Emily: Hello, everyone. Thanks so much for joining us for today’s webinar 4 Must Do’s for Math Instruction. Before we begin, I’ll review just a few quick housekeeping items. We will be accepting questions throughout the webinar for a Q&A period at the end of the presentation. Please send us your questions as they come up for you using the questions feature in your control panel, you’ll type your question into the top box and click send and I’ll receive that question and I’ll put it into the queue to be answered.
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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. Just keep an eye on your email tomorrow for links to access those materials, we’ll also post the recording to the CORE website at corelearn.com. Sometimes those emails do end up in a junk or a spam folder, so be sure to check those tomorrow afternoon if you’re looking for the email and don’t see it in your inbox or you can head on over to the CORE website and find the materials there.
Now, let’s get started. I’m pleased to welcome today’s speaker, Dean Ballard, who is director for mathematics with CORE. Dean holds a master’s 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. 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’s a member of the National Council of Teachers of Mathematics, National Council of Supervisors of Mathematics and the California Math Council. Dean has a lot of information to share with us today, so I’m going to go ahead and turn things over to him so we can dive right in.
Dean Ballard: Thank you Emily, and good afternoon everyone. Thank you for joining the … Thank you for joining the webinar everyone. Thank you, Emily. So today the essential questions that I wanted to explore today are how do explicit instructional techniques apply to both direct and inquiry-based instruction? Why is student discourse important and what are tips for generating meaningful discourse? Why is vocabulary important in math and what are strategies for building math vocabulary? And what are some easy to use differentiation techniques and how these help students?
And just by the way, just wanted to mention that I know that there are a lot more than just these four there’s more to good teaching in mathematics and just these four things. But I only have a one hour webinar here to work with and then I had to pick four things to talk about. So I chose these four, which are really important things.
Right, so let’s kick it off here, while we’re going through ideas and samples of lessons today and different slides, I want you to think about the following conclusions derived from a meta study of research around the question. What is it about teaching that influences learning?
According to the meta-analysis of research, Hiebert and Grouws concluded that two things make the biggest difference overall. And that teaching attends explicitly to concepts, to connections among mathematical facts, procedures, and ideas. And the students are regularly engaged in struggling or wrestling with important mathematical ideas. That students expend effort to make sense of mathematics, to figure something out that it’s not immediately apparent that students are solving problems that are within their reach.
Hiebert and Grouws further explained that attending explicitly to concepts means treating mathematical connections in an explicit and public way. And this can take many pedagogical forms including discussing the mathematical meaning that underlies procedures, asking questions about the ways in which different strategies for solving problems are similar to and different from each other, and reminding students that what the main point of the lesson is and how this point fits within the current sequence of lessons and ideas.
Hiebert and Grouws also state that both student centered and teacher centered approaches to instruction can provide targeted and structured activities in which students, with the teacher guidance grapple with challenging concepts and problems. In fact, providing sufficient boundaries and direction so that student learning is centered on important mathematical concepts does require some amount of teacher guidance.
We’re going to do something a little different here, this today start to work off and that is, we’re gonna do some math jokes. And you have to figure out what math term is the answer to each of these questions? And I’ll give you a few seconds as I’m reading them over to read them over and think about it. So what type of gin do mathematicians prefer in mixed drinks? What do little acorns say when they realize they are all grown up? What should one do when it rains? What animal is known as the king of the beasts? And the morning after the angle had too much to drink, what did it feel like?
Some of you may have figured these out and some of you may have figured some of them out, and some of you may not have figured any of them out, which means you are a mixed group of math students just like I always have in a classroom. I could scaffold this activity a bit in a way that may help you some, I could show you a list of terms and tell you the five terms that answer those questions are on this list.
I could actually scaffold this a little further, reduce the number down to seven terms and tell you the fiber within these seven. I could even eliminate the two that are not on here and say, “These are the five terms, just match them to the ones that go with,” which would still leave you with needing to explain to me why they make sense. So I might leave you that bit of thinking still to do on your own.
So, just to make sure everybody is on the same page with the answers, I’m going to go through these real quick. What type of gin do mathematicians prefer in mixed drinks? Well, that’s origin. What do little acorns say when they realize they’re all grown up? Geometry. What should one do when it rains? Go inside. What animal is known as the king of the beasts? The lion. And the morning after the angle had too much to drink, what did it feel like? It felt like a rectangle.
All right, let’s step back for a moment here to imagine this as a math lesson or a task in a lesson. I can imagine presenting this task as an inquiry type task in which you’re given the problems to work on with a minimal bit of a direction to start with, I asked you to talk with a partner and brainstormed some ideas around each question. Even as an inquiry task, I can still scaffold at different levels depending on your needs, even at the last bit, I was showing five answer choices, I still had … You have to explain why they made sense.
I could also see approaching this completely differently from a direct instructional approach. I begin by showing the first joke and telling my students which term is the answer and explain why. I know it doesn’t seem like there’s a big a-ha there, the Joy, the punchline seems diminished when I do that. But then I’d give the class the next one for guided practice and I provided maybe two or three terms such as geometry and Algebra, choose from and probably let them talk in Paris for a moment to see what they think is the answer and why, and then process that as a whole group.
And then I’d get the class, the other three to practice on independently, providing some answer choices to choose from. Within both those approaches, students are still engaged, the content is taught, math connections are made explicit and students are grappling with thinking through some of the problems in the connections at least.
The key to success is the use of explicit instructional techniques for both methods. So now, let’s look at what I’m talking about when I talk about explicit instruction. Explicit instruction is a systematic instructional approach that includes a set of delivery and design procedures derived from effective schools research. It’s an unambiguous and direct approach to teaching that incorporates instruction, design and delivery.
