Cari Knopp: Okay, my name is Cari Knopp. I teach at Clovis East, Clovis Unified School District.
Okay. What's really important to me is that the kids understand. Not just that they can plug and chug, not just that they can do the problems that I give them, but that they understand why they're doing them. That they understand that there's a reason behind, that there's an application. And um that's a hard thing as a high school math teacher because so many of the kids just want you to tell them how to do it. Tell me the uh tell me the way to do it. Give me the formula, and then we're done. And um I struggle with that a lot and the kids struggle with the fact that I'm saying, Well, why? Well, why? And make them understand definitions. I make them talk to me about stuff. So part of my philosophy as a math teacher is to say, This is how you do it, and then this is why you do it, and this is how its going to apply later on in your life, or this is where it comes from. And no, not everybody is going to be an engineer, but even going to the grocery store involved proportions, so that's kind of where I come from in math.
Okay. Well, I just finished teaching geometry this summer and um basically what I did with the kids is a lot of times they'd say to me, Well, why do we need to know this? Why do we need to know that an isosceles triangle has 2 edges that are the same, or why do we need to know that an equilateral triangle has 3 edges and 3 equal angles? And we started to go through it and I said, Well, think about all the time what structures are used in bridges all the time? And all of a sudden they were like, Oh, they're triangles. And we talked about, you know, there would be strongest because it the isosceles triangle would be strongest because the sides support and they're equal. And then we went from there and I said, Well, and if you're an engineer you need to know that stuff, or if you're going to build stuff, or if you're going to go into construction all of that stuff is really important. And we went that way. And then I told them examples of times when engineers haven't paid as much attention to those kind of structures, or even rounding errors and how this causes bridges to fall down, and stuff like that.
There actually was there's and I can't remember it so I don't know if you you know but there was an actual example where the engineers had built, I believe it was a mall in Minneapolis, I believe, where they had built a walkway. And because they hadn't taken into account the structural differences, it actually fell, and it was all a mathematical thing they hadn't taken into account. And so I kind of tell my kids that story a little bit. Kind of say, This is important and it you need to pay attention now because it was a rounding error that caused it. It was something they shouldn't have learned in junior high or high school.'
I think I'm really lucky. I think that um I I find it really easy and incorporate the math standards. Um, sometimes it's really easy to just start and say, Okay, well this is the unit I want to teach or this is the unit out of the book. These are the standards it encompasses, and then say, What can I do to supplement that? Um, I find books, I find units; I find activities. I'm really creative so we always do something fun. We're making games, we're doing art work. I have an entire shelf that's full of art stuff and art supplies so that we're always making chapter summaries. So I'm able to tie the um, the standards in, but I'm also able to do lots of hands on creative activities with the kids, and they kind of forget that they're doing pure math. That they're writing sentences, or they're drawing pictures, or they're making posters for the classroom. So I think that's how I'm able to do the to incorporate standards into my day to day lessons and make sure that I'm covering what the state wants me to cover, but I'm also doing something fun and and that the kids are enjoying.
Okay. Um, well literacy was always really important to me. Reading was always really important to me. I love books. I love to read. I have an entire bookshelf full of books that I still haven't read because I buy them and I hoard them. But I started doing things I noticed about the last 20 minutes of my class my kids were not paying attention. I was on a 2-hour block. It's hard to teach math, and I went and I bought some interesting fact books, and I started doing that the last 20 minutes. Hey, guess where French fries come from? And we would talk about that. And all of a sudden the kids started really to pay attention, and they were excited to hear about that stuff. And it was really easy to take that and bridge it into scientific um discoveries and say, This is where the light bulb came from, or This is where post-it notes came from. Things that just happened that really involved math and science, and the kids really got interested. And then it turned into Hey, read us a book. And I went and I started finding math literature, different math textbooks that would support what I was teaching, and um and I started reading them kind of interspersed. The kids really enjoyed story time and they started paying attention, and sometimes it's the best behaved my kids really are. They're quiet, they're attentive, and they enjoy it.
Most bookstores have a a lit a small math section. It's normally not very large, but um I was able to find some really good ones just even starting there. Um, um there are plenty of picture books, if you look at the titles. Um, there's one titled The Library and Who Measured, and I'm sorry I don't know who um authored any of my books, of course. I was thinking about bringing them, but um, there's a couple of books that are titled SIR Cumference and circumference is the SIR and then cumference is in circles. And they tell they deal with um geometry, radius, um and so forth, diameter. And they really deal with um geometric terms and how the kids can remember them, and they remember the stories. And I know those are available. There's um Anna's Multiplying Jar. There's tons of tons of titles if you just start typing in books. And a lot of the um a lot of the people who work in children's books sections really know and they can help you find them too.
