Skim Read Chapter 7 ( I know most of this is above the level of math that you teach)
Reflect on the following questions and comment below:
1. When you were in school did you ever learn these tricks? What is the effect of learning these tricks on higher level learning/understanding and rigor?
2. Comment on a colleagues post to engage in a meaningful conversation.
Oh boy…… I was in school a very long time ago when I learned these topics. But I do remember 7.6 “FOIL” very well. I have to say I never learned to concept of solving these problems, so I just used FOIL because that was what the teacher taught me. That is probably why I have no idea how to solve it or what it means today because I learned the trick, not the concept. I have to say, that is how I remember most math classes. Some teachers would try to teach the real concept, but it took a long time, we didn’t have the background knowledge from previous years to draw from. It was like they gave up to make everyone’s life easier and said “Ok, just follow this trick and we can move on”. In the end, the student losses. I guess when I think about it, I may have been better in math if I learned no tricks, just all concepts from the early years. Is that realistic and could it be done?
ReplyDeleteAudrey,
DeleteI too remember FOIL and know that I could probably jump right into solving the equation now but like you stated would not know why. Great point..
Audrey...I believe we are on our way to moving past using tricks with the implementation of the CCLS and the math modules! I found that if I choose to share a trick with my students, it comes after all the conceptual understanding and lessons have taken place. Perhaps tricks can be used more as a checking tool for students.
DeleteI agree, as time passes since the onset of common core I have found it easier each year. I do the same thing and share tricks after a lesson and for the same reason, a checking tool. And then my students will respond with "why didn't you show us that first?" Ugh!They keep it entertaining.
DeleteWhen I was in school I did learn some tricks but I remember the tricks were added to the lesson. I was fortunate to have very good teachers in high school and that was part of the reason I chose to become a math teacher originally. Math made sense to me when I was in high school. I remember taking Calc and saying finally this makes sense, see being a visual learner I enjoyed calc since it was applied math. Seeing things like FOIL still being taught is I feel, two fold. So many teachers probably do not use FOIL as a supplement to understanding, instead they probably just use it to teach the entire concept. For this reason I think it will have a negative effect on learning/understanding as well as rigor. The author trying different principles on his students is a great idea to me to show for instance why OLIF and FOIL both work. So many times we teach, (this is the way) and not (this is the way because this isn't the way) which can really add to understanding.
ReplyDeleteMatt~ It is great to hear that part of the reason you went into math was because of good teachers!! I love that, and don't we all hope a former student would say that? And I agree, as with most of the tricks, just teaching the tricks and not the entire concept would have a negative effect, and ultimately make math harder for students as they move up in grades.
DeleteMatt,
DeleteYour comment about "understanding" made me think about the responsibility we have as teachers to clearly present our lessons so students don't have a fear of learning or a hatred for math because they don't understand. I would like to hear ideas about how we can accomplish this. I think making learning fun is key. Can we play a game that teaches a concept, a video clip showing a real world application of the mathematical concept. I think the math taught in vocational classes allows for that real world AHA this has a use versus doing math problem after problem just because I need this for my diploma.
Peter,
DeleteThat's a great thought on implementing the AHA moment I made a note to include this in my future plans. All too often they think the "Science" teacher is coming in, instead of the "Math Applied using Science Teacher" is coming in that's where I want to get them to...
I don't think I was ever taught how to foil with actual values. I was taught how to distribute but I just trusted that is worked. I see it now very clearly if you use values like in the fix for 7.6.
ReplyDeleteI also never used the long division process for dividing polynomials. I think that is a great idea as well because it also shows how the terms have to stay grouped together throughout each operation. the x or the -3 didn't just go off on their own and start divided through with each term.
My daughter who is in college now used FOIL. My son who is in algebra II right now uses the array model. At first it confused me until I realized how it was organizing how he multiplied "each by each". My son missed 2 weeks of school because of pneumonia. I tried to support him by watching the online lesson videos he missed in school. When dividing polynomials they factored the polynomials and used a number divided by itself is one to simplify. It worked, but I really like seeing the long division process used for polynomials. It makes sense too. My son had trouble understanding why he was "crossing out". I would take everything back to whole numbers to give him examples of why something worked with variables as well. Boy that was a stressful time for us both. I didn't remember anything. No tricks. However I was able to use my basic knowledge of algebra from teaching 6th grade to reason through problems. What I didn't know, he did.
DeleteI commend you for helping your son! I think another math disservice for students is that once they pass the primary years of schooling parents are not able to help their children with their math homework. I hope that doesn't come across as undermining the intelligence of parents but it is commonly known that the better part of the average population has trouble in basic mathematics. I think this is why "tricks" have always helped teachers, parents, friends, family etc through helping students to pass their necessary classes for graduation. And as we are noticing in these higher chapters, the math is getting tougher and the tricks are getting more common. I'd like to nix the tricks but the students are getting the tricks at home when they need homework help. I know my step son is getting taught using some tricks by dad for his 3rd grade math. His dad knows how to do it but he feels as though if his son is getting a correct answer then that is awesome... Which it is but... he has no idea why he is getting a correct answer. So I try to step in from what we have learned here and what I know as a math teacher in order to correct this pattern.
DeleteAs a student, I definitely had exposure to and used some of the tricks in this chapter. For me, I think the tricks just served as a “trick”. I feel I did understand much of the conceptual understanding of upper level math because I was interested. Where I feel silly is with the 4th/5th grade math I have been teaching for the past several years. I thought I understood math very well, but implementing the CCLS, using the modules, and watching concepts spiral for the students gives me a better understanding of lower level math.