From this, we see that explicit instruction is systematic instruction and related to both lesson design and lesson delivery. With the two approaches I described with the math jokes, the lesson was systematic in both approaches. I had a plan from where I wanted students to go. I prepared the materials to address the needs and get students moving in the right direction and I prepared to make those connections clear at the end.
However, there’s more to it. Explicit instruction includes these four less than design elements. Instruction focuses, focuses on critical content. We teach skills, strategies, vocabulary, terms, concepts, rules and facts that will empower students in the future.
Fortunately, standards and or curricula are designed to do the heavy lifting here for us, targeting and sequencing lessons. So there’s a greater focus on the most critical content as well as on other related and necessary content. Secondly, break down complex skills and strategies. Skills, strategies and concepts are sequenced logically, complex skills and strategies are broken down into smaller instructional units that are manageable. Easier skills before harder skills, high frequency skills before low frequency skills. Pre-requisites first.
Next, provide systematic instruction, careful planning goes into constructing a lesson. Lessons are on topic well-sequenced. Students know what is expected, including targeted outcomes as well as how information or skills will be presented. Before the lesson review, relevant information, linking new skills with past concepts. And this pre-lesson work is also an opportunity for the teacher to verify the students had the pre-requisite skills, and include in this section that skills and the decisions needed to complete a task or modeled and clarified, such as using clear, concise language, giving clear directions for guided work, independent work or investigations and providing examples and non-examples.
And finally for the design of the lesson, provide judicious practice, provide guided practice to build initial success and confidence and distributed and cumulative practice to support retention, automaticity where needed, proficiency and independence with a concept or a skill.
Also, lesson delivery, explicit instruction includes these five lesson delivery elements. Frequent responses, having students respond frequently, helps them focus on the lesson content provides opportunities for student elaboration and keep students active and attentive. This also supports teachers in monitoring and learning, checking for understanding, monitor student performance closely, watch and listen, verify, understanding and adjust instruction as needed.
Provide immediate affirmation and corrective feedback. Immediate feedback of student work and student responses supports a high rate of success. So students are not continuing to practice the wrong methods, while it is often very useful to use errors as an effective source for learning through classroom dialogue. Allowing students to practice the wrong method can just ingrain the wrong method.
Deliver the lesson at a brisk pace. Pace will vary with the size and difficulty of tasks, the learners and on task behavior. However, lesson still should be at somewhat brisk pace, still allotting some time for thinking and processing, of course.
The desired pace is just one that’s not too slow, you don’t want students getting bored but not so quick that students can’t keep up. And then help students organize their knowledge, organizing information and making connections apparent and explicit help students use, retrieve, integrate new information. This means making explicit how ideas in math that are linked together and build on one another. So explicit instruction consists of these two big components, the design of the lesson, how the content is organized and accessible to students, and the delivery of the instruction.
Well next I’m going to discuss connections between explicit instruction, direct instruction and inquiry-based learning. And after that we’ll look at some sample lessons to highlight elements of explicit instruction with both direct instruction and inquiry-based learning.
Well, now let’s dig into this idea of direct instruction and inquiry-based instruction or investigations, explorations or discovery. There are a lot of different names for similar ideas around inquiry-based instruction. Explicit instruction as I’ve just reviewed is a set of instructional techniques that work effectively. One of my key points to you today is that these techniques should be applied to both direct instruction and to inquiry-based instruction.
I found that a lack of focus, clear connections and lesson pacing often where inquiry or discovery fall short, and a lack of student engagement and activity in the lesson pacing where I found direct instruction to fall short. I suggest that when we see direct instruction that is not working and when we see inquiry-based learning that is not working. It is most often because implementation of these methods is not following the general evidence-based techniques of explicit instruction. I know the short descriptions I’ve given to both direct instruction and inquiry-based instruction do neither of them justice.
I know proponents of either method will tell us that much more is involved and that is just it, much more is involved in that more is essentially explicit instruction for both. As we look into this, a little more today, I’ll illustrate, with examples of lessons, what I’m talking about with explicit instruction, direct instruction and with inquiry based instruction.
Keep in mind it’s not an either or, it has been a controversial issue and this seems to have been the centerpiece of the math wars in the past about whether classroom instruction should be more teacher directed or more student-centered. These terms do encompass a wide array of meetings with teacher directed instruction can range from a highly scripted direct instructional approach to a very interactive lecture style.
And a student center instruction can arrange from students having primary responsibility for their own mathematics learning with a wide open discovery activity to a very highly structured activity in group work. But it’s not an either or situation, all encompassing recommendations that instruction should be entirely student centered or teacher directed are not supported by research and yet, we also do know the teachers are often left wondering about, well then in that case, when do I organize instruction one way or the other, whether a certain topics are taught more effectively with one approach or the other, and whether certain students benefit from one approach or the other.
The research as a mixed bag on this, it’s an inconclusive picture of the relative effects of each type of instruction and there are studies on each side that show significant student gains. In fact, Hattie, Fisher, and Frey note that both methods are actually effective, and effect size above 0.4 is one that is considered effective. Effect sizes for direct instruction and for dialogic instruction are shown on the slide here.
And by the way, dialogic instruction correlates to dependence on student dialogue, investigations in inquiry. The authors point out that this doesn’t mean teachers should choose one approach over another, it should never be an either or situation, both dialogic and direct instruction have a role to play throughout the learning process but in different ways. Both of these are well above 0.4 so we would consider both to be very effective. However, I want you to keep in mind, this is also based on the definitions of direct instruction and dialogic instruction that Hattie, Fisher, and Frey worked with. So I want to look at those definitions next.
