Supporting Students Struggling in Algebra I Episode 5

Supporting Students Struggling in Algebra I – Episode 5 Developing Algebraic Reasoning in Students with Difficulties in Mathematics Including Students with Learning Disabilities

JC Sanders: Hi, welcome to another episode of supporting student struggling at algebra one. I’m JC Sanders.

Virginia: I’m Virginia Keasler.

JC Sanders: We are excited to be here. Once again, we have a special guest. What we’ll be talking about today is characteristics of students with learning disabilities in math and what the research says about building great and developing great algebraic reasoning.

Virginia: How do we address that JC?

JC Sanders: We are going to find out how to address that and how to support those students. I’m really excited about this. This is my passion Virginia. This is what I enjoy the most. Talking about students that have disabilities, that are struggling and how we can help them in math, by bringing in a relevance piece, so we can catch everybody’s attention to our podcast today.

I wanted just to mention that the author we have today, she has numerous papers and one of the things that came out in one of her papers that she mentioned was that the national assessment of educational progress of mathematics, indicated that students with disabilities, including LD, learning disabilities, continue to lag significantly behind their typically achieving peers. I thought that was really significant. That just came out in 2013. I feel like that really brings to light really why we are talking in this whole podcast, right Virginia?

Virginia: Right, because it is an ongoing problem and we want to be able to help every kind in the classroom. Yes, for sure and I think teachers struggle with that and sometimes become frustrated

JC Sanders: Let’s go ahead and talk about our guest we have today. I’m excited to be introducing Dr. Dianne Bryant today. Dr. Bryant is a professor in the department of special education at The University of Texas at Austin. She holds the Mollie Villeret Davis professorship in learning disabilities and is project director of the mathematics institute for learning disabilities and difficulties and the Meadow Center for Preventing Educational Risk.

She also is currently the principal investigator on project AIM, which is an algebra readiness project, which is funded by the Institute of Educational Sciences. Dr. Bryant as well is a co-editor in chief of the learning disability quarterly, publishing numerous articles on instructional strategies … Assistive technology for students with learning disabilities. She is co-author of text books as well as educational test. Welcome dr. Bryant.

Dr. Bryant: Thank you. I’m very happy to be here.

Virginia: Let’s just jump right in. How about helping us with some of our listeners or not necessarily at the SPED world. Can you tell us some common characteristics of students with LD in math?

Dr. Bryant: Sure, that’s a really good question. There are several characteristics related to students with LD in math. I’m just going to give some examples. The first one that comes to mind relates to executive functioning. That has to do with the brain and processing information. For a number of students with learning disabilities. The research has shown that there are difficulties in the area of working memory, which obviously is going to affect their ability to recall important information, procedural steps et cetera. That’s one characteristic that comes to mind. Some of the other characteristics related to education include struggling with basic skills such as multiplication facts.

We had a focus group with high school teachers and that was one of the things that they mentioned is that they wish students would come to high school, ninth grade algebra in this case, with more knowledge and proficiency with multiplication facts because that’s such an important skill. Another characteristic that comes to mind is weak conceptual understanding. This is huge. There is just not enough time spent for students with learning disabilities on developing and helping them to understand concepts. That’s critically important. Also, they have difficulties with what’s called number sense. Number sense refers to things like the magnitude of numbers, understanding patterns, understanding place value.

Both weak conceptual understanding and number sense, we see this in middle school and high school students as well as at the elementary level. Another, in fact, I can give you an example, if that helps with the conceptual understanding and number sense. Specifically, for example, students might say one fifth is greater than one half. Thinking that the five in the denominator is greater than the two in the denominator. Therefore, one fifth has to be greater than one half. It’s just that number sense and that conceptual understanding that’s problem.

Virginia: I was just talking to a teacher yesterday who, a high school algebra teacher and he just was lamenting the fact that he could say five out of ten and they would say that’s one half but if they were given five divided by ten as a problem and asked to reduce it, they had no understanding of that.

Dr. Bryant: That’s a good example of the difficulty with number sense for sure. Another characteristic is that a lot of kids don’t understand the properties, specifically distributive property. You know how important that is when it comes to algebra and solving equations, algebraic expressions and equations. That’s a really critical idea that teachers need to be aware of. I was in a high school class and the teacher was using the words distributive property. I was sitting there thinking, “I wonder if some of the students with learning disabilities, and maybe some of the other students, if they understand the distributive property, what that means.” If it’s something that they learned in elementary or middle schools or didn’t learn.