Okay. Um, I I I have the scholastic magazines. We actually um my school subscribes to the scholastic math magazines and I think that they made them from grades 6 to 8, and so we have those. But in addition to those, we have um I have children's literature, I have adult books that I keep in my room just in case the kids want to read them. The Lord of the Rings was really popular this year and so um I was able to go ahead and purchase the set of those and then keep them on the shelf, and when the kids are done with tests and they don't have anything to do it's Here, read this. Try this out. So I keep that stuff in my room. Um, sometimes I go clip magazine articles that are particularly interesting. When the stock market changed from fractions to decimals we kind of put that up so the kids would be able to look at that and say, Oh look, this involves math and it was in the newspaper. Anything I can get my hands on is up. Um, lots of posters that involved reading. Um, I have a big poster that says When are we ever going to have to do this? And the kids really have to read and look at it and say, Oh look, I want to be a scientist, or I want to be a pharmacist, or I want to be a construction worker. This is the kind of math that I need, and all of that is reading and all of that supports their um learning.
Sometimes it's really easy. Um, some of the books really support what I'm teaching. For example, the geometry books, um the circumference books, really support those um those units in geometry that we talk about. Um, there are other books um that I can base an entire lesson off of. Um, there's a book by Demi called One Thousand Grains of Rice that really supports exponential growth so when I come to that section I can use that book. Or just as a fun little mini lesson or a mini unit. If the kids need a break and we've come to the end of a chapter and I don't want to start a new one until the next week I can find a book and say, Well, let's do a lesson off of this. One of my very favorite lessons that I do with the kids is I spend about 3 weeks and I read them a whole bunch of different books. I read them the circumference book, sometimes I read them the Demi book, and I read them there's a Twizzlers book and an M&M counting book, and I read them all of those books. Different age levels, different groups, and I say to them, Okay, now you're going to write your own book. And that's my we do that every year I do that with my kids. They write their own math book. And I'm always amazed at the books that I get. They do amazing things with them. Last year I had a student who actually wrote a book um about a line of symmetry and an artist, and it was an amazing book. She I have the kids write, they illustrate. I went down and I laminated them and bound them for the kids and they were just ecstatic about the project and so that's kind of a way that I um that I can incorporate literature in my classroom.
I think that um in order to even look at your math textbook you have to read it and then and you have to understand what the words mean. And it's not enough to just go to a dictionary and look them all up and says, Well, this one means this, and this one means this and this one means this. And if that wasn't an effective way of teaching then they wouldn't need me because there's there's a section in the front of every chapter that says This is how you do it. Here's 5 examples. Plug and chug. That's the kids don't learn anything that way. And so I think that that learning and literacy go together hand in hand. You can't learn anything without reading and if you don't understand what you're reading you're not going to learn anything.
One of the most important things that I really learned or one of the things that has been so important to me is the fact that I have kids who struggle. I have students who hate math. They come in every day, they hate being there, we do lessons out of the book, we do stuff out of the book, and the only time they're really alert, the only time they're excited to be there, the only time they're happy to be there, or paying attention, or being active is when a book comes out, when they're able to express themselves um writing, reading, drawing. Whenever they're able to do it, whenever they're able to say, Oh, I really like this book, those are the those are the times that I say, You know what, I really reached that child. I was really an effective teacher today. That child will always remember what this means because this book helped, because they wrote this. Because they took some ownership of their own education and it wasn't just me making them (INAUDIBLE) memorizations and some plug and chug problems.
Um, I think it's a great way to reach kids. It's a way to come at it a different a different way. Not all kids are going to be able to just sit there and say, Okay, show me what to do. I can do it now. And it's a way to say, Okay, let's start all over. Instead of Let me show you, or Let me do 5 examples and now you've got it, let's let's have you think about it this way. Some kids really they need that um I don't know what the word I'm looking for is they need that different motive of learning. They need that um that active participation, or they need that um I don't know what I'm looking for now.
Um, I make it sound like this is just fun and and that's what we're doing is I'm I'm out to make entertain the kids, but um it's really important to connect kids a different way. For those who are struggling, those who don't understand, it's important to say, You know what, it's okay that you don't understand when I'm writing examples on the board, but how about we try it this direction, how about we read something. How about I try and reach you a different way, because not all kids learn the same way. They have different modalities of learning and it's important to me that I reach all those kids.