Emily: The teacher decides the learning intentions and success criteria make them transparent to the students, demonstrates them by modeling, evaluates if they understand what they have been told by checking for understanding and retelling them what they have told by tying it all together with closure.
Dean Ballard: Well, we see here a high correlation between this description and the list of explicit instructional techniques we saw earlier.
Emily: In the dialogic model, students must a) actively engage in new mathematics, persevering to solve novel problems; b) participate in a discourse of conjecture, explanation, and argumentation; c) engage in generalization and abstraction, developing efficient problem-solving strategies and relating their ideas to conventional procedures and to achieve fluency with these skills; d) engage in some amount of practice.
Dean Ballard: So this description from researchers, Munter, Stein, and Smith, is the description Hattie, Fisher, and Frey worked with. You may notice that this description seems a little less overtly connected to the list of explicit instructional techniques that was then, then what’s the direct instruction description on the previous slide.
However, if you know your mathematical practice standards that are a part of every state standards and part of the common core, you’ll also recognize that this description very closely aligns to those math practice standards. And there are certainly similarities between this description of dialogic instruction to the list of explicit instructional techniques. And we’ll argue and demonstrates the parts that seem to be missing are actually embedded and essential for making this dialogic construction work.
And by the way, I just want to mention that according to research from the TIMSS, video study in 1999 that in the United States, teachers tend to transform inquiry-based lessons into direct instruction. And I still see this all the time. Unfortunately, what we practice seems linked to how we learned ourselves, and let me point out how ironic it is that in a country in which math phobia seems to be a cultural norm, we continue to teach the way we were taught. And what is even more unfortunate as you will see when we look more closely at both good direct instruction and good inquiry-based learning, is that both of these highly effective techniques are often not what we’re seeing in classrooms.
And certainly not what I experienced as a student or it was taught in my teacher credential classes. So what I’d like us to do today is to put aside any bias that we have towards methodologies at this time, and consider the idea that both direct instruction and inquiry-based instruction can be very effective. It’s just a matter of doing each well and that’s the right time. So next I want to look at some lessons modeling this.
Daniel Willingham says that, and he’s a cognitive scientist, works a lot of mathematics. That memory is a residue of thought, we remember best that which we really put thought into. So the more I can get students to think about the math we’re focused on, the more likely they are to retain that information. But if the information is beyond their reach, then they won’t be retaining any of it. So my challenge with whatever method I use to get students to do some important thinking around the mathematics is to get them to actually be thinking about it.
What I have to make sure is that the table is properly set, that the food’s digestible, that they can get access to it. So here is a lesson in which students are to learn how to determine the area of triangles. Diagrams such as the one shown on the slide provide an excellent opportunity for students to see the connections between the areas of triangles and the areas of rectangles to fit exactly around those triangles or what we call the rectangle circumscribed the triangles.
On the next several slides, I’ll outline examples of what I consider a poor and good direct instruction and poor and good inquiry-based instruction. And an example of what I would consider poor direct instruction would be, if I were to just say to kids, well, diagramming this on the board for them to see or on a document camera, the area of triangles is computed by the formula one half base times height. The base can be any side of the triangle you choose, the height is an altitude or line from the vertex office at the base, directly to the base and perpendicular to the base.
Here are some examples of computing the area of triangles. So then I would show students examples, doing the math myself and telling students to take notes. Then I might have the students turn to a certain page in their book and compute some areas of triangles on the page for practice. So this is a very straightforward example, I tell the information to students, I’m modeling on the board or screen as students take notes then assigned some related practice problems. Pretty much me doing all the talking and the thinking.
Okay, so that’s not what I consider a good example of direct instruction. There are many important elements of explicit instruction that are missing. Here, on the next slide is another approach that could be taken with direct instruction that does incorporate the elements of explicit instruction. And by the way, the different font colors signify nothing in particular about what’s being said, they’re just there to alternate the text. So if you’re trying to follow along, it might be easier for you to read it.
So good direct instructional model, for a warmup of the class I would have the students solve some rectangle area problems and we would process those solutions together. And then with a handout on the document camera that the students also have, I would say the students, the goal today is for you to learn how to determine the area of triangles and connect this to what you know about finding the areas of rectangles.
All I have students determine the area that our triangle by accounting squares and next I would have them share out as a whole class and I’d verify with students the area that they’ve encountered or that they’ve got that right. Next, I described the circumscribed rectangle around this triangle and have the students determine the area of the rectangle using the formula they know that that base times height formula.
And then we have a whole class share and clarify as needed. Next, I have students determine the areas of the second rectangle or triangle. Then again, we have a whole class year and verification of those areas. And then I could ask students, look back at our two rectangle triangle sets and what do you notice about the areas? Have students then talk to a partner for a minute and share some ideas out whole class.
Then we have a whole class discussion I suggest, I confirm and help verify is needed that the triangles are half the areas of the related rectangles. I show the class how to write this as a formula, we discuss connections of the triangle area formula to the rectangle area formula and connect back to the diagram.
I then have students use the formula to determine the areas of a third set of diagrams and to verify the triangle area by counting the squares if needed, and to see that the formula comes out with a good answer, a reasonable answer, a direct answer. And then I would sign another problem with the practice, process, whole class, check for understanding and then assign some classes for students to practice on.
This time, we see the goal clearly said at the start. Greater student involvement in engagement, students doing some of the thinking, connecting ideas together, having students use prior knowledge to connect to new learning. The lesson is systematic and organized to keep moving forward step-by-step at a good pace and frequent student responses are required. Student performance is monitored with immediate feedback provided in practices included.
