Watch this hour-long recorded webinar about how you can make math more accessible to every middle and high school student in the classroom, particularly English Learners, and raise student achievement.

Webinar Transcript

Emily: Hello, everyone. Thank you for joining us for today’s webinar: Break Down Language Barriers in Math for English Learners. Before we begin, I’d like to review just a few quick housekeeping items.

Emily: We will be accepting questions throughout the webinar, but we’ll be holding them for a Q&A period at the end of the presentation. Please go ahead and send your questions in as they come up for you, using the questions feature in your control panel. You’ll just type your question into the top box and then click send. I’ll receive the question, and I’ll put it into the queue to be answered by our speaker at the end of the presentation.

Emily: If you experience any technical difficulties during the webinar, please use that same questions feature to get my attention, and I’ll do my best to resolve the problem for you. We will be sharing a recording of the webinar with you as well as a copy of the slide deck once we wrap up here, so keep an eye on your emails either this afternoon or first thing tomorrow for details about how to access those materials. You’re going to see shortly that we have a lot of information, and a lot of resources to cover, so you’ll definitely want to go and download that slide deck tomorrow.

Emily: Now, let’s get started. I’m pleased to welcome today’s speaker, Dean Ballard. Dean is the director of mathematics with the Consortium for Reaching Excellence in Education, also known as CORE. He holds a master’s degree in Math Education from Sonoma State University and secondary teaching credentials for both mathematics and English.

Emily: Over the last 12 years, Dean has specialized in professional development for those elementary and secondary math teachers. This work has included the formation of state math exams, coordinating the creation of high school math standards, writing math courses, and directing math programs for the pre-college department at Sonoma State.

Emily: Over the last six years, Dean has focused on writing, editing, and facilitating math professional development for both online and face to face work with teachers. Dean has 20 years of experience in the classroom teaching all levels of math, from 5th grade through AP calculus. He’s a member of the National Council of Teachers of Mathematics, National Council Supervisors of Mathematics, and The California Math Council. We’re really excited to have Dean with us today to share all of this information. So, I’m going to go ahead and turn the program over to him now.

Dean Ballard: All right. Thank you Emily. Welcome everyone. I’m happy to be here with you, and just as a note about CORE, that’s who’s sponsoring this, we’re a professional learning services company that provides professional services for math and ELA. We do both workshops and in class coaching work with teachers and instructional leaders. Just a reminder also, that if you want a PDF of any of the activities that I share during the webinar, feel free to email me and I’ll send you a PDF of those activities. Just tell me which ones you want, and that’ll take care of it. My email address is is at the footer at the bottom of each slide, so you can get it anytime you want.

Dean Ballard: All right and with that we’ll kind of get going here. Some of the things that we’ve seen as consultants, myself and the other consultants with CORE are that schools are both struggling with not only struggling students but student who are having trouble with the language, and these could be EL students or just a lot of regular students. But most teachers are looking for best techniques and ideas for advancing the mathematical knowledge of all their students including these EL students and the struggling learners as well. Some of the ground, we’re going to cover a lot of ground in this hour and it’s going to include the specific challenges, looking at the specific challenges that EL student face in math. Proven research-based strategies for addressing those challenges, and models for applying specific techniques effectively in the classroom.

Dean Ballard: Now this map which comes from NCES which is the National Center for Educational Statistics an represents their 2014 data which is their most recent data, shows the relative percentages of EL students in each state, and you can see there are many states with a very high percentage of EL students such as California which actually leads that nation with over 22% of our students identified as English learners. I realize that if you’re attending this webinar you’re likely already somebody who recognizes the need for improving our instruction in math, especially with regards to meeting the needs of English learners. However, I want to briefly give you a little bit of data on the next couple of slides to help one, reinforce those convictions, and also to give you some statistics that your colleagues may be interested in, and to emphasize that we’re not alone. That this is a nationwide challenge we’re all trying to address. We’re in it together, trying to find the best ways to meet the needs of all students and to improve learning.

Dean Ballard: In general I can tell you the good news is that most of the strategies and techniques I’ll be sharing with you in the next hour not only benefit EL students, but benefit most if not all students. There’s a lot of strategies that we’ll talk about and share, and a couple of them will be just for EL students, but most of them will benefit everyone. We need to keep in mind that mathematics really is a second language for everyone. Some students are very proficient with this language, but some are not, and the language continues to develop on a yearly basis. There are many new math terms to learn and related concepts to master every single year, and the pace just never really seems to let up.

Dean Ballard: So look here is a chart of data that I got from the California Department of Education on our Smarter Balanced results. Smarter Balance is the test California uses for a state wide testing. As I mentioned, California has over 22% of our K-12 students that are identified as English learners, and as you can see from this data set, looking at the performance of EL students we see that they’re doing not to well. It’s clear that they’re struggling a lot, at least on this assessment. Of course the assessment is not the end all and be all, but it is absolutely true that the language and reading are playing a big role in the math sections on most state assessments these days. Test writers I know are trying to both have problems and some sort of context and include more reliable assessing of conceptual knowledge and problem solving ability along with assessing for skill and procedural fluency. I applaud the assessments that better reinforce the balance instruction we’re trying to emphasize in the classroom, but I also recognize the inherent challenges this means for everyone. It means that more language dependent math tests are coming, and that we’re here.

Dean Ballard: In reality, I think that this really simply mirrors the fact that our instruction really is very language dependent. On this next table, we further illustrate the gap between English learners and other subsets of our students. The graph compares English learner students with subsets of English only student as well as comparing them to economically disadvantages students and to non-economically disadvantaged students. When we look at it, you can see that the lowest set of bars, sort of the yellow ones, those are the English learner students and by a wide margin they’re the lowest performing subgroup in California and I would guess that there’s a similar pattern that would emerge if we looked at the data from most states.

