I just don't get it. Adding and subtracting polynomials is basically adding like terms. How can they get to this point in Algebra 2 and not be able to combine like terms correctly? I have several students who tell me that 3x^2 + 6x^2 is 9x^4. I still have students, in spite of having calculators available, who cannot add or subtract integers properly. I have students who will take an expression in parentheses such as (6x^2 + 10x - 3) and try to combine it into one term such as 13x^3.
I understand there can be confusion keeping straight what to do with the properties of exponents. I'm multiplying (4x^3y)(2x^2y^4) and you want me to add exponents but multiply the numbers? When I divide you want me to subtract the exponents? When I raise something to a power I'm supposed to multiply exponents? It can be a little confusing. But when I explained it one on one to some students the day before and gave them as many hints as I could to help them keep it straight - remember, you are not doing the same operation to the exponents as you do to the numbers; notice that the operation you are doing is one step lower on the order of operations list - they still don't remember it the next day.
And those are just the graded learning targets. On the feedback learning targets, multiplying wasn't horrendous, but there are still some errors that I'm not happy about seeing them. On the factoring problems, most students didn't even attempt them and of those who did, about half (that I've seen) didn't have much of a clue. It's like what we've done for the previous 2 days totally was a waste of time.
This week is the end of the grading period. I can pretty much tell what's going to happen. The students who care about their grades will want to re-assess the learning targets to bring up their grades. From the surveys I gave them after their test (same one as the Advanced Algebra 2 students with many surprisingly similar results as far as the comments went - but that's another post), I can see that they do care - about their grades. That, of course, means between Wednesday and Thursday there will be a bunch of students who want to come in to re-assess so they can get ___ grade. The problem is, they don't have the skill. Some of them do, but for the most part, my students don't have it. The students who really don't have it most likely won't re-assess. They will continue to not get it. Since math builds its knowledge on previously known things, as the year continues, those students will continue to struggle and this issue will compound.
So, I am sitting here on Saturday morning pondering the same question I was yesterday as I drove home from school - now what? I know that if the students don't have the skills I tested them on Friday, there are going to be places they struggle the rest of the year. If they don't understand the properties of exponents and adding and subtracting polynomials, trying to do multiplication of polynomials is difficult. If they don't understand how multiplication of polynomials works, how are they going to factor them? And let's not even get into the implications for later topics (solving quadratics by factoring, other polynomial skills, exponentials, etc.). I thought by doing the stations activities that it would help their understanding but obviously if it did, it didn't stick in their brains.
I have a four day week when we return. Do I go back over these skills and provide the opportunity to re-assess in class for the whole class? How do I go back over these skills? I have already taught, provided in-class activities, put together 2 screencasts (exponent rules and +/- polynomials) for them to be able to view outside of class, plus helped them in class as they asked for it. I'm kind of at my wits end - I'm frustrated that they aren't doing preparation they should be. I'm sure part of it is my fault - I don't check their homework and I don't go around and mark whether they've done it. I had good intentions at the beginning of the year, but it takes time to do that at the beginning of the period and in spite of my best intentions, many students spend the time talking instead of working through the warm ups like they should. I suppose I ought to collect those too.
Add on top of that a fairly busy week this week outside of school. It's not like I have hours and hours of time to prepare a whole lot of things for them. Probably one of the more frustrating things right now is that I spent 6-7 hours putting together those stations activities I blogged about earlier and now that they've had the assessment on it, I feel like it was somewhat wasted time. They still don't have the skill. I'm not really feeling like putting in a ton of effort on something they a) aren't going to "care" about and b) won't really put forth the effort they should to do. But I know as a veteran teacher that I will always put in way more effort than they will. I also know that it doesn't change that I need to do something.
So, I guess I'm doing two things here. First, I'm venting. I'm frustrated and I need to vent (not that any of you have ever been there before....). Second, I'm looking for suggestions. What would you do? Would you give re-assessments in class this week? What would you do to make sure students understood these two concepts? I probably only have 1 or 2 days tops to review with them (again) before the end of the grading period if I'm going to do anything as far as re-assessments. I'm just stumped at the moment. Please comment and help. Thanks in advance!
