Flipped Classroom: Math 9 – Polynomials Unit

This little project started out as an idea for my Math Education (EMTH) class. Little did I know, it seemed to tie three of my classes (all of my ed classes this semester) together.

This page is a collection of everything that brought this project together for me. Feel free to use, modify, comment on, contemplate, rework, and ponder anything and everything!

Overview

This unit is designed for a grade nine math class, based on the Saskatchewan Curriculum (see below for the link). It is the polynomials unit, but entirely flipped. I created short videos for students to watch at home to give them the jist of each topic, along with inquiry-based worksheets to work on during class. The idea being that students do all the “learning” (or what would normally go on in class) at home for homework, and then come to class to do what they would typically do at home in class. In class, they will have access to a their peers, me, answer keys, computers, the internet, or anything else they can get their hands on, rather than being stuck at home with no one around to help out. The benefit is two-fold. The students have a better learning community, and I, as the teacher, get to spend more time with each student so that I can better know where they are at in their learning. Knowing this can help me customize their learning as much as possible.

The other part of this unit is the assessment. Initially, I decided to go with a learning contract that uses standards-based grading (based on the outcomes and indicators from the curriculum documents). It also included a What Can You Do With This summary lesson and  unit project that tied the whole thing together. Later, I added a different option: assessment through learning. This is an “exam” (I hesitate to use that word because it really isn’t, it’s more of an assessment that looks sort of like a test but really isn’t) that guides students through factoring. While factoring isn’t in the grade nine curriculum, following each step outlined in the process shows that the student can demonstrate all the necessary skills for this unit. It also gives a good introduction to material they will see next year. This helps to avoid the “I’ve never seen this before — it must be really, really hard!” mentality that so often blocks student learning. This type of assessment helps make the grade nine to grade ten transition more smooth. It helps explicitly show flow and continuity of subject matter. At the very least, it lets a student go home and say, “Guess what, Mom/Dad/Parent/Guardian/Dog/Brother/Sister/Grandma/Grandpa/Next Door Neighbour/Etc.? I did grade ten math today!”

Where I stand now that this project is nearing completion is a recommended blend of the two. I’m less enamored with the final project. I think the assessment through learning would be more practical — less class time, more beneficial to the learning continuum, and slightly more traditional (so it helps with preparing students for “real exams” that they will inevitably encounter). So to blend the two, I recommend the learning contract with SBG, but the ATL as the unit wrap-up. It shouldn’t be worth very much of their grade, since it isn’t designed to be a high-stakes test. It’s just a way to pull together knowledge. I’m not even sure if it should be graded. Grades are just icing on the learning cake — I care way more about the cake itself right now. Aside from the learning contract, you won’t find any grading in this project. Just assessment and feedback here.

Flipped Classroom Website

Math 9 – Polynomials: This is the Sophia website for the flipped classroom. It is where my future students will go to watch the videos. It also includes every worksheet for in class, which are available for download directly on the site. Worksheets are also available below.

Resource Guide: This is a quick overview, with curriculum ties, and includes a timeline.

Worksheets

Lesson 1 – Language of Math

Lesson 2 – Terms and Definitions

Lesson 3 – Like Terms

Lesson 4 – Intro to Alge-Tiles

Lesson 5 – Adding Polynomials

Lesson 6 – Opposite Polynomials

Lesson 7 – Subtracting Polynomials

Lesson 8 – Multiplying Monomials by Monomials

Lesson 9 – Multiplying a Polynomial by a Monomial

Lesson 10 – Dividing a Monomial by a Monomial

Lesson 11 – Dividing a Polynomial by a Monomial

Assessment Models

Quiz-Quiz-Test Model (QQTM)

The exams/quizzes I looked at were based on the common exam from the high school I interned at. Therefore, I am not posting them as they may potentially be used in the future.

