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Response: Differentiating Algebra Instruction

Stephanie asked:

How do I differentiate Common core algebra for students who are more than 3 years below grade level and still expected to pass an Algebra regents (state test) to graduate from high school? My students receive one 45 minute math class a day and no resource room.

Differentiation means a lot of different things to lots of people, and it also varies across subjects.  Today's post will explore what it might look like in algebra.

Wendy Jennings, Yvelyne Germain-McCarthy, Billy Bender, Derek Cabrera, and Ed Thomas contribute their thoughts, as do readers.  You can listen to a ten-minute conversation I had Yvelyne and Wendy on my BAM! Radio Show. You can also find a list of, and links to, previous shows here.

You might also want to explore previous columns here on Differentiating Instruction.  In addition, you might fine this collection of math resources useful.

Response From Yvelyne Germain-McCarthy

Dr. Yvelyne Germain-McCarthy is author of Bringing the Common Core Math Standards to Life: Exemplary Practices from Middle Schools and Bringing the Common Core Math Standards to Life: Exemplary Practices from High Schools. She is currently writing a third on diverse students and mathematics:

This is indeed a challenge. A good initial assessment is, of course, necessary to clearly identify areas where students most need help. Common Core (CC) identifies critical skills that students need to progress, so base your assessment on CC  to determine which skills absolutely have to be re-taught. Another critical component is awareness and appreciation of your students' learning styles, so that re-teaching of concepts and strategies is approached through learning modalities that work best for them.

A critical question is: How do we help struggling students fill learning gaps while moving forward?  

A. Try the following whole-class strategies 3-4 days per week:

(1) Start with a warm-up having 2 or 3 review problems based on your initial assessment.

(2) It is important to make problem solving the heart of mathematics so that students do not ask, "Why do I need to know this?". Therefore, make the last warm-up problem a challenge related to a subject of interest to your students for which they do not yet have the skills to complete. Those skills are what the day's lesson will teach, as students explore ways to solve the problem. Such problems draw students into doing mathematics and when they reach an impasse, then they have reasons and motivation for learning.  Such problems also provide differentiation opportunities through appropriate levels of challenge because of multiple entry points or paths to solutions. During an exploration process, provide direct teaching to those needing it through guided questions. Require that students summarize in their own words and in writing what they learned and what is still confusing; use that to guide future lessons.  Have a whole-class summary and be sure students write and save notes on important concepts. Explicit details for this Launch-Explore-Summarize Teaching Model, are given here.

(3) For HW, Include review problems targeted to needs from your initial assessment. Many struggling students don't do homework, but assign a few anyway because some will try.

(4) Use alternative assessments and include mini-final assessments with problems from current and past chapters. The current mini-final should weigh more than the others.

B. For 1-2 days, try a focused differentiation to address needs from initial assessment. Have stations targeting different skills/concepts through interesting structured activities. Include one for students to create assessment questions or to help each other.

Finally, it is an important and worthwhile investment to get at least one other teacher to work with you to get assistance in finding strategies and problems of interest to YOUR students. And remember that teachers are not miracle workers. Just do your best, based on best practices.


Response From Wendy Jennings

Wendy Jennings has been a mathematics educator in Sacramento City Unified School District for the past 13 years. She has degrees in Anthropology (BA), Mathematics (BS), and Human Behavior (MA). She also works with the UC Davis C-STEM center on the integration of computer programming and robotics into mathematics curriculum:

Many people would think that this scenario is more the exception than the norm, but it really isn't. The last time I had a situation like this, I was fortunate enough to have a teaching aide to work with them one on one with basic math skills until they could handle the Algebra. If you have no support, there are several options open to you. Below are the options I use most often:

1. See if there are any senior students who are ahead in credits that would be willing to be a peer tutor in your class. Often, administrators may see this as a non-academic option for that senior but if you do, indeed, use them to work one-on-one with that student, not only can you offer the "letter of recommendation" carrot but the student can also list that experience on their resume. If this is an ELL class, a peer tutor who was also fluent in the student's native language would be the best option, although they should instruct/tutor as much as possible in English as I assume the exam is written in English.