Dean Ballard: All right so let’s look at some of the challenges that we’re talking about for English learners. Limited background knowledge is a factor. Some EL’s may lack the basic mathematic skills and ability to grasp new math concepts as they’re being taught. However, we also have to be careful not to assume that this is an issue with all EL students, that it’s a language, or a mathematics issue, that’s what we’re truly trying to figure out. Often it takes more investigation and time to assess the math preparedness of EL students as opposed to it being only language only challenges. Limited language knowledge, likely also includes limited math vocabulary knowledge, but students may have a rich or a deep math background from their native countries. And of course, the limited knowledge about English language makes it more difficult to understand what’s being read and what’s being heard because it no only is the mathematical language students are struggling with, but it’s the language that even surrounds it.

Dean Ballard: Cultural differences make a difference for kids. Mathematics itself is often considered a universal language where numbers connect people regardless of culture, religion, age or gender. However, learning experiences vary by country as well as individually. Some EL students may have little or not experience working in cooperative groups or sharing or discussing solutions to problems. Some symbols have different meanings such as commas and decimal points, and mathematical concepts can differ in the way their approached, and often frequently, especially when expressing currency values or measurement or temperature, those thing are different in other countries. I mean, just think about standard measurement versus metric measurement. This can impeded EL’s understanding of the material being taught or slow it down.

Dean Ballard: Early in the school year teachers can survey their students and learn a little bit about their backgrounds in order to more effectively address those needs and better yet, to know which students can be called on to share some cultural variations because this kind of a discussion can be a rich discussion that deepens all students understanding of the mathematics.

Dean Ballard: Linguistics also plays a role. Every day language is very different from academic language, and EL’s experience acquisition difficulties when trying to understand and apply these differences. Some of these challenges are understanding mathematics vocabulary, that it’s difficult to decode and specific to mathematics. Associating mathematic symbols with concepts in the language used to express those concepts. Grasping the complex and difficult structure passive voice that’s really used all the time in word problems. And complex phrases, strings of words to use to create complex phrases with specific meanings such as square root or measure of central tendency.

Dean Ballard: Here’s some additional challenges to talk about. Polysemous words, these are words that have the same spellings and pronunciations, but the meanings are different based on the context. For example, table, put dinner on the table, put data in the table. Two different meanings of table. Think about the word operation. It’s a medical procedure, it’s also a mathematical procedure. In this case for polysemous words, the meanings are different from each other in context, but there’s a meaning, an underlying relationship between those meanings. In other words, homonyms for instance, where there is no underlying connection between meanings of the words that sound the same. There are a lot of words that have the same sound, and may even have the same spelling, but who’s different meanings have to be gathered from the context.

Dean Ballard: For example, sense, cents, and since, or left versus right versus left over. I mean think about sentence like: I sense that you have 50 cents in your pocket since I can hear the money jingling. Or, Thomas has five cups of flour to his left, after making some cookies only two and one third cups of flour are left over, how many cups did he use? So again, the context in exact same sounding words or the exact same word in spelling and sound are used for different meanings even in math classes. It’s not even just that it’s another class. So, it adds a layer of confusion for all kids, especially EL students.

Dean Ballard: What’s important for us, are not really these labels, polysemous words, homonyms, for these different types of words, but it’s the recognition of the challenges these words present for most students and especially for EL students. The instruction of specific vocabulary in math is actually crucial because vocabulary knowledge correlates with mathematics reading comprehension. Yet I know for many of us, especially the math teachers, and this is me I’m talking about here, the idea of vocabulary instruction feels more like added work on top of a curricula that really already seems impossible to complete in a lot of classrooms. Especially where we have struggling learners. But it is in fact one of the tools, one of the ways in which we can help that instruction.

Dean Ballard: Other things that challenge students, syntax and semantics. I’ll talk, I’ll give you some more examples of syntactic challenges on the next slide, but I just want to mention that this is how we look at sentences to see if they’re correct. That the syntax has to do with the correct structure of a sentence, for instance when we know that a sentence has to have verb and a noun in it, and semantics refers to the meaning of the words themselves, which can be affected by the way the sentence is set up, by the structure or the context. And in language, for a sentence to be considered valid, it has to be correct both syntactically and semantically. It has to make sense as well. For example, here’s a sentence: A square has volume of 20 cm. Well, syntactically that sentence is perfectly fine, however, it’s nonsensical because squares don’t have volume, and if they did have volume it would be cm cubed and not just centimeters. So it’s possible for sentence to have the right structure but not have the right meaning.

Dean Ballard: Another thing, translating words into symbols. This is really a huge area. We know word problems and the language intensive tasks are very challenging for most students, but even short phrases are challenging, especially when an attempt for a word by word translation leads to incorrect symbolic representations. For example, consider: x is 5 less than the number y. Well the first, if we were to do a word by word translation and the first temptation for a student might be to just try to write a symbolic representation and in the order it’s given they might write: x=5-y, however that would be incorrect. The correct representation would be: x=y-5. So, these challenges for students, you know the words, connecting the words, word problems themselves, requiring students to read, to comprehend the text, to identify the question in there, then to create a numerical equation, then to solve the equation and check it back and see if it makes sense, there’s a lot of reading and understanding about the written content of word problems that’s a challenge for most students, especially EL students.

Dean Ballard: If all of this sounds like challenges, like I said for most students, not just EL students, then you’d be exactly right. These challenges for the most part are greater for EL students, but they do represent part of the same list of language struggles with mathematics that all students must work through. This means that time spent adjusting instruction to increase the access and success with math for EL students, will do the same for most students.

Dean Ballard: All right so I mentioned I would give you some more challenging stuff, features real briefly here, let me show you some of these. We have long dense noun phrases such as the volume of a rectangular prism with side 8, 10, and 12 cm. Classifying adjectives that precede a noun, such as prime number, rectangular prism. Qualifiers that come after a noun. A number which can be divided by one and itself, how familiar does that sound? Just sounds like straight out of a word problem. Conjunctions are very important, if, when, therefore, given, assume, they play a very big role.

