Emily: Hello, everyone. Thank you for joining us for today’s webinar, Supporting Struggling Math Students. Before we begin, I’ll review just a few quick housekeeping items. We’ll accept questions throughout the webinar for a Q & A period at the end of the presentation, but please go ahead and send in your questions as they come up for you, using the questions feature in your control panel. Just type your question into the top box and click send. I’ll receive your question and I’ll put it into the queue to be answered at the end of the presentation.
Emily: If you have 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. Dean’s got a lot of great resources and ideas for you. So, don’t worry too much about scribbling notes because you will get that PowerPoint deck after the webinar concludes. Just keep an eye on our email tomorrow for details about how to access the recording and the PowerPoint slides.
Emily: Now, let’s get started. I’m pleased to welcome today’s speaker, Dean Ballard, Director of Mathematics, with the Consortium for Reaching Excellence in Education, also known as CORE. Dean holds a Masters degree in Math Education from Sonoma State University, and secondary teaching credentials for both mathematics and English. Over the last 12 years Dean has specialized in professional development for both 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 fifth grade through AP calculus. He is a member of the National Council of Teachers of Mathematics, National Council of Supervisors of Mathematics, and the California Math Council. He’s just an exceptional math teacher, and he is here to share a lot of great strategies and information with us today about how to better support our struggling math students.
Emily: So, I’m going to go ahead and turn the program over to him now.
Dean Ballard: Thank you, Emily. Good afternoon, everyone. I appreciate you taking time out of your busy schedules to attend our webinar today. The main takeaways I’m hoping to provide everyone are to learn more about how to plan for and us time for intervention, what the focus of intervention time should be, foundational knowledge or today’s lesson or tomorrow’s lesson, how and what types of assessments to use, what materials to use for interventions, and how much fluency work to include.
Dean Ballard: Throughout the presentation I’ll include, I’ll describe 10 myths related to math interventions. Calling them myths, I’m probably going too far, but the point I’ll be making each time is that there are common beliefs that get misconstrued, misunderstood, or misused, however, each has some truth to them work recognizing. From that truth we can springboard into other insights into what isn’t true, what is also important to keep in mind, or what may even be a bigger truth to keep in mind.
Dean Ballard: So, to begin with, I want to lay a quick foundation for math intervention by briefly reviewing some key ideas related to multi-tiered systems of support or MTSS, in response to intervention, also known as RTI. This brings us to our first myth. The first myth was a prevalent belief for many decades, but has been soundly disproven by research. Myth number one, students learn at different rates so they will catch up later.
Dean Ballard: While it’s true that not all students are alike and there are variations in learning rates, what is perhaps more true or more important to understand, is that students who fall behind are likely to stay behind if they’re not provided support. The earlier the support to catch up, the better. The further behind a student is, the more intensive the support needs to be. Benchmarks for reaching and math are set at levels that all students (except possibly those with learning disabilities) can and should attain.
Dean Ballard: There’s ample research to back this up, such as studies from ACT, Geary, Hoard, Nugent, and Bailey, Shaywitz , and others. Keep in mind, kindergarten and first grade are not too early to identify and address learning deficits.
Dean Ballard: Because identifying and addressing learning needs at all grade levels is important, I want to briefly call attention to practices of multi-tiered systems of support in response to intervention. According to McIntosh and Goodman, MTSS is a coherent, strategically combined system, meant to address multiple domains or content areas in education. MTSS is important as a systems focus within a district or a school on meeting the academic and behavioral needs of all students. I’m not going to delve deeply into MTSS since that’s not the focus of this webinar, however, it’s good to recognize that what we’re talking about in this webinar is part of MTSS and RTI.
Dean Ballard: It’s worth just taking a moment to review key components of RTI, because these heavily impact the success of math support for struggling math students or in math interventions. Leadership and a team that plans, monitors, and drives an intervention system are critical for school-wide and district-wide success. One of my colleagues at CORE was just telling me about two different schools she was working with, and how the leadership at one school was focused and drove an emphasis on supporting struggling learners in both math and reading, versus another school in the same district with leadership that, at most, provided suggestions to staff on interventions.
Dean Ballard: The results have been very clear. The school with the focused support on addressing learning gaps is far out performing the school that was driven by suggestions for interventions. Additionally, research has proven the schools that succeed with RTI have leadership RTI teams that monitor and manage intervention systems within the school, and use data to drive decision making with generally three tiers of instructional support.
Dean Ballard: Those three tiers of instructional support include Tier 1, which is the core instruction itself, which can include some differentiated support within the core instruction to address immediate or emerging gaps and misconceptions. Tier 2, which is small group instruction, is instruction designed to address more persistent gaps, very weak or missing important prior knowledge, or continued and deepening struggles with understanding core instruction. Tier 2 can take place within core classrooms or it can be additional time in a support class.
Dean Ballard: Tier 3 addresses severe learning needs, typically students who are two or more years behind in key areas in mathematical understandings or skills. Tier three requires working with students outside the core instruction, and in extreme cases may even replace the core instruction with what is called an intervention core class.
Dean Ballard: My final words on RTI itself are to mention the four main steps in the RTI process. The MTS/RTI team may first explore and identify as a school-wide needs, resources, and personnel that are likely to be part of the system [practicum 00:07:00] and support. Then a plan is put together for how to identify and place students, along with what assessment systems and data will be collected, how it’ll be used. The plan includes what interventions to be put in place, what content will be taught, who will the teachers be, and how students and programs will be assessed.
Dean Ballard: Additionally, the plan includes how interventions will be implemented, such as you might just do one or two classes to start with, and then expand from there, or the whole school will implement it incrementally. Then the plan is implemented. Finally, interventions much be sustained with on-going support and be monitored. Part of sustaining a system is to continually monitor its effectiveness. We want to know where the interventions, where they’re working and where they’re not working, so we can see how to improve them. We should monitor if students are closing gaps and if they are able to progress out of the interventions, or if at best, students are just treading water and not actually catching up to grade level proficiency.