And algebra is it's kind of almost easier in the sense that algebra is one of those things where you're solving for x'. You're setting up a proportion, you're setting up an equation, and that's something you do at the grocery store every day. You look at Is this can of soup bigger then this can of soup and um is this, you know, per ounce? Is this price better then this price? And I sit there at the grocery store and I do that, you know, and kids need to learn how to do that too because that's an important thing to know. They don't want to over pry pay for anything. They don't want to pay a price that's going to be more. So I think that's an important connection to algebra, is just even that solving for x and setting up proportions.
Um, we do a lot of stock market stuff and that's that's (interruption). Um, we do a unit on the stock market. I have a book about the stock market and um I was really fortunate, I taught at a school that had um internet access so the kids so they could they had laptops and so we were able to get online every day. But for the kids who were non laptop, I would bring the newspaper and we would look up their stock prices every day, and we were able to um to read articles about the stock market, to watch it as it fell, to watch it as it rose, and to talk about what that really meant, and from there we were able to make predictions, we were able to buy stock and sell stock. And we did that, of course, all hypothetically, but it really gave the kids uh a way of understanding what their parents were talking about when they went home, and they were able to go home and have conversations with their parents about the stock market. And there are lots of activities and Find-a-word, and we did all sorts of What does this word really mean? What does this word really mean? What does it mean when the stock market is bearish? What does it mean when its not? So the kids were really able to understand adult terms and sit down and talk to their parents about a really a really adult topic. And the the parents loved it.
I try to be really active with my parents. I try to um I tell them the first day of um school when they come to Open House, Back to School Night, I say, You know what? We're going to do some things that are non-traditional. We're going to we're going to read some books and your kids are going to come home and they're going to say, Guess what I made in math class today, and they're going to come home with all sorts of projects and things, and you're going to be able to come in and see your students work on the walls because this is a math room and that shows that they're learning. It shows me that they're learning. It shows you that they're learning. It's more then just pencil and paper. And my parents really enjoy that. I have many parents who come back at the end of the year and say, This is the first time my student has enjoyed math. This is the first time my student has enjoyed coming to your class. What do you do in here? And I point them to the walls or I point them to the projects and I say, This is what we do. Your student enjoys being here because they're learning. Because it is fun and it is important to be here, but your student is enjoying it because they finally understand.'
Okay, this reminds me of another algebra problem, actually. Um, one of the problems that we do is we set up an amusement park problem with the kids and we say uh, very much like going to any kind of a fair there's an entrance fee of let's say $5.00 plus $2.00 a ride, or you can pay an entrance fee of $15.00 and $.50 a ride. Where is it better? How many rides is it better to go with Plan A versus Plan B, which is a problem that a lot of people encounter at any kind of amusement park very simply put. And so the kids really work that problem and they figure out point of intersection of lines, they work with that, and um one of the things that I do with it is we set up an entire problem. We work through a problem like that. Depending on the level, sometimes I just say to the kids, Okay, well now that I've got you started you finish the problem. They make big poster boards describing they they restate the problem, they show me their work, they come back with a solution. Um, for older kids I would work through that problem with them in a classroom situation and say, Now you make up your own problem. Now you come up with a sol you know, whether you're going to Sea World, whether you're going to Great America. Wherever you're going you come up with that problem. I had a girl who actually came up with she was buying dogs and so, you know, you had to pay so much for the um to walk into the door, and then so much per puppy. And it was a very cute promise. She she totally made it her own and she took a lot of ownership of her problem, and um she enjoyed doing it. So we make posters, we do lots of um activities where you do chapter summaries. We um sometimes I have the kids state a problem, they have to solve it, they have to put that um, we make mobiles that hang from the ceiling. All sorts of stuff that remind them when they're looking around the classroom what they've done, what they've learned, and there's a sense of pride once they look at that stuff and say, Hey, I did that. I learned that.'
Um, one of the things that I do is I would start at the beginning of the year and I would start walking them through the problems, probably saying, Okay, well here's how we start a word problem. Most textbooks have word problems in them and they lend themselves to projects or to more in depth research if you're willing to spend the time. So we spend the time and I start out my kids at the beginning at the year working step by step. This is what I expect. These are the steps I expect to see. This is the way I expect it done, and as we move on through the year the kids take ownership of it themselves. They do it themselves. They know what I expect and they do it, and by the end of the year I can assign a problem and say, I like you to do something like #71, but I would like you to take it this direction, and the kids can do that all on their own. They can write their own problems. They can come back and say, Okay, well this is the kind of thing that Ms. Knopp is looking for, and they do it, and they do wonderful jobs at it.