I guess another characteristic to be mindful of is difficulty with generalization of information. That is so important because a lot of our learning is based on understanding patterns and how they apply to similar types of problems which would be generalizations. Then another characteristic that comes to mind is the need for repetition and multiple opportunities for practice. That students with learning S-3Algebra_episode5 Page 2 of 12 disabilities need definitely many practice opportunities to help them really learn the concepts.

Virginia: Teachers will say, “I need to give them some more practice but I’m going to give them numerous problems to practice on.” Maybe they are going to say, “I’m going to have you go home and work ten problems.” Then they feel like doing that will solve the problem. What do you say to those teachers?

Dr. Bryant: That’s a really good question. I think one of the things is to be sure that the students understand the concepts, or the procedural steps that are involved in the problems. We don’t have to overdo the practice. I’m not talking about a lot of worksheets and that sort of thing. Practice that gives them different ways to think about problems, to represent problems, the multiple representations. Those are the kinds of things that are important.

JC Sanders: I think that actually just goes back to our last episode. We were talking with Russel Ralston about practice as well. He brought in the idea of instead of mass practice but distributive practice, so that students are getting to see these things in multiple times, multiple opportunities to work on them, rather than just worksheet after worksheet of the exact same thing, stop and then walks you over the exact same thing. The more of a distributive practice that we are kind of, that’s spiraling as well, looking at how there is multiple representations and getting to practice those things throughout instead of just in silos.

Virginia: I think he was talking about doing practice in different ways as well, not just by getting another worksheet and some teachers think that if they just keep putting those problems in front of them, then they’ll eventually somehow by osmosis be able to get them. A lot of math teachers learned that way, it’s hard for them to break away from that and think. “I learned that way, why can’t they?” It is kind of a flag when kids show up without multiplication facts. That kind of tells you what they were unable to learn it in the normal …

JC Sanders: The typical way.

Virginia: Way that we think of, the typical way.

Dr. Bryant: The distributive property. I’m glad you brought that up. There is a lot of research to support distributive practice so that students keep coming back to these concepts and skills to get the extra practice, to get the review, the cumulative review. That’s a really good point.

JC Sanders: Go on with the more, instead of characteristics, talk to us about maybe some students’ misconceptions that you see out there.

Dr. Bryant: That’s a great topic. Misconceptions can occur at the elementary level, the middle school level and the high school level. Just thinking about some misconceptions related S-3Algebra_episode5 Page 3 of 12 to the math, sometimes students are confused when working with formulas because finding the missing value requires them to analyze the relationships expressed in the formula. There may be some misunderstandings about how do you go about using the formula to solve problems. Another misconception that comes to mind is related to algebras, writing equations or inequalities to represent word problems.

Often times students might say something similar to and n equals rope. Indicating that the variable is an object and not a quantity. That’s something to be mindful of. Another misconception that comes to mind is that students often mistake and will think that the solution to an equation is the constant that comes to the right of the equal sign.

That’s a huge problem. We see it at the elementary level as well and it just seems to be something that lingers into middle school and high school where students might see the equal sign interpreted as operational. Here is the equal sign, here comes the answer, rather than seeing the equal sign meaning that both sides of the equation have to be the same, balanced. That’s a relational interpretation.

JC Sanders: That’s one of those rules that kind of expires, right, that we talk about, we are trying to work with elementary and middle school teachers to help them understand that we don’t want to lean towards the equal means answer. Because that’s something that when they get into algebra it is all about balance.

Virginia: There are procedural ways that you see in text books or whatever they want you to take the variables and kind of identify what they are as part of the problem solving. Then maybe a teacher would have a misconception of their own, where they are saying, n is equal to rope or whatever, and maybe get away from that because just myself hadn’t thought of that until today. Can you tell us about students who are needing additional practice? I think we kind of covered or talked a bit about that?

JC Sanders: We did.

Virginia: What is your experience observing and working with classroom teachers and any other misconceptions when they are providing practice for their students?