Dean Ballard: Those important components of a RTI process bring us to the key areas I’m going to address throughout the rest of this webinar, as they relate to providing support for struggling students in math. I’ll talk about who are the students that need and will receive added support. How do we decide on what to teach? What is the content focus for added support or intervention? When to teach? Where is the time in the schedule for providing added support? How to teach? What are the basic instructional techniques to use for different levels of support? Who are the teachers that will provide the added support? When should we use a publisher intervention curriculum program? When that should be used versus other resources? When are the best intervention materials to use in different situations? Why we need to assess, review, and revise intervention programs.
Dean Ballard: Alright. A common misunderstanding we see is the belief that the benchmark test will tell us what students need to learn. Benchmark assessments like MAP and NWEA are excellent at identifying students who have gaps or are at risk of falling behind, and monitoring overall growth for the year. They may also inform the intensity of the intervention needed, based on how far behind the student appears to be. And some general areas for instructional focus may also be identified. For example, NWEA provides a continuum of learning, they call it, that identifies a list of concepts and skills students at each level are likely ready to learn.
Dean Ballard: However, even NWEA rit scores and NWEA’s continuum of learning statements are not individualized diagnostics. Benchmark assessments are not intended as diagnostics. They’re more like thermometers telling us something’s wrong, rather than a blood test where we try to identify what the specific problem is. Research recommends students in tier three should be given diagnostic assessments that identify specific key learning gaps that need to be addressed. Benchmark assessments also do not provide the type of on-going progress monitoring students in tiers two and three intervention need, so that instruction can be adjusted as needed in time for it to be effective.
Dean Ballard: Determining … Sorry. The … I just want to give this quick overview of the assessments from which we collect our data. A summative assessment such as statewide tests provide end of year results. Many schools use these as one data point for identifying students in need of support for the following year. Screening and benchmarking assessments are typically given three times a year at the beginning of the year to identify where students are, to start with, and which students are at risk. At the approximate mid-point of the year also to provide a mid-year progress report, so that if things are not going well there may be still be time to do something about it this year. And at the end of the year to identify progress or the lack thereof if that’s the case for each student for the year.
Dean Ballard: Progress monitoring provides weekly, bi-weekly, or monthly progress data on students in intervention. Diagnostic assessments provide specific information on students’ individual needs. For example, identifying that a student may understand addition but not subtraction. Teacher input. Based on observing and teaching specific students also further informs which students need added support, and the type of support they may need. We also look at how students are doing in core instruction. Struggling with core instruction may signal knowledge gaps. This almost always warrants further investigation to identify the root causes for the student’s challenges. We want to make sure that we identify the true underlying causes, be it a lack of knowledge, behavior, language, or disability.
Dean Ballard: Then determining what to teach students during intervention time is important. There is very limited time for interventions and students are already struggling, and frankly not every gap can be addressed. Many students have gaps in multiple, if not all, math domains, such as numbers and operations, operations with algebraic thinking, measurement in geometry, data, statistics. Students need focused support in the foundations that are key for continued progress in mathematics. And of course, we can say that every grade level lays some foundation for the next grade level in all of these domains. However, some domains are more critical for grade level progress than others.
Dean Ballard: So, the next myth I’m addressing is the idea that each teacher can best figure out what to teach during intervention time. The truth here is that for less intensive levels of intervention, teachers are often in a great position to identify the learning needs. This is typically the case where the same teachers, the one who will be addressing those needs, or the needs are related to more immediate gaps or misconceptions connected to the current lessons. However, for tier two and especially for tier three, data and research provide valuable guidance on what to teach. Significant gaps exist and certain knowledge needs to be prioritized. This is what we’re going to look at next.
Dean Ballard: The areas in which we have seen success with teachers taking the lead in determining what to teach, when providing additional support, include, as just mentioned, identifying and addressing the gaps, misconceptions, or other struggles with learning, related to recent or current lessons. Then also providing extra help on classwork or homework and firming up the foundation for the next lesson. The two ways we’ve seen this work effectively, most effectively through, are with additional support within core instruction, such as extended or added time in the elementary grades, a block or longer periods in secondary grades, or teachers creatively finding intervention or added support time during the regular scheduled math time, such as on catch up days or what’s called flex days or work station days or during lessons in which the whole lesson time was not needed.
Dean Ballard: Sometimes there are days in which the whole class is getting added practice, or some of the class may work on an extension or application problem while the teacher spends time with a selected small group of students to work on addressing a common gap or a misconception. Teachers also employ strategic use of warm-up problems and fluency activities. Students may also be assigned via computer program that targets specific need.
Dean Ballard: We’ve also seen the successful use of an additional math support class in the middle or high schools that is aligned to core instruction. These classes are math time in addition to core class time that selected students take in place of an elective, usually. At these schools, success they have has come part from a high level of coordination we’ve seen between what is taught in the core class and providing additional direct support for these lessons in the math support class. The support is based on reviewing key examples, concepts or skills, going over assigned problems individually or group work … these are related to the lesson they had in their core class … and are laying the foundation for and previewing the next lesson.
Dean Ballard: However, for more intensive interventions, both more intensive tier two, such as an additional support class for students who are struggling a lot, and tier three, research has provided some key areas for focus. Here are three recommendations from the What Works Clearinghouse in the Institute of Education Sciences practice guide titled, “Assisting students struggling with mathematics: Response to intervention for elementary and middle schools.” These recommendations are based on a meta analysis of research to hone in on what is most important. For early grades, the in-depth learning and a meaning and operations with whole numbers is critical. For upper elementary and middle school, an important focus should be on work with rational numbers. That is fractions and decimals. It includes both positive and negative numbers.