2. If the student has access to the internet from home, assign video on the internet, like those provided by Khan Academy, that are relevant to the Algebra skills needed along with extra worksheets for practice. You can use these worksheets in lieu of other class assignments or provide them as extra credit. Also, many of the Khan Academy or even Youtube videos allow for subtitles in different languages if the student is not proficient in English.

3. If there are no senior peer tutors available and the student does not have internet access, you can always take one of your high performing students and have them work with this student on the necessary skills. I often offer extra credit or I will excuse them from the assignments they will miss because they are tutoring. In fact, studies show that the tutor herself will also get more depth of knowledge out of helping another student. When I do assign extra credit, I occasionally tie it into how well the student receiving the tutoring performs on an assessment of my choosing, depending on the tutor.

4. Sometimes, at least in California, the public libraries or Boys and Girls Clubs offer free tutoring after school for adolescents and teens. This could be an option if the student feels comfortable reaching out to the community.

There are many other options, of course, but I usually try one of these first since they are the most accommodating. Other options could be to reach out to the local college for a civic minded student to come and tutor (maybe an education student?) or to work with the student after school. Unfortunately, if the student if unwilling to put in his or her own time, then you are very limited in your options.


Response From Billy Bender

Dr. Billy Bender is an international leader in the instructional strategies area, including differentiated instruction, project based learning, response to intervention, and technology in the classroom. He has been in public education for 36 years, having taught in middle school, and at Rutgers University and the University of Georgia. He has written 24 books in education, many of which are very well received by educators around the world:

In my work on differentiated instructional models, I've discussed four models, two tradition models (traditional lesson plan modification and learning centers), and two recent models (flipped classes, and project based learning). Using these models, here is a "typical" differentiated lesson for a 45 differentiated Algebra class.

When a teacher begins initial instruction, he or she should initiate three different 15 minute activities; partner instruction, video-based self-instruction, and teacher explanation. For partner instruction, I like to see maybe 1/3 of the students paired up with the specific task of learning the new content together. Students pay attention to peer partners much more than teachers in traditional "teacher-taught" whole-class lessons, so this strategy will increase achievement over time.

Self-instruction during the initial instruction phase is the basis for the flipped class model. The teacher should flip the learning and assign 1/3 of the students go online for their initial instruction. SAS Curricular Pathways, YouTube, and TeacherTube are no cost options for online learning. Finally, teachers should selectively identify students who seem to understand better when they are provided initial instruction directly from the teacher. The teacher can then lead a 15 minute traditional lesson for them.

After this 15 minute initial instruction, I'd suggest a five minute "quick check" quiz to assure that every student "gets it." Students need to know their comprehension will be assessed often, and this quick check procedure will help.

Next, I suggest individual skill building activities two or three days each week for 15 minutes. This addresses the needs of students who are several grade levels behind, since the skills stressed will be at each student's specific level. For this, I am an advocate of Khan Academy. It is free, covers Algebra content, and can help students across the achievement spectrum. For the Algebra student who is functioning at a three or four year deficit, Khan Academy can provide skill instruction to "fill the gaps" in the student's understanding. Teachers must fully implement Khan Academy. Using only the videos is NOT the same as implementing Khan Academy. Full implementation requires that all students be set up with their individual "Knowledge Map" and the individual online exercises. Teachers should set themselves up as each teachers "Coach" for each student, and also encourage students to do Khan Academy work outside of class time. Surprisingly, many students will do so!

On days when skill building is not done, a project based learning activity is recommended. I'm an advocate of student choice of projects because the higher student engagement in project based learning leads to increased achievement. Thus, I suggest some project time every other day.

This mix of instructional activities will result in a highly differentiated Algebra class, and teachers will engage in much less "traditional, whole class" instruction. Further, this will increase achievement overall since students will be much more engaged, even though they will not all complete the same lesson activities. In that sense, differentiated instruction is the future of effective instruction in the Algebra class.