So important mathematical connections are made explicit and students are grappling with important mathematical ideas themselves as well. All right, so now I’d like to look at an example of what I would consider a poor inquiry-based lesson. I put a copy of the handout that students each also have of the three triangles on a grid, on the document camera to show the class. And I say to the class, you have a paper with some triangles and rectangles drawn on them. Determine the areas of the rectangles and triangles and write about what you discover or any patterns you notice. After students work for a while, I have students present their answers, acknowledging strategies for determining the area of the triangles and assign some additional work.
So we don’t see yet in that a goal, a verification of a prior knowledge and explicitly connecting important concepts together for this lesson. So what would I do? Well a good inquiry-based lesson I think would include these elements. Again for a warmup, like with the direct instruction, I’d have the class solves and rectangle area problems. We’d process those together. With a copy of the handout on the document camera, I begin again and the lesson just as I did with the direct instruction model stating the goal of the lesson and then going over the first triangle rectangle set with students doing much of the work as before and I’m clarifying, correcting as needed.
Then I started the inquiry portion of the lesson, telling students that there are four more sets of triangles with circumscribed rectangles on index cards that I’ve given to each group. Each group members to take one, copy it onto their graph paper and determine the areas of the triangle and the rectangle.
After about four minutes, I have students rotate the cards around their group, draw their new triangle and rectangle on graph paper and determine the areas from this second card. After another four minutes, I stopped the class, give them a couple of minutes to compare areas with the group members sitting next to them. Then we check and verify these areas whole class to make sure everyone has the right information to work with for the next step.
Next, I tell students to look for patterns in the areas of the triangles and rectangles and write a conjecture about the areas of the triangles. I give students a couple of warm as they write their conjectures and I walk around and assist as needed. Then students share and discuss their conjectures in their groups. I walk around, listen, look and pick two or three conjectures to share with the class based on how these conjectures can move the whole class forward in the learning goal.
I help connect the conjectures to the work and guide the conjectures into a formula for finding the area of the triangle, making sure this final piece is clear and emphasize for all students to hear and see. Next, I have students turn and talk to a partner for 30 to 60 seconds about how the area formula for a triangle relates to the area formula for a rectangle, and why this makes sense.
So some students will then share that whole class, I clarify, confirm and connect, making sure a clear connection is stated for all students to hear. The remainder of the lesson is pretty much similar to the direct instruction model about the practice that happens. So, now we see with this version of the lesson, a more explicit version of an inquiry-based lesson.
The goal is clearly set at the start. There was a lot of student involvement and engagement throughout the lesson with students doing a lot of the thinking in this case and connecting ideas together, especially with the four tasks on index cards they worked with in their groups to come up with a conjecture about the area of triangles.
However, I laid the foundation for this work by first reviewing area of rectangles with the warmup problems, and working on the first rectangle triangle problem together whole class so that all students were clear about where they were going and what they were going to do with the problems on those index cards.
I made sure they were sent in that direction to be able to connect it to the prior knowledge. In this way, the lesson is systematic and organized to keep moving forward step by step at a good pace. Notice how I paste each part also four minutes for the index card problem, four minutes for the next card problem. Then added directions for checking their work within their groups and checking this whole class before going to the next step in the inquiry process because it was important to have the right information before writing a conjecture.
I split up the part where they talk to each other about any patterns they see within asking for a conjecture. Again, keeping control of the pacing and moving it forward for kids. I selected the work to be shared whole class based on that work that I saw that would move the class toward the learning goal, and then I made sure that the important concept is clear for the whole class. So once again using good explicit instructional techniques with an inquiry-based lesson. Important mathematical connections are made explicit and students are grappling with important mathematical ideas themselves.
And I just wanted to add one other note about stating objectives at the beginning of an inquiry-based lesson, and this is because sometimes I hear from people that they feel that stating the objective first in an investigation activity is giving away the punchline and I don’t agree. I think it’s stating or defining a goal of a lesson is a strong evidence-based technique, no matter the lesson.
I think it just imagine it, I suppose I’m having elementary students discover the community property of addition. That three plus five is the same as five plus three. Sure, I wouldn’t state the objective as today you will explore how the order of adding numbers does not matter. Rather, I would state the objective as today you’ll discover or explore a new property about how addition problems are related to each other. So then I’m not giving away what they’re supposed to figure out, but students are now have an idea that we’re looking at addition problems with the purpose of discovering some sort of new connection. And the lesson has direction for kids at this point.
So between these two lessons, the direct instruction model and the inquiry-based model, the ones that I considered good models, there’s a great deal of similarity and that’s because they’re both using explicit teaching techniques as listed on the chart here. These are the nine techniques that were listed earlier on the slides.
Well, there are challenges of course with this, so I wanted to list a few of the challenges with inquiry-based instructions that I see in classrooms. It’s important for students to grapple with the mathematics themselves of course and whether that’s for a few minutes or for the majority of the class period.
However, the task must be at a level that’s appropriate for students, so the task is doable. Requisite prior knowledge needs to be in place and students need to know what they’re to do in investigation. They should be grappling with the mathematics, not with figuring out what they’re supposed to be doing. And we have to be prepared with questions, guidance and be versatile in our own understanding of the mathematics.
This is so that we can provide support that leads students from their work and understanding toward the objective of the lesson. And student work typically shows incomplete understanding, and we must clarify the meaning, make connections between the methods, and show a mathematical pathway from where students are to where they want them to be. Even if we are having students help explain those connections, usually students are looking to us to verify that, that what was said was right and which part of that is important for them to take away.