Dean Ballard: This chart shows some examples of words in some of the categories I just described, along with some other types of challenging words. Take a moment to look it over, this chart, there’s double meanings, homophones, multiple terms for the same idea, small words or phrases, some unique terms in mathematics that are complicated, hard to pronounce. Similar sounding words are a challenge. Tens, and tenths, sixty and sixteen, then and than. So, this is just a sampling. This isn’t an exhaustive list by any means, but it’s a good sampling of the kind of challenging words and meanings that the students are facing daily in a math class.

Dean Ballard: All right, now that we looked at some of those challenges, let’s talk about some recommendations around this. Focus on students’ mathematical reasoning, not the accuracy they’re using in the language itself. Remember it’s importance in meeting not the syntax. Focus on the mathematical discourse practices, not the language as words, or grammar again, but are kids discussing the math ideas in the problems? Recognize the complexity of language in math classrooms. Essentially, what I’ve been talking about on these last few slides, and treat everyday language as a resource, not as an obstacle. See differences in culture and meanings as opportunities to elaborate, to compare, to have discussions about important math vocabulary and ideas. Essentially uncover the mathematics in what students say and do, which is what all of these recommendations are really about.

Dean Ballard: Additionally, some other recommendations that we’ve found, some techniques that we’ve found very important and very helpful are to make vocabulary and concepts explicit and visual. That repetition is a good thing. Use scaffolds such as sentence frames, partnering, allowing students to talk in their native language to each other to translate ideas, to use engaging activities, and to create, manage and process opportunities for students to talk about the math with each other. On this slide, a lot of these strategies, I would say that the third bullet really gives you the most specific kinds of techniques that we’re going to share, and I’m going to share some more specific techniques along the way because I realize that ultimately it’s the specific teaching techniques that you can take into the classroom that make the biggest difference for you.

Dean Ballard: All right, so we’re going to move on, and I’m going to get into a math problem with you. This problem and the next two that I’m going to share with you are featured in the books CORE put out called Spend Some Time with 1 to 9. The problems are actually available to you in a free sampler that you can get as well, that you can download from our website. These are activities that we created to provide some engaging problems that build number sense through problem solving and mathematical reasoning. This is in itself modeling a couple of techniques for addressing language challenges. That of using gauging problems, and also the idea of minimizing the language demands in a problem, while still maximizing mathematical reasoning.

Dean Ballard: While taking you through a couple of these math tasks I’ll model and discuss techniques for helping students overcome language barriers. With this particular problem I’m going to first go through the directions like I would with a class of students. And then, we’ll process it together. I’ll have you actually input into the chat, or the question window some of your thoughts, and then I’ll give you a few minutes to try that math challenge yourself actually, and then we’ll briefly process some of your answers as well.

Dean Ballard: So, going through the math problem: Create equations for the digits 1-9. Create as many equations as you can using these digits and when I say digits versus numbers, I want you to think about the digits are like letters, and numbers are like words, just like we make words using the letters, we make numbers using the digits. The digits 1, 2, 3, 4, 5, 6, 7, 8 and 9, and 0. So, we’re going to use digits, those digits 1-9 to make some equations. So they’ll be creating numbers and equations, and we’re going to see how many of the digits we can use. For example, 8 divided by 4 = 5-3. Here’s an equation, so it’s equal, and let’s see if it’s actually a true equation. 8 divided by 4 is 2, 5-3 is 2, so we have 2=2 so we have a good equation, it works. I’ve used 4 of the digits. 3, 4, 5, and 8 creating numbers and we have a good equation.

Dean Ballard: Here’s another example. 6 x 7 = 42. Here we have again an equation that’s true. 6 x 7 does equal 42. I’ve used 4, I’ve used some of the digits, I’ve used four of them, 1, 2, 3, 4 to create 6, 7, and 42. Another example. 2 to the 3rd power equals 8. Again, a true equation, it is an equation that is true, and I’ve used some of the digits. I’ve used 2, 3, and 8 to create that equation. Final example here of ones that work. I think I’ve gotten in this equation to use all of the digits. 1, 2, 3, 4, 5, 6, 7, 8, and 9 so I used all the digits in this case to create numbers in an equation that’s true. Let’s see if it’s true. 7 x 5 is 35 + 8 is 43 – quantity 6 + 1 is seven, so we have 43 – 7 makes 36. Over here we have 29 + 3 makes 32. 32 + 4 makes 36. So we have 36 = 36, so it’s an equation, it works, it’s great.

Dean Ballard: Now you noticed I did not use any digit more than once within the same equation. Now I can re-use the digits on new equations, but within each equation I did not use the digit more than once. So you never see here, like the 3 only used one time here, 6 never used more than once here, none of these digits are repeats. None of these cases are the digits repeated. So you can’t do something like this, this does not work. 8 divided by 4 + 3 = 7 – 3 + 1. The equation is true, 8 divided by 4 is 2 + 3 makes 5. 7 – 3 + 1 makes 5, that’s 4 + 1 makes 5, so we have an equation, and it is true. 5 = 5, but I used 3 two different times, so I’m not allowed to do that. I’m also, you might have noticed, I did not use 0. 0 is not from 1-9.

Dean Ballard: So, even though it’s a digit it’s not one of the digits we agreed on using here, so you can’t do something like this. 16 divided by 2 is 8, and 40 divided by, let’s see 8 – 3 is 5 so 40 divided by 5 is 8, so we have 8 =8, that’s good, that’s the equation it works, all that’s good except I used 0, not allowed to do that. I’m crossing that out. So you can use any math operation you want. You can use add, subtract, multiply, divide, or exponents. Whatever you can to make an equation use some or all of the digits, don’t use 0, don’t use a digit more than once within an equation, and you see how many you can create. Be as creative as you can.

