Differentiate Instruction for All Students

By Savannah Hill, RME Professional Development Coordinator

In the article 7 Steps to High-End Learning in Teaching Children Mathematics, M. Katherine Gavin and Karen G. Moylan describe seven steps to help teachers present differentiated instruction to all students. They have been working with the National Science Foundation on Project M² to research how this is best executed in the classroom.

From Microsoft Office

From field-testing high-level, differentiated geometry and measurement curriculum units on diverse populations of kindergartners, first graders, and second graders, they have found that all students are capable of developing deep understanding when each lesson is differentiated to accommodate the variety of student abilities, interests, and prior experiences.

They share seven steps teachers can implement in their classroom to help differentiate instruction for all students.

1. Select an appropriate task. Since research has shown that students are able to justify thinking at high levels, begin with an advanced concept in order to allow for opportunities to differentiate and support students.

2. Increase expectations for all students. Provide concepts that will challenge all students. Allow for activities that challenge high learners and can be differentiated with scaffolding for those who may need extra support. The National Council of Teachers of Mathematics (NCTM) advocates that challenging mathematics curriculum should be provided for young students (2002). Their study found large gains from pre- to post-testing on all students and found that they significantly outscored a comparison group.

3. Facilitate class discussions about the concepts. Encourage students to justify their reasoning and generate classroom discussion in order to help students work to understand each other’s ideas and come to a conclusion on correct answers. This will allow teachers to gain insight about students thinking, any misconceptions they may hold, and in turn, allow the teacher to better differentiate instruction.

4. Encourage all students 
to communicate their thinking 
in writing. When students create written representations of their work, such as in words, pictures, or tables, they are challenged to explain their thinking in ways that others can understand. This can also allow the teacher to have insight on the students’ thought process. Teach your students to practice “writing” out their math by having group responses on a question and then have students write with partners or independently, scaffolding where needed.

5. Offer additional support. Teachers can create “hint” cards to differentiate instruction for those students who are struggling when working a problem. The card, which teachers can drop off on a desk as students work, can include a definition with a picture, a question to connect prior concepts to current ones, or a way of modifying the activity.

6. Provide extended challenges. Teachers can also create challenge cards that can be shared in the same way as hint cards, but challenges those high performing students.

7. Use formative assessment to inform instruction. Make sure to analyze students’ thinking in case instruction needs to be adjusted to correct misconceptions before giving any final assessment. Gavin and Moylan suggest using open-ended questions focusing on the essential math concepts.

They encourage teachers to start small. Pick one or two current lessons and differentiate instruction using the hint or challenge cards. After you have tried it once, reflect on how it went, and try again!

Summing it All Up 
Differentiation is a way for all students to access high-level mathematics. It can easily be done in your classroom one step at a time. Check out their full article in the October issue of Teaching Children Mathematics and see examples of how they executed these steps in their research.

What are ways you have differentiated instruction for all students in your classroom?

Gavin, M. K. & Moylan, K. G. (2012). 7 steps to high-end learning. Teaching Children Mathematics, 19(3). 184 – 192.

National Council of Teachers of Mathematics (NCTM). 2000. Principles and Standards for School Mathematics. Reston, VA: NCTM.

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Twisting, Turning, Doodling, and Games

By Erica Simon, RME Project Specialist

Middle school math students are voracious doodlers, as any teacher of the 11-13 year old set will tell you. But what if all those doodles could teach you about mathematical concepts? What if doodles made math cool?

Sierpinski’s Triangle and Candy Corn

A series of You Tube videos has introduced the world to Vi Hart, the young creative mathmusician, as she calls herself. Although not formally trained in mathematics, but music rather, she LOVES math! And by making it fun with doodles (graph theory), stars (geometry and polygons), and triangles (Sierpinski’s Triangle and candy corn), Vi Hart has engaged young boys and girls to think about prime numbers, binary trees, and fractal patterns and see them as cool. “It’s mathematics that anyone can do.” Said Ms. Hart (New York Times, Jan. 17, 2011).

Ms. Hart does all this doodling with a pencil and a sharpie and records herself “thinking aloud.” Although her musings seems rather stream of consciousness, she is teaching her viewers to make connections and see the elegance of mathematics. She creatively and comically humanizes the father of mathematics theory and proportions (Pythogoras hated beans!) but helps the viewer see the logic behind what the squares in that formula a²+b²=c² really mean.

Click below for the YouTube video about Pythagoras.
What was up with Pythagoras?

Although we never see anything beyond the hands of Ms. Hart as she is exploring the wonders of Sierpinski’s Triangle and using candy corns to explain the theory, the viewer listens to a young, confident, and gifted mathematician who is infectious about her fascination with the ways in which mathematics surrounds us. And by showing us only her hands, the viewer feels confident that they can do this too!

Did you know about Borromean rings?

Click below for the YouTube video about Borromean rings.
Doodling in Math Class: Snakes and Graphs

Now you do. And you can draw cool snakes!

Share Vi Hart's videos with your students and let us know what you think! vihart.com

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What is STEM and Why is it Important?

By Toni Buttner, RME Assistant Director
 
Definition of STEM: Science Technology Engineering Math Problem: 80% of the fastest growing occupations in the United States depend upon the mastery of mathematics and scientific knowledge and skills. Demand far outmatches supply, AND, students are not equipped to satisfy this growing need. This infographic taken from Edutopia explains why a STEM education is important in today's society.

