Here you can post a discussion topic, an issue, or a problem; solve a problem posted by someone else; or a substantive comment to a solution posted by another student in this section.
You need at least five times active participation in this Discussion Board. Hope this section will bring you up for an active collaborative discussion.
EDSC 353: Teaching Secondary Geometry 
I couldn’t find the “Demo Problems”
In class, I thought Mokter said there would be some demo problems posted, in addition to the “real” ones. I was expecting to use these, to get an idea as to how he would like to see us deal with questions such as how to specify unlabeled lines or points, as in Problem #A1.
As I uploaded actual problems, deleted the demo problem.
~MMH
The textbook seems to give many “real life” examples about how geometry relates to our world. High school students, in particular, want to know, “When will I use this information outside of class?” How would you answer this question? What examples would you give?
I have always wanted to make my own pool table so i can use my geometry skills to help build one. To make the middle pockets i would find the mid point between the two corner pockets from the same side.
I might give one example of how it relates to the real world, but then have the students brainstorm and come up with examples on their own. That way the students are a bit more involved.
My HS Geometry teacher found a common ground with me and my friends. We all skateboarded so he helped us design a halfpipe and several ramps. We needed to understand the radius of a circle and how a triangle was the best support for the ramps. designing the halfpipe ended up being a pretty cool project. Building it was better.
The examples i would give is obviously construction workers, architects, etc. I would also tell the athletes that are in the class that they use geometry in their sports as well.
There are so many examples here. Simple things people do around the house like hanging pictures and tiling floor. My dad made a picnic table for us when my brother and I were little and this took a lot of geometry. Putting in a pool would also involve a lot of geometry.
I just thought about skiing. Boots and bindings are built (in part) to keep your legs at certain angles relative to the skis, in the hopes of minimizing injury and making skiing easier. So if kids ever want to build ski equipment… In fact, just about all sports equipment is based on precise geometry. Surfboards, footballs, etc…Also, what if the yellow center lines on the road weren’t parallel…might get pretty confusing. In fact, I would tell them that geometry is deeply rooted in most of the common sense ways that we shape our world.
There are a lot of jobs that require an understanding of geometry. People who work in construction in particular, such as architects, engineers, draftsmen, contractors, and carpenters rely on geometry daily. Geometry is also used in jobs involved with design such as interior design, graphic design, etc. I know there are many more examples. I think I might have students do a research assignment to discover on their own how geometry is used in a variety of jobs.
Would the research assignment include the type of geometry and application. I think that this is a brilliant idea. Especially if each student can see the topics and the exact geometry involved they will want to learn because now it is important and not just stupid stuff.
It’s also a great way to differentiate your normal instruction and create a deeper understanding for the students. Sounds like it would be a very beneficial assignment.
With a dad who was an architect for 22 years, and two uncles who are engineers, I have to say that we’re teaching a fallacy if we tell our students that these guys use geometry on the regular. They don’t, they use trig. If they need to make some crazy curved surface, they use continuous mathementics. That age old geometry textbook diagram of a house’s roof beams that draws triangles over it and tells students that architects use geometry has always driven me nuts. This is deceptive, and does not encouragean understanding for the nuances and demands of design.
And again, that isn’t what geometry education should be about. It is probably the least relevant mathmatical subject on a concrete level, and if students are taught otherwise than they could be doing much better things with that school year. However, the reason geometry exists in our curriculum is because it teaches abstraction, logic, and relation better than any other high school math subject, and does it’s inherently visual nature makes it much more accessible for students.
I really like this idea! It might be neat for them to see how geometry is used in their own household too. I remember last summer my dad was putting in this fake tile floor and he had to measure out everything ahead of time to buy the correct amount of tile pieces. Then he had to cut and fit everything together. Even hanging a picture so it is straight involves geometry.
Sam, I like your idea of having your students do a research assignment on how geometry is used in the real world: I think that assignment could be done across the board with the different levels and subjects of math (algebra, trigonometry, probability and statistics, etc.). In another one of our required education classes I created a “Formula Sheet” rubric, which the students fill out each time a different formula is given. On that sheet, students fill out the name of the formula, where is it found (chapter or page in the book), the actual formula itself, an example of how it is used, and when and why we use it. Through this sheet, I believe students are given the opportunity to practically apply the math they are learning and discover why we use it.
I think it’s important to give students the opportunity to discover on their own how math applies to their everyday lives, instead of us just telling them. Another way to accomplish this goal is through graphic organizers, specifically concept maps (bubble maps). For example, doing a concept map on squares and rectangles and having a section for “real world examples.” Little activities such as this allow students to constantly be aware of how math applies to their daily lives.
Scooter, I really like your idea of the formula sheet with all the different sections for students to write in. It is a great way for students to figure out for themselves the question they always ask “When am I ever going to use this in real life?” I think when students realize just how important math is in just about any field of work they will begin to understand why teachers stress their understanding of material so much. I once had a math teacher who had a poster in the back of her class that had every part of our Algebra 2 class on it as well as a job/activity that required those skills. Whenever a student asked that question she sent them to look. I think that was clever but I honestly would have preferred your “Formula Sheet”
ps…I may have to steal that idea for future use!
This is somewhat delayed reply to “Wally”. I think there are any number of measurement, surveying and building examples available, that relate to home ownership and construction trades, as well as careers in Physics or Engineering. One I saw in another class involved a community project to revamp a local park (surveying, measurement). Another one I saw was the use of simple area and volume formulas to calculate heat and cooling requirements as part of Project Recharge (Environmental Science). Both of these were very “hands-on” and fun for Middle and High Schoolers.
So I came across the story of Eratosthenes while reading a Carl Sagan book recently (yes, I’m a dork), and thought I would share how he calculated the circumference of the earth 2300 years ago using the very geometry that we have been using. But then I realized that it would be a fun challenge to start a discussion with. NO CHEATING (i.e. looking it up on the internet). The only tool Eratosthenes had was a straight stick. And he knew that in the town of Syrene, a known distance due south, on June 21, at noon, there was absolutely no shadow from a stick stuck straight in the ground. Given that info, how did Eratosthenes calculate the circumference of the earth without leaving Alexandria. (Maybe most of you already know this story, but if you do, please don’t be a spoiler!). Hint: Because the sun is so far away, by the time light rays hit the earth they are running parallel to each other).
Dear Ricky –
I don’t think you give quite enough information here. Eratosthenes also measured the (angular) difference of the sun from the zenith (directly overhead) in Alexandria, at noon, June 21 and determined that it was non-zero.
Don’t mean to be a complainer, but it’s awful hard to comment on people’s solutions to the problems, when only 1 is solved by Wednesday afternoon.
Heron, yeah, at first I put that in there, but then worried I was making it too easy! So I took it out. So you have posted the first piece to the puzzle. To be more specific, when Eratosthenes stuck a stick straight into the ground (in Alexandria) on June 21 at noon he noticed that the angle that the shadow formed was 7 degrees. What would he have to do next?
What is your strategy for solving nth term problems?
Geometry is a branch of math that is all around us. Has there ever been a time where you have used geometry in a real life situation when you were not planning to? For example, I once had to use it on a camping trip to build a sort of shelter because someone had forgotten our tent. Have any of you had any similar experiences?
You can definitely say there are many situations on having to use geometric strategies in real life! (or maybe I just complicate simple things I do from day to day)
Like just the other day, I was trying to hang up a picture in a very specific place. Therefore I had to measure the frame, find the midpoint and put the nail at the midpoint of where i want the frame to sit exactly (slight OCD i know) But who would have thought that was all geometry!
I definitely feel that Geometry is one of those subjects that can easily be applied to real life situations. This may make it not as hard to teach to students if we are able to link the mathematics to the real world. I know that the biggest battle will be convincing those students who struggle with math, that they do need to learn it and will use it later in life. I used Geometry the other day when I was trying to center my television that is mounted on the wall. Similar to the picture frame, if I just found the midpoint between two endpoints then I would know where to place the middle of my television.
I know what you mean! I have always loved geometry because it is math that you can visualize and even hold in your hand. It helps to learn concepts through hands-on activities so you can see why things work the way they do rather than just being told about certain abstract concepts. I noticed using geometry concepts when hanging my mirror and even when I was helping my dad put down the “easiest flooring ever” as it claimed to be. This kit gave you different tiles in certain shapes and sizes and you had to fit it together like a puzzle so that it fit as your kitchen floor. It is neat how geometry seems to be everywhere when you look for it and I think one key to teaching this concept is to help students see it everywhere too.
In my previous career in construction management I was able to quickly check if 2 walls were square by measuring out a 3-4-5 rt triangle. there were several “quick” checks that I was able to use to make sure we were on track.
This is one of the best applications of geometry in real life I’ve ever heard. Its useful, its smart, and its not reaching too far like so many other “examples” I’ve seen in the past.