Dean Ballard: All right, so, everybody, all my webinar friends here that’s the end of the instructions with the class. What I’d like to think about right now is think about was there one thing I did that stood out to you with those directions that you thought was good in terms of making the directions understandable for students? I’d like you type in that one thing in the… And Emily help me is that the chat, or the question window they’re using? Emily?

Emily: Yes, sorry. It’s the question window.

Dean Ballard: The question window.

Emily: Yes.

Dean Ballard: Okay, so you have a window called the question window. Enter in there whatever you saw that was a good method I used to make the instructions understandable. Type that into the question window, and if you have also I have an idea for something that would be another way or a better way or another suggestion for making it more understandable to get something I didn’t do, share that as well. We’re all interested in always getting better ideas. I’ll give you a minute to share your thoughts in the question window, and then we’ll move along. All right so you’ve got one minute.

Emily: So we’ve gotten quite a few good responses. Many people said and congratulated you on providing good examples, and non-examples as well. The metaphor you provided between digits and letters was good, as well as the non-examples were very important several people felt. You broke the directions down into small parts and used very plain and expressive language. Again, defining what a digit was, was really essential and important to the task. You gave explicit directions, but I wish you would have done that first.

Dean Ballard: Mm-hmm (affirmative).

Emily: You pointed to each number to show that it was used only once.

Dean Ballard: Mm-hmm (affirmative).

Emily: Some of the examples coming in are 8 + 1 = 3 x 3. And 7 – 4 = 3 x 1.

Dean Ballard: Oh okay. Hang on, thank you Emily.

Emily: Okay.

Dean Ballard: So we’ll get to that in a minute. Great, I appreciate those comments and even the suggestion about being a little more explicit earlier on. I also want to mention that the pacing is difficult on here, I don’t know if I was going too fast or too slow, I’m sure for you it seemed pretty slow in the webinar, but again, I know you’re all thinking about in a class of students what would be the right pace? And, the pacing always depends on the class. I know that as you’re looking at a class of students and you’re reading their faces and interacting where you’re asking questions along the way it’s not just always you talking, that gives you an idea of the pace or when you might have to add another example or clarify an example even more or non-example.

Dean Ballard: I want to also point out the use of visually demonstrating the non-examples that are not correct by crossing them out. That I’m doing that as a way to provide a visual confirmation that when I say this is not right, that there’s an understanding and it’s a very clear understanding. If I’m in a classroom doing this, you would see me gesturing to the problems, you see me pointing to them. I mean I tried to use my mouse arrow to point on the screen which I think you saw some of, but I would do even more of that in the classroom. Pointing here and there and emphasizing things just as I was trying to do a little bit on the screen. Essentially I’m doing everything I can to be visually, orally, and in writing very clear and I’m connecting all three methods of communication together to give students as much opportunity as I can for them to succeed with the task.

Dean Ballard: All right, let’s see here. I wanted to mention too, that I’m going back to a slide we just had before this. The point of what I’m doing with this task is trying to show how to make the vocabulary and concepts explicit and visual. I think the visual is an important component. I have a good use of repetition, where saying things over again, showing multiple examples, using the word digits and equation again and again, and trying to have an engaging activity that the students can get into, but yet still has them working on a task that they can do.

Dean Ballard: So as I mentioned I was going to let you try to work on the task a bit yourself. So I would like you to take a minute or two and try to do the task yourself. Come up with some equations. I know Emily was already reading that some of you were, and enter again into the question box your own equations. Try to use as many digits as you can, or be as creative as you want. Use many operations, whatever you want to do, and while you’re doing that I’ll actually be…. You’ll see my screen change a little bit so that there’s not a lot of dead time here, I’ll actually be creating another equation at the same time that you’re working on yours.

Dean Ballard: So go ahead and take a minute or two and try to come up with some equations and put them in the question window. Thanks.

Emily: We’ve got quite a few that have come in.

Dean Ballard: Oh go ahead.

Emily: All right.

Dean Ballard: I haven’t got mine to work yet. I’m still struggling with this particular one I decided to start off on the wrong foot I think. So, let me go back to our regular screen and share some of those Emily.

Emily: So we’ve got 9 divided by 3 = 2 + 1. 8 + 1 = 3 x 3. 7 – 4 = 3 x 1. Then we get into some bigger numbers here. 98 – 76 = 54 – 32 x 1.

Dean Ballard: Wow.

Emily: Yeah.

Dean Ballard: That sounded like it used all the digits.

Emily: Oh yeah, let me count. 1, 2, 3, 4, yeah. In reverse order, that was hugely impressive.

Dean Ballard: That’s yeah, that’s cool.

Emily: Very cool. 9 divided by 3 + 4 = 7. Yeah so we had a good selection there.

Dean Ballard: All right, well great. Thank you everyone for participating and putting in some of those equations. I hope that as you can see with a classroom of students of course we would give them much more time to work on it. In fact, I’ve seen students take at least half a period or more and then we kind of change it up. I would include some sharing throughout the class to help students get some more ideas by seeing other students work. They even compete in groups to see who can get the longest equation with the most digits or even the most operations. There are a lot of different types of extensions you can create with this type of a problem, such as you know you could just include 0. Say “Here’s another digit you can use.” You can require a certain number of operations to be used within an equation. Maybe the ones you had been working on. Even square root. You can use negative numbers if you’ve been working on those.

Dean Ballard: Include a variable even in the equation and say that “The value of the variable has to work out so that it’s one of the digits, one of the digits that’s not already in the equation.” So you could do a lot of different things, you can play with it and have kids really be challenged to build their number sense, but still while they’re having to think and problem solve at the same time. The initial directions when you think about it were really quite short in this problem. They were just create equations using the digits 1-9, and you cannot use a digit more than once, and you can’t use 0. So, they’re very short directions, but yet there’s a lot of elaboration that went into trying to make it very clear as you saw. Everything else that happened was just clarifying with examples and non-examples what those directions meant, so that every student had a chance to get started. And again, I was trying to connect visually, orally and written communication all together to improve the likelihood that all students get it, and be able to get started and working on it.