Dean Ballard: Additionally, struggling learners in math need work with solving word problems. I don’t think that’s news to anybody. Specifically with strategies and techniques for solving word problems, such as recognizing underlying structures that identify the types of problems and operations they may needed, and visually representing those problems.
Dean Ballard: Students with significant gaps almost always have fluency gaps that get in the way of learning. Because students who are spending cognitive energy deciphering basic facts and procedures during lessons that assume this knowledge have less cognitive to the energy to put towards learning that new concept. So, therefore, some regular work on fluency is also important for success.
Dean Ballard: While we’re at it, let’s also look at the recommendations from the National Mathematics Advisory Panel in their final report from 2008. Here are three recommendations made in that report for the most important foundations for algebra. Again, their study of research on mathematics provides insight into what should be the main focus of instruction in our intensive intervention classes. The three key foundations for algebra identified in their report are fluency with whole numbers, fluency with fractions, including positive and negative numbers, and some particular aspects of geometry and measurement, namely area, perimeter, volume, and similar triangles.
Dean Ballard: Alright, finally, I want to share one other person’s insights into providing added support for struggling math students. Marilyn Burns is a well known and respected math writer and materials developer. So, let’s take a moment to read her statement in an education leadership article from 2007.
Emily: In developing intervention instruction, I have reaffirmed my longtime commitment to helping students learn facts and skills, the basics of arithmetic. But I’ve also reaffirmed that the basics of number and operation for all students, including those who struggle, must address all three aspects of numerical proficiency; computation, number sense, and problem solving.
Emily: Only when the basics include understanding, as well as skill proficiency, will all students learn what they need for their continued success.
Dean Ballard: Thank you, Emily, for reading that. So, from many sources we see a convergence on some key topics for instruction and intervention classes. Certainly math teachers looking at class and school data can add and customize based on their students’ needs, but we should keep these recommendations from research and math experts in mind so that with the limited time we have for intervention, we use it as wisely and beneficially as possible for the kids.
Dean Ballard: So, in summary, here’s a list of math topics for content focus interventions under the what we should teach category. You see I’ve added for grades six to nine that ratio and proportions and solving equations are two additional [inaudible 00:18:49] that kids often struggle with, that are part of a critical foundation for almost all the math that kids learn beyond sixth grade. At CORE, we’ve been working throughout the country with struggling math learners for the last 10 years, and these two areas along with work on rational numbers, I can tell you are typical areas of struggle that are part of the really necessary foundation for math success.
Dean Ballard: One additional note that’s on here, that last bullet, from the Center on Instruction, the National Center on Response to Intervention and the National High School Center, they all recommend that for tier two in high school that we pre-teach and re-teach concepts from the core curriculum. This can be done using supplements to the curriculum and or the core material itself. But remember, that’s for tier two in high school, not tier three, which even in high school, tier three definitely requires a focused instruction on foundation knowledge.
Dean Ballard: Well, we saw from both the IES guide and suggestions from Marilyn Burns that fluency with facts, number facts, is part of a necessary foundation for mathematics. I know at grade levels there are many types of fluency building activities, but it’s a myth that the only or best types are just straight up drills, flash cards or 50 problem practice sheets. Practice sheets and drills do provide one good type of practice called rote practice. It takes many repetitions for something to be retained when we depend primarily on rote practice. But again, it is one important piece of the learning pathway. But one very important truth is that there are many great fluency building activities that also build deeper number sense.
Dean Ballard: These are not drill worksheets. These involve what is called a lab rehearsal. Students are practicing with facts and procedures while simultaneously having to think about mathematics, about mathematical connections or solving a related problem or puzzle. For example, there are websites devoted to the challenge called Make 24. In the Make 24 Challenge, students are given four numbers, such as one, two, four, and 12. They have to use any combination of operations to get a result of 24 with those four numbers.
Dean Ballard: So, for example, I could take the quantity of 12 minus four, times the quantity of two plus one, which gives me eight times three. That equals 24. Or I could take 12 times four times one over two or one half. That gives me 48 times one half. That also equals 24. So, in having to find one or more ways to combine these numbers to make 24, students are using and practicing multiple math facts without being aware of it, while at the same time they’re thinking of mathematical connections between numbers, operations, and combinations of numbers.
Dean Ballard: The cognitive scientist, Daniel Willingham, who does a lot of work around education, says that, “Memory is the residue of thought.” We remember best what we actually put thought into. David Sousa, in his book, “How The Brain Learns Mathematics,” describes both rote and elaborate rehearsal as important. He emphasizes that without rehearsal there is almost no transfer of knowledge to long term memory. So, rote practice is good but elaborate rehearsal or practice that includes thinking about numbers, operations, and connections is also good, if not better.
Dean Ballard: By the way, for more on fluency activities that build number sense, you can go to our CORE website and look up the webinar we had last fall on Mathematical Fluency and Number Sense: Techniques, Access and Sustainability for All Students.
Dean Ballard: Well, a huge issue in many schools is when to provide intervention for students. There are many scheduling challenges with this for middle, high school, and for elementary school, where a lot of other things are required to be done during the week and during the day. So, it’s a mistake to let ourselves believe that all intervention needs can be taken care of through and during core instruction, or with just some tutoring after school. It is true that added support can and is provided in those situations. Teachers scaffold and differentiate, use various activities throughout the lesson to provide direct support for the lesson. Tutoring time, such as after school tutoring, is also a typical way some students get needed and added help. Students who need a little help can do well with this type of support.
Dean Ballard: However, it’s important to realize that these types of support will rarely move the needle on students who have significant gaps. Students with big gaps, consistent misunderstandings, and persistent comprehension issues need added, focused help. They need regular small group or individual instruction focused on those key areas of need.