How about this one: by using the converse of the Pythagorean Theorem, you can tell when a post, or tent pole is vertical (right angle with the ground). You know the length of the post (L), and the distance from the base (D), and using the PT, you know there’s exactly 1 right triangle whose hypotenuse (H, distance from top of post to the ground) such that L**2 + D**2 = H**2.
This is a reply to an earlier post by “Scooter” regarding the “Formula Sheet” Rubric. What I great idea for encouraging students to develop a more comprehensive understanding of math formulas. As part of a class project, students could bind all the Formula Sheets they have completed, using the school’s spiral binding machine. This way students would have a reference guide of formulas, that they could use for future math classes as well as everyday real world problems they might encounter.
I was required by my geometry teacher to have a small notebook to write all the formulas, proofs, postulates etc.. It made it very easy to reference while working on practice problems.
I also had the same thing! I think it is important in order to keep students organized that you require a notebook where they are to keep all of their notes. I think that if you just give students pieces of paper to fill in they will eventually lose them.
I have heard many students say that Geometry was the hardest math class that they had to take. Many times students are great at algebra but have trouble understanding geometry. Did any of you experience this? And if so, is there any specific topic or section that you are most nervous to teach? Was there one area that you really struggled with?
When I took geometry I struggled with proofs. It was my first experience with that kind of problem solving and it took me a while to understand how to do it. I am hoping that this class will give me ideas on how to teach proofs effectively to my students.
Jane Doe,
I loved geometry in high school. So far I don’t think I have run into a math subject that didn’t agree with me or me with it…until now! I just go my first test score for Calc 2 and was a bit disappointed by it. Well, that is putting it mildly. Anyway, I think Geometry would be fun to teach as it seems like it could be more of a crafty subject, which is to mean, a subject you can use a lot of arts and crafts with. I like hands on learning.
Convention vs Validity
As teachers, I think we need to be cognizant of what signifies validity in a mathematical statement and what is mere, possibly local, convention. For instance, as a solution to the equation 3x = 21, by convention, we usually write x = 7; however, the statement 7 = x is equally true, or valid, and is easily seen to represent the same fact. When speaking of triangle equality or congruence, we say “if 2 angles and an included side are equal, the 2 triangles are equal, or congruent”, or “if the 3 sides of the triangles are equal, the 2 triangles are equal, or congruent”, or “if 2 triangles can be flipped and/or rotated so that the corresponding sides and angles match, they are congruent” . By convention, when writing this symbolically, we say “Triangle ABC (congruence symbol) Triangle DEF” only with AB=DE, BC=EF, and CA=FD. In some of the A+ lessons, in the PLATO lab, where I tutor, the “standard form” of a 2-variable, linear equation is given as “Ax + By = C”, and an equation of the form “Ax + By + C = 0” is considered a wrong answer, even though the second form is by far the most commonly given definition of “standard”. I think this sort of thing is one of the main reasons that students get frustrated with Mathematics.
Some notation, such as ordering the pair (x,y) always in the same fashion, cannot be whimsically altered, without destroying the meaning of “point”. In the same way, “blue” cannot be interpreted as “brown”, without destroying meaning. When is this point of “definition”, as opposed to “style”, or “convention”, reached? I maintain that it occurs precisely when there is not enough information left to determine validity, if the convention is ignored.
One of the best methods I’ve ever seen for breaking this slavery to convention is to use foriegn characters instead of letters for variables. Students get so bogged down in the never ending quest to find x. Instead, I’m an advocate of at least occasionally providing students with something along the lines of
0 = 3@ + 9$ – 6; find @ when $ = 2
or even using Japanese or Thai characters that none of the students have seen before (although using some of the simpler characters that are easy to copy is usually a good idea)
I think that this is a good idea, because it truly illustrates one of the more important concepts of mathematics- that the symbol itself holds no real truth, but rather that the operation is transforming values into something else, that function is being applied to the value. Multiplication is not so much the operation which is represented by the symbol “x”, but rather the operation which takes 3 to 15, 2 to 26? If students are only briefly introduced to new symbols to represent operations and variables, their understanding of abstraction is extended that much further to help them visualize more complex ideologies.
So I was recently talking with a friend about her practicum experience and how horrible her lead teacher is. She was describing to me how the teacher is going through some serious life issues and then claims certain students just CANNOT be helped by any one and that they will never make it in life. I guess what I want to know is:
How can a teacher keep his/her self from letting their personal life affect how they treat their students and prevent oneself from becoming overwhelmed? Is there really a student who is hopeless?
I don’t think that there is ever a way to separate the personal from the public and even if one could I don’t think that it’d be a good idea. Our personal experiences help us feel compassion and also help us tell when people are using their own personal troubles as excuses. It helps us realize that people are indeed human and that there are going to be people who find things to be easier than they are to us and also harder.
There is also no student who is helpless. There may be students who are harder to reach or have a disability, but there is hope and the only way is to work through it is to try to use our own experiences and the experiences of others to encourage and inspire the student until they realize that they can do it and consequently try – even the tiniest bit.
It’s simple enough to say that we just need to keep our personal and professional lives separate from one another. However, one must remember that we are still human. As a coach, I’ve figured out how to check my personal life/issues at the door in order to set my mind to doing the best job I can for the kids on a daily basis, focusing solely on the task at hand. I can only assume that the same attitude can be applied to teaching as well. Given that, it still sounds to me like this teacher is somewhat overwhelmed. I don’t know how long this teacher has been in the field, but it sounds like some sort of sabbatical might be needed in order to get their head and priorities/issues straightened out. Regarding the second point, I really don’t want to sound like a jerk, but I believe a lot of young teachers are naive to believe that they will able to get through to and help every single solitary student that comes their way. In reality, this just simply isn’t the case. You will encounter kids that just simply do not care, no matter how you approach them. It’s very sad to think about, but when you look at the real world, we’ve all seen/encountered adults that we just see as hopeless. Well, at one point they were kids too. All we can do is try our best and hardest to get through to as many as we possibly can.
Walter,
I agree with you completely. As the ‘professional’ you have to check your problems at the door and try to do the best for your students any way you can and not let your personal problems get the best of you and cloud your judgment as a teacher.
You may think it sounds harsh but you are just being realistic. We all want to be that one teacher who actually gets through to that one troublesome student but in reality we may not. I have always held the belief “never give up on a student/child” and I still do but there comes a time when they just don’t respond to our assistance anymore. Now, I am not saying to stop trying because that is just wrong to fully give up but you may just have to back off a bit and just go about your business in hopes they will turn around and ask you for that help on their own because they realize it is there. Your last sentence about trying our best and hardest to get through to as many as we possibly can is the truth. We are only human and we can only do so much. Thanks for your input!
Winnifred,
Teachers are human just like everybody else. So they have their good days and their bad days. However, teachers still work in a professional working environment and they should not let their personal issues clash with work.
When I am having a really bad day and need to go to work. I take a breath and say to myself, “there is nothing i can do about it right now so just go to work and do your job”. If i let myself think about my problems at work my work performance goes down.
A teacher cannot take their frustrations out on their students. It is not the failing student’s fault your personal life sucks. Yes, It is hard not to let your emotions get the best of you however, sometimes you just have to suck it up.
No student is ever hopeless. If a teacher thinks that then they are hopeless. I feel if you wanted to be a teacher, then you made a commitment to help EVERY student no matter what. If that student seems hopeless then maybe the real thing they need is a little more guidance than usual.
I would mostly agree, but there are exceptions. I went to elementary and middle school with a kid named Anthony, and he was hopeless as a student. Not for lack of intelligence, the kid was actually quite smart, but he was like the Joker in the Dark Night: he had no response to incentive, positive or negative, he had no discerning targets for his misbehavior. Instead, he just wanted to disrupt the class in every way possible. We’re talking about a student who studied the nuances of WCSD conduct codes, teacher agreements, etc. just so he could exploit the flaws in the system. I know of three teachers who retired early after having him as a student. He made sure he was hopeless, and was more volatile the more a teacher tried to reach out to him.
Fortunately, I think students like this are about as rare as Will Hunting.
I completely agree with your statement. For some students the teacher is all they have in life in terms of guidance and motivation. It is our jobs as teachers to help every student achieve. Yes, it does take work from the student, but if we give up on them then how are doing our jobs. We are the adults and we must remember what it’s like to be students. If we give them respect and show them that they care, hopefully they will take our classes seriously and work hard to do well.
Unless you’re a robot, I don’t see how you can completely compartmentalize your life. That being said, most people are asked to do this to some extent in most workplaces. For instance, psychologists must focus on their patients’ needs, rather than venting about their own issues. It’s considered poor form to cry in business meetings, even if your dog just died. It’s called “Professionalism”.
Oh, I completely agree. But what can you do to prevent yourself from telling students they don’t know anything and are dumb and just overwhelming yourself. I talk to more and more people who claim the teachers they are overwhelmed and I don’t want to turn into a teacher who can’t separate her personal life from her professional life. I mean, do you assign less homework so you have less grading to do or ….?