Dean Ballard: All right, on the next problem I’m going to look at another idea for… It’s called providing a mirror problem for scaffolding the instruction with kids. So here’s another example, this one’s called Spend Some Radical Time with 1 to 9. So, we have the problem: Place any of the digits from the set above. And we see that set is 2, 4, 6, and 8. Into the blank boxes in each inequality shown to the right to make the statement true. Now let me just, as in a side letter right now, I’m not going to have you actually solve this problem, I’m just going to discuss it and then show the point I want to make with it in showing another mirror problem, a problem like it afterwards.

Dean Ballard: All right, so this one we’ve got a statement here. I’ve got, as you can see, I’ve got a set of inequalities over to the right, and I have an example here, again we’re talking about digits as we talked about digits and numbers before. I would go over that again. That these are inequalities because it’s less than something times the square root of 5 is less than the square root of 30 something is less than the square root of something with a 7, like 67, 47, 27, 87, whatever number you put in there, would make that value. So, we’re also not allowed to use any of the digits more than once within the same statement, and we’re not using calculators.

Dean Ballard: So this is of course not the initial, not the first time we’ve seen square roots, we’ve done a little bit of work on square roots already, with radicals. This is a chance to see, can we build on what our conceptual understanding, what is our understanding of square roots? Can we build on that or practice with it or stretch it a little bit here? Now, here’s an example of filling it in with the numbers 2, 6, and 8. So I put a 2 where the digit sign is, so I put a 2 here with the square root of 5, an 8 with 3 and a 6 with the 7, and this actually is true. So I actually got solution here for a. That we can look at together as a class. And then what I do with the class is they would want to know “Well how did I get there?” “What thinking could I use to figure out these answers since I can’t use a calculator?” So I would like to look at that with kids.

Dean Ballard: First I’d look at the easy part, the 38 and the 67. That since 38 is less than 67, we know that the square root of 38 is less than the square root of 67, so that was kind of easy to fil those two in and get an answer that works for those two. Then I had to figure out the other part. How do I know that 2 x the square root of 5 is less than the square root of 38? So, that’s going to take a little more thinking. That’s not quite as easy to figure out. Well first of all we can look at the square root of 38, and then compare it to the square root of 36. Square root of 36, we know is 6, and since 38 is more than 36, the square root of 38, that’s going to be a little bit more than 6. Main thing is, it’s more than 6. We don’t know exactly what it is, but we know it’s more than 6.

Dean Ballard: Let’s look at the 2 times the square root of 5. Well, square root of 5 happens to be between the square root of 4 and the square root of 9, now why would I care or even think to use those two? Well, those happen to be easy ones because I can actually, without a calculator, figure out that the square root of 4 is 2, and the square root of 9 is 3. So, I didn’t want to, you know, square root of 5 is also greater than the square root of 3, and same with the square root of 2, but I don’t know what those are without a calculator. I don’t have those memorized, so these were good ones because I could figure out exactly what they were.

Dean Ballard: Now, I’m really trying to figure out 2 times the square root of 5, not just square root of 5, so let’s put a 2 in there, and since I’m going to multiple the square root of 5 by 2, I’ve got to multiply the other guys by 2. So 2 times 2, 2 times square root of 4, and 2 times 3. So 2 times the square root of 5 is between 2 times the square root of 2, and 2 times the square root of 3, just like square root of 5 is between 2 and 3. So, this is going to help because I can actually do a little bit of math here and say “Oh, then 2 times the square root of 5 is between 4 and 6. 2 times 2 is 4, 2 times 3 is 6. Well, golly that means 2 times the square root of 5 is less than six.” If you remember we just talked about how the square root of 38 is more than 6. So, then I know that 2 times the square root of 5 is less than the square root of 38.

Dean Ballard: Didn’t have to use a calculator, just used my understanding of square roots and I’m able to figure that out. So, it’s a little bit of a challenge. It’s kind of experimenting, it’s kind of trial and error, using what you know about the square roots. You can figure out some things that will work for the other equations. So, all my webinar friends out there, that would at that point having done one worked example with the class, where I’ve been very clear about the logic, the thinking that I went through to get there. So that all student’s kind of have an idea that there is some thinking involved, it’s not just guess and check and hope you get it right.

Dean Ballard: That they can start to put that in, and I would have them then try number b. come up with an answer and we would do that together. Then we might do c. together or some students might work on c. and not everybody. But I want to get at least two worked examples up on the board together as a class so everybody can see the logic of putting some of this together and how you can do it. Convincing everybody not only that it’s how to do it, but that it’s doable for them. Then after doing that, they start to look at some things that don’t work. So I want to emphasize that it says you have to have an inequality that is a true statement, and so I might show an example like this where it’s not true. That 4 times the square root of 5 is not less than the square root of 32, so that you’re not allowed to do it, even though we only used digits 4, 2, and 8 like we’re supposed to. This particular order or putting them in there doesn’t work. So, that’s a no, you can’t do that.

Dean Ballard: Also, you can’t use the same digit more than once. So I would show an example like that. Say “Ah ha, I can’t do that.” Even though this whole statement, this inequality is true the way we’ve written it, because I used 6 twice, I’m no allowed to do that, I can’t count that one. There’s actually more than one solution for some of these, as you might have guessed that for a. the solution I gave was not the only one and students can come up with another one. All right, after all those directions then, highlight again one of the answers that works, maybe have them find a second one. “Oh yay we got one that works”. Then we’re going to move on to the actually problem that this all came from.