Dean Ballard: In elementary classrooms teachers often use extended time, 75 to 90 minutes of class time is spent for math during each day. However, the biggest pitfall we see with this is teachers using the entire time, the 75 to 90 minutes, for one single but continuous math lesson. Even when the time is not self continuous, like maybe 60 minutes now and then 15 minutes later in the day, we see teachers using that whole time for just that one lesson. Student attention and retention diminishes with each passing minute. Return on instructional time is almost nil during the last part of these lessons. We have not yet witnessed students who are not getting it in the first 50 to 60 minutes having any aha moments in the last 15 to 30 minutes.
Dean Ballard: In these situations … Though I will say, teachers do usually recognize there are many and significant learning gaps that need to be addressed, but they are uncertain about how best to do so. What they are certain about is that they’re required to complete approximately one lesson per day. So, the lesson becomes the driver for the whole time. The lesson is a good driver for an approximate amount of time. Let’s say, 50 to 60 minutes. However, if necessary, lessons needs to be pruned to what is most essential so they can be completed within that time, that 50 to 60 minutes, and the rest of the extended math time needs to be used to address learning gaps.
Dean Ballard: It doesn’t need to be the last part of the math time, even. It can be the first 20 to 30 minutes, be used for small group interventions for some students while others are doing extensions or other work. We’ve seen teachers very effectively use programs on computers to have students rotate onto and off computers while the teachers works with selected students. The key is you have students with learning gaps. Use the extended learning time for additional support for those students, not continued core instruction. Because the extra support is what many students really need.
Dean Ballard: Generally, we find three types of support for general education students, all of which are effective when done well with the appropriate students. As mentioned before, in core class teachers use differentiations, scaffolding, warm-ups, and individual help while the rest of the class has independent practice. This is helpful for students needing some just in time support for recent or current or the next lesson. As also mentioned previously, many schools provide and assign students to a math support class in addition to a core math class. The extra math class is either focused on direct support for core instruction or it’s focused on foundational knowledge, as we described, or both, depending on the needs of the students.
Dean Ballard: One of the biggest [hesitancies 00:26:33] we see with this idea is that it does usually require using up an elective period of the students. So, let me just say that I think that enjoyment of an elective class often is just a short term gain, while success in math is a long term gain for many students. Despite the resistance of students to giving up elective time, we’ve seen this option be very successful. This is for students with significant gaps so the students also realize they need help. Instruction in these classes is very focused, either in direct coordination with the core instruction, and or depending on the needs of the students, on the foundations we described. I just want to add that as a math educator for over 30 years, I can tell you I’ve seen so many students who appear to be in excruciating pain during their math class.
Dean Ballard: I’ll say that even in some of my math classes, and I do a lot to try to engage kids and make it meaningful for them. But unfortunately, I’ve seen so many kids have … They just resign themselves to sit through that pain for one hour every single day. Perhaps all the way through 11th or 12th grade! I mean, that’s years of this, where they just know that every hour, that one hour every day, it’s going to be so painful for them. However, if through our added support, students could gain some success, believe in their ability to learn math, they will gain as much, if not more, from this support than they will from almost any elective class.
Dean Ballard: If nothing else, if closing that knowledge gap makes that one math hour out every day less painful, there’s a big win for the kid. The added confidence with math rather than a complex about math, this kid takes with him or into the world beyond school, is an immeasurable gain.
Dean Ballard: Finally, for students with at least two or more years behind, it is sometimes necessary to provide an intervention math class that actually replaces the core math class completely. Learning math is so sequential. Students who are way behind may not have the knowledge necessary to learn grade level mathematics. Think of it like learning a language in school. It doesn’t matter your age or what grade you’re in. You don’t jump to French five without first taking most of, if not all of, French one through four. You would just be completely overwhelmed, frustrated, feel defeated. Learn next to nothing and likely become a behavior problem. Gee, does that sound like any math students you’ve seen?
Dean Ballard: However, there’s also little evidence showing students ever get out of these lower tracks. So, assigning students to these types of classes should be done cautiously, and we really have to do more to help students succeed in moving beyond these classes.
Dean Ballard: Next, let’s look at effective teaching techniques and how these are similar to and different from the techniques that may be used for core instruction. One common complaint from teachers is that students come into their math classes unprepared, because students don’t know their math facts and are not fluent in basic procedures. This often leads to a common refrain we hear, and this learns to our myth number seven. Students just need lots and lots of practice with math facts. Well, of course it is true. Students do need plenty of practice, both immediate and distributed. This is even more true for students struggling in math. As we’ve seen in intervention, what we have them practice on is a subset of earlier math concepts, the key concepts that are most meaningful for helping these students move forward.
Dean Ballard: However, it is just as important to recognize that students rarely need extra support simply because they don’t have all their math facts memorized. As described in research and by math experts, these students still need understanding of concepts, fluency with math, and ability to apply the math. The key difference is in how we narrow that focus on specific concepts for these students.
Dean Ballard: Additionally, with more focus for struggling students, we need to make math connections very explicit. Take a moment here to read this statement from Marilyn Burns.
Emily: Students who need intervention instruction typically fail to look for relationships or make connections among mathematical ideas on their own. They need help building new learning on what they already know.
Dean Ballard: Thanks again, Emily, for reading that. So, it’s important for students to see connections between and within mathematical ideas. Conceptual understanding is the foundation for fact and procedural fluency, just as fluency is part of a foundation for continued learning and math ideas. However, students in interventions struggle to see those connections on their own or on their first pass through a concept. Therefore, we as teachers need to provide more explicit direct instruction that attends to these connections.