To Winnifred:
I think that it is not necessarily an issue of separating your personal life from your professional life but rather being able to balance the two of them. Students like knowing that their teachers are real people with lives outside of the classroom. It makes it easier for students to relate to teachers when they know that teachers really are just like them. Putting aside your personal life when you enter the classroom is not necessarily the best thing to do and is not the easiest thing to do either. However, this does not mean that you should dump all of your problems on your students. You still need to maintain professionalism. High school students are old enough to realize when you aren’t having the best day. Maybe being honest with them is the best way to handle a situation where you are having a hard time. They may even be able to support you by being more obedient or more respectful.
Regarding being overwhelmed, less homework may work as long as your students don’t suffer because of it. Being good about staying on top of grading assignments and creating lesson plans is the best way to not get overwhelmed. And as you become a more experienced teacher, the grading and planing will get easier.
Jane Doe, I agree that there needs to be a balance. Kids like to know the teacher and get along as well. My government teacher in high school did a great job of teaching and being himself in the classroom. Some days the teacher might have a bad day but if the kids can see that they might cheer him up. My teacher coached soccer so when they lost he was not in the best of moods but we learned to adjust and still make class worth while. So there definitely needs to be a balance of the two.
I agree with Jane Doe. You don’t always have to separate your personal and professional life but there should be boundaries you never cross. Students do like knowing their teachers are real people…I remember whenever some students would get excited and also weirded out when they would see their teacher at a grocery store or something. Students do realize teachers have problems but when they go into class they are expecting to be taught rather than to be scowled at because the teacher is in a bad mood. If you are a teacher and are having a bad day, it shows during your instruction if you can’t separate the two.
I like how Jane Doe says to just be honest with them. If they realize you are having a bad day maybe they won’t test the limits as much and respect that you are in a tough spot and just behave. Although, you may have those students who just don’t care what kind of mood you are in and will misbehave anyway.
Also, as Jane Doe said, less homework may work but if you lower your expectations of your students because you don’t have time to dedicate to them (which is basically your job) they will suffer. They won’t learn as much and will suffer if you suddenly bring your expectations back up and they aren’t at the level you want them to be, Lesson plans are supposed to be your best friend. They help you stay on track and make sure everything is getting done as planned. You may not even use an entire lesson plan but rather a guideline for the day to help you get through. However, as a teacher the grading will never go away and you do have to plan for it whether or not you are having a rough minute. It’s inevitable.
I personally think this teacher may be facing, or be nearing “burnout”. There are ways of reaching students, however, if the students see her frustration with them, they may start to think the same way about themselves. Any teacher that tells ANY student they are dumb should not be teaching. We become teachers to find anyway possible to reach each and every student one way or another. As teacher we are supposed to be encouraging students to live up to their full potenial, not give up on them and kick them to the curb. Assigning less homework is not an option either, especially in math!! If this teacher is that overwhelmed, she could apply for a teachers aid or take a leave of abesence. I honestly think this teacher needs to take some time off to remember why he/she wanted to teach in the first place; and if they can not remember, then maybe they need to find a new job!
I was thinking the same thing, if in fact you feel that you can no longer keep yourself together during class then you may need to look for work elsewhere. Or at least take off a year or two and sort out your own issues. Hopefully the school principal noticed this behavior during one of his or her evaluations. When your personal life is affecting the education of your students it is absolutely necessary for you to seek help.
Response to Jane Doe
I think I was one of these struggling students that over achieved in calculus, and algebra and had my biggest struggles with geometry and trig. Now that I am older, I have a better understanding for it, and definitely do not have an fear of teaching it with the exception of proofs. The reason I like math is because there is always a set way (steps) to do a problem, and generally you always get ONE answer that you can double check and see if you are right. When it comes to proofs, there are many ways to attempt a problem and I think that is where I struggle. I like to have a set way of attempting a problem. I think when I teach geometry, I will definitely try to create real life application and use multiple strategies with my students while learning geometry, to hopefully prevent more students from being lost.
Does anyone else have trouble with proofs (of any kind)?
I also have a problem like yours. I feel like I just don’t know where to start and then I confuse myself and make things much harder than they should be. When I took geometry in high school, we never did proofs. I had no idea what a proof was until I started calc 1 last fall. And creating real life applications for any math class is the most important thing a teacher can do to help students understand the material.
I agree about the ties to “real life”. The number 1 (non-technical) question I’m asked by my tutees is “Why do I need to know this?” I think an emphasis on “real-world” applications needs to start much earlier than high school. By the time students are juniors, they’ve already made a lot of their decisions about whether Math is meaningful for them.
I completely agree that math needs to be made relevant and meaningful to students much earlier on. I was fortunate enough to have teachers from third grade on who made the effort to give us word problems and even showing the connections across content area. This was enough to get me interested and confident in my mathematical abilities. For both seventh and eighth grade I was moved up to integrated one and two and had the most amazing teacher. She encouraged us to find outside problems and bring them in and was always willing to work one on one with students until they understood the concept. It is this sturdy mathematical foundation that has set me up for success and enjoyment in math. This is actually one reason I plan to teach middle school mathematics because I believe that these are such formative years where students are building up the base knowledge and overall interest in mathematics that is necessary for everything else. I feel like even if you have a bad teacher or less engaging classwork later on but had the initial understanding and interest, you are much better off and have the tools to be able to succeed.
Perhaps one of the most difficult parts about being an educator is knowing how to help students do more than work, but to learn; that is, motivate students to want to learn what is offered to them. although many students have the thirst for knowledge, many turn away from the most valuable thing one can possess- an education. As mathematics teachers, we have a particularly difficult time offering motivation to students, and it can be difficult for us to let them know how important math is.
I mean, mathematics is applicable in so many ways- I feel that the first step for any student is recognizing the value of knowledge- period. Once a student understands that, they can have so much more potential. It’s so difficult for teachers to not only teach the how something is important, but why something is important. If we, as teachers can understand this… well, that has to be everything that we need to continue to pursue helping others.
I could not agree with you more about it being difficult as an educator to know how to help students do more than work. I think you hit the nail on the head with the word “Work”. Work is the perfect word for what happens to students who are given a large amount of homework which repeats the same operation over and over again. Understanding the topic is so much more important than cranking out solutions. I believe as educators we can teach students how to be good learners, not just in one math class, but in life. As an aspiring math teacher I feel tying every lesson to the real world is a way to accomplish this goal. When working with a function in Algebra it is so easy for students to say, “I will never use this”. When that happens and nothing substantial is said in return the student will believe they are right and tune out. In school I remember having faith in the teachers that what they were teaching me was going to be worth it, but not all students have this faith. As a teacher I am going to try and have a thoughtful answer every time I hear someone say, “When will I ever use this.” Explaining a real life situation where what we are learning that day could be used will help not only them but other students who hear the conversation that this work is not in vain. In addition, having thoughtful problems to solve that involve real life situations using the lessons of the week will reiterate my statements throughout the week that the lesson is worth understanding.
In every math textbook I’ve ever had, the chapter starts out with the numbers, and word problems don’t come about till the very end. Usually we never get to all of them. Maybe word problems should be at the front of each chapter. Learn context first, then there is a reason to learn the formulas.
That’s not a terrible idea. However, students greatest struggles generally come with word problems and how to generate and formulate given the information they are presented. Given that this is the case, how could we expect our students to process the information and generate the formulas without prior experience simply processing the number problems and gaining that grasp of how the formulas work? Not only would the students lack a fluent understanding of how the formulas work, they would be handed a puzzle with limited knowledge of where the pieces are supposed to fit.
Another problem I would have with giving word problem first is, what about the students who do not speak English fluently? If they cannot read the question properly, you cannot expect them to give you the right answer unless someone translated for them. With “number” problems and formulas first, they might be able to scan a problem and pull out the important numbers and deduct where that number goes in the formula.
I agree with Winnifred and Walter…word problems are the most challenging for students, even for myself. However, I also agree with Ricky Martin. Providing students with context and meaning behind the math is necessary to ensure comprehension of the tools and formulas. So, I think presenting the students with a real life situation and/or word problem in the beginning of the lecture, but not revealing necessarily how to solve it until the end will keep their interest, challenge them to find connections between the math/formulas and real-life situations, and provided clarity on how the math is applied and how to carry out applications. In summation, the teacher would present the problem at the beginning of class, introduce the math and formulas for the section (referring back to the word problem periodically throughout the lecture), and end the lecture by demonstrating how to solve the problem using the math that was just taught. That way the students have real world application throughout the lecture and are able to see how to apply math to solve problems.
I understand the argument that learning the formulas first makes it easier to figure out the word problems. But I still think there is something to be said for trying to get the students to “discover” the formulas on their own by giving them time to work with a real life scenario. This way, they not only learn the formulas, but they also get comfortable figuring out the formulas as they need them in life. Not just memorizing. But I agree that we still need to teach them the formulas, they won’t all just “discover” them on their own. So a little balanced approach.