Dean Ballard: You notice how similar it looks to the one we just had. Again, I’m not going to ask you to do this problem so don’t bother trying to… You know, you can play with, but we’re not going to be working on it and sharing those answers out, because the main point is that now we’re going to give students independent work time, individually to work on what looks like the same problem, but actually they’re some differences that you can see. We have a different set of numbers at the top. 3, 5, 7, and 9. And even if you look closely at the inequalities to the right, as similar as they look, if you had them side by side with the other ones there’s some subtle differences so that the actual numbers not only are different, of course we have even numbers in those inequalities to start with, were odd numbers before. But, there’s other things moved around a little bit so that the students still have to reason through it quite a bit. It’s not just that they’re going to copy the strategy straight across and be able to put the numbers in and say “Oh I put the numbers in this order last time. I can do exactly the same thing here.

Dean Ballard: It’s not going to work that way, and that’s deliberate. I wanted to give a worked example that gave them a leg up on the problem, but I didn’t want to do all the thinking for them. I want students to still have plenty of thinking and mental math for them to work on, on the second version. That’s really the key with worked examples is making sure that we are providing clarity about the directions, clarity about the concepts, and the vocabulary, but not to provide all the mathematical reasoning. We want students to still provide plenty of that themselves. However, we also want all students to be able to have an idea about how to work on the task and be able to get started with it. So, our attention to the language, to clarifying examples, to connecting visuals with the language, and to do the things like I even give the students opportunity to have a partner and to talk about the directions in their native language if they need to in order to translate it and understand it are important ideas to help all students understand, especially EL student, and to get started on tasks.

Dean Ballard: so, again this ideas was, even with a complicated task, being able to create a second version that has some differences that allow you the opportunity to give some very explicit instruction using worked examples is a good technique especially, so students all are very clear about what to do, but then cut them loose on some stuff that really still requires them to put some thinking into it. Knowing that now I’ve got some idea about how to get started on it.

Dean Ballard: All right next I want to get into another technique I’m going to look at with another problem. This problem’s called Best Deal. The problem is: One store is having a 50% off sale. Another store has a 40% discount with an additional 15% off of the sale price. Which sale should you take advantage of if you want the best reduction on a sweater that costs $68.79?

Dean Ballard: Well I would have the students, basically the problem with them is I’d have them guess which is a better deal and then record that guess and then share that guess and their reasoning for it with another student. I don’t know if you’ve seen stuff from Dan Meyer, he promotes this idea a lot of students taking a guess at first to kind of get their commitment, or to get them engaged in a task because they’ve already kind of put some skin in the game there by making a guess, and the idea isn’t here that they calculate it out, it’s that they just kind of commit themselves to something. It only takes a minute, they take a minute to make a guess and then a minute to share their guess and why they think that’s a good guess, and I find that students, sometimes they’re a little bit wild with their guesses of course, you know like “Well, I just think that always you add.”

Dean Ballard: But they over time start to get more strategizing, they reason a little bit about what they’re seeing in the task, and even for this task, I could see students saying “Well, I looked at it and I see that 40% plus 15% makes, if you were to add them it’s 55% and that’s more than 50%, but I’m pretty sure you’re trying to trick us Mr.Ballard, so I’m going to go with the first store, because somehow I think that, that 40% and 15% really don’t go together like I think.”

Dean Ballard: And you know what, if a student says that I’m very happy, because yeah they’re just looking for a trick, but guess what? First of all they know 40% and 15%{ together makes 55% and that’s more off, but they know that there might be something going on in this problem where somehow it doesn’t work exactly that way, so now that’s what they’re looking for to happen. Of course you know, that’s exactly what’s going to happen in this problem. That’s what the problem is doing, and so they’re already looking for that thing to happen, and that’s a great thing to have a student doing. Is they’re kind of got an idea and they’re kind of, now they’re going to get it you know? So, that’s one of the things I love about this guessing thing.

Dean Ballard: Anyways, so then students will solve the problem, show their work, and have a partner maybe share their work with a partner and stuff like that. What I want to talk about with this idea of using a sentence frame to help you get the problem going with kids, and that’s just something where you kind of give some of the language around what you want them to do and leave some blanks for them. Here’s a sentence frame here that’s: I think the 50% discount is blank than the 40% discount plus the 15% off the sale price because. So they might say I think the 50% discount is better, or not as good as or the same as, and then they have to explain why. I know that this seems like “Well, it’s not that much you know, you’re giving a little bit of a sentence thing there for them and some blanks.”

Dean Ballard: But if you think about it, the blanks are actually the very thinking that we want students to put into the problem. That’s the actual stuff that they’re working on. It’s not the language around it, and so what we’re providing is a little bit of a scaffold here, a framework so they’re not having to focus so much on the language, and the language kind of helps them focus their attention on where they need it to be, and they feel like they’re getting a little bit of help here in trying to make their explanation, and we do want students explaining their work and showing us what they’re doing.

Dean Ballard: Also want to keep in mind that this sort of thing I recognize that this technique, these sentence frames, they are a scaffold. Meaning that eventually we’re going to take them away. We want students to develop the ability to write out the whole sentences themselves and develop their communication skills, but I’m going to get them there, I’m going to get them to that point by providing different models like this, different kinds of examples to help them think about and understand what it looks like. Here’s a few more sentence frames to look at. Just briefly skim those over. I wouldn’t give them all to kids at once of course. I’d just take one of these and give them to the class and say “Okay, you’re going to fill this in. Here’s the sentence. Write the sentence out then where the blank is, you may have five words, you may have one word, you may have 20 words, but you’ve got to complete it.” Especially after because, that’s a very typical part to put in one of these for math because that’s what we’re looking for is that explanation piece.

Dean Ballard: Anyway, so sentence frames a very good tool to help student with the language, to help them focus on the part of the mathematics and still see models of the language that they’re going to be using. Next I wanted to share real briefly here, this idea of graphic organizers. This is a Frayer Model for vocabulary and I know many of you may have seen this before. It’s been around a long time, and there’s a lot of different variations that you can see with this, but it works really well. These charts help organize the understanding and ideas about a single word or a phrase. In this example, the word is fraction, and in this version, the vocabulary term is in the middle, and I’ve seen a lot of times where there’s some different kinds of versions where the term is at the top of the chart. Here you see over to the top left you’re writing in a definition. In the top right you write facts or characteristics about a fraction. In the lower left some examples. In the lower right some non-examples, and I’ve seen some other kinds of categories that you might put in those quadrants as well.