Dean Ballard: We still need to require students to think and reason about math and make some connections, but as teachers, we’ll do more of the heavy lifting than we typically do with a Tier 1 instruction. For example, suppose students are making a common error when adding fractions with unlike denominators, such as one half plus one fourth. They incorrectly add the numerators and the denominators to get the wrong answer of two sixths, rather than three fourths. To address this misconception about adding fractions, rather than simply reviewing the correct procedure again and again, which students in intervention will have likely seen multiple times already, a better approach is to return to the conceptual level and the important maths concepts this is built on.
Dean Ballard: Use a number line to start with, to visually show adding one fourth and one half. Provide a number line, such as this shown on the slides, and have students label the parts on the number line. Then have students justify the placement of one fourth and one third and one half. Make this clear for all the students, where those go and why they go where they go. Provide bars to visualize the size of the fractions on the number line. Show students how to make the drawing on their own number line that illustrate one fourth plus one half, then have them determine the value of these two bars when they’re put together, where they end up on the number line.
Dean Ballard: Have students locate their answer of two sixths on the number line and compare this to the visual of three fourths. We do that because this helps convince students that their answer … Hmm, maybe it really is wrong. Because they see that the answer should be three fourths. This provides a nice visual proof for that. Misconceptions can be like weeds. They’re very hard to get rid of. So, you really have to take the extra step to help students see that, “Oh! That answer was wrong. It’s not just because I have a procedure that gets a different answer. It’s because it really doesn’t look right. I mean, there’s a reason for it.”
Dean Ballard: So, we want to convince students about that. And next, remind students about adding whole numbers, and how fractions are like adding whole numbers. We add like units together. We add the ones to the ones, the tens to the tens, the hundreds to hundreds. Then we explain that one fourth and one half are not like units. They’re not ones ….. They have different denominators. You have to make them like first. So, in order to add them together you have to get them to be the same unit, just like we do with whole numbers. So, first we change one or both fractions so they have the same denominator, because the denominator is the unit of the fraction.
Dean Ballard: Now we move to the procedure for adding the fractions. We changed one half to two fourths, and then we add one fourth and two fourths to get three fourths. Have students identify this on the number line and compare it to the visual model. This kind of explicit instruction combined with student interactions around the connections within the math, allows students to see why a common error is wrong and why the correct procedure makes sense.
Dean Ballard: Next we would model and provide guided practice on a couple more like problems, and then have students practice this skill. Practice should include having students illustrate on number lines a couple of problems on their own to solidify the connections between the procedure and the concepts, in their minds. Then students can practice the procedures many times to build fluency with it.
Dean Ballard: So, you see from this, we include some student thinking and reasoning into this sequence of instruction by doing things like, we provide a number line without numbers and have students fill those in. Have students think about where those numbers are. Have them justify the placement of the numbers, one fourth and one half and one third on the number line. We ask them to figure out where the two sixths goes on the number line. Compare that to the answer of three fourths. We have students make a drawing of their own, of these. Let them see and make that illustration to show what the value of the fractions are and what they are when they’re put together.
Dean Ballard: So, we can be very explicit about the math. We can scaffold the problem to give access to all students and ask targeted questioned within the framework, that ask students to do some thinking that lays the foundation for the mathematical connections. The math connections that are going to be made clear, most likely by us. By connecting visual models to math ideas, we help students see and understand concepts, and see the sense in those give procedures. Connecting to prior knowledge, such as connecting fractions, operations to whole numbers, also reinforces the consistent and connected nature of mathematics.
Dean Ballard: To reinforce what I’ve been saying about instructional techniques, here are some evidence based recommendations, again, from the What Works Clearinghouse and the IES guide on instruction of students struggling with math.
Dean Ballard: Instruction should be explicit and systematic. Provide models of problem solving. Verbalize our thought processes. Provide guided practice, corrective feedback. Frequent cumulative review. Intervention materials should include work with visual representations of mathematical ideas, and include motivational strategies in tier two and three interventions. Additionally, here are six recommendations from the RTI Network on instruction for intervention. Instruction should be explicit. Instruction should ease the learning challenge or the cognitive load on students. For example, with the fraction problem, I provided a number line with tick marks matching what was needed, but not a bunch of other tick marks on there. I still ask students to label the number line and justify their labeling, but I reduce the overall challenge, the cognitive load, by taking the first two steps for them.
Dean Ballard: Provide strong conceptual basis for procedures that are taught, just as that model with the fraction addition problem. Provide plenty of practice. Again, straightforward practice is important. I want to also reemphasize though that over the last few years, many excellent resources, including our own book, CORE’s own book, titled, “Spend Some Time With One To Nine,” provide practice that includes building deeper number sense. Incorporate plenty of both rote and elaborate practice. Thinking back to that fraction addition problem again, at the end I mentioned that one of the things to have students do was to have them create the number line visual models for a couple of fraction addition problems on their own, and connect it to solving the problem with the correct procedure.
Dean Ballard: This is that elaborate practice piece. In this case, would be followed by many more fraction addition problems for students to simply solve procedurally without their drawings. That would be the rote practice.
Dean Ballard: Cumulative review is important. Research clearly shows that without review we forget things. It’s just a natural act of the brain. Finally, do things to motivate kids to maintain attention and work hard. Have students collaborate. Recognize the hard work and effort. Continually express confidence in them and in their ability to understand math, as long as they work at it. Most of all, help them succeed. Make sure they see their own success, because nothing builds confidence and motivates more than success.
Dean Ballard: Now I want to talk about who are the teachers that provide intervention support. To start with, because the most common resource for intervention in math has become computer programs and online resources, I want to talk about the computer as the teacher. Although varied teachers and school leaders say it out loud, at least not to use, we often see actions that indicate many educators are secretly hoping that the computer programming is the intervention teacher. You know what? I cannot deny, nor do I want to, that there are many really good computer based intervention systems and online resources today that provide diagnostic assessing, progress monitoring, individualized and adaptive instructing, fluency building activities, and even rewards.