As far as folks that don’t speak english, I don’t think we should let language barriers keep us from teaching practical math. People in a classroom need to all be speaking the same language (no matter what language it is) for learning to be maximized. Even in math. You’re right that you can sort of get around it when your just using numbers, you’re gonna need a translator either way if the kid doesn’t understand what you’re saying.
I think the reason the chapters start out with numbers and lead into word problems is because the formulas and numbers are what your main lesson is, not necessarily the word problem. If they didn’t get the numbers and formulas first then they wouldn’t know where to begin with the word problems. Usually, if chapters are like that, numbers first and word problems second, the teacher does a better job of incorporating all the material. Most teachers generally include word problems on assignments as well as number problems. I think if you never get to the word problems you teacher isn’t doing a good job of planning out his/her lesson to include all material covered in the chapter. You could also do it where you work with numbers and formulas, then do word problems, and then continue on to do more numbers and formulas that way your lesson goes full circle and everything was incorporated and students won’t miss out on useful knowledge. Also, it is unfortunate if your teacher doesn’t get to word problems because in like we aren’t given formulas and numbers to solve but rather a real-life situation which is like a word problem. Or, you may have a later teacher who loves word problems but your previous teacher never got to them so you are lacking for your current class.
I just think it would be difficult to change the order of the text book. If “every math textbook” you have looked at is organized the same it must be working somehow.
So in our classroom on the back wall are some book for the Interactive Math. Program (IMP) where is is most all word problems. I used these books in my high school math classes until I got to Pre-Calc. If you get a chance to look at them you should. They are all word problems. You will find very little traditional “number” problems in these books. If you can’t look at the books you can check out the website for the program (http://www.mathimp.org/ ). I thought it was hard, but not nearly as hard as the students who did not understand the academic language of math.
Point is, I think the books can be structured in any way, but it is up to the teacher to cover the material and give the students enough practice to understand/ master the material.
Good point, Sr. Martin,
I’m tired of “word problems” being relegated to their own, neglected corner of the classroom. Sure, they’re more difficult, maybe even more so for the teacher than for the students because of their capacity for divergence. If we plan on teaching to the test, then yes, “word problems” are a thorn in our side and aren’t going to do much. But if we are really focused on fostering critical and analytical thinking skills in our students, we need to double down on these problems, focusing on actual application, asking the right questions, and put the follow-the-equation problems into the supplemental role they’re really meant for.
This might be useful for some students, but I agree that the ELL students are going to struggle and even some other students who are not doing well with the material. Maybe word problems should not always be placed at the end, in their own section, but instead spaced evenly throughout a chapter. This would be good to also help the students gain a conceptual understanding so they are not just memorizing the processes that they must take to solve the problems.
I have personally met and worked with foreign students who don’t know much english and have recently come to the US to start school. It would be a much smarter approach to put problems and formulas because I know that it’s much easier for them to understand that rather than a complex word problem.
Tastes great! Less filling! Life is a word problem. I think the underlying problem here is that we are required to drill students to score high on multiple choice tests, not instill an ability to do Math in daily life.
In math I think it is especially important to show students that math really does apply to every day life. In order to keep this in mind I think that word problems are a very important aspect when testing children in math at a young age. It will help diffuse some of the regular questions math teachers get, such as, “When will I ever use this again?”
I thought of an interesting topic that was brought up in my education class regarding the future of all teacher’s. It is amazing to me that they are considering making the pay scale equal for people with all different educational backgrounds. It seems as though getting a masters will not be worth it due to the fact that they may no be paying us more? How are teachers going to be motivated to further their education if there is no increase in pay?
I can say that for myself I am not even sure that with a pay increase I’d want to get a masters degree. In this economy, there is a large amount of people (a lot of my older friends included) who are “overqualified” for most jobs and are therefore not being hired. If what you say comes to fruition I think that there would, indeed, be very little incentive to go further into one’s education to get higher degrees. I suppose that for some, self satisfaction would be an incentive, but internal motivation can only go so far in a world that demands a capitalistic mentality. Money is the great necessary evil and if furthering one’s education can’t get one farther up on the pay scale, then I don’t know what would encourage the vast majority of American teachers to get more esteemed degrees.
The question really depends on what kind of incentives they put in its place. If you get a pay increase based on how your students do on tests you might want to learn more effective ways to teach. I agree that if they take away the pay increases for having a higher degree it will decrease the number of people seeking higher degrees. However, I think that eventually a new type of incentive system will come along and when it does people will do what they need to do in order to qualify for it.
I think a lot of it depends upon your personal aspirations moving forward in your career. If you are content with teaching at the secondary level for the duration of your career, then there may not be much incentive to further your education for any reason other than for your own self satisfaction. However, if you aspire to be a principal or something of the sort in the future, then getting your masters in administration would be the path for you. Likewise, if you would like to pursue teaching at the college level, then furthering your mathematical education would prove to be a worthwhile venture. I think it comes down to each individual’s personal goals and desires. However, I do agree that there is no incentive provided for teachers to further their education and remain in the secondary teaching field.
There really wouldn’t be any motivation for me. I know that education is important and that being educated is key when you are striving to educate others but why would I increase the possibility of debt (because the cost of education continues to increase) to NOT get paid more in the end? It seems illogical.
My step-father was teaching for the outreach Master in Teaching program at Leslie College which is a travelling college outside of its brick and mortar location. He would be flown to a city to teach classes on two weekends with online assignments due between the weekends and the students would all be working toward a master degree in education. This was all around the country. He was laid off for this year due to low interest in the program. This of course was due most of the states offering no incentive for teachers to get a Masters’ Degree. Personally I am surprised NV is still offering the monetary incentive and hope to finish my degree while it is still in effect. In my step-fathers home state of Indiana the state is justifying this by saying that teachers should still want to get a masters degree so they will be better teachers. Then they have added the stipulation that pay raises come from student performance and the teacher being labeled by their school as an excellent teacher. What defines an excellent teacher you ask? Well just that, student performance but the catch is that each year only 20% of the faculty at a given school can be designated as excellent. Therefore even if all of the teachers are excelling only 20% of them will get rewarded for their efforts. This is troublesome and the question I ask is who makes the final decision. Is it the principle? And if so if I take him/her golfing every weekend do my chances of being labeled excellent increase.
I think they will do it because they want to move forward into administration. Idaho already does this to their teachers. It’s supposed to be temporary, but it probably won’t be. I think if you want to better yourself and what you could offer the students you should keep going forward in graduate school, but it certainly isn’t a necessity to teach.
Two things. Why is it that in all American organizations I can think of, managers/administrators get paid some multiple of what a worker directly involved in the production of goods and services can hope for? On the extra pay for grad degree – many (most?) education undergrads take essentially the same courses as graduate students, who don’t already have an education degree.
I personally want to earn a masters degree. Not for the pay benefit, however, for the chance to move into teaching college math. The accountablilty that goes along within the high school setting will probably frustate me into burnout. If I have the option to tell a student to leave my classroom that has no intrest in being there in the first place, I would be a much happier teacher. To second my desire to teach college math would be dealing with the parents. I know what kind of parent I am personally, yet parents tend to blame the teacher not the child. In some instances, yes the teacher can be at fault. Whether teaching style vs. learning style or where ever the struggles may lie; teachers can become the scapegoats for many failing or struggling students. So for these reasons earning a masters degree will help me save my sanity!
It does seem like there could be some serious advantages to teaching college. I would prefer a junior college, as I am not very interested in research and being published. I just want teach. The only thing that I might miss if I taught at a junior college instead of a high school would be the possibility of coaching and helping clubs, etc… The community aspect of it.
You’re so right. I’ve had to deal with parents as a coach this year, and so many parents have such a difficult time with the idea that their child is not the little angel that they think they are. And most of the time the students play to this, because they know that even if they don’t do the right thing, or do all their work their parent will defend them.
I don’t really understand what you are asking Heron but I agree with mathgirl. I also want to get my masters and then hopefully phd in Math, so that I can teach at a university level. It’s not for the extra income, its just for the simple fact that I want to try teaching high school and college and see which I like better. I think after doing so many years of high school, I might be ready to move on and actually teach to those students who ENJOY math and chose to take it, rather than just fulfilling a requirement.
I completely agree. I doubt any of us are going into teaching with money being the first thing they expect. I’m doing it to make some kind of impact on the students and because I enjoy math. I also would like to continue on and get a masters and perhaps a phd, not necessarily for the job advancement or money, but because I would like to continue developing my own understanding in something I have a passion for.