Dr. Bryant: Sure. The practice and misconceptions related to it is something we hear a lot from teachers. We’ve already covered the distributive practice and that’s a good thing to do. Rather than just extra worksheets. We’ve talked about that but I think the other misconception is that if we put kids on computers and they are doing whatever the program might be, that that’ll give them all the practice they need.

Sometimes we forget that some of the software that they might be using or the apps that they might be using on mobile devices like an iPad for example. Maybe those particular software, programs, don’t contain some of those essential characteristics for helping kids to learn more, to practice more. Some of the evidence based features that we would hope that are part of the software and the apps. That’s one other thing that comes to mind. S-3Algebra_episode5 Page 4 of 12 Another thing, often times we hear teachers say that they don’t have time to go back and work on skills that have not been mastered from previous grades. We get that. we can understand that point of view. Sometimes those prerequisite skills do have to be focused on perhaps at the beginning of the year, perhaps at the beginning of each chapter, that students are going to be learning, that content. Students have to have the prerequisite knowledge in order to benefit. They might need more practice.

JC Sanders: Because if we don’t go back and address those, the gaps only get larger.

Dr. Bryant: That’s right.

Virginia: I also have a question about that. You’ll have teachers who’ll say, “At the beginning of the year I’m going to review all of these things for four or five days, and then I’m going to start my regular curriculum.” Then others will … Then they will think that because they did that that they have covered all these prerequisite skills or reviewed them or something.

JC Sanders: Checked that box.

Virginia: Checked it off. Then they feel disappointed as they go through the year because then they realize, “I reviewed this at the beginning of the year but the kids still don’t know how to do that.” How would you overcome that as a teacher?

Dr. Bryant: That’s a really good question. I think covering all the of the prereqs at the beginning of the year. That’s a place to start. Going back to the characteristics of students with learning disabilities, with the working memory. They may not remember the content that was covered at the beginning of the year. Making the content prerequisites, closely linked to the new concepts that are going to be taught, that would be a good time to review some of the prereqs and then to move into the new content. You can do through a warming up. For example, activity to focus on those prereqs.

JC Sanders: I think also teachers always trying to figure out that homework situation is, why can’t you bring up some of those prerequisite skills from past into their homework to see, like that little formative assessment? When they come back the next day we are seeing like, “Are my students ready to move on to this new concept, what they are lacking, where are their gaps before I move on?” Spending some of that time there. We know we always have classroom time, where else can we find time to find out what kids know and they don’t know?

Virginia: That was something that came out in another conversation, I had with some teachers. They actually said that the kids last year were not able to really conquer the word problems and all the types of problems that we see on the star. If this year coming up they wanted to maybe do some prerequisite type of skills at the time that they were working on it and constantly be spiraling those in, and trying to see if that would work. Their idea was, let’s give them some kind of a pre … A few problems before on the test, S-3Algebra_episode5 Page 5 of 12 give them a few problems that kind of have to do with parallel and perpendicular lines but haven’t been in a word problem yet.

Then that gets them to thinking a little bit about parallel and perpendicular lines and easier problem. Then later when they get to the harder problems then they kind of feel successful with that concept and can do better on that word problem. Then the thought was to bring that problem back later again, without those prerequisite kind of problems, to see if they could retain that. Is that a good way to think about it?

Dr. Bryant: I think that make a lot of sense. Where you keep spiraling in JC as you were talking about this idea of content, skills, just keep coming back to it. We just can’t take for granted that because where the students are now in ninth grade they’ve learned all of the material before, we have to pay attention to those prereqs.

JC Sanders: That’s right, I think that idea, it’s almost like priming the pump, that’s kind of what I was thinking when you have kids kind of thinking about those things in a little simpler form and then we add on.

Virginia: When we are making our test, we think we have to make it just like the start test, every test they take. Then the kids sometimes fail miserably on those tests throughout the whole year. By the end of the year they are just thinking, “I’m never going to master this topic. Sometimes we are not encouraged to put skill based questions on a test, just to kind of help the kids work into those harder problems. I just wondered about that.

JC Sanders: Let’s just into some approaches that you and your team have found when you are doing your research, to help build some algebraic thinking for our students, especially students with learning disabilities.