And we have to periodically check for understanding and adjust instructional pacing, increasing or decreasing scaffolding as needed or plan adjustments into the next lesson. And of course there are challenges with direct instruction. And to say the least, we need to balance teacher talk with student interactions.
There are many methods I’ve illustrated in the examples where a back and forth during direct instruction serves well to keep students engaged and attentive. We have to provide questions that are challenging at an appropriate level for students in order to get students to think about important mathematical ideas.
Again, students look to us to verify what is most important and to make clear to math connections. And of course we also have to check for student understanding, we have to monitor what’s being understood by kids. So when to use direct instruction and when to use inquiry-based instruction? Well, that’s a great question and I don’t have a complete answer, but I have a couple of guidelines that may help. I see inquiry-based instruction as especially powerful and successful in situations where a logical and sensible connection in mathematics is something students can see on their own.
For example, if before teaching before about solving equations, I were to do a mental problem solving activity with students where I say, “I have a number, I multiply it by six and I get 96, what is my number?” Students are able to see themselves that they have to work backwards using the opposite or inverse operation to get to the beginning number. With this type of realization, students are better ready for my direct instruction that follows on how that connects to solving equations.
So if an idea is not too complex, students can make connections themselves. Other cases where I find inquiry works well or where patterns can lead to a concept. We saw this with the area of triangles connecting to the area of rectangles. Having students make these connections by generalizing from patterns is an important part of understanding mathematics. So, seeing how ideas connect and build is really important for kids, and where patterns and a fairly accessible logical connection do not provide good accessible routes to a math concept, I still often find that a short investigation or some sort of problem to solve can be very useful.
It can often at least prime the pump or it gets students in a better position to understand the concept I’m going to explain our model through direct instruction, or at the very least it, it could make students more interested in what I have to teach them because it will show them how to solve something they couldn’t otherwise solve or how to solve it in a more easily and more quickly.
Sometimes, after getting some information, redirect instruction, students can go to work investigating, talking to one another, or using some hands on materials to see the next level of connections and to verify a model in different ways what I’ve just explained. Well, in many cases though direct instruction, it seems the best route to me for understanding for students. And that’s of course if it is done using explicit instructional techniques that I’ve discussed in modeling the lessons.
Sometimes a little direct instruction at the start of an activity is important to lay a foundation or review important prior knowledge, I modeled this with the triangle area lesson where I first reviewed area of rectangles with both the direct instruction and an inquiry-based lesson models. I find many concepts are too complex or abstract for students to get to on their own, especially with limited class time.
For example, the process of adding fractions with unlike denominators. I certainly can use hands on models and diagrams to make sense of the idea, but I find good direct instruction is necessary. Often vocabulary terms and symbols simply need to be explained to students. For example, how a student’s going to know the names of place values or even the existence of place value if we don’t introduce and explain it? How are students going to know the top number fractions called the numerator, or that’s a symbol for a square root? What that means?
So direct instruction is often very important for connecting math ideas together concisely, clearly and coherently for students regardless of the type of lesson that has taken place before it. And students look to us as experts to at the very least confirm what it is that they’ve shared or heard from each other or seen that is the main idea, and is the key takeaway from the lesson. We have to make sure that one way or another as Hiebert and Grouws say, “The mathematical connections are made explicit.”
And I also find that with both inquiry and direct instruction, students seeing examples of applying ideas is important. In inquiry type lesson an example maybe at the beginning to get on the right track or at the end to see a discovered idea fully and clearly applied and typically work through the explaining examples falls into direct instruction.
All right, I want to share with you a bit more finally of what Hattie, Fisher, and Frey say about this conundrum of direct instruction versus inquiry-based learning because I really think they hit the nail on the head here. Emily? Sorry.
Emily: Yeah. When we talk about high-quality instruction, we’re always asked the chicken-and-egg question; “Which comes first to the mathematics lessons start with teacher-led instruction or with students attempting to solve problems on their own?” Our answer, it depends. It depends on the learning intention, it depends on the expectation, it depends on students background knowledge, it depends on students’ cognitive, social and emotional development and readiness.
Dean Ballard: Yeah, so it does depend. I wanted to mention also that one important aspect of explicit instruction that I’ve highlighted in both direct instruction and inquiry examples are having students talk to each other about the mathematics. And this is a key type of frequent student interactions, it’s called for it an explicit instruction that this is the idea of math discourse, which is our next topic.
Mathematics discourse is defined as communication that centers on making meaning a mathematical concepts. And here’s some key features of that math discourse. Students should explain their reasoning, mistakes are not covered up but serve as an opportunities to examine reasoning and to deepen everyone’s analysis of problems. And the final authority of whether something is both correct and sensible lies in the logic and structure of the mathematical reasoning which students are to engaging and the teacher needs to promote, to communicate, assist, clarify, correct, and connect this reasoning is needed.
Also, wanted to point out that with just a quick note here about discourse that it doesn’t require being completely fluent in English to participate in mathematical discourse. Research shows that ELs even as they are learning English, can participate in discussions where they grapple with important mathematical content, instruction for this population shouldn’t emphasize low level language skills over opportunities to actually actively communicate about mathematical ideas.
And there are many techniques for assisting ELs students with full participation in math classes, and most of these techniques benefit most, if not all students actually, whether they’re native English speakers or not. Here are three resources for such techniques, I won’t be able to give you time during the presentation to copy these down. However, let me remind you, you have access to the slides and you’ll be able to get them from that access or you can just email me and I’ll be happy to send this to you.
All right, so math discourse, it’s something that I find very interesting and from my experience teaching math, science and English courses that rings true. And that is a talking about math in many ways is more challenging for students because it’s a different kind of talk to students typically experience in other content areas. In many content areas the point of discussion is often to express, support on opinions and ideas. How were a math the point of discussion? The focus for students is usually to determine and improve a correct answer or to recognize and describe mathematical relationships?