ReplyDeleteJanelle- I agree that the implementation of the CCLS has forced teachers, especially the elementary teachers that are not certified in math, to really understand the concept. Many teachers are realizing that thy never fully understood the concept themselves.
DeleteI had a teacher in high school who was the "trick" master. She had a trick for everything. I liked this teacher, and I have adopted many of her tricks for my own classes. In this chapter, I have used rise/run, All students take calc, foil, inside is opposite, slide & divide (barrow and payback), and synthetic division in my classroom, or while tutoring other students. I have tried to get away from many of these, and after reading this book, I will make great efforts to move further away. When I taught Alg. 2/trig, I religiously used All students take calc, in addition to the QRS method (Quadrant, reference angle, sign). I actually used to tell my students who had a different teacher that they were doing it the hard way (drawing the triangles). I am tutoring some students in Alg 2, and have adopted the triangle method because it is a much better way to show the students the concepts instead of a trick. Learning these tricks greatly reduces the actual knowledge learned so that skills can be applied to problem solve and think critically. There are many tricks out there, but not for everything. At some point, we just have to think!
ReplyDeleteI not only learned many of these tricks, I relied on them to get me through some of my higher level math courses in high school. I was a very weak math student. I like to think that I am a more effective teacher because of it. I totally understand, “I don’t get it.” I think it has forced me to go deeper within a math concept, even before the implementation of the CCLS. I also think this is why I don't use as many of the tricks as some other teacher might use them. I think a lot of understanding is lost when teachers use tricks. There really is no rigor.
ReplyDeleteHigh school was a long time ago. I really don't remember any of these tricks except the ones I saw my own kids do on their homework. My daughter who is a self proclaimed math and science geek loves the tricks and doesn't know why we wouldn't teach them. However, she understands the math behind them so for her they simply seem like short cuts.
ReplyDeleteI remember loving algebra because it made sense. I had a list of rules and exact orders for things to be solved and I could do it. I didn't know why, but I just memorized. I do remember thinking that math does not make sense when I took calc I and II in college. I remember being given a bunch of formulas but I always felt so confused as to when to use them and I still don't have a clue of what I was solving or why. It was like a big puzzle and when I came to a solution I had no idea what it was...even if it was correct or why anyone would ever need such math. I did receive B's for the class, but I have no idea what I was doing.
Now I always try to make sure my students know why someone would use the math I teach in everyday life. I try to help them reason with real life situations. I have also found that any of the concepts I found difficult, my students tell me are easy. For example mean absolute deviation I found tough at first. I had taken the extra time to teach myself and how to break it down to be understandable and so for them it seemed "easy".
High school was a long time ago. I really don't remember any of these tricks except the ones I saw my own kids do on their homework. My daughter who is a self proclaimed math and science geek loves the tricks and doesn't know why we wouldn't teach them. However, she understands the math behind them so for her they simply seem like short cuts.
ReplyDeleteI remember loving algebra because it made sense. I had a list of rules and exact orders for things to be solved and I could do it. I didn't know why, but I just memorized. I do remember thinking that math does not make sense when I took calc I and II in college. I remember being given a bunch of formulas but I always felt so confused as to when to use them and I still don't have a clue of what I was solving or why. It was like a big puzzle and when I came to a solution I had no idea what it was...even if it was correct or why anyone would ever need such math. I did receive B's for the class, but I have no idea what I was doing.
Now I always try to make sure my students know why someone would use the math I teach in everyday life. I try to help them reason with real life situations. I have also found that any of the concepts I found difficult, my students tell me are easy. For example mean absolute deviation I found tough at first. I had taken the extra time to teach myself and how to break it down to be understandable and so for them it seemed "easy".
Carol: I think thats a great way to look at it.. a trick as a short cut once the concept is learned!
ReplyDeleteI totally agree. I think tricks have a place. They just can't replace the understanding of the concept.
DeleteThis is totally what I have been thinking about tricks the entire time! I'm glad to hear that others have that same feeling.
DeleteFOIL is a trick that I remember from high school which was a long time ago. I do believe that when I was in high school we were taught to follow steps and were not brought to the level that is expected of students today. I am not a proponent of "trick teaching" and I know a couple people have mentioned in comments, the subject of time. How much time is acceptable to spend on a concept? And what do you do with the students who truly understand initially or can see it abstractly without concrete models or examples?
ReplyDeleteIt has been a very long time since I was in high school, but I do remember a few of these tricks. I remember rise over run, and in fact, my daughter still uses that one in 10th grade and FOIL. Maybe there is something to be said about the fact that I no longer remember the tricks, and in addition, no longer really remember how to do any of that math anymore. That should be an AHA moment for me because I did very well in high school math, and now I am unable to recall any of it!! Either it is because I haven’t used that type of math for many years, or I was only taught the tricks and not the concepts.
ReplyDeleteWhen I was in school I learned Y=mx +b . All I knew was b was the unknown what we had to find. Sadly math didnt have much meaning to me until I saw the application in science solving for x to find a solution to a physics problem. I think math is a very abstract concept. Its easy to teach to students who easily comprehend math but how about reaching those who hate it because they dont see the value or application ? I think the FIX could be better fixed if we show real world problems.
ReplyDeleteDr J
DeleteI totally agree with you. Sometimes its hard for the students to see the real world value of math. If you ask a kid now how much math they use from when they were in school they would say none. Just the way it is
When I was in school I did learn some of these tricks. I also think that it had ham-stringed me as well. I could do the math if i had the trick with me but the minute you took the trick away I couldn't remember how to solve the problems. Its probably because I didn't fully understand how to work my way through the problem. I had the option to drop math after 10th grade and go to vocational. At the time it was a huge relief but now I regret it.
ReplyDelete