8 comments:
I don't know how much help I will be but I can sympathize with your situation. When I taught Algebra 1 and we were working with polynomials I tried to make it as concrete as I could. With like terms I would replace the variables with shapes or colors to show them what to combine. Expand the notations so they can "see" all the parts, especially when dividing terms with exponents, they will see the reason for subtracting the exponents. Show more "why" and less "how" they will develop the how on their own if they know why. I would spread polynomials out all across my board and put boxes around the terms so they can see the parts and process it better. Hope this helps. Good luck!
Lisa,
Sorry to hear how frustrating this is. FWIW, this sounds like a combination of "discouraged math learner" dispositions combined with very weak study skills. It sounds like your lower-skilled students don't know HOW to focus in on what they need to practice. They're in a defensive crouch and they're not able to inquire into what they need to do
Here's what I would do (I've found variations on this to work with many discouraged groups):
I would break each skill (i.e., both skills) into 4 one-question quizzes -- one for each day's class. I'd keep these all at the basic/proficient level.
Then each day, as soon as they come into class, I would give them the 1QQ as their warm-up, then collect them and use this "My Favorite 'No'" discussion & remediation technique I discovered via Kate (see f(t) for a great discussion about this):
https://www.teachingchannel.org/videos/my-favorite-no
You sort the quizzes into a "yes" pile and a "no" pile. Then all discussion revolves around YOUR favorite 'no.' It stays anonymous and you copy it onto the board or onto a fresh card/sheet on the document camera. What did this person do right? What did this person do that I loved?
Then once they've got the hang of analyzing the positive, AND ONLY THEN do you start asking about the mistake(s), i.e., where did this person veer off into a ditch?
After this whole targeted remediation discussion, you give them a fresh SAME type of problem to try and capture their learning. And you collect that one too.
AFTER THIS ROUND, you do another 1QQ round on the OTHER skill. And finish out the whole pattern.
THE KEY THING IS THIS: you will NOT reveal anybody's scores on anything until Friday (or Monday). That way you don't have to deal with the whiny "please please please let me remediate this skill now). So everybody has to focus in class on the skill and the discussion. THAT'S IT. No exceptions.
Since you're using SBG, you only need to catch each student in the act of doing the problem right once for this first round. But chances are, after repeating the discussion daily for four straight days, most students will start to get the hang of it.
With discouraged math learners, I have found that the only way to get them out of their defensive crouch posture is to get them to experience the turnaround process to success. By Algebra 2, they all know they can't find it because they have no clue what they are looking for. Your job in this specialized, targeted short week is to give them a massively scaffolded experience of succeeding.
Then once EVERYBODY has gotten there, you celebrate as a class. This is because they need to understand that nobody succeeds until everybody succeeds.
Hope this is helpful. It has worked wonders with my previous Algebra 2 classes, but it takes the patience of Job.
- Elizabeth (aka @cheesemonkeysf on Twitter)
Is the bigger problem that they don't know how to answer these questions correctly or that they haven't prepared adequately?
If the most important thing here is to have them understand the concepts, I would hold off on assigning a grade in the gradebook. You have the information that you need: they don't get it and they need more time, reteaching, etc. I would spend time helping them conceptually understand the ideas (I like what Matt said about shapes... I might also look at what happens when plugging numbers into the expressions). The other option is to create lists of steps that are catchy (easy to memorize) so that they know what to do in each situation. This just feels like an end run against understanding though.
However, if the bigger issue is their lack of preparation, I would input the grades and have a conversation with the class about good student habits and what is required from them to be successful. Then I would move on to new topics, while require tutoring from the students who need to clear up misunderstandings so that they aren't handicapped on future topics. I don't know what is available for you, but I can force students to come in during homeroom to get some help when they need it.
I agree with Elizabeth on the idea of a good way to work with discouraged learners. What a cool way to go about revising a topic.