Students feel there’s a right and a wrong answer and they need to find the right answer. It’s not about opinion. Well, in many ways this is true. However, there are also many different ways often to approach a problem. Sometimes there are different answers and in many discussions about math connections, there are several good observations to be made, different ways to generalize, and often that’s several bits and pieces from different students that builds the mathematical objective. But all that being said, it is still much more true in math and in any other content area that discussions have this general flavor of seeking that right answer to them.
Writing, oops, sorry, we’re not to the writing yet. Higher order questions, generally these are the kinds of challenge students to provide more information to think about the math, to make connections, to discuss those connections and getting students to justify and explain their answers with kinds of like what if questions or compare, contrast ideas together. It’s a really important technique, so keep in mind that again, the type of questions we asks makes a big difference.
And it’s not only for a big difference for mathematical discussions but a big difference when students are doing writing because writing is another form of math discourse and through writing students are having an internal discourse about the mathematics, and it’s important and necessary for them to do this kind of writing where they are thinking through the math. And this can include like quick writes or I think write pair share technique where students are writing about their ideas.
And by enlarge the kinds of prompts that are good for promoting verbal discourse are good for written discourse. The writing may be something that is to be turned in or maybe a proceeding in discussion with another student or group students or whole class. It may fall from a discussion after ideas had been discussed and clarified, in Teach Like a Champion, Doug Lemov described six benefits that can be derived from such writing activities.
These are listed on this slide, it’s all started to click through them. Writing improves, thinking and understanding, processing thoughts in writing refines them, writing promotes prompts or the writing promotes that challenging that students have is through the prompts that we give them. So students can think about what the important ideas of the math. As a matter of fact, students remember twice as much of what they learned when they write it down.
I want you to consider, it’s also when students are taking notes, that while they’re taking notes sometimes that can be a distraction if they’re not understanding what’s being said. So it’s important to pause at different points when we’re talking and having students takes notes so they can have time to take notes and have time to process it.
Every student participates, every student when they’re writing is allowed to have an opportunity to have an internal discussion about the math without having to put his or her hand up to be called on. And when students are writing, we can select effective responses to share whole class. This means that we can walk around and see some different things and share it, because we have examples from everybody, or we can have students doing the pair sharing after the writing.
Which means cold call also works well here because every student has at a chance to do some writing about the mathematics. So every student has something to share, especially if we also match that with a little bit of pair talk. And teachers can use the prompts that are being given to students to guide students towards what was most important things to think about in the math.
Well that being said about both verbal and written discourse, here are seven steps on the slide for initiating, managing and concluding the discourse. Initiate with a question or prompted, it’s focused on processes and or outcomes that promote DOK or depth of knowledge levels two and three. These are thinking and reasoning and connecting versus repeating facts or practicing procedures. Focus on the why something works the way it works. Why is the formula for a triangle half the formula for a rectangle? Provide time to think.
I almost always preceded student, to student discourse with brief individual think time, and manage time to discuss, provide some time limits for discussions. This really helps everyone, manage the process for sharing and connecting ideas. You need to decide how ideas will be shared, perhaps which ideas and why. It’s not a matter of being fair or giving everyone a chance to share everyone over time, over many lessons gets a chance to share, but with each specific discussion you only have so much time to work with in class, so use it wisely.
Make the mathematical connections explicit. If I’m listening as a student and I hear some ideas being shared and you’re asking clarifying questions, I’m starting to lose what was the main point, which thing that was shared was the thing that I’m supposed to take away from. So the teachers is responsible to make sure that the key mathematical connections are clear to everyone. And always ask, why does this make sense?
Ultimately, this is what the whole discussion is trying to get at mathematically. Some of these ideas, these seven steps can be combined into a single step or altogether this process may only take five or six minutes for a brief short pair share, or it could take 45 minutes if connect to an activity and depending on the situation and the concepts being explored or worked with in the classroom.
And this chart lists many techniques that can be used to promote and manage discourse. On the left are ideas for promoting discourse with suggestions for question types, and on the right are ideas for managing student talk while this is happening. I was just mentioned, most good questioning strategies linked to one central idea around mathematical reasoning and that is, why does this make sense? A good prompt is only a third of the process though.
You still need to actually get students to talk about the idea and that’s we’re putting them in a good position to have something to say is important with the work that proceeds this, or the type of question that’s asked. Giving a reasonable amount of time for the discourse, or having students write about it first. Monitoring that discourse and providing added prompts and scaffolds as needed to help it move along as needed.
Along with this or the specific moves you see listed on the right side of the slide. And remember, that after initiating discourse with clear directions and a good question prompt and managing the discourse using multiple strategies that include circulating around the room while students are working, there’s the critical last piece that is to conclude the discourse by processing student ideas and making the mathematical concepts and ideas clear.
All right, so from discourse, we get to vocabulary because discourse naturally intersect with math vocabulary, the two go hand in hand. Most vocabulary in math falls into what we call tier three words. These are words specific to a certain field and in our case specific to mathematics. Tier three words are usually best learned in context, not by front loading a chapter or a section with a whole vocabulary list and creating a glossary to start the chapter with.
Students may create their own glossaries or vocabulary charts or you may have a word wall, but the timing’s important. Teach and emphasize math vocabulary as it rises within the context of relevant lessons and concepts. Why is vocabulary important in math? Well, reading comprehension does positively affect achievement and arithmetic and problem solving. And as previously stated, vocabulary instruction should focus on specific words that are important to what students are learning at the time.