Well, and it is the textbook is an important part it is an important part for me. It's a starting place. It really is. It's a place where I start every day and I say, Okay, this is what I'm going to teach today, but where can I go from here? Because it's not an ending point, it's not enough. It really isn't enough. It's not enough in terms for any teacher. It shouldn't be because if that was the case they wouldn't need. I tell my kids all the time, If all you needed was someone to stand up here and lecture to you, they would have videotaped a teacher 10 years, 15, 20 years ago, plugged them in the videotape and we they wouldn't need us. They wouldn't pay me. I wouldn't be cost efficient. I always tell my kids that. It's an important part. And so I start with the textbook and I say, What do I have that I can work with? The internet is a fabulous resource. You know, you can go on there and find anything that you wanted to that ties, and have the kids do it. It's really easy to have the kids do that.
Well, like any good math teacher I give lots of tests and quizzes, but along the way projects, chapter summaries, um they're a great, great way of assessing what the kids know, and really if they can solve a word problem, and they can make a project out of it and they explain how come 7 tickets is the best for this answer, and for this answer, um they can tell me if they can tell me why 7 tickets is the best for this can I start all over? Thank you.
Um, I do give lots of formal assessments, lots of tests, lots of quizzes, but in addition to that, any kind of chapter summary, any type of review material where the kids are doing word problems, where they're answering questions and then they're explaining their answers. They're not just getting the answer of 7', but they're saying Seven is the answer because of... and this reason, tells me that they're understand, that they really understand, that its really important to them and that they finally figured out why it works, not just that it works, and that's a really important way that I assess my kids. I ask them why all the time. That's my favorite question, Well, why does that work? Why do you do it that way? Why is that the right answer? And if they can't tell me why, then I know that either I didn't do a good job teaching or they didn't do a good job learning and we have to go back and we have to start all over. Um, I do lots of different types of assessments in terms of I have them do peer assessments with each other, they quiz each other, they review with each other. Board games are a great way for them to do review and assess. Um, I do a project where my kids write their own board games instead of doing formal review at the end of the semester. Um, we make our own board game and then they play them. So they do that project instead of 60 worksheets, or 15 worksheets, or how ever many you would do at the end of a semester. They make their own board game. They look they write their own questions, they write their own rules, they make their own boards. I give them paper, and crayons, and markers, and rulers, and all sorts of stuff I bring in. And at the end of it the kids really have learned so much. They've had to write their own questions, they've had to answer their own questions, and then they turn around and they play them. And they're proud of their games, they like playing their games, and it's a great way to review.
I always try to give a warm up. I try to give word problems because those tend to be the things that students in mathematics the most. In giving them 2 word problems to work on when they first walk in makes them very comfortable with them. There are questions they're going to see on any kind of national test, there are questions they're going to see on my tests, there are questions they're going to see throughout life because life is full of word problems, not full of numbers, and plug and chug. And so I give them a couple of warm-up questions. As I'm doing that they know they have their homework on their desks. I come around and I check their homework. And they know that that's the way we start every period. I set that from the first day of school we do that, and the kids get use to that. And if I forget, or I get busy, or I'm distracted Aren't you going to check our homework? Where's our word problems? And sometimes I found even to ask a student to turn on the overhead projector and have those warm up problems right there, the kids get right to work, they get right thinking about math, and they get ready. They're practicing reading, they're practicing writing. Of course, they need to answer their word problems in full and complete sentences. All that stuff is going to support literacy in my classroom, but it also supports math every day.
I'm very fortunate in that I get to use a program that um ties big themes into each unit. Um, one of the themes for a unit was um going camping. And so the unit was all about tree growth and we were able to chart tree growth in algebraic sentences using that kind of tree growth. We had to buy food to go camping, we had to plan our camping trip, we had to hike up hills, so we were able to use elevation, positive and negative numbers. But you could do that even if you're um book didn't allow that. You just need to be creative in what you think. Positive and negative numbers, elevation, talk about hiking, talk about camping. The kids really get a kick out of that. It's easy to make a big bulletin board and say, We're here. Here's zero. Here's sea level. And say, Okay, now if we go up we hike 10 feet. Are we going positive or negative? We're going scuba diving 10 feet, positive or negative? And really tying in real life applications to really simple terms will help the kids really think about positive and negative numbers, something that seems very difficult for even junior high and high school kids to really understand.
It would definitely have to be no plug and chug. That's not the right way to teach. It's not an effective way to teach. If it was kids would learn. The biggest complaint I have day after day is my kids don't know their multiplication facts. They get to high school and they don't know their multiplication facts and every teacher says, Well, give them a worksheet on it. But that doesn't work. That's not the way to do it. They need to understand why. They need to understand that it's grouping. They need to see the connection. They need to see the picture. And no plug and chug problems.