Dr. Bryant: They were three main big ideas that I want to talk about in relation to that question. The first big idea is, reversibility questioning. This idea of just the concept of reversibility. What that means is that questions are changed, the direction of the students, really is intended to change the direction of student thinking with reversibility questions. In other words, reversibility questions give the students the answer and then ask them to create the problem, which is a twist.

JC Sanders: I love that.

Dr. Bryant: For example, a question might be, what are two integers whose sum is negative eleven? Usually it’s those parts of the problem are given to students and they have to find the answer. This is a twist. It is the reverse in terms of the kind of question that students are being asked to solve. It’s really fun because we’ve seen teachers present those types of problems and questions. Then they put the kids into smaller groups with problems where they have to think from a reverse sort of thinking process and how the students work together and come up with some ideas. Then they have to talk to each other about, does that makes sense, does that equal negative eleven? That’s one big idea.

S-3Algebra_episode5 Page 6 of 12 JC Sanders: I love that also, I just wanted to throw in because I feel like that’s helping us look towards that math is not about finding an answer, math is about problem solving. I really love that idea.

Dr. Bryant: A second big idea that comes to mind is flexibility and questions related to flexibility. These types of questions support students’ development of multiple ways of finding relationships among problems and their solutions and solution methods. For example, teachers can ask students to show on their whiteboard a decimal that is greater than nine tens but less than 1.0.

Everyone has to share, they work together and everyone has to share at least one decimal number and when it comes time for students to provide their example, if their idea has already been presented, then they have to be prepared to give another idea. They have to be a little flexible in terms of their thinking about, “This answer has already been given. Now I need to find or have ready actually another response.”

An example which might be helpful is asking students to represent or shade six tenths in two different ways using a ten by ten grid. Just thinking about the notion of flexibility and that some students might do it one way, other students might do it in a different way. Having students share how they went about shading that grid is good to show how flexible the thinking is.

Then the third big idea would be generalization. Generalization questioning. The aim is to create statements about patterns observed within particular problem classes, so that students can use them to predict answers or check the reasonableness of their responses. Let me give you an example, going back to the flexibility example with the decimals and nine tenths and so forth. Teachers can help students develop generalizations regarding place value because that’s really when you are talking about nine tens and 1.0 and finding numbers between those two numbers.

You really do have to think about place value. There is a possible generalization that can be arrived at from those types of problems. One thing that we’ve done in our research is when we come up with generalizations, the students come up with generalizations. We have a section in their booklet, the student’s booklet, it’s in the back. They are to keep track to write the generalization and then they have to draw a picture, a pictorial representation and then they have to come up with a numerical representation. It might be an equation or something of that nature. So that they can really see, “How does this generalization work, what does it look like?” It’s been very helpful we think for the kids.

Virginia: That sounds like there would be a lot of benefits to this kind of approach. I’m just wondering, what do you think the benefits are?

Dr. Bryant: That’s a really good question. I think that it helps students to develop their thinking processes, and to help them develop their conceptual understanding. To be able to verbalize some of their findings. We know that it’s important for students in math to talk about the math, to justify it, to explain it. All of these things really help to promote S-3Algebra_episode5 Page 7 of 12 conceptual understanding flexibility, reversibility, generalization, really do focus on building conceptual understanding. Of course, conceptual understanding, procedural knowledge, they just go hand in hand. I think these three big ideas can help students benefit more from instruction.

Virginia: It goes back to; I remember having students who always wanted to know why we were doing something. Why is it we are doing this? Then they have the students who say, “I don’t care why we are doing it, just let’ do it.” There is always those two sides of the fence. It sounds like some of the things you are having them do would kind of solve some of their problems.

Dr. Bryant: Sometimes we run into students who say, “I already know how to do that.” Fine, they have learned a procedure or a trick for solving the problem. We say, “Time out. Let’s talk about some ways to look at these numbers, to look at these concepts, and to build some of that conceptual understanding.” It takes a little bit of time for students to come around to … We are going to talk about math perhaps in a little different way. Most often than not, students like it. By the time they get through the modules that we’ve developed, they are beginning to get it and they understand why it’s important.

Virginia: That’s great.

JC Sanders: That is great. I’m excited about those approaches. Tell us, and for the teachers out there. How do they get started with some of these things? I want to bring in these types of questioning with their verbal, with their generalization and their flexible thinking and things. What are some tips?