And be aware that math vocabulary is confusing for many students for several reasons and this affects their understanding of what they read in here in a math class. Here are several of the challenges with math terms, double meanings, words mean different things in math versus non-mathematical context such as a table of data versus a kitchen table.
Multiple terms, more than one word can be used to describe the same concepts such as sum, the total, or altogether. Nested vocabulary, some words require understanding of other math terms in order to be understood. For instance, numerator as part of the understanding fractions and symbol intensity math is full of symbols and graphic representations that carry as much weight as words. Symbols such as parenthesis, signs of operations and angle notations.
And the constant mixture of symbols with words in math makes it a whole another level of having to comprehend with students who are reading when they read it. Homophones or many math words that sound like different non-math words. For example, some and sum, some people figured out that the sum of the numbers is 25. And find the small words. Many small words make a big difference in meaning such as of, a, or, and. Here’s a chart that has a sampling of words and several of the categories mentioned on the previous slide. And it helps for us to be aware that these words and potentially do provide potential issues for students.
And if we can anticipate confusion around these, that can help kids and help us be looking out for that and make be careful helping point out these words, especially when we’re using them, especially for EL students and struggling learners. And at the bottom of the slide in the last two categories are some additional types of challenges besides those that were just mentioned, unique terms that happened in math, words that have only meaning in math such as hypotenuse, parallelogram, coefficient, quadratic, you know, those words that just roll off the end of your tongue.
And similar sounding words like tens and tenths. So several challenges with math words. One of the things we need to do with math language though is we have to make it visible in the classroom, such as emphasizing words we’re putting them on the board and connecting them clearly to the current work being done in the math lesson. And there are additional ways that are excellent techniques for doing, working with math language as well in the classroom, making it visible. Such as using a math word wall, these are a place in the classroom where key terms are posted to remind students of the terms, often a definition or illustrations included. The Math Word Wall is refreshed often to keep it current with the worst, most relevant to the current lessons.
Anchor Charts take word walls a couple of steps further, a word or set of related words or illustrator defined are shown in content and/or showing us some examples. And there’s a Frayer Model charts, these are often made on a single piece of paper. These are charts that student can create or help create and can even use as part of a glossary of math terms.
On the next slide is an example of a Frayer chart by the way, so let’s look at that. Frayer charts are typically have four sections on them for a student or a group of students to fill in. One section is for defining the term and another section the student provides an example or illustration of the term. In the third section, facts or characteristics of the term have shown. And the fourth section the student shows some non-examples of the term. Making this can be a great review activity for individual students or even for groups of students. One idea is as a group activity is to have each students start with a blank chart in a different math term, each student feels in one section of the chart, then passes his or her chart to the next person in the group.
And each student in that group didn’t fills in another section of the chart that he or she just was handed. And we continue passing the chart around until all four sections of the chart had been filled in. Now, the great idea is to take two or more completed charts and cut them up into four sections and then the mix the pieces together and have students resort them, figuring out which pieces that relate to which terms. So there are many great ways to use Frayer Model type charts.
Now, another important thing about math language is we need to have students use the language. So, one of the examples I want to show with this is, is this one here, then you’ll recognize this as a middle school example. But what I’m going to explain works with any grade level, you just have to insert the concept and then related terms that apply to your students. And I emphasize that we have to work hard to get students to use the terms because it just won’t magically happen.
The math terms, like I said, they’re unfamiliar to students and they just don’t roll off the tongue, you know. So, take this case students see an equation to solve and we start brainstorming it together a list of terms that are related to this, such as coefficient, variable, constant, expression equation, inverse operation, distributive property.
We connect the terms of the diagram, then we have students work on solving an equation, explaining to each other how they solve their equations. But we require that the students, when they’re explaining to each other, use two or three or more of the terms listed on the diagram. And you can even have a third student observe the payer and check off the vocabulary is being used and is being used correctly. Then have students switch roles, so each student has a chance to explain or listen and checkoff vocabulary, because math terms are so intricately linked to math concepts.
Learning and using math language reinforces learning and understanding math concepts, and recall that one of the tenants of explicit instruction is helping students organize information. Well, math vocabulary is an important part of organizing math ideas in our minds that connecting them together.
So just to wrap up the vocabulary part here, here are four main points that I wanted to make about math vocabulary. Know the challenges with math vocabulary, teach math vocabulary in context, make math language visible, and require the students use the math language. That brings us to our next topic, last topic was a differentiation. I just want to say, well, knowledge is built a different pace and varying degrees of depth with each student. That doesn’t mean we need a different lesson for each student. It’s really about how we deliver the lessons.
Differentiation is the consistent use of variety of instructional approaches to modify content, process and our products in response to the learning, readiness and interest of academically diverse students. The why for differentiation is embedded in this definition. The needs of academically diverse learners. And Hattie and both of his books recommends differentiating instruction, both the visible learning for mathematics and visible learning for teachers. Doug Lemov recommends it in Teach Like a Champion. Tomlinson a leading research in this area who taught for 20 years in middle school has written extensively about the importance of differentiation.
And there are many others resources that highly recommend different forms of differentiation. We know we are constantly adjusting instruction and differentiating and how we provide varying levels of support to students based on their individual needs. And how it’s related to how we adjust the content process and products.
Adjusting content is about scaffolding depth into our application, it’s not about moving ahead in a lesson or sequences or further behind in the lesson. Think in terms of adjusting the complexity of the problems such as using more friendly numbers or asking deeper level questions. Adjusting process refers to adjusting the strategies students use to access information, such as emphasizing different representations, such as using manipulatives or drawings or working with a partner or peer tutoring, working individually or in small groups with additional help from a teacher. Adjusting product means allowing them some flexibility in which problem students solve and how understanding and solutions, or justifications are demonstrated.