I am responding to the previous discussion about introducing word problems involving real-world applications earlier in a lesson. I think introducing a mathematical concept or formula through presenting an application/word problem for students to solve has some powerful advantages. If students are allowed to discover concepts and applicable formulas on their own, they are much more likely to understand what the concepts or formulas represent, and they are much more likely to retain this information. Of course it’s the teacher’s responsibility to provide students with the background necessary to do this successfully. Students need to show evidence that they have mastered previous concepts and formulas, before moving on to a new topic. One way to ensure this would be for every discovery lesson to be followed by a formal, teacher-led lesson or discussion to make sure every student understands the concepts being covered. I think the self-discovery approach in the context of a group activity, followed by teacher-led instruction would help enhance the learning of all students including ESL students or students who have learning difficulties.
How many of you are planning on using more hands on techniques for lessons rather than text book work? What do you think is gained by hands on lesson planning? How will it help your students grasp the concepts you are teaching? What kinds of lessons have you already thought of that incorporate this type of lesson?
I plan to do a bit of both, I think I’ll have them “discover” the relationships between the various things we’re working with and then have the students work the problems out of the book with what they learned in the hands on portion. I think, at least for me, the most helpful thing is repetition.
I think that having them discover the relationships hands on will help them to visualize what they did to find the relationship if (probably when) they forget it later on. Since they discovered it themselves, hopefully they can do it again.
I haven’t really given that last question much thought because I want to teach algebra or calculus so I’d have to be a bit more creative in coming up with something that will make those subjects more hands on; maybe something with an actual balance (to balance the equation when manipulating it) or maybe use a “dis-integration” ray when they have to do differentiation. *it’s a funny joke, get it? 🙂 I couldn’t help myself*
I do think that hands on learning is important though, it’ll solidify their knowledge and give them confidence.
I plan on using a lot of hands on techniques in my classroom. First of all, hands on techniques help ELL’s and since I am becoming a teacher in Nevada it is almost guaranteed that I will have ELL’s in my classroom. Second of all, not all people can learn from lecture, there are many people who learn through hands on experiences. Hands on activities help students visualize the concept they are trying to learn.
I developed a few lessons last semester that are hands on. One was a math bingo game that was pretty cool. I had different bingo cards that were passed out to all the students. Then randomly I would pull an equation and the students would have to find the missing variable. The answer was somewhere on their bingo card. The process is proceeded until someone gets a bingo. It’s great for review!
I plan to use hands-on techniques in my classroom, but only if those hands are on a touchscreen, or a smartboard, or something else that can truly express the interrelated nature of these topics. This is esspecially imperative for teaching geometry, trig, and statistics. (something I think every high school should offer and encourage taking) I think that “conventional” hands on methods are actually hurting math education, because they present the topic as being static and unrelated to anything outside that single example. Cutting out paper triangles, using wooden block shapes, and counting M&M’s seems good enough, but it fails to engage all but the students who are most adept at abstraction, in seeing that these quantities can transition, either largely or by the smallest shift, but the relationships and principles remain the same.
I’m not talking about activInspire, or PowerPoint, I’m talking about truly great visual tools like this: http://www.gapminder.org/world/#$majorMode=chart$is;shi=t;ly=2003;lb=f;il=t;fs=11;al=30;stl=t;st=t;nsl=t;se=t$wst;tts=C$ts;sp=5.59290322580644;ti=2010$zpv;v=0$inc_x;mmid=XCOORDS;iid=phAwcNAVuyj1jiMAkmq1iMg;by=ind$inc_y;mmid=YCOORDS;iid=phAwcNAVuyj2tPLxKvvnNPA;by=ind$inc_s;uniValue=8.21;iid=phAwcNAVuyj0XOoBL_n5tAQ;by=ind$inc_c;uniValue=255;gid=CATID0;by=grp$map_x;scale=log;dataMin=295;dataMax=79210$map_y;scale=lin;dataMin=19;dataMax=86$map_s;sma=49;smi=2.65$cd;bd=0$inds=
I absolutely LOVE hands on learning. I think I will use it as much as I possibly can when I start teaching. Allowing students to use manipulatives in math gives them one more tool to build their schema and making it easier to recall later for a test or when they are describing the material to someone else. One of the problems I’ve had with it though, is making sure there is some kind of filler for the transitions and that all of the students are actually engaged. When everyone is engaged, everyone should be learning; making the lesson a success.
The biggest thing you have to make sure when doing hands on learning is that you do not just do something for the sake of doing it. You can’t have them do something that doesn’t relate to the material, and there has to be learning taking place. And you have to make sure that all of the students are accountable to do the work. Make sure they turn something in, not just a hands on project with no accountability.
In response to the post about using hands on approaches in our teaching I would have to say that this is a great idea. I also think that it is very realistic to use hands on approaches in subjects like geometry but what about other areas of math? I have had issues in making lesson plans in algebra or calculus that involve a lot of hands on work. It is hard use manipulatives in geometry and lower level math but what happens when you get to algebra? Does anyone have any ideas for hands on methods of teaching these types of subjects?
I think for subjects like algebra you can have hands-on approaches that doesn’t necessarily mean manipulatives. Anything “hands-on” can be something like giving students a project about loans and living on their own after high school. They would have to find the loans and the interest rate as well as figure out food cost,and typical prices of bills and rent. I think it may not use manipulatives but it is hands-on because they have to do all the work and the research. Another benefit is it does play into real-life so they learn to do these things before they get out there for real.
I have one idea for a hands-on activity for algebra involving slopes. In one of my education classes a girl incorporated this activity into one of her lesson plans and I thought it was a great idea.
When studying graphs and slopes, take your kids outside and have them calculate the slope of the bleachers (or any other objects that have slope). Give them tape measurers and something to record their results and let them have at it.
I know this doesn’t provide you with a whole lot of ideas for hands-on activities for algebra, but at least you have one more than you did before 🙂
What are some ways that you would like to see geometry taught using software for a touch screen media tablet? What are some benefits or challenges you can see with that type of platform? Any lessons in particular that you think could be more beneficial on a touch interface than with a mouse/keyboard or pencil/paper/construction-tools?
So, I had this idea of sharing a riddle or math joke with my students every morning, or at least every Friday. With that said, I need more jokes. So, does any body have any good math riddles or jokes they would like to share?
My math teacher told this one the other day…
“I’m going to prove that the number 2 is odd…”
List of prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, etc…
“2 is the only even prime number… isn’t that odd?”
It’s silly, but it made me smile!
Its been my experience that the best math jokes just present themselves in the material. You could try something like http://xkcd.com/, but you really need to be halfway through a college degree to get a good deal of their humor.
I have an interesting conundrum. My practicum teacher is a proponent of the “drill and kill” school of instruction. Pretty much every day, from bell to bell, he’s working problems. The students take notes or do assignments from worksheets, with more problems. The notebooks are checked and graded frequently, along with homework. The busy-ness of his class seems to be a major component of his (effective) behavior management.
The school does quite well on it’s standardized tests, makes AYP every year, and is a school of choice for students from other schools in the district, so this looks like good policy to administrators. The students find it tedious, as you might expect, and interesting only from the standpoint of grades. Any suggestions?
Personally, I like it for the most part. In my experience a lot of practice is what made me, as a student, remember what to do. I think that spicing it up a little is necessary, but I think a lot of drill work is necessary and beneficial. I think that it is also reflective of the real work environment where they will be evaluated often and harshly (depending on where they work). I think it’s an important life skill to get used to as well as being a good way to practice. As they say, “Practice makes perfect”…even if it is often boring 🙂
I too like the traditional “drill and kill” technique. However, I was able to learn with that method and enjoy it. In turn, I was successful and now want to teach math. The point being is maybe my brain is “wired” for this type of learning; however, not every child is or will be. We all learn in different ways, and we need to be sensitive to this. Even if it is “working” in the eyes of administrators, we should never give up on creative ways to teach the material.
I also learned well with the “kill and drill” technique, and I think, with math in particular, a lot of practice solving problems is necessary for successful learning. However, sometimes I would learn how to solve the problems correctly and do well on tests and homework assignments, but not really understand what it was I was doing. This caused me problems later on in more advanced math classes, because I had not really developed a deep understanding of the concepts. This is why I think the most effective approach is to find a balance between using self-discovery investigations with students and providing sufficient time for them to practice solving the problems once a concept is learned. I think a combination of both approaches is going to lead to better results as far as students learning math.
After doing research for other classes I discovered that students that are taught using this method lack a deep, comprehensive understanding of the subject matter. Even if it raised test scores, it fails to help students truly grasp the key concepts. It may have worked for some of us but in general it does nothing but help students pass tests. I think that teachers who want to raise test scores must promote deep conceptual understanding of the subject matter. Getting practice in math does not need to be drill and kill.
There is a lot of pressure on teachers and administrators these days to make sure students do well on the state assessments. If a teacher’s students are performing well on the tests, it seems unlikely that you would be able to convince him to change his way. However, it seems to me that a few hands-on activities thrown in once and a while would not only break the tedium, but could also lead to a deeper understanding of the concepts the students are learning. I think this approach would promote more student interest in learning the subject, which could ultimately lead to them performing even better on tests.