Dr. Bryant: That’s a great question. One tip that comes to mind is for teachers to model, how to respond to these three big idea questions that I was talking about. Really, one of the best ways to do that is by making their thinking processes visible to the students. Going through a problem that involves reversibility or flexibility or whatever for the teacher to work the problem, to model it, and then to articulate out loud what they are thinking to help students see that, or to think about that thinking process that’s involved.

JC Sanders: I have to add to that, I love that idea that for teachers to think about they may need to practice before they introduce. They may need to be thinking about some of these, ask themselves those reversibility questions. Plan it out themselves, work it out and talk out to themselves possibly, rehearse almost because when you get in there and all of a sudden students bring up things that you didn’t think about, that puts you a little bit on the spot as a teacher. I would highly encourage teachers when you want to get started with this, please do but don’t forget to practice yourself.

Dr. Bryant: I think that’s great advice. It may be uncomfortable for teachers.

JC Sanders: Sometimes little change is hard but it can be good.

Dr. Bryant: It benefits the student and I think it’s not only students with learning disabilities but it’s S-3Algebra_episode5 Page 8 of 12 students who are low performers, who don’t have identified learning disabilities and even some of the typically achieving students. They may have gaps in their knowledge as well. Really, having the teacher model and think out loud, why he or she is doing something? I think that’s really critical.

Virginia: Just, I know the first time I heard think aloud. I wasn’t sure what’s think aloud, I don’t know what that is. Can you just say what, if I’m a teacher and I’m thinking aloud, I know that makes perfect sense to me now. I think teachers still may stumble over the idea. What do you mean thinking out loud?

Dr. Bryant: If there is a problem, when there is a problem to solve, the teacher can just start by saying, “We are going to solve this problem. It’s a reversibility type problem. I’m going to solve it and give examples to answer the reversibility question. I’m going to tell you how I’m thinking to arrive at my answers.” For example, step one, this is what I’m thinking.

Now what do I have to think about in terms of finding for example a number between nine tenths and 1.0? What do I have to rule out, what do I have to, what are some non- examples? Why is this particular number that’s not in this area that we are talking about, it’s not between nine tenths and 1.0 and why? Giving non examples and then certainly giving examples of why the teacher arrives at a particular answer. I think that’s helpful.

Virginia: I guess what you are doing then at that time is you are not really asking the kids what the answer to that is, you are just kind of modeling as they would go through it but you are thinking, you are talking about it. It sounds like the same things that I would be asking the kids sometimes. Instead of that, I wouldn’t be asking for responses. I’m asking for my own responses in my own hand.

Dr. Bryant: That’s right.

Virginia: That makes perfect sense.

JC Sanders: To say out loud, now I’m going to start here. Because I know this about the problem, I remember this in the class we talked about, and telling, just sharing with the kids. “My brain goes to hear this strategy first; my brain goes to this strategy next.”

Dr. Bryant: I think that’s a good idea as to not only the strategy, in addition to the strategy, but maybe tying in some of that prerequisite knowledge that would be really important. That’s another way to get that prereq knowledge activated as students are engaged in this thinking process, in a different way that they may be accustomed to now.

JC Sanders: We talked about think alouds but I know you have a few other tips for teachers.

Dr. Bryant: Another one is to provide prompts when students get stuck. We know students, the thinking gets stuck along the way for whatever reason. They don’t remember a step or S-3Algebra_episode5 Page 9 of 12 they don’t remember a particular concept, going back to working memory. Just one quick tip would be to come up with an acronym. If it’s a procedural step is to come up with an acronym or the first letter or the letter for the acronym signifies something. What am I supposed to remember, what am I supposed to do?

Acronyms can be very helpful and they are highly supported in the literature as being effective. Another tip that comes to mind is think, pair, share, that strategy. I think most teachers are familiar with think, pair, share. In terms of the three questions, the reversibility, flexibility and generalization. It’s going to take some time because this may be something new for students that they will not only need the think aloud, they may also need some prompting, but they are going to need time to think. Teachers need to think, it’s okay to give them more time, and just to put that in their planning for that class period.