I run into those issues as well and also use ideas of fruit or shapes instead of variables as a starting point. I have found students much less likely to say 2 apples + 2 apples is 4 apples squared than with variables and once they get the idea, it can transfer more easily.
Dvora @teachdig
One thing that helped my students with adding polynomials is to think of each term as a family. So then adding would be like having a family reunion where you have to get each family together. If they don't have the same 'last name'(i.e. x, x^2, x^3, x^4) then they can't be combined. My students were taught that way in middle school and continue to use it throughout Algebra I and II. I ask them things like can we add 6x^2 + 10x - 3 and get 13x^3? And when they say no I ask why and they tell me matter-of-factly 'because they don't have the same last name'.
I too was going to suggest the My Favorite No activity but why not turn it into a whole class activity? Create a worksheet based on the problems they got wrong on your quiz and put them in partners. Have them compete with each other to see which team can find all the errors first. But make sure you count and know how many errors they are looking for. If you started class by modeling this for them through the My Favorite No activity, they should be clear on what to do. That way you can clarify individual errors in a non-threatening way. Mimi just wrote a great post about this process. http://is.gd/bpbETt
I also agree with Andy. You have the information telling you that they don't understand so I would hold off on the gradebook for now.
Again, I would suggest asking the students why they are struggling with these concepts and ask for their suggestions on how you should reteach them. They may not have an answer but they may have insight.
One thing I forgot to say in my previous post is, I think you're coming up against the limits of their study skills/practice skills.
The discouraged learners honestly don't know (a) how to practice a skill, and (b) what mastery/success in a math skill would actually feel like.
They've been navigating without a guide, without a map, and without any experience. But now they have you.
As Dan Pink says in Drive, intrinsic motivation requires autonomy, mastery, and purpose. They understand autonomy, and they understand that they need to graduate. But many, if not most of them, have no experience in mastering something.
I found that my Algebra 2 students got quite 'hooked' on practicing through to mastery once they had some experience of what it takes for them personally to get there. I'm seeing the same thing with my Algebra 1 students this year.
If you can give them a taste of that experience, you will be saving yourself a lot of aggravation as well as putting them onto the path towards success.
- Elizabeth (aka @cheesemonkeysf on Twitter)
Thanks everyone for your comments - I do appreciate it! Here's what I decided to do:
http://oldmathdognewtricks.blogspot.com/2012/01/plan.html
I am unaware of what technology you have in your classroom, but share your struggles with polynomials nonetheless. Matt commented about writing expanded expressions with the properties of exponents portion. I did a simple example, like x^3 * x^4, followed by something involving x^14. The students were excited to tell me the shortcut/property because they were impatient with the idea of expanding whatever I dreamt up next. Of course, no one had their book open yet, so we gave credit to the student who "invented" the property, the (your student's name here) property. It has stuck with them, now that we are mid-unit.
Another comment said to celebrate as a group. One milestone could lead to another. Reward their mastery of properties of exponents by bringing brownies for each class period in different sized pans but with the same number of slices. A brownie mix costs little more than a dollar but would produce the same amount of batter. This would be a great application later in the chapter with polynomial equations, because inevitably some student will complain that their piece was smaller than someone from 2nd hour or whatever. Find the volume of each pan occupied by batter, then divide by x slices, or (x+3) slices and confirm they were same volumes or different volumes (while confirming or denying the myth that teachers have a favorite class). Actually, I stopped by our local pizza shop and asked for a couple unfolded boxes to take to class and help students visualize the construction of a box and the side relationships as well as box volume. Hoping to draw it into a function like V=x*(16 - 2x)^2 for a template that is 35 in. by 16 in. or so. Could be good but obviously relies on the visual aid of the unused pizza box.
Anyhow, there's my two cents. Like a decent stand up comic, it is beneficial to establish some sort of inside joke earlier in the "show" to refer back to in order to see that they can extend the connection of what was said earlier. Mhope something I mentioned helps.
Scott Keltner
Eudora, Kansas
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