I agree. This helps to solidify the information a teacher is communicating and gives the students real examples to confirm their learning. Book work is tedious and can bore students to sleep.
Other hands-on activities are very beneficial when trying to show how the information is relevant to everyday life. The biggest challenges in my opinion seem to be the length of the class and the policies of the school. It’s hard to make time to implement other activities into the math curriculum, at least when you first start as a teacher.
Hands-on projects and activities is a great way i think to get students to understand what they are doing. I do more hands-on things now that i am in college than i did when i was in high school. I wish i could go back to my geometry class i took and do some of the activities that we have done in this class.
Because we were given the question creation assignment, I have been thinking about the most effective way to create testing questions for students. Is it better to create questions directly based on homework assignments? Or should they be more challenging and combine multiple concepts? Or should they prepare them for standardized tests? Obviously, we can choose to combine the three, but are there advantages to one vs. another? Thoughts?
I guess it depends on what kind of testing you are conducting. If it’s a regular test for a grade, I don’t think the questions should be harder than the homework/classwork that they’ve seen. I think the time to pose more challenging questions is during class and homework. During tests, I think the questions should be as straightforward as possible, just to make it a fair assessment of the students.
I agree with what you are saying. Maybe a challenge question could be considered for extra credit, etc.
I agree, challene questions for extra credit is a great idea! I also agree with Ricky Martin: test questions should be a fair assessment and should be straightforward. There are a lot of students who get test anxiety and just can’t seem to perform well on tests. However, we still want to challenge our students, so allowing them to experiment with more abstract problems would be beneficial to do as a class or with partners so that they are able to think through the problems together, using what they’ve learned from the material.
On the flip side, I don’t think test questions should be a piece of cake. My suggestion would be to give students challenging homework problems or classwork and then put a similar challenging problem on the test. That way they’ve already been exposed to it and have seen how to work through it.
I agree. Making test questions simple doesn’t help the students at all. Expose students to some challenging questions and they will be able to approach the other questions with confidence.
I am currently in Calc 2 and my teacher creates test questions based on the homework assignments. This is beneficial to me, as a student, because it is easy to know what to study. Given this is a more difficult class, I think it is appropriate for our topics. I also believe that it makes her job easier by creating less work. I don’t know if I necessarily would create a test based on standardized tests, but possibly would consider those type of questions for a quiz. I just think that too much of what teachers teach on is based on the standardized tests. I would like to know if students are understanding the topics based on the county/state/country standards that I have to follow as a teacher and what I deem worthy as an educator.
I definitely agree with how beneficial it is to be tested on something that we have actually worked with. How your teacher tests based on homework assignments.
Lately I’ve been helping my younger family members with their math homework & I’ve noticed that they tend to throw in a question that the kids have to learn how to do themselves. They come to me saying “I don’t understand this question”, me countering with “Did you learn this in class?”, and they always respond “No.”
I found it frustrating in a way that they send kids home with problems that leave a door wide open that leave the kids confused. And correlating it back to how beneficial your teacher is, what my little cousins have to deal with is not beneficial to them, because it’s either they figure it out themselves (mind you they don’t have a book to bring home with them) or get marked down on their h.w. assignment.
Part of the higher level thinking that is promoted with Bloom’s Taxonomy is difficult for teachers to promote with Math as their subject. Most math problems are designed around students regurgitating information on the page, such as plugging numbers into a formula, or rules. Often times students become frustrated in Math becuase questions are at a higher level of thinking, such as a word problem. Maybe this is where the frustrations that your cousin’s are feeling arise from? They may have learned the information at a lower level, but now are being asked to critical think, which is difficult when it comes to Math.
I think that it would be great for a teacher to tackle these kinds of higher level thinking questions in class…when there is a little extra time. Or maybe giving extra credit on them if a student is able to do them at home.
I believe that test questions, especially in Math, should be base on homework assignment because that is what has been taught and studied. I think it would be unfair for a teacher to test on subjects which were taught in class but not reviewed in homework assignments. That is not to say that some teachers create tests this way and in essence surprise the students with certain questions that were not on the homework to see if they were paying attention in class. I have recently taken Calculus I, II, III to gain the approval of the state to teach secondary mathematics and the homework as well as the review sheets given by my teachers were essential to me passing the classes.
I don’t know if a teacher can base an entire test on the type of questions in the standardized tests because the standardized tests span such a wide range of topics, an entire year of math teaching. During my practicum I was with a middle grade teacher who had mostly 8th grade students with one of the classes being Algebra I. During the month prior to the Standardized Tests the teacher would gradually spend more and more time quizzing the students on typical standardized test questions, which can be had from previous tests through the district. He would put the question on the overhead and have the students call out the answers and do a quick review. Another way would be to make a quiz worth points each week that had these types of questions on it and was tied to that week’s curriculum. The students would then be very used to answering those types of questions and when the standardized test came along they wouldn’t be overwhelmed.
The Van Hiele model suggests that students’ geometrical understanding progresses through various levels, which cannot be skipped. There is a lot of research that supports this theory, and has found that most students enter high school geometry with a low Van Hiele level of understanding. How would you make sure that your students will make the right connections to the content especially the concept of proof if they cannot possibly understand it since writing formal proofs requires at least a van Hiele level 4 (formal deduction). How do we get them to this level with time constraints such as getting through the regular curriculum and preparations for the Proficiency Exam.
I can’t think of a single type of systematic knowledge (math, sciences, etc.) that doesn’t build on itself in inclusive layers. Of course, your students’ understanding is never going to be uniform, and for a great teacher will be only normally distributed around your current content. The key is to be mindful of those layers of understanding, and make sure to contain elements of each in your instruction. That keeps the current material accessible to students that may be a little behind, and even if they aren’t completely up to speed, they can be working towards developing those fundamentals at the same time. This is one of the many reasons that I’m an advocate for tying current curriculum into past lessons, as well as foreshadowing what’s to come.
I agree that testing a student on something that you have worked on in class helps a lot. One thing that i have notice when i am helping my little cousin do his math homework is that some of the problems on the homework don’t really go with the notes that he had taken is class. I think if a teacher is going to assign a problem on a homework and not give some kind of an example of how to solve it that they are actually hurting the kids.
Although I completely agree that the best way to scaffold a student for tests is to give them example problems, I also believe that some questions should be given to enhance critical thinking. For example, combination problems ,(where they have to do a set of exercises to get the final product), is a great way to enhance critical thinking. As long as you ,as a teacher, are scaffolding the students and giving them the proper techniques and information to solve the problem, then putting different pieces of what the students learned together shouldn’t be that hard. This allows the students to understand WHY and HOW they are completing the problem. There are many times if you give a student exact exercises that will show up on a test, they will go home, memorize the steps and complete the problem.But if you give them a different TYPE of problem, with the same way to solve, they might not recognize it and understand. Therefore, the student really hasn’t learned. I am guilty of this. In my Math 310-Analysis class, he gives us example problems of what will show up on the quizzes and because I don’t really understand what’s going on in the class, I go home and memorize the steps and hope the exact type of problem shows up on the quiz. So therefore, I am not really learning or understanding anything.
I have encountered what I think is a huge issue with teaching math. Since working in the tutoring center, I have seen many students that are taking the same course but from different teachers. Each teacher has different objectives for their own courses. I agree that looking at the same problems in different manners can help our students, but for one teacher to say that one concept is more important than another, and other teachers doing the same, is harmful to students understanding mathematical concepts. Should there be (or at least I think there should be) a conceptual guidling that all teachers must follow on what they teach? Not on how, but on what. All students take the same classes and to not teach them the same concepts accross the board means they will move up in their classes with different, sometimes harmfull, skill sets.
**guideline** not guidling
This is part of why NCTM ( and standards) are around in the first place. Every teacher has a different teaching style and we cannot expect them to teach the exact same way. It would prevent teachers from teaching well if they cannot teach the way they are comfortable. I do, however, think some teachers get over excited about some material and not excited about other parts and that causes them to put a different emphasis on what and how they teach the material. Having a guideline is also one of the other reasons a lot school have text books.
If all the math teachers use the text book to teach, then the students are all “learning the same material”. The downfall though is that text books aren’t enough. So teacher’s use supplemental materials that differ. Some teachers use hands on activities and others use worksheets…
A lot of school (especially Title 1 schools) will also give you “maps to teach” or guidelines for teaching because you have to teach exactly what is going to be on the standardized test. What helps the students remember what has been taught though is HOW they learn that material. Washoe County also has “Power Standards” instead of the regular state and county standards that schools have to make curriculum for. Teachers in the same subjects are working together to create the guidelines for each week and have to instruct on the same material each day. There is a lot changing about how things were. And college is WAY different from how the HS and MS work when it comes to lectures, class work, and homework.