Dean Ballard: Computer programs are getting better each year, and they do provide a great resource for addressing needs of struggling students. However, teachers teach. It’s vital that we understand this. The computer programs are aids in the classroom. Computers are great tools when used appropriately. Because computers are used so frequently within intervention, I’m going to go over just a few pros and cons and dos and don’ts based on our experiences at schools all across the country. Again, that’s because computers really are so prevalent now with math intervention.
Dean Ballard: You know, Khan Academy really revolutionized our thinking around computer instruction for math. Computers went from a resource to be seen as perhaps a teacher that can teach at all. However, keep in mind that just because a program can provide all the aspects of assessment, instruction, review, practice for all concepts to be taught, doesn’t mean students benefit from trying to get everything from a computer. Individualized instruction is great. Glossy eyed tapping on the keyboard is not so great. Selecting and keying in answers on a computer is one level of practice in assessment. That’s good. However, having students actually explain and diagram their work is another level of thinking.
Dean Ballard: So, here are some things that we’ve seen that do not work. Unmonitored use of computers. Students will quickly figure out how to get [inaudible 00:40:08] tabs and screens, mindlessly tap out incorrect answers until the right one pops up, or just sit and stare at the computer. When we’re teaching in front of a room, we don’t have our backs turned to the room for a reason. We know we need to continually monitor students. Being in school is just not a natural state, and students are generally here because the adults in their lives believe they need to be. So, we as teachers need to continually confirm to students that their attention to learning is important. Our attention to students does motivate kids, and just because they’re on computers, they still need to know we’re checking in on their progress and work because it’s important.
Dean Ballard: Unlimited use of computers does not work. By this I mean students are left on the computer for 50 to 60 minutes, for the whole class time. We’ve found time and again that the typical maximum time in one sitting that students remain focused on the computer is around 20 to 30 minutes. Past this they are spacing out and losing interest. Learning seems to grind to a halt.
Dean Ballard: Paraprofessionals put in charge of the room. It’s not the best idea. This is nothing against paraprofessionals. Paraprofessionals can be great classroom assistants, but they shouldn’t be tasked with doing things that they’re not trained to do, nor that they’re paid to do. They’re not there to run the whole class. They’re rarely trained or equipped to manage a class on their own, and even more rarely have the math expertise to address the learning gaps for struggling students. Consider the fraction addition problem we just did earlier. I’ve seen parents help students with this, but frankly it’s always been [there 00:41:47] to simply reiterate and help them practice the correct procedure with no conceptual knowledge or math connections included.
Dean Ballard: And I don’t fault the parent, because this is what he or she remembers from the math they took and they’re trying to help the student, and the student wants that help. “What am I supposed to do? How do I do it right?” And so, that’s the help that they can get, typically, in those situations. But students in intervention classes, they need really good math instruction. So, besides the parent, they need the teacher there to be the teacher.
Dean Ballard: So, what have we seen that does work? Well, like I just said, the teacher is always in charge. The teacher is teaching. The teacher is managing and guiding instruction, and monitoring and motivating progress. This includes monitor computer learning, keeping an eye on what students are doing and how they’re progressing on the computer, just as you would with paper and pencil. Walk around and see what’s going on. Let students know that on your teacher dashboard, you’re also checking on their progress as well.
Dean Ballard: Limit the computer time. As I mentioned, we find 20 to 30 minutes to be a good rule of thumb for maximum time on the computer in one sitting. Many computer based programs are actually starting to include that as part of the expected routine when using the program, such as Math 180.
Dean Ballard: Use time for small group work. While some students are on the computer, many teachers have found this an ideal time to work with selected students in small groups. This allows the teacher to provide the instruction that can really benefit those kids. Teachers often rotate students with half the class on the computer for 25 minutes, while the other half is working on some paper and pencil activity. The teacher works individually or in small groups with some of those students who are not on the computer. Then after 25 minutes the students rotate. Those that were on the computer go to pencil and paper activity, while those that were not on the computer now get on the computer. Teachers during extended time in core instruction also often use computers in the same way. Although computers can get boring, that’s why I say provide a mix of instruction, rotate groups on and off, as I’ve talked about here.
Dean Ballard: Lastly here, train students on the use of the computers. Students need to learn the routines you expect for use on the computers like anything else. Time spent training students on their routines is time well spent and saved over the long run.
Dean Ballard: Oh! I forgot. I had one more thing I wanted to mention here. Students can keep a learning log or journal for their computer work. Students write down a brief explanation and example of one main thing they learned during each session they were on the computer. Think of this as the exit ticket for the computer work. This provides you with insight into their progress, provides another level of accountability for the kids, and most importantly, provides students an opportunity to review and deepen their knowledge so they’re more likely to retain what they have learned. It also provides you, students, and their parents a type of record of their learning. Everyone can see as you turn the pages on that journal, the progress they’ve been making over time.
Dean Ballard: So, next I want to talk about, just kind of wrapping up this part on who will do the interventions. Well, as you can see, my answer is teachers do. I’m big on teachers working with small groups and organizing their instructions so they can do this kind of intervention kids need. Keep in mind that research on RTI assumes small group work for tier two and small group and individual work for tier three. I know it’s not always possible, especially when the intervention class is a big class, but even with lots of students, use of stations, computers, independent work time, these kinds of things help provide opportunities to also provide small group instruction. We just have to organize our classroom in order to do it.
Dean Ballard: And of course, as stated in previous slides, computers and pairs are resources. We just have to use them wisely. Now, about the materials to be used for interventions. Unfortunately, we’ve been seeing some floundering intervention instruction where teachers do not have specific materials to use for their intervention classes. Part of what feeds into this circumstance are two commonly held beliefs. Myth number nine, teachers themselves can create the intervention materials and or program, and myth number 10, re-teaching and intervention can easily be done using the core program materials. Just use lessons from prior grades.