Being in classrooms and working with teachers and students, another tip that comes to mind is creating a safe environment. I think teachers really strive to make the classroom safe and comfortable for all students. Keeping in mind the student with LD, they really are going to benefit from very explicit directions on how to discuss a response. Going back to think aloud, that would be a good way to help them understand what the steps are, understand what the thinking is. Then also how do I want, me as teacher … How do I want you to respond? That would be something else to keep in mind.

Again it’s creating the safe environment so students with LD know what the expectations are and how to approach some of these problems. Then, how do you share, how do you talk to your partner and then how do you share with the class? Students with LD may need some guidance with all of those pieces. I think that really helps to promote a safe environment for students with LD, as well as low achievers as well.

Another tip that comes to mind is consistency. Being consistent in a daily math routine. That helps students a lot. It’s predictable, they know it’s coming up. Including these types of questions, these three areas that I’ve been talking about as part of the daily routine, they could be integrated into the warm up. It doesn’t have to be all three everywhere.

Integrating them into the warm up, certainly is part of the lesson, especially with the generalization. Students really need to help to arrive at generalizations. Those patterns, so that they can use them when they go on to more advanced mathematics. Then formative or summative questions for assessment, for determining, are students getting it? That would be really important.

JC Sanders: I think bringing in a daily routine is a great point because I can kind of foresee that some of these questionings are going to be a little challenging. Students are going to be a little challenged. Teachers maybe a little challenged maybe, and if we don’t put it in a daily routine and we bring it up just once or twice, then it’s going to be a bigger ordeal and maybe the teacher is going to be a little, “This isn’t working.” If we weave it in a little bit S-3Algebra_episode5 Page 10 of 12 more often, everybody gets used to it. It’s just a routine and we think this way in our classroom and that’s not a big deal and it’s not such a stopper of our learning.

Virginia: If I’m an LD kid I might, I want to feel like I can offer some information, right, wrong or indifferent without having any repercussions in the classroom. I don’t want to be that person who doesn’t know the answer. I want to be like everyone else. Working in groups or pairs or something like that that gives them the opportunity really shine a little bit more and not just feel like I’m intimidated, I’m never going to answer a question.

JC Sanders: That’s fantastic, what a lot of great information? Let’s see if I can kind of wrap it up and summarize for us. We talked about the students with learning disabilities in math do have struggles with working memory. We want to see what we can do to help support that and that these questionings and opportunities can help students with their working memory and developing their conceptual understanding as well as their procedural fluency which would be great, goes hand in hand there.

Teachers need to think about how they can provide multiple practice and build these into their routine of their classroom, so that students are seeing it in multiple representations. Teachers can start using these approaches, this reversibility, flexibility and generalization to help foster algebraic thinking in their students in all ages and all ability levels, not just for algebra students but throughout the school.

The type of questions allow students to see that relationship between concepts and skills and help them build. I really like this in the article … I saw this in the article and that students, give students mathematical prowess. Is that right, prowess? Which is the power to attack problems and the confidence to engage in solution process.

I love that. I want all my students to have mathematical prowess. We are going to put this article that we’ve been talking about today. We’ll put a link here at the podcast. If you are listening to the podcast, you hopefully can see that link and any other resources that we come up with. We’ll add those in. Thanks Dr. Bryant for coming and talking with us about this wonderful information.

Dr. Bryant: Thank you so much for inviting me. I really appreciate it.

JC Sanders: Virginia, we are almost wrapped put with the podcast series.

Virginia: I know. It’s sad.

JC Sanders: I can’t believe it. We are going to try to get together one more time, Virginia and I to do one final podcast before we sign off. I hope that that podcast will get to wrap up all the podcasts we’ve had and kind of review all the great stuff we learnt from all the wonderful guests we’ve had on our podcasts. I really would like us to talk about, Virginia as many intervention product that she has created that is fantastic. It brings in all of these things as well. S-3Algebra_episode5 Page 11 of 12 Virginia: The cool thing about all of these podcasts, I’ve learned something myself and cool thing is that they all seem to be saying the same thing. It’s like I can think back to something you are saying to something that Dr. Debbie Junk said, or what some of our other people that we’ve had. It’s been really interesting for me and informative.

JC Sanders: We hope everybody feels the same way. Thanks a lot for listening and we’ll catch on next time. bye.

Virginia: Bye.

S-3Algebra_episode5 Page 12 of 12