Creativity is always a good thing to have within the lessons being taught. With Halloween being the most recent celebration, what ways could we incorporate Halloween in the classroom? Lines of symmetry on a bat ot candy? The angles of a candy corn and maybe even solving the sides of each color on a candy corn? How about thanksgiving?
I think we use geometry a lot when carving pumpkins for jack-o-lanterns. traditionally we use triangles for facial features. when we use rotations and and symmetry to make the face balance. For Thanksgiving we use recipes that feature measure of volume.
This isn’t really about Geometry, but I thought it was so cool, I really wanted to share it with the class:
http://www.learner.org/teacherslab/math/patterns/mystery/
This is an applet which requires students to guess what operation is required to change 1 number to another – it will drive you crazy!
Interesting. I’ve been on this site for a good fifteen minutes now. It’s pretty addictive. It would be great to use as an activity in a classroom to get students to think a bit harder about how the operations work.
It is quite neat, I approve 🙂
What a cool website. It is definitely challenging and fun for students to do. The most challenging thing as a math teacher is getting students to have fun with math and keep interest. Incorportating games and fun activities that are relative are definitely beneficial to all.
When I was answering my problem I almost over looked the fact that to change a cubic inch to a cubic foot is most than just dividing by 12, I know that I can’t be the only person who does silly things like this now and again and I know that students do things like this all the time. What suggestions do you have for students that will help them remember to really think about something before starting a problem? Another scenario would be performing calculations before changing all the units to be the same. Suggestions? Tips? Any “that was me” stories?
Re: twoqayl
I remember always being disappointed when I made stupid mistakes like not converting measurements or reading a number wrong. It stinks when you get a problem wrong just because you made a small error. I think that teachers should remind their students to read problems carefully and not just jump right in and start solving. Also, teacher’s should not test students on anything that they have not been shown. For example, if the teacher never gives examples of problems with different measurements, students shouldn’t be expected to use conversions on tests or quizzes. Students often forget information from past learning and need to be reminded before they are tested on it.
Any suggestions for how to make sure students read carefully and don’t make silly mistakes?
To Scooter:
A funny quote you could write on the board everyday for your students is.
“There are three types of Mathematicians in the world: one’s that can count, and one’s that can’t”
Jane Doe:
I agree with you that students like to know that their teachers are REAL people, but it is a matter of how to balance both their personal and professional lives. Teacher’s need to know when it is appropriate the leave their home problems at the door. They are going to create a more professional and laid back learning environment for their students if they know when to separate work and home.
Definitely agree with both of you, I had a spanish teacher back in high school that talked about how much her husband sucked. Found out (from students that have her now) they got a divorce. Clearly she didn’t learn to keep home & work separate.
Then I had some teachers I practically no nothing about, but they are the most amazing teachers! Still keep in touch with them, but yet still know nothing about their personal life.
Should teachers grade on process or production?
Should teachers award full points on assignments if students do the problems right but make simple math errors or should teachers only grade according to getting the right answers? In life, students need to be able to check their work and correct mistakes but should we as teachers give them lower scores on assignments to teach them to proof read their work? Or should we be lenient and just make sure they know how to do their work and not give them lower scores based on math errors?
I actually think that teachers should do “partial credit”. Make it clear that you have to show your work, or if you dont have to show work make it a simple multiple choice answer and have the question worth less points.
But for the questions that are required to show you work when making an answer key, have the points designated almost like check points, to see if they know what they are really doing. If the teacher is persistent on explaining on specific ways and what they need to see on the test will help the kids out.
My only concern would be if a student figures out an answer in a completely different way but gets the correct answer, how many points should be awarded?
If they can ( and do ) show how they got the answer, then full credit should be given. Why should they have to solve the problem in only one way if that way is hard to them or doesn’t make sense? They shouldn’t. They should be allowed to use a method that makes sense to them (and that be shown and explained to others).
I would recommend having a rubric that clearly states how many point a student gets for doing the work completely ( solving it for the correct answer and showing HOW they solved it). If they don’t do all this, they know where and why they have lost points. It goes with what you said about making an answer key and the “check points” but something more general the students can keep with them when doing the homework and something the parents can see when their child brings home a homework that only has maybe half credit for getting the answers right but not showing any work.
I like math because it is (for the most part) objective. When you start getting into these details about partial credit, it can open the door for subjectivity. I think it’s best to have a check point system like eamesor suggested, especially for the more involved problems. This can help keep consistency from student to student. I also like the idea of having multiple choice questions mixed in to limit the subjectivity in grading.
I think that the best math tests are the ones with a multiple choice section as well as a free response section. The multiple choice section would feature questions that are more calculation driven and the free response would feature questions that had more steps and could be more easily graded on a partial credit curriculum. It is important that students learn to follow their questions through and not merely rely on partial credit to get by.
I think that teachers should mark studetns off but then they should allow them the opportunity to correct their mistakes. In life a lot of times when people make mistakes they are allowed to, and in many cases expected to, correct their mistakes so I think we should treat students the same way. Really though, you can justify any method you prefer so as a teacher it really comes down to what you feel comfortable doing.
Jane,
I think it’s a mistake to award full points for work that doesn’t lead to a correct answer, BUT I think it’s a bigger mistake to give a zero to a student who clearly understood the method and analysis required for a problem. I think a tougher question is, “How much credit do I give for a correct answer, with no work shown?”
I honestly think that it just depends on the teacher. We all have our preferences about what is partial credit or what is no work shown. Math I think is one of those subjects that is hard to grade because you know what you are looking for compared to someone else.
That’s the million dollar question, isn’t it? The amount of zero/partial/full credit will vary from teacher to teacher depending on his/her belief system on the matter. Personally, I am all for partial credit. If a student shows a solid understanding of the concepts and methods, but simply misses the correct answer due a computational error (i.e. missing a negative or something of the like), as we’ve all done before, they deserve a majority of points because of their UNDERSTANDING of the problem. However, the question of whether they deserve credit for a correct answer without work is another on entirely. Depending on the level of math, there are some problems that simply require little to no work to solve. These are essentially “all or nothing” problems. However, if the problem does require any sort of work to solve, I have no problem requiring my students to show work or risk losing points despite a right answer. By doing this, I will have no problem assigning odd problems as well. That way, they are privy to the answers in the back. However, not showing their work, or providing incorrect work with the correct answer will not net them points.
I think partial credit should be given for showing correct work but obtaining the wrong answer. If the student understands the concept and did all the work correctly but accidentally made a subtraction error and obtained the wrong answer, i do not believe they should get a zero on the entire problem. i think they should get some credit for doing the work correctly. However, if both the work and answer is wrong, then yes giving no points is obvious.
I agree that the subject of partial credit depends a lot on the teacher but I also think that it should have a lot to do with the students. If the student is going to be an accountant or something that requires exact answers or something like that then there should be a strong emphasis on getting the correct answer. If the student is going to be in other fields then often the important thing is just that they know HOW to do the problem not that they can arrive at the answer.
In regards to the comments above about keeping personal life and professional life separate: I agree that there should definitely be a balance between knowing your students and letting them know you and in letting them get too involved in your life or vice versa. I think this is especially important because there are a lot of cases where teachers get themselves into trouble by getting to involved with students.
I think with social media outlets like Facebook, it is easier for students to see your personal life on display and it sometimes gets teachers in trouble. I think teachers should give themselves two options. One: do not let any student become your Friend on Facebook. Or Two: If you allow your students to be friends with you on Facebook use it as an outlet for helping your students by answering questions or providing feedback. Whichever option you choose, i do not believe you should use Facebook to bitch about your life…when you become a teacher you are a professional and should not post stupid status updates.
Has anybody done any work with the Promethian Smartboards? I’m doing a practicum in a math classrrom now, and the teacher uses the Smartboard. Maybe i’m old fashioned, but i feel like the Smartboard isn’t very time efficient. I think that a whiteboard is a little more effective. Does anybody have any benefits to the Smartboard?
One of the things I like about the smartboard is you can save your notes (whether hand written or typed) and you can also use templates on it that the textbooks have given you. It also has an undo in case you erase something you didn’t mean to, whereas on the white board once it’s erased it’s gone for good. I’ve talked with teachers who save their daily notes and sometimes email them to students who are absent because a classmates notes are not always accurate.
I think that one of the major benefits to smart boards, especially for geography, is the ability to insert specific shapes. I know that when I personally draw on the board it can get a little sloppy cramped if I did not draw a diagram big enough or something like that. So I think being able to insert shapes and manipulate them the way you want helps your notes look more professional and easy to understand. I know when I was in school and even now if the diagram is drawn poorly I have a harder time making sense of it so that is why I think it is so beneficial to be able to do that.