  • The QQTM is what you would see in a typical classroom, where each unit has two quizzes followed by an exam/test at the end of the chapter. This is not really assessment as much as it is evaluation. I examined the two quizzes and test that I gave during my internship. While I wrote the quizzes, the school I was at had common (read: standardized, see Joe Bower) exams for each chapter. I had a bit of freedom to alter the exam for my class, but even then, it wasn’t good enough. I found that I was baffled every time my students wrote their exams. A few of them would surprise me (sometimes for better, but more often for worse). That’s because I wasn’t assessing their knowledge. I was evaluating them on how well they could study the nigh before.
  • QQTM does have some benefits. It helps students prepare for college/university where high-stakes testing is common place. Learning how to write an exam is a necessary skill for universities today. However, I would like to argue that if you understand a concept, regardless of how it is presented, you’ll do well. The other benefit is that some students do really well on high-stakes testing. They thrive on a few pressure filled moments, rather than frequent assessment. Again, I’ll argue that this is only for some students. Additionally, frequent feedback and assessment shouldn’t be stressful. The stress is cause primarily by a desire to get a good grade.  You and I both know that good grades don’t necessarily mean good learning.
  • QQTM has more drawbacks than I’ll ever be able to cover. The biggest drawback is that it has very infrequent, if any, feedback to the student about how his/her learning is progressing. Even if the student wasn’t grades-focused, he or she would still most likely be going into the exam with material that he/she has never received feedback on. The quizzes can only cover so much. Aside from that, when a student receives a poor grade test or quiz, the first (and most common) response is, “I better do well on the next one.” This is flawed thinking — the material that needs the work is the quiz/test, not the next chapter. Since math tends to build on itself, not fully understanding one topic inherently makes understanding the next topic fully nearly impossible. This thinking is very grades-oriented, rather than learning oriented. Again, good grades don’t necessarily mean good learning.

Learning Contract and Standards-Based Grading (LCSBG)

Polynomials Learning Contract

Interior Design Project

  • LCSBG was the first assessment model I chose for this project. I knew that there was no way I could possibly assess my students using QQTM, since I wasn’t teaching to a quiz or test, and nor did I want to be. After a lot of searching, thinking, and research, I decided to use a learning contract, along with Standards-Based Grading. Combining the two only seemed natural. Each indicator in the Saskatchewan Curriculum is on the learning contract. After the student successfully shows me  twice (through a quick assessment — a few questions, asking the student to create and solve a question, asking the student to explain a concept to me, etc.), the can check off that specific indicator. Each attempt is evaluated on a rubric to appease the grading gods. The scale goes from 1 – 5, with 1 being very little understanding and 5 be complete understanding. The highest mark from the attempts is used in my grade book. For a better synopsis, see the EMTH Reflection link below.
  • During my critical project for Moral Ed, I did make a small shift in my plans. Part of SBG as I’ve come to know it implies that I can do a check-up on a topic at any point in time. In other words, I can ask any student on November 27th to do a check up on the indicator they completed on September 8th. While I’m not convinced that final exams are a good idea, I do understand they are part of the current model for high school math programs. Instead of having a killer math final that my students spend a day or two cramming for, I can periodically do check-ups throughout the semester, building toward their math final mark. Come the end of the semester, I theoretically will only have a few indicators to assess, and could easily put them on an assessment, build them into a project, or find something else useful to do with them.
  • The benefits of LCSBG are vast, so I’ll pick the highlights for me. Firstly, students get constant feedback. When used in a flipped classroom, I can be free to help out students, give them feedback and get feedback for myself. Each student can be working at his or her own pace, so there is little pressure to “keep up,” all while minimizing the stress of homework. If a student feels that he/she understands a topic, they can ask for an assessment. I can assess at any time — morning, noon, or night — because they are short and sweet. Because students need to repeat their assessment for me, they will theoretically show authentic learning. Also, since this model is student-driven, each student is accountable for and in charge of his/her learning. Part of this appeal of this model is how transparent it is — if a parent is checking on a student’s grades, each item is listed as an indicator. They will see that their child has a 3 in “can identify a binomial in a set of polynomials” and a five in “can multiply a monomial by a monomial,” for instance. This says a whole lot more than 60% on “Quiz #1” and 85% on “Quiz #2.”
  • As always, there are drawbacks to LCSBG. The biggest drawback is that is isn’t what students/parents are used to. Initially, grades may drop, but that is all in the favour of authentic learning. There will be a “learning-how-to-learn-instead-of-study/cram” learning curve initially. Parents may complain that this isn’t what “real” math classrooms do. It’s a big change, and with change comes resistance. Another drawback is that there isn’t any high-stakes testing. While students who understand material will do well on any assessment, they might have exam anxiety once university midterms hit. Practicing writing exams has benefits for those students who plan to attend university. Lastly, and certainly least, is that this model is more work for teachers. There are more assessments, more paperwork to keep track of, more marks going into your gradebook (and then changing again). However, as I see it, it’s worth the extra time and effort.