Dean Ballard: There’s a lot of variations. You can do a lot of different things, you can do with this kind of a tool, or this kind of activity. You can of course have students individually create their own chart from scratch. You just give them the word in the middle. You may have the whole class working on the same word, or students may be working on a variety of words in the class. Students can work in pairs or small groups to create a single chart.

Dean Ballard: Sorry cup fell down there.

Dean Ballard: Another wrinkle you can put into it is you can have students start a chart. So, one student might have the word and then the other one fill out the definition, and then they pass it to the right, and the next student has to fill out another quadrant, and then they pass it on, and the next student has to fill out one. So every student starts one and then they keep rotating around and each student has to fill in another piece of the one they’re getting, and each one has a different word on it. Students can also make a chart and leave the word out of the middle, and then trade charts with each other and they have to figure out what the word is that goes in the middle. You can create a bunch of charts, and again don’t put the word in the middle, and you can cut them up into four pieces, and you can create to have different charts from different words cut up in pieces and mix them all together, and then give them to students, and they have to sort them out and figure out which pieces go together.

Dean Ballard: So a lot of different ways to do activities with this, and the main point is that working on the key of math vocabulary is really important. There are a lot of different strategies, a lot of different techniques and activities you can do like this that are really cool. A lot of different ways you can mix this one up even, and they all help with the vocabulary, but it’s also important for us to know, especially as math teachers I think it’s important because we have so much to do, is that simultaneously when we’re working on the vocabulary in a math class, we’re really also working on deepening, refreshing, reviewing, having students practice with the math concepts associated with those terms, because in math, all of the vocabulary that we work with practically are what we call tier three words, and those are words that are connected to mathematical concepts, mathematical context. So, our vocabulary’s very tied to concepts and any time we’re working on vocabulary is almost synonymous as that we’re also working with the concepts as well. So, we’re helping students with both at the same time.

Dean Ballard: Another strategy. A turn and talk, which you’ve heard about, and here’s a kind of a different approach to it. The Agree/Disagree turn and talk. So, look at this problem: Michael says the graph of the function y = 2x squared intersects the y-axis but not the x-asis, another student Janice says “Well I think the graph of the function y = 2x squared intersects both the y-axis and the x-axis.” I’ve had it many times, you probably know it. You’ve done something, you’ve probably even taught the lesson already to kids, and then you put an example up on the board or something like this, or some problem, and you get the answer, and you kind of think “Okay we got this one. This is the simple part and everybody’s going to get it.” And then you look and somebody says one thing and then another student says another thing, and you look at the class, and you look in their eyes, and they’re looking at you like they’re waiting for you to tell them, which of those two students was right. Because, even after everything you’ve done, they don’t know. They’re just looking for confirmation from you.

Dean Ballard: So then you realize, “Oh my goodness, I thought we got this.” And that’s a great time to say “Okay everybody, I want you to turn to your partner and decide who you agree with, Michael or Janice, and come up with a reason. You’ve got two minutes with your partner to figure out who you agree with and why and then I’m going to call on a few of you to share out.” Now this utilizes like I said that technique called a quick turn and talk with students to resolve and justify their thinking, and these are great for all students, including EL students because the time or amount of discussion is very limited here, just two minutes or one minute [inaudible 00:52:08]. Even you can take moment prior to that to make sure there’s clarity about the situation, about the language, anything to help everybody…. They understand what the task is, and then it’s often easier for students to share from a discussion with a partner than being responsible to share on one’s own.

Dean Ballard: For EL students this turn and talk provides a chance to practice language as well as then to prepare for language use, because then they might share out and then they can share out, they don’t necessarily have to share their own thinking, they can say “Well my partner, or we,” Sort of take some of the burden off of them as well.

Dean Ballard: All right, another thing I really love working with kids and then I see a lot of great stuff happening are matching activities. I know we’re getting short of time here, but I want to go over a couple with you real quick. I’ll give you a couple examples where kids can sort things out and match them up, and it can work really well with language. That third bullet there is really important, that students keep their own records individually so that they’re writing things down themselves because a lot of times we have group work and then everybody kind of walks away, and they don’t have anything, and then we don’t know whether, you know, was everybody engaged, and having individual’s be accountable in some way is really important I think.

Dean Ballard: Here’s one example of a matching activity. You have the Cue cards on side on the right in salmon color, you have the symbol cards on the left. This isn’t the whole activity. The whole activity actually has about 25 Cue cards and about 12 symbol cards that all mach up somehow. So, there’s more than one Cue card that goes with the symbol card. The idea is student have to… You get a couple of students, and you give them an envelope that has all these little pieces of paper, slips of paper or cards with this in, and they have to sort them out and match them up together. The idea is that there are so many really tricky terms in mathematics, especially in Algebra, we have all these different ways of saying something and then you have to represent it symbolically, it’s really hard.

Dean Ballard: You look at the ones on this slide right here and think about how similar they are. n less than 8, n is less than 8, that’s a completely different statement, and it sounds so much the same. When you see it in writing and say it, then you try to match it with symbolic notation you get a chance to work in a less stressful way or trying to actually have a problem to solve when you’re doing some matching like this. I’ve seen that this really can help students, and I’ve seen them actually really enjoy doing the matching, because it’s not so stressful, and yet they’re really doing some important work with the mathematics, because it’s a really important scaffolding step for them. Every student has to see, think, and write, because I have them record stuff, these ideas, and the connections that are being made between them. So it’s really good.