I worked the smart board in my practicum class last spring and thought it was great. I hate the smell of the pens and the mess created by the whiteboard and thought it was a great alternative. I still saw use for the whiteboards in the same classrooms, however. On the smart board you can have prepared files that, like an overhead projector, appear on the screen instantly. You would have to write out the formula you were working on, or draw the x and y axis of a graph, or draw a circle for a pie chart and the document you use could have it already done. Then in this way you can write over the top of the pre-drawn graph of pie chart and easily erase without the mess. The board is smaller than the whiteboard which can spread across the front of a class but in this way the students have to concentrate in one area. Each page before you erase can be saved in case someone misses the notes, if you are so kind. The issue of back to the class is still an issue unless, like in my class the teacher had a tablet to write on which went directly to the screen. I think if you had one in your class and practiced with it you would begin to really enjoy its advantages. At first, being a new teacher it may make things seem more time consuming but it probably isn’t and once you get the hang of it could be very beneficial.
I have worked with smartboards in my middle school practicum. I actually used it for one of my lessons. I think that it provides engagement for the students but sometimes I felt that is was hard to fit all of my notes on a single page. You are able to flip between pages, but it is still hard to go back and forth sometimes. Does anyone feel that even when a teacher is using the board, it is hard to not keep your back to your students when you are writing? I almost prefer an overhead so that I am able to visualize the entire class at all times.
There’s a hand held board that some schools get with the smart boards. If the school doesn’t order them, the teachers can order them on their own. It’s like writing on a tablet that doesn’t have a screen (or the screen is the smart board). It is WAY cool because you can walk around the class with it or let the students use it while doing a class problem. I don’t remember what it’s called, but it’s definitely out there and very helpful 😀
I’ve never had the opportunity to use a smartboard. Being left-handed, it’s more difficult to write on a standard white board, as I have to be more careful so as to not erase what I just wrote. I really like using the Elmo/overhead, as I can avoid this problem, while also keeping a better eye on the class to prevent any misbehaving. You also have to take into account the kind of school you’ll be teaching at. Especially early on in your career, you’re more likely to get your jobs at the lower income schools, as the veteran teachers will have the first dibs for the more “desirable” job openings. Teaching at a school like Sparks or Hug makes it less likely that you’ll have much of any modern technology in your classroom, as opposed to teaching at a Spanish Springs or Damonte Ranch.
The reason i posted this question is because I am doing practicum work at Hug this semester, and both classrooms I work in have the smartboards. Also, you brought up the point of teaching at a lower income school early in your career. One of my practicum teachers is a veteran teacher and has teached for over 20 years. She has taught at McQueen, Reno, and now Hug. She said that she is so fortunate to teach at Hug, because the students are much more likely to allow you into their lives and create a connection. At the upper income schools, there is not this opportunity. Kid’s don’t allow you into their lives as much.
I have the same issue. I also feel that presenting problems on PowerPoint presentations doesn’t allow students to see all the steps to solving the problem. The next line just magically pops up completed when you click the button. I think it is much more beneficial to see the steps written out, number for number, letter for letter so that they themselves see this is a fluid process. With the slides, it appears a choppy process that is taken in chunks.
Giggles- I agree with you. When i was in high school i liked seeing all the steps written out. A power point is okay here and there but to perform every lessons through a power point is not beneficial for the students.
I think for math it is important for the students to see the work and i always thought the smart board was great because even if you were behind in the work you could still see what is written while the class goes on. When I used the smart board i hit extend page so there was no flipping of pages for each example. So you get all of one problem on one page of notes and no need to look back.
I’ve never really thought about the disadvantages of using a white board before. When I was in school that’s all we had so I just figured that that’s what was always used. I guess many classes are using smart boards now though. I definitely do see the disadvantages of using a white board. As a teacher, having your back to the student during most of the class period gives students the opportunity to misbehave. If a teacher is not looking at the students they are more prone to do things that they shouldn’t be doing, such as texting or talking to each other. I think that smart boards are a little better than white boards because there are ways to use the board so that the teacher does not have to write on it and turn away from the students. However, a better solution would be to have a handheld device that allows you to write on it and it appears on the smart board. In high school, I had a teacher who had something like this and she never had trouble with student misbehavior. She was able to write things on the board while facing the class. I think this was a great solution to the white board problem.
My cousins are both freshmen in high school and have a teacher that set up a website for their class. She posts all notes and assignments on the website along with tutorials for students who are interested or just need some extra help. My aunt told me about this website and I thought it would be a good thing to share on this blog since it directly applies to this class and teaching geometry. I have heard that it is a good idea to set up web pages for your students but I have never actually seen it in practice. Let me know what you think. http://www.sites.google.com/site/missnickellshomepage/honors-geometry
Very cool website. I know a lot of college professors do this for their math classes as well, and i know it is very helpful. Different students learn in different ways, and having these options available are great.
That website is really awesome. My sister’s math teacher had a website when she was in high school but it was nowhere near as organized as this one. His was baiscally just for posting homework and getting email, this one looks like it would be really helpful for students. I think if your a teacher who decides to create a website then you should go all out like this person did to make the most of it for the students.
This is a well organized thoughtful use of a webpage for a math class. I like how it is organized the notes from each class are given for the students review. I believe by looking at the notes that they are the pages directly from the Active board/Smart board class session. Looking at the way the words are written with not the best smoothness I assume this is the case. It looks like she starts with a pdf on the board with preprinted questions and topics for the notes and then writes things in as she goes. This is how I envision using the active board in class. It has some of the notes written already and then you fill in the blanks. This way you can be standing towards the class more and can concentrate on the topics and not just making sure you write everything down. If you look further into the more recent files the notes have not been added but the original pdf with the outline is there. This shows how it is time consuming for the teacher and keeping up with the latest notes could be delayed. I do worry in High School for the students without computers and/or internet access. I believe the legally the teacher may have to make all of this information available to the students on paper as well. Even if it is not the law one should not assume everyone has internet access all of the time. That way no one is left all of the students could still get the material whenever they needed it. Great website for helping students review the material and know understand the topics of the class.
Another really cool website for both teachers, parents, students, and well anyone is khanacademy.org That guy have the answers to everything. How to file taxes, geomerty, physics, calculus, anything someone could need help doing.
the math nerd that I am went to the website and spent approximately a half hour watching various videos about how to work out problems or why they were the way they were. Another cool math nerd video, http://www.metacafe.com/watch/2069228/las_mejores_matematicas_de_los_mayas
check it out!
That video was pretty interesting. I had no idea that you could do math like that. It is a little confusing and maybe not the best approach for teaching studetns but for math nerds it is really cool.
I just checked out the khanacademy.org website and it was really cool. That website really does have the answers to almost everything. I also really like the website wolframalpha.com. I have used it a lot in my calc 2 class!
Great point, Mr. Wayne. I didn’t think about having the ability to use shapes. However I think that you can probably do the same thing with a PowerPoint or other type of presentation if you have prepared well enough and planned in advance.
Mathgirl- is Khanacadamy the site where the guy has tutorials for all different subjects? A lady I work with was trying to explain it to me. She says he has thousands of videos that are very helpful to students. I will have to ask her if this is the same site.
The Khan Academy is an incredible site! I would highly recommend everyone to watch at least one video- if not to get help on a subject, to at least see how they react to getting online instruction. Personally, I’ve found the site to be helpful to me for my schoolwork- their descriptions are thorough and they offer aid in a variety of subjects, applying to a plethora of fields.
This is actually a great opportunity to relate our discussion to our class; as classroom embrace technology, it’s important to question what role the internet in education, and, more importantly, how significant disconnected instruction to students is to education in the twenty-first century. In many cases, I have heard a student say that they’ve learned more about a class through YouTube videos then they have through classroom lectures. Often, algorithms are learned over online demonstration and theoretical description is lacking… but usually, this description is not needed by the students anyway.
Websites like the Khan Academy, Standford’s Home Page, and TED all provide online instruction to students in a way which is informal, direct, and, most importantly, personal. Because this instruction is provided to students in a comfortable, convenient and concise, many a student will learn better in this environment than in a typical classroom setting. We should not deny the effectiveness of these types of media and what they can give to students who find themselves struggling in school.
Moreover, the accessibility of these tools illustrates one of the most important features it offers- the supplementary instruction that is offered online utilizes students talent with web-browsing and surfing, which is indeed a skill which defines our generation and provides with so much information. The amount of education that can harvested from the internet is immense, but usually this crop is in the depths of the Information Age. However, if utilized correctly, students could find that their new classroom could be found within their computer, and, indeed, the very face of education might be altered.
So one thing I have noticed while working on the take home final is that many math majors find the simplicity of this math frustrating. I actually had a math tutor freak out and get mad because they could not understand how to slove one of the problems without using higher levels of math. To watch this, while extremely immature, was quite funny. It is amazing how much we forget in our years of moving up in the levels.
Geometry assignments demand precision and clarity. For students facing challenges, Geometry Assignment Help offers tailored assistance. From basic concepts to intricate proofs, expert guidance ensures thorough understanding and polished submissions. Don’t let geometric complexities hinder your progress; seek support to navigate shapes, angles, and theorems with confidence. With the right help, mastering geometry becomes achievable and rewarding