Assessment Through Learning Model (ATLM)

Polynomials: Factoring Assessment (Assessment Through Learning)

  • I wasn’t entirely convinced that LCSBG was the magic solution I’ve been searching for. It’s quite utopian, in fact. I struggled to find a happy medium that provided the stability of the QQTM with the authentic learning of the LCSBG. Somewhere in the midst of the last few weeks, I had an “Aha!” moment. What if learning was going on during the assessment. Now, this is no new concept. Assessment as learning works for this all the time. What I was looking for didn’t quite fall into assessment as learning. It wasn’t even close to assessment of  learning and it didn’t quite make the cut for assessment for learning, at least as I’ve come to know the three. So, I decided to add my own preposition to the mix with assessment through learning. Let me explain: as I’ve said before, math builds on itself. I noticed a huge disconnect in learning between the grade nine and grade ten students. During my internship, I saw the grade ten students struggle with factoring. I then saw them struggle with exponent laws, the very same thing I was teaching to the grade nines an hour before. They didn’t understand the exponent laws from grade nine, so all of a sudden, the grade ten math was nearly impossible. Like any good intern, I retaught the exponent laws in a hurry (who has extra time? It had to be quick!), and they were well on their way again. The grade ten students who struggled were the ones who still didn’t understand the exponent laws. Needless to say, I drilled those exponent laws into the grade nine students!
    • This is where my inspiration came from. What if I could connect the learning somehow? I took this and ran with it, only this time with polynomials. One of the biggest stumbling blocks for the grade tens was learning how to factor. They seemed to be frightened by all the variables, and were quite nervous about multiplying, adding, subtracting, and dividing polynomials. Instead of letting the grade nines think that their unit with polynomials is where it ends, why not connect the learning by showing them a preview of what’s ahead? I took a look at a couple of different factoring methods, and voila, they incorporated exactly what the grade nines were learning. My prof Rick always said that in math, we don’t really learning anything new, we just combine stuff we already know (I think he said it a tad more eloquently though). Factoring is just combining the properties of polynomials that the grade nines are learning about.
    • How it works: I would give this assessment just as any teacher would give a test; although, I might argue that I would give it to students when they were ready, not necessarily on a specific date or all at the same time, but that’s a different issue. The assessment has very specific instructions. I give the students an expression that they will factor. I break down the factoring method into very small and easy steps. Each step covers off different indicators that I expect the students to be able to do. Within the first question on factoring by finding a common factor,  I have completely assessed what it took two exams in my internship to assess. Just for good measure, I gave a second question using factoring by grouping.
    • What it does: It gives me insight into what the students understand from the current unit, and it gives the students a preview of what they will be learning/why it is important to retain this learning. When they see factoring next year, they won’t be nearly as intimidated by it. They’ll be able to say they’ve seen it before, and it (hopefully) wasn’t too bad!
  • ATLM has so many benefits — it enables the stability of having a test at the end of a unit with the authentic learning of using something and building on it. I won’t rehash everything from above, but I’m quite curious to see it within a real classroom. It helps students to see the scaffolding that math requires, and then successfully put their knowledge to work. Wouldn’t it be cool to go home after school and say “I did grade ten math today, and it wasn’t so bad!” when Mom asks how your day went?
  • ATLM has some disadvantages too. The biggest disadvantage right now is that I can’t find any research on this. Perhaps I’m looking in the wrong places, but so far I’ve found nothing. (If you find some, please let me know in the comments!). Justifying this kind of assessment without research wouldn’t be easy until there’s been some success with it. Another drawback is that I’m not sure how it would fit into my classroom. Depending on the students/class it could fall closer to the QQTM, where I would have a couple of mini-assessments that build on knowledge similar to this one; alternatively, it could work really well with LCSBG, where I would give this as a unit summarizer instead of the project.

Reflections

In-Process

Flipped Classroom – EMTH Reflection

Flipped Classroom – ECMP Reflection – This is my final evaluation reflection for my entire Technology in the Classroom course, but my brief reflections about the flipped classrooms appear under “Teach Us Something” and “Create Your Own Assignment.”

Flipped Classroom – Moral Ed Reflection – This includes my video reflection, as well as few other written comments on this project and the process.

Direct Resources

These are the articles that I used that directly pertained to this project. I will not deny that I have done a ton of reading of tweets (particularly in under #flipped and #flipclass), reading other not-so-related articles, as well as held many conversations with fellow students, profs, family, and teenagers (who were surprisingly helpful), all of which helped me to come to the conclusions that I have so far. Below are simply the most relevant pieces pertaining to this specific project. This list will grow as this project continues to evolve.

Saskatchewan Curriculum

The Trouble with Rubrics (Alfie Kohn)

Assessment System Overview (Mr. Schwenn)

Abolishing Grading (Joe Bower)

Think Thank Thunk: Standards Based Grading (Shawn Cornally)

How Math Must Assess (Dan Meyer)

It Works: Virtual Manipulatives (Alge-Tiles)

Leesburg Today: Schools Push Toward Tech-Savvy Education

Educator, Learner (Brian Bennett)



4 Comments

4 thoughts on “Flipped Classroom: Math 9 – Polynomials Unit

  1. Pingback: And So It Ends… But Not Really. | A Slice of Sara

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