Response From Derek Cabrera

Derek Cabrera, PhD, is an internationally recognized expert in cognition, systems, and learning, and teaches at Cornell University. His theoretical models of "systems thinking" have made impact worldwide as the basis for individual learning and educational change as well as organizational learning and design. Dr. Cabrera is currently senior faculty at Cabrera Research Lab in Ithaca, New York. He is the author of Thinking at Every Desk: Four Simple Skills to Transform Your Classroom. Watch Dr. Cabrera's TED Talk on how thinking works:

Differentiation of Common Core algebra for students at various grade levels is a challenge. The first thing an educator must do to succeed is know that, ironically, differentiation actually requires looking for structural similarities in the topical area, rather than differences in content taught to different levels of learners. For example, in algebra in particular, there is a set of structural components that are universal to topical content. If we focus efforts on teaching these fundamental structures, which are used with varied levels of content, we can effectively teach all of our students skills at the appropriate level.

For example, in an algebraic equation, there are variables, numbers, and operators. All algebraic equations are comprised of these parts. Therefore, developing a deep understanding of each of these parts and the special relationships among them (manifested by mathematical operators) is required to truly differentiate Common Core algebra across levels.

The problem with the way we teach mathematical concepts is that we focus more on equations than on the ideas they represent. If I give you the equation for surface tension dynamics as [email protected]}, you will immediately think it makes no sense based on your existing knowledge of mathematics. And it doesn't make sense because we just made it up, and it is no more than a string of meaningless symbols. This is the way children often experience mathematics.

If we give students the equation for velocity, which is v=d/t, all they see is something akin to what you saw in the string of symbols above. It means nothing to them.

For understanding to occur, we must deconstruct the ideas represented by the string of symbols that make up the formula. In the case of v=d/t, we must help students realize that this equation is the definition of velocity (or v). We know this because one of the symbols in the equation is an equal sign (=), which is essentially saying that the thing on the left side is the same as the thing on the right side. Now we need to go deeper and figure out what the symbols on the right mean. There are parts, where d stands for distance and t stands for time. Those parts are related by the symbol for division (/). So, v is a relationship that needs to be solved by dividing the numerator (the top) by the denominator (the bottom).

Of course, the problem that many children have with algebra is that the little letters represent some actual numbers. Which number they represent depends on all the other numbers, so they vary, which is why we call those little letters variables. So the equation describes a particular relationship between numbers. If the d-number (or distance) is 5 miles and the t-number (or time) is 1 hour, then 5 divided by 1 = 5mph, which is the speed (or velocity).

The process of differentiating common core algebra for a range of learners demands an effort to explicate the universal structural relationships among the parts of an equation that allow all levels of learners to achieve content mastery.



Response From Ed Thomas

Dr. Ed Thomas is founder and president of Dimension 2000, a professional development organization that works with teachers of mathematics. He has over 30 years of experience as a mathematics teacher from the secondary to undergraduate levels:

Before moving into strategies for differentiated instruction, I would start by inventorying the missing skills and understandings that are causing students to remain below grade level. This can be accomplished by administering a concise Algebra I pre-requisite skills test. The test doesn't have to be long--twenty thoughtful questions can help diagnose students' needs.

Once you have a better idea of which skills need more attention, allotting just five minutes at the start of class to post a simple problem is a great way to address students' needs and build underdeveloped skills. Imagine posting this problem: The length of a rectangle is 14 units. While the width is unknown, the perimeter of the rectangle is 49 units. Find the width of the rectangle.  

Within the five-minute challenge, have students work individually for two minutes and then share their ideas with a partner during the next minute. For the final two minutes, collect students' ideas and explain how to solve the problem. Note that these pre-requisite skill problems do not need to directly connect with the lesson of the day. It is important that these first five minutes be used to help students close their personal math-achievement gaps, which will have a tremendous payoff throughout the school year.

Regarding differentiated mathematics instruction, I've found that Graduated Difficulty (Thomas, Brunsting, & Warrick, 2010) and Task Rotation (Silver, Brunsting, Walsh, & Thomas, 2012) are two powerful techniques for helping all students succeed with algebra.