Dean Ballard: All right, one other one I want to share along with this matching sort of thing is this polygon activity. I was going to have you maybe play with me once, but we don’t really have time for that, but let me explain it to you. I give this to kids, and it’s another really great one where they work three different levels. First, they work as a partner, it’s one kid against another kid, and one kid then, or me or whoever it was, just takes one of these secretly. Which shape am I going to pick secretly? Then the other student has to guess, which one they picked, but they an only ask yes, no questions.

Dean Ballard: And the question has to be related to the characteristics of the shape. You have to ask like, “Well does it have four sides?” You can’t as “Is it in the third row, in the fourth column?” You can’t ask questions like that. It has to be about a characteristic. Have the students write down the questions they asked, and the answers they get, so that they not only say it, hear it, but they’re reading it and writing at the same time, that record’s going to help them in the next round because they kind of remember these questions they’ve asked.

Dean Ballard: Because, then after they play it with each other a couple of times, you know take turns who’s guessing and who’s picking the polygon, then I have pairs of students pair up and make a team of two, and play against another team of two. So they have played this level, team of two against a team of two, and what happens here is then after they play that a little bit, then I have a team of two get together with another team of two and make a team of four, and they play against another team of four. So we get 45 minutes of actual work around this, and what’s happening all that time because I want to make sure that… I say maybe 40 minutes because I want to make sure I have ten, fifteen minutes at the end of class to really process some of the strategies they’re using.

Dean Ballard: But, that’s what happens is, once the kids then start to pair up and get into teams, they talk about their strategies, which questions to ask the other team or, which figure should we pick because it’s harder because of these characteristics. What’s going on here, is there a language being used around the mathematics. They’re using the vocabulary, they’ve had to write it down, they’ve had to think about it individually so there wasn’t pressure of a team. And then, they got a chance to get together with that partner they were playing with and be a team with somebody else. All that time there’s a whole bunch of interaction around the vocabulary like I said, and then that discourse happens as students are talking about it and competing against each other and having fun with it. That’s the idea, is that there’s an engaging activity, there’s a language work around it, there’s an opportunity for them to talk about the math, to hear it and write it, and to really stress the vocabulary is important.

Dean Ballard: All right, so, again there’s some other things we could talk about. Math reading of text is really hard. Math texts are so dense and I’m going to quickly jump through a few slides here with you. It’s the most dense text, but there’s reasons why it’s important for kids to learn to read the math text and one is that it’s the one text in school that’s most similar to technical manuals that they’re likely to read outside of school in workplaces and stuff like that. Besides, it helps them to have resource in school. They can preview the vocabulary, sometimes looking at the bolded words and get an idea of what you’re going to cover, they can preview and skim over the examples, not to understand them, but again they can look at the examples and see “Oh this is what we’re going to learn today.” They can look at how examples are changing from one to the next and see “Oh this is how the lesson’s going.” Again, they’re just skimming because they’re not having to understand it yet, they’re just getting a picture of what’s going on, and realizing that it’s all there in their book.

Dean Ballard: Anticipation guides are another cool thing. That’s what this is, an anticipation guide where you write in a few questions before students read a section of the book or go through a lesson with you or a set of lessons, and the, Me column they check off whether they think it’s true or false. And then after the lesson or after they do some reading of a textbook or look through a page in the book to see they know what things they’re looking for because you’ve written them down here as anticipation statements, then they write down their answer after doing some investigation and compare what was their guess before, and what was their answer afterwards. But, they know exactly what they’re looking for, and it keeps them focused.

Dean Ballard: All right, word problem strategies real quick I wanted to share with you that a couple of things you may not have thought about before. Reading the last sentence first sometimes. In math word problems the point of the problem is the last sentence, typically. It’s unlike anything else they read where the point of a paragraph is the first sentence, in a math word problem, the point is the last sentence, and that tells you what everything else is for. Another thing you can do is have them read a problem to understand it, not to solve it. Which means you create a mirror version of that problem, and you tell them “You’re not going to solve this problem. If you solve it I don’t care, I don’t care what the answer is, you’ll get no credit for solving it. We’re only here to talk about what is the problem about, and what would be your strategy to solve it. And then after we do this one, I’m going to give you another problem where the numbers might be different, and then you get to solve it, and then I’ll care about it.”

Dean Ballard: But first of all, just give them a problem where there’s no credit for solving it. That way they really focus on just understanding it.

Dean Ballard: All right, so I’ll wrap up here real quick at the end. What we have done today so far. We provided clear directions and instruction, modeled that. Used and connect visual, oral, and written instruction and directions and to get it all tied together with kids, so they see and hear and read what’s going on. I’ll be very clear and repetitious with key vocabulary. A focus on some instruction on the vocabulary, using Frayer Models or other charts or other strategies. Use scaffolds such as sentence frames, partnering, allowing students to talk and translate for each other in their native language. Focus on the mathematical meaning students are communicating about not the syntax. Utilize cultural differences as opportunities for discussing and comparing meanings and interpretations. Recognize the complexity of language in math classrooms. Use engaging activities. Require writing along with oral communication from kids and create, manage, and process opportunities for students talking about the math with each other.

Dean Ballard: All right, I’m done, with two minutes over, thank you for hanging in there everybody. [inaudible 01:00:49] right to the end. Emily.

Emily: You did great. You packed it in, I wish we had more time. But, thank you for all of this, and as a reminder to everyone we are going to send you copy of the recording and these slides, because there’s a lot of good resources in them. So, look out in your email box for links to those either later this evening or first thing tomorrow morning. I guess we’ll sign off now, but do be sure to check out the free resources at corelearn.com and if you’ve got any questions for Dean and the CORE team feel free to reach out directly to him. I can tell you from personal experience he is always happy to help, and loves chatting with educators and seeing how CORE can support the work that you do every day. So have a great evening everyone and thanks so much for joining us.

Dean Ballard: Thank you everyone, thank you Emily.