Dean Ballard: So, yes. It’s true that for support directly linked to core instruction or current lessons, teachers can use a variety of resources they draw from, including the core program and their own experiences and things that they know about. However, we can say that for more intensive support to address significant learning deficits, teachers do not have the time or expertise to develop teaching materials. Students in these situations may be two or more years behind in key math areas and need more than a diet of worksheets or homework help for today’s lesson. They need well designed math support and materials that address those needs. Just the grade by grade development of key ideas that’s within a core program, that doesn’t help these students catch up because those lessons are designed for a regular paced core instruction, for students that are on pace at their grade level, not for students who have these gaps.
Dean Ballard: There are many material resources that can be used for math intervention, as mentioned, in some cases the tier one core materials are a good resource. There are a lot of good supplementary resources available as well, which teachers use, which again, are appropriate in some cases for the less intensive needs. Teachers can piece together resources based on what they’re targeting with students. There’s supplementary resources we see teachers taking advantage of that are not full intervention programs, such as assigning things from Khan Academy or Reflex Math, MobyMax, ArcAdemics.com, KenKenPuzzles, worksheets, practice problems, released items from assessment sources, fluency activities that also build number sense, such as Spend Time with 1 to 9, and extra resources included in the core programs often. There’s others. So, there’s a lot of supplemental resources to draw upon in the right situations.
Dean Ballard: There are also full intervention programs that provide a coherent focused curriculum for intervention. There are computer based programs, pencil and paper based programs, and there’s some that include both. Look for programs that mostly focus on key areas for math intervention, as described earlier, and provide for the type of instruction described earlier. Don’t expect a perfect program though. None exist. As always, the teacher will need to be in charge of that instruction and make adjustments as needed.
Dean Ballard: Lastly, I want to emphasize the importance of assessing both student progress and program progress. Often we struggle so much to get a system in place and to collect student data, and then use the data to make decisions about individual students, which is all good and right. However, that data on student progress is also vital for informing us about our interventions themselves. Are we doing a good job? Is it the right intervention? My assumption is that nothing is perfect and whenever a new system is put in place, such as interventions might be, even less is perfect. So, I’m going to want to revise and improve my program, and the evidence I need to help me is the data on students. So, use it.
Dean Ballard: Some typical things to look at for changing or adjusting based on the results you’re getting from the students are frequency of interventions. How often do students get the extra help? Is it three times a week? Five times a week? Once a week? Is it working that way? Intensity. The time and the group size. How much help are students getting in each session? How long is the intervention time? How many students are getting the intervention at the same time by the same teacher? Materials. Are the materials an issue, or is it more about learning how to use the materials? Pacing. Are we trying to do too much too fast? Are we going to slow? We don’t want the students falling asleep and we don’t want them being totally confused because we’re going to fast. Personnel. Are the right instructors providing the intervention?
Dean Ballard: Well, there you have it. We looked at 10 different myths revolving around math intervention. I know, yes, calling the myths was probably unfair since each of them has some truth to them. However, I wanted to show how those ideas I sided as myths are often misconstrued or over-applied, and to highlight important points about identifying which students need added support. What math is most important to teach to these students? When are the opportunities in the school day to provide that extra support? What are the best evidence based practices to use for interventions? Who are the intervention teachers? What materials can and should be used for intervention, and the importance of not only progress monitoring students, but to apply that data to progress monitor our interventions also so we can revise them and make them more effective.
Dean Ballard: So, that about wraps it up for me. So, I’m going to turn it back over to Emily right here.
Emily: Great. Thank you so much, Dean. You shared a lot of information and great strategies with all of us, any many of you did ask if you’d be receiving the PowerPoint slide as well as the recording, and the answer to that is yes. You will definitely get all of this information that Dean has shared. We will send out an email tomorrow with a link to access all of that and we will also post it to the CORE website, which corelearn.com. It’s there on your screen there in the footer of the slide.
Emily: So, before we begin the Q & A, I did just want to thank CORE for sponsoring today’s webinar. As Dean mentioned, he works regularly with schools and educators to help build their instructional skills around math, and to help implement evidence based instructional practices with their chosen programs. Working together, CORE helps schools and districts and implement a plan to raise achievement for all students, including English language learners and those struggling to master math.
Emily: So, you can look for more information, again, on the corelearn.com website. While you’re there, you can also get some free activities from samplers of the Spend Some Time With 1 to 9 supplemental materials that Dean referenced a couple times throughout the presentation. That URL is also on your screen and we’ll be providing it in the email you receive tomorrow.
Emily: So, we do have a few minutes for some questions, so we’ll go ahead and dive into those. One that has come in is do you have any recommendations for specific diagnostic measures that teachers or tutors can use to guide their instruction?
Dean Ballard: In math, we are like, a decade behind reading in this area, both in research and in developing materials for diagnostics. Probably some to look at are like, Easy CBM or the Dibble [Sets 00:52:47] that’s now part of Dibbles. Dibbles, which has always been providing reading diagnostics. They now provide math. We’ve heard some good things about looking into iReady. So, we’re going to start looking into that one as well.
Dean Ballard: Most computer programs now that have intervention, they have the diagnostic built into it, so it’s just part of what they’re doing. So, there’s … I can say there’s limited right now, but keep looking for those things to be developed.
Emily: Great. Do you have recommendations in terms of an ideal group size for small group interventions?
Dean Ballard: Yeah. If you’re working with a group directly, so like, you’re sitting at a table and you have students there, we’re talking about three to four students, maybe five at the most, just working directly with the teacher on one thing, and if we’re talking about a classroom, then it really depends on what are the other resources you have. If they’re not … If you don’t have a lot of other resources to use, besides maybe a program and some other things that you have, then maybe 13 to 15 is probably an ideal size for a math intervention class.