It’s important to keep in mind that differentiation is about finding a balance between the need, a responding to individual differences among students and the need to keep lessons manageable by not trying to individualize instruction for every learner in the classroom. How do you recommend focusing on just one area at a time if you’re new to differentiating? Just either content, process or product. There are challenges with differentiation, time, how will it affect the time of the lesson, a formative assessment, how do we know or do we know what the gaps are that existence students and what are we going to do with that knowledge once we have it, and resources, what resources do we have for differentiation?
Well, here’s some techniques that don’t add a lot of times to the lesson, questioning. The types of questions we ask, a variety at your fingertips that you can walk around and be ready to ask deeper level questions for some students and more scaffold requesting for other students. Choices and menus, give students some choice in which problems to solve. Suppose a worksheet or a page in the book has 20 problems divided into three sets their problems, A, B and C.
Tell students they need to solve 14 of the problems but every problem said, B counts as two problems and every problem and said, C counts as three problems, or something like that. And group and pair work, have students working with a partner is a national differentiating taking place when students are helping each other. And calling all students to share, choose who to call on sometimes based on giving students a chance to succeed because you’ve walked around and seen the work they’re doing. Sometimes one student can share the first step, which may be all he or she has done so far, but it’s correct than another student can build on this.
Here’s some of the techniques for differentiating add time to your lesson. Peer tutoring, research shows that students can be very helpful to each other, setting aside some time for students to get help from each other does take time but it’s very helpful. Small group instruction, work with a small group to provide additional support related to current lessons as needed. Flex days are days where students I’d be doing catch-up work and might be doing extension activities but it might give you a chance to work with a small group to provide additional supports that they need.
And station rotation days for students to rotate around in groups throughout the day or there might be two to four stations with students building independence in these stations, you also get a chance to again work with a small group. And finally, there are other resources such as computer programs that provide built-in differentiation, providing practice and lessons based on students demonstrated knowledge and mastery.
Well, those were it. So, here are some of the, what we’ve done today, that about wraps it up. But I know in some areas we just really just scratched the surface, but the main ideas connecting explicit instruction, instructional techniques to both direct and inquiry-based instruction, the importance of and techniques for student discourse, why vocabulary is important and the four key ideas to keep in mind that I mentioned. And some differentiation techniques at work with some that had little to no time to lessen and others, “Well they admittedly do take up quite a bit of time in a lesson.”
So that wraps this up and I’ll turn it back to Emily for any questions and answers session and to just kind of pull it all together.
Emily: Yeah, thank you, Dean. You did a great job getting through that super quickly. And folks, if you weren’t able to keep up and take all the notes you wanted, you will get a copy of the PowerPoint deck and the recording tomorrow by email or on the Core website. So don’t worry if you didn’t quite get all those notes taken, we definitely went at a quick pace but lots of good information there. I do want to thank today’s webinar’s sponsor, which is Core. And Core is dedicated to supporting educators as they build their math instruction skills, and Core has expert knowledge and experience assisting schools with implementing evidence-based instructional practices within their chosen math programs.
So if you’re interested in learning more about how Core and Dean and his team can help you implement some of the information he shared today and more effectively implement your curriculum, please reach out to Core at corelearn.com, you’ll find more information about those services and also a variety of ways to get in touch with us.
You also might want to check out the three samplers of “Spend Some Time with 1 to 9”, which is a great product put out by Core and it’s available, as I said, for free at that URL on the screen, and we’ll also send that by email tomorrow.
We are right up against our time, but we’ll go ahead and take one question that we have and then we’ll sign off. Were you suggested to verify or check for understanding, what methods do you recommend for doing this?
Dean Ballard: Well, one of the methods that we tried to get teachers to use this, not during the lesson, so at the end of the lesson just to getting an extra ticket. But during the lesson, several key things to do. One is, of course walking around and listening in on what students are saying and doing and seeing what they’re writing. When you ask a question rather than cold call … I mean, rather than taking a hands raised, cold call on students but I generally want to do that when there’s like us and they’re in pairs and then I can ask, have them talk to each other first so that I make sure they kind of have something to share.
Other strategies are, as they’re working in groups and talking to each other then they can help each other understand stuff. And then again, you’re walking around and listening in on those conversations, excuse me. So during the lesson, it’s really more dependent on the teacher to actually walk around and see what’s being said and listening in when asking students to answer a question whole class or walk into a group and asking students to answer a question in the group. Are they actually getting it? The least effective method of course is calling on raised hands.
Because then, you know, you’re only getting a few who got it, it allows you, of course, to move your lesson because you’ve got a few people who answered your question, so you can keep going but you really have no idea whether it’s just those two or three or the rest of the class where they are at. So I know that’s probably not that much more enlightening than what you’ve probably already were thinking. But really as I said, it’s dependent on the teacher too, to walk around and see what’s going on with the students.
Emily: I think that was helpful given our time, and certainly Core is a great resource for lots of questions like this. So again, feel free to reach out, check out the website and see what they offer and feel free to reach out to Dean there, at the information on the screen.
So thanks again everyone for joining us today. If you’ll just take a moment when you log off from the webinar to answer the five questions that will pop up with a survey, we’d really appreciate that. And one of the questions is to provide input on what other topics you might be interested in hearing webinars about, and we are in the process of planning our back to school webinars series.
So any ideas you have, we would welcome and hopefully we can accommodate as we head into the fall. Have a great evening everyone and we hope you’ll join us again on an upcoming webinar.