that I will provide some open-ended prompts that the students can respond to, so that they have a jumping-off point for organic thoughts to develop. For instance:
- I noticed ...
- I wonder ...
- I learned ...
- I wish ...
I also really like the Talking Points activity from MathMinds and plan to modify it to make it chemistry specific.
Another idea I've been toying with is some sort of homework assignment that addresses a few or all of the 5 Questions to Ask Your Students To Start the School Year from @gcouros but my problem with this is that I don't want it to JUST be about rapport building, it needs to address or be linked an aspect of science (or science learning) specifically... to be determined.
2 - Setting the tone for the rest of the year.
We will, presumably, be doing chemistry so I would like to begin the first day with a demonstration, or an activity related to the nature and processes of science. Some ideas I would like to try:
Stacking Cups (Dan Meyer) - related to concepts of measurement, accuracy, precision, and estimation
Candle Light Activity (Art of Teaching Science) - importance of observation (qualitative and quantitative) in science, making inferences and predictions, chemical and physical properties
Ira Remsen Demo (Michael Morgan) - observation, predictions, inferences, chemical safety, chemical reactions
I believe that it is important to talk about my expectations and what students can expect out of the class, however, what I DON'T want to do is just read the syllabus on the first day. A prof once suggested just letting the students read the syllabus at home and talk about it the following day so they can ask questions about what they read, or doing a quiz if necessary about the content in the syllabus.
First Day Plan (rough draft):
1) Greet students at the door
2) Have an activity for them to get started with on their desk (either to quietly read the syllabus or fill out a Who I Am handout)
3) Introductions myself and the course
4) Student introductions + talking points
5) Do some science!
6) Dismissal routine
My first day experiences thus far have been pretty nerve-racking and exciting. I'm slowly learning to strike the right balance between talking about rules and procedures to relinquishing control, and giving voice to the students. This is particularly difficult in a room full of ELL students, but once they gain confidence in their ability to speak and be heard, I found that they had a lot to contribute. With international schools, it is usually the case that the students are well acquainted with each other already, so usually the introductions are more for the teacher rather than the students. Even though students may already know each other, however, my role as a teacher to facilitate a safe and positive community cannot be ignored. This was made prevalent to me in Korea when I realized that students still felt unwilling to work with particular classmates even though they had been in the same classes for years. Regardless of country, language, or culture, my biggest take away for the first day of the school year is to BUILD RELATIONSHIPS and ESTABLISH COMMUNITY. I will keep this in mind as I continue to plan for my first day of school in China this school year!
I'm participating in the #sundayfunday blogging initiative within the #MTBoS community.
More info here.
1. Build a thinking classroom.
This isn't a new goal for me, but something I'm always trying to do better. In teacher's college, I was introduced to the phrase "Explore First, Explain Later" in my Introduction to Biology Teaching class and this is something I try to incorporate into my math and science classes every single day. The concept is self-explanatory; students are given a chance to explore, investigate, and uncover ideas within a particular topic or concept prior to taking formalized notes. This teaching methodology is congruent to the constructivist theory of learning which states that "that learning is an active, contextualized process of constructing knowledge rather than acquiring it" (learningtheories.com).
"Exploration" can take many forms; investigation, experiments, noticing and wondering... however, something I'm keen on devoting more time to in my planning and lessons is developing the question. Daniel T. Willingham writes about this in his book Why Don't Students Like School, "Sometimes I think that we, as teachers, are so eager to get to the answers that we do not devote sufficient time to developing the question." I've really been following Dan Meyer's lead on how to do this; his blog post on "The Three Acts of a Mathematical Story" are a good place to start.
Peter Liljedahl also hosts a free webinar on how to build a thinking classroom, available here.
2. Get students to talk more.
It is so easy to just fall into a routine of lecturing/note-taking followed by independent (usually textbook) work, but I eventually want to create an environment in which students manage themselves. This begins by getting them to talk more, exchange ideas, and share what they already know. Some things I'm excited about trying in my classroom are Stand and Talks (Sara VDW), and talking points (adapted from Lyn Dawes).
3. Do fewer things better.
When I first started my student teaching, it consumed my life. Go to school, plan for the next day, sleep, and repeat. I stopped exercising, watching TV, hanging out with my friends... and basically anything that was not work-related. I could've used an old lesson plan my associate teacher has taught before; I could've downloaded lesson resources online; or I could have picked one really good question and focus the class on that for the entire period. There were a million things I could have done better, but no. Instead, I scoured dozens of sites for lesson ideas, worksheets, and activities before creating my own unique cocktail using an amalgamation of the best ideas I had gathered. I made my own worksheets and presentations because I wanted things done in my own exact, particular way. Planning a single lesson would take me hours - this is not sustainable!
I know better, so I'm going to do better this year. Angela Watson's keynote presentation for the Build Math Minds Virtual Summit really helped me refocus and re-evaluate my priorities. I'm going to invest my energy in doing the stuff that matters, and NOT because:
Instead I'll only commit my energy to doing something if:
Three things I'm going to start doing now to achieve this goal:
1) Manage my time by setting a timer for the tasks that need to get done, and stick to it. Whatever gets done during that time doesn't have to be perfect or have beautiful fonts and layouts, it just needs to be good enough.
2) Reduce my workload by only formally assessing student work if I believe it is a TRUE reflection of student learning.
3) Increase efficiency by delegating tasks to students, like self-marking formative assessments.
International math educator who writes, occasionally.