I've always been a secondary teacher. I've taught 7th through 12th grade students and I use literature that ranges from books that I would read to my 18 month old niece all the way up to books that I would read in college. Um, the kids need a variety and sometimes the picture books that seem the most juvenile are the ones that reach the most kids. Um, their the their the ones that reach the kids who are at a calculus level, all the way down to the kids that are at a basic math level. Different books, different meanings, different times, and it gives the kids an idea you know what especially when I do the project that I do when they write their own books. They don't have to write a college textbook. They can write a counting book, a number book, as long as they can tell me why it's important. They can really apply what they know.
Math in itself is a second language. It's a it's a language that a lot of people struggle with whether they are native English speakers or not. Um, that's kind of the struggle as a math teacher is that students struggle with it and it is its own language, um and so students struggle in terms of the even the language we use we have words, for example, the middle numbers, the median in math. We can't just say middle, we have to come up with its own name, and so that makes it a second language for native English speakers, as well as second language speakers. And so what I do with my second language speakers is the books help, the pictures help, examples, um and making sure that I'm not just writing through an example. I'm drawing pictures. I'm making sure that those students are really aware of what's happening, and I think the picture books only help. I think that that would be a great time to say, Even though this is only a counting book, here's what's going to help you, or This is a book about adding and subtracting, but the picture and the story are going to help second language learners really learn really think about what the application is.
Math is a universal language. It that's kind of it's the it's the language that everybody speaks in terms of writing. We all use the same numbers. No matter what you call them in French, in English, in Spanish, we all write the same numbers. Um, even if you do a problem a different way there are different ways to do multiplication, we all get the same answer in that sense, and so it really unifies a classroom sometimes.
Part of what I do is is I spend a lot of time showing the kids I spend a lot of time saying, Okay, the word median, this is really what it means. Now let's practice what it means. And I'll say to them over and over again when I'm giving definitions, especially when I'm giving definitions when there's another definition embedded in it. And I say to the kids, Okay, so it's the median of the line segment. What's a line segment? And we start we go over definitions, and not just repetition, but so that the kids can say, This is a line segment. I say, Draw me a line segment. Show me in the classroom. Point one out. Because there are all sorts of things. Parallel lines are all over the classroom. Geometry lends itself easy to that, um but even in terms of mean, medium, mode range. There are tons of activities you can do with the kids that really get them thinking about it. I do an M&M activity in terms of teaching um measures of central tendency where we count out What is the mode? It's the mode. So let's find the mode. And by the actual the kids actually counting out each M&M saying, The color we have the most of is yellow. That's the mode. They really understand. They remember. Anything hands on helps those kids with their definitions.
I love math. I loved solving (INAUDIBLE). I loved the theorems. It all made sense in my brain and um and I loved it, but I noticed that a lot of my peers didn't. And um when I went to college I took math. I have my degree in math. I'm very scientific. I actually have a Bachelor of Science degree, but I didn't like the science part of it. I didn't like the proof writing. And when I came back and I said wanted to be a teacher um in California you have to have a Bachelor of Science degree to teach upper levels and when I said I wanted to go to the high school they said, Great, you can teach upper math, and I said, No, I want to teach the struggling students. Um, my favorite thing in life is to see the light bulb go over someone's head and say, I finally get it. I finally understand, um because I believe that all students can do math. I believe its not it's not a struggle. You've just got to reach them. I think everyone is capable. I think everyone can do it. But it's important to me that you need to try. And if your first attempt doesn't make it, you need to come at it a different way, which is probably how I got started in in books, and literature, and activities, projects, things like that.
Well, I think that it's important that um nobody failed. You know, whether it was the teacher, the student who was not doing what they needed to do, somewhere in the middle there was a communication problem and and you just need to come at it a different way. I think teachers tend get very defensive. I did it. You know, I I did the best I could do. It was the student's fault. But it's not. And it's not the student's fault. It's not the teacher's fault. It's just there's a different learning style, there's a block. Sometimes it's just a matter of trust. The student's been very unsuccessful and all of a sudden we want them to be successful. And if you can reach that student and make the trust, kind of bridge that gap, they'll be willing to try and really be successful.
I think that a lot of students have been in a classroom where they've struggled at mathematics. They've been very unsuccessful, they've felt like failures, and all of a sudden, um they just don't want to do it anymore. It's not good for their self-esteem. It's not good for them. And if you can get a student and say, I believe in you, you can do this, and not just say it but really show them that you believe in them, really actively say, No, no, no, no, you can do it and I'm going to help you do it. And if you can get a student to say, Okay, then I'm going to try, you'll be very successful as a mathematics teacher.