Graduated Difficulty helps ensure that all students are engaged in the learning process--the entire class works on the same content while individual students have the opportunity to choose their level of challenge. To implement this strategy, start by posting three tasks on the board and noting that each task will be more challenging than the previous:

I)  Solve for x:  6x + 12 = 30

II) Solve for x:  5x  - 15 = 45

III) Solve for x:  2(x + 1) + 10 = 22

Invite students to survey the tasks and independently choose the highest level task that they think they can complete successfully. All students will be working with math problems that connect with their personal levels of skill and understanding. Students who successfully solve a lower level problem should be encouraged to try the next level. Students who successfully solve the highest level problem can be asked to coach other students who still may need help.

Task Rotation incorporates four tasks designed to speak to these styles of thinking and learning: Mastery, Understanding, Self-Expressive, and Interpersonal. All four tasks support the same mathematical content, and you can have students complete specified tasks, choose their own tasks, or combine choice with specificity. For example, this Task Rotation works well if all students are asked to complete the Mastery (I) task plus one remaining task of their choice (II, III, or IV):

I) (Mastery)

Solve each equation for x.

  1.  8x - 12 = 28
  2.  3x + 16 = 25
  3.  10x + 9 = 49

II) (Understanding)

The following equations "prove" that 1 = 2.  What went wrong? Review the equations, then find and correct the error(s).

let x = 2                            given

then 2x = 4                      multiply both sides by 2

x - 2 = 0                            subtract 2, both sides equal 1

2x - 4 = 0                         subtract 4, both sides equal 2

2x - 4 = x - 2  transitive property

2(x - 1) = (x - 2)            factor

2 = 1                                   divide both sides by (x - 1)

III) (Self-Expressive)

Create your own real-world phenomenon or situation that can be modeled by an equation. Be prepared to share your model with the class.

IV) (Interpersonal)

Work with a partner to solve the following problem:

Alex bought three slices of pizza, a side salad, and a large drink.  The total bill was $11.75.  Knowing that the salad and drink cost $3.75 and $2.00, respectively, determine the cost of one slice of pizza.  

After solving the first Mastery problem, students will be drawn to one of the remaining problems. By offering students different styles of tasks, you can engage all students and give them the chance to demonstrate what they've learned in their own way.

Even with limited instructional time, the five-minute skill-building problems, together with regular use of differentiated instruction techniques like Graduated Difficulty and Task Rotation, can help all students increase their chances of success in algebra.



Response From Readers

Julie Boatright:

I ask the same question. I teach Algebra I to new arrivals that are ELL level 1s. Many are SIFEs (students with interrupted formal education) and the majority of my students begin the school year at 2nd and 3rd grade level proficiency in mathematics. (I want to be clear here. That is the majority, and some of the students in my high school level class are below 2nd grade level proficiency.) These students still need to develop basic number sense and learn the core pre-algebra skills in the 6th - 8th grade common core curriculum before realistically doing common core high school algebra. This goes beyond scaffolding and differentiating instruction. These students need the opportunity to learn the prerequisite skills. They need to be in smaller classes with more than 45 minutes of mathematics, and they need a high school math class that will address their academic needs in mathematics before sitting in a common core algebra I class. Yes, all students should have access to grade level content, but sitting in a class in which the content is not in your zone of proximal development does constitute "access." It is painful and not fair to struggling students.

My students can learn, but they cannot magically develop six to seven years of mathematical understanding because I scaffold or differentiate high school algebra I content.

I would like to learn more about what high schools with SIFEs and other students far below grade level proficiency are doing, especially if they are trying to meet those students needs before they sit in a common core algebra classroom.

Kevin W. Jones:

I have the answer to Differentiating Algebra 1 - it's called AMPED, Algebra 1 in Manufacturing Processes, Entrepreneurship and Design. It's an aligned curriculum developed from the same team that created Geometry in Construction. A very successful program in it's own right. Google "Geometry in Construction" to learn more about both.

I'm pursuing this option at my school as an alternative to Double Period algebra because I question the efficacy of doubling the time doing the same thing. Time needs to be doubled productively.

Thanks to Yvelyne, Wendy, Billy, Derek, and Ed, and to readers, for their contributions!

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