Dean Ballard: When it gets above that you really start to have to have maybe, like I said, a computer intervention program or some other resources, so half the class can be on that resource while you’re able to pick up and work with small groups or individual kids. So, what we’re seeing a lot of us schools where they have a lot of students who need help. So, these intervention classes get really too big. That’s just a matter of resources. You know, how many … The school only has so many resources available to them, so many teachers and stuff like that. So, that becomes a challenge.
Dean Ballard: But like I said, it’s a matter of organizing the class around those resources and figuring out, “Okay, if half the class will be here working on this, where they can work independently, then I can work with these students individually or in small groups.”
Emily: Great. That was very helpful. So, another question that has come in is in elementary school in particular, we see that many of our teachers aren’t comfortable with interventions. They, in fact, sort of act like you describes the para acting. Do you recommend that students even in tier two of intervention should work with math specialists?
Dean Ballard: Well, I would say that in those cases, the people who most need to work with that math specialist are probably the teachers. I say that because those teachers are providing both tier one and tier two instruction. Tier one, if they’re struggling to provide conceptual understanding with tier two, I worry that they might be struggling to do that with tier one as well. I know that for … I know a lot of elementary school teachers who just honestly say, “Well, math wasn’t … I didn’t really like math, myself. I basically remember the procedures but I’m not that great at it.”
Dean Ballard: So, they struggle to do more than really focus on procedures. So, that would be my first … If you have a math specialist, the teachers themselves need to spend some time with that specialist as well, on maybe just some key areas of focus. Because it’s going to get … That math specialist would be just spread too thin trying to reach all the kids. So, that would be my first recommendation.
Dean Ballard: Other than that, looking at materials, helping the teacher start looking at materials would be an important thing, so they can see it there. So, I’m really not getting away from saying the teacher’s still going to have to do it.
Emily: The teacher needs to learn the math first.
Dean Ballard: Yeah. Now, there can be a situation where those students are pulled out or they have another class or they have some other situation where they get together as a small group with a math specialist. In that case, I could see that working. That all, again, depends on how many students we’re talking about and how many math specialists you have.
Emily: Great. Let’s see. Do you have suggestions for high school students who are more than three years behind in math?
Dean Ballard: Yeah. Well, this really gets to … And I have a lot of experience teaching those kids myself. We used to have math A, math B, and math C that were all pre-algebra classes. We had, you know, ninth through 10th and 12th graders in those classes. It’s just … What I’ve found to be most successful, first, you really got to believe in their ability to understand if they work at it. I’ve always found that to be true, that they do understand math well enough to see what’s going on. You’ve got to get back to those basic foundations for algebra for them. Provide a pathway to get through algebra and geometry in high school. You’ve got to help them see, “Okay, if you get this course then you can get that under your belt, you’ll be able to succeed I algebra and then be able to succeed in geometry and beyond.”
Dean Ballard: But if they’re just staying … If they’re that far behind and they’re still in an algebra class, that’s, you know … I find that that’s very difficult. I find the kids getting very, you know, behavioral problems because they don’t know what’s going on. Like I said, I also see that kids get stuck at that lower track when they get placed in that class, they identify themselves as being, you know, poor at math and this is their class, then they don’t work hard in that class, and they never seem to get the knowledge that’s the foundation for math. So, they continue to perform and look like this for years behind, and they never get out of it.
Dean Ballard: So, it really is a scenario we have to do a lot more work at in helping students in those classes have that foundation and be motivated to work and understand it. Part of that, like I said, goes towards a lot of teacher belief in those kids, and being honest with them about, “Okay, here’s what you can do.” But I want to also mention that, as I said in the webinar, success breeds success. If kids can find themselves being successful at some level, then they start to believe in it and that starts to motivate. So, you really got to find something where they can succeed in the math, make sure they understand and see that, and start to build on that.
Emily: Yeah. That was a large question, but you gave some good suggestions as a starting place. I’ve got another large question for you. We really are at the end of our time. I’m going to go ahead and ask this one, and see if we can just squeeze it in real quick.
Emily: We are providing 30 minutes of daily math intervention at our elementary school. We wrestle with that question about helping students with the concept currently being taught in core versus trying to fill in prerequisite gaps versus pre-teaching the coming concept that will be taught in core class. Do you have any, you know, super quick advice on that?
Dean Ballard: Yeah. Well, I think that you’re probably going to try to find that you have to divide that time up. You have to, first of all, figure out which students have really significant gaps that have to be addressed. I mean, if they’re in fifth grade working on fractions and they don’t understand what a fraction is, you might have to … You have to work on that.
Dean Ballard: But at the same time, there’s continuing with core instruction and they may, you know, that may be part of that, but that time [inaudible 01:00:00], okay, what was the essential idea or the foundation that’s most important for that core idea that they learned in the lesson, and do a little bit of work around that. So, I think for those students, because they’re really in both camps, a lot of them are, where they’re continuing with core instruction, as they’re going to do in elementary school, but they have big gaps. So, they probably need a little bit of both. Or I say a lot of the gaps, if those gaps are big, but still need a little bit of support for their core instruction.
Emily: Great. Well, I’m sorry we were not able to get to all of the questions that came in, but we will share these with Dean. Again, you guys will get the PowerPoint and recording so that you can spend a little more time with this.
Emily: And again, I’ll remind you, speaking of spending time, to take advantage of the free activities that are available in the Spend Some Time With 1 to 9 samplers that you will receive a link to in the follow-up email that you’ll get tomorrow.
Emily: Thank you again, Dean, for sharing all of this. Thanks to everyone for joining us today. Have a great afternoon or evening, depending on what time zone you are in.
Dean Ballard: Hey! Thank you and glad everybody joined. Have a good week!