From Edutopia.org/STEM-Strategies

Solution: It depends. (Did you expect a clear cut solution?!) This alarm has been sounded and many have answered the call. One event in particular hallmarked the kick off of a solution, the 2012 US News STEM Solutions Summit held here in Dallas in June. This was the first conference of it’s kind in which leaders from K-12 institutions, community colleges, universities and private sector industries came together for three days to begin the discussion of how to interest, retain, and graduate more STEM-proficient students in the U.S. and get them hired.

Hands down, one of the key take-aways agreed upon was: we need partnerships across all constituents. K-12 is beginning to talk to institutions of higher learning to find out what students need to have mastered to be successful in college; meanwhile, the institutions of higher learning are talking to companies to find out what characteristics they are looking for in employees they need to hire - this is called the STEM pipeline. The transition between kindergarten through career, or some call it, birth to career, is a holistic approach of finding out where we are losing students when it comes to proficiency in the STEM fields of science, technology, engineering and math. Statistics say students begin to mentally drop out by 5th grade and then legally drop out in 9th grade, which is just one of the many ‘leaks’ in the pipeline. The Education Policy and Leadership department of Southern Methodist University tackled this topic specifically at their 2012 annual conference ‘Transitions’.

Get Involved: Do you want to keep up with what is going on in STEM in the state of Texas or nationally? A few great places to start are Educate Texas, Innovate-Educate and STEM Connector.

Have you found a great STEM solution or encountered a STEM success in your classroom? If so, please leave a comment to help others towards the goal of proficient students and connecting them with great jobs!

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Engaged Learning: Thinking Outside the App

By Dawn Woods, RME Elementary Mathematics Coordinator

Technology, an essential tool for learning mathematics in the 21st century, not only stimulates students’ interests, but also maximizes understanding and proficiency in mathematics. As touch devises, iPads, iPods, iPhones, Kindles and other tablets, become a standard tool in the classroom, it is noteworthy to realize that these tools can be more than a way to access apps that support curriculum content or as an e-reader. These devices can be creative tools that enable teachers and students to learn beyond the walls of the school.

As a classroom teacher, armed with a single iPad and a few iPods, I found that myself, as well as my students loved the ease of operating these tools because of their intuitiveness! We found that we needed very little instruction on how to operate the tools; all we needed was a willingness to play and explore. Since my classroom “touch device” supply was limited, I began to think of innovative ways to use the iPad and iPods. I discovered that these tools easily incorporate into a learning center, as a productivity tool where partners and trios create a multi-media product, and as a presentation tool for students and myself. Eventually, I adapted a BYOD (Bring Your Own Device) policy, which permitted students to bring in their own personal devices to use and manage.

Through my own iPod and iPad explorations, I have discovered that these tools not only deliver content in an interactive and differentiated way but also allow my kinesthetic and visual learners to manipulate the content, providing individualized and engaging instruction. The iPad also lends itself to inquiry or problem-based learning where students are engaged in authentic learning activities based on interest.

Since there are more that 100,000 iPad Apps, and countless other apps that work on a variety of platforms, deciding on where to get started is very overwhelming. A rule that I keep in mind as I evaluate apps is to “think outside the app” or how can the app help students as they consume essential knowledge, collaborate with others, and/or create a product? For example, if I wanted apps to support my instructional goals, I would look for ones that students could use on their own during center or independent activities. Some of my favorite instructional apps are Khan Academy (videos and challenges aligned with instructional goals), Algebra Touch (algebraic lessons and skill practice), Pearl Diver (conceptual number line activity), Pick A Path (activity that tests skills with powers of ten, negative numbers, fractions, and decimals), and Equivalent Fractions by NCTM (activity that builds conceptual understanding of equivalent fractions).

From Discovery Education

There are many great apps like Skype and Twitter that enable students share work. These are both free and easy ways for your classroom to meet people, talk to experts, collaborate and share ideas with peers, and create experiences with others. Another example is Dropbox. This is a free service that allows photo, document, and video sharing. You can save a document to a Dropbox folder and it is accessible on any computer or mobile device. You can also share the folder with others so that they can access documents for collaboration. Pair this app with CloudOn and you can edit documents, anywhere at anytime.

Apps can create multi-media products. ShowMe Interactive Whiteboard allows users to record voice-over whiteboard tutorials and share them online. Here, students could be the teacher, sharing how to solve a problem, then post it to the classroom website. Diptic enables users to combine photos to make new images, while VoiceThread enables the user to create and share conversations around documents, videos, and diagrams.

Summing it All Up
Touch devices such as iPads and iPods are awesome tools for the 21st century classroom. With 100,000 plus apps to choose from, users are challenged to find the best way to consume knowledge, collaborate with others, and create products that fit their individual needs. Using these tools in the classroom maximizes the potential of technology, enabling teachers to develop students’ understanding and proficiency in mathematics.

So, how are you going to “think outside the app” in your classroom?

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Key Priorities for Implementing Change

By Sharri Zachary, RME Mathematics Research Coordinator
 
From a classroom perspective, implementing change for the improvement of math requires that we, as teachers,
1.    Remain Flexible
2.    Reflect on Current Practices
3.    Build Relationships

One thing about TEACHING…Every school year is different!

As we work toward improving the outcomes in our math classrooms, one of our key priorities has to be that we plan to incorporate any changes or additions that are made at the State level into our curriculum and that curriculum should then drive our instruction. The State standards are our “non-negotiables.”  They tell us what students should be able to do upon completion of a grade level.

The flexibility comes in being able to adjust your lesson plans and calendar which is not always easy to do when you have assemblies, pep rallies, and other “special” bell schedules that (while important) take away from your instructional time.  This is why it is important to remain flexible and have a backup plan.  Even with all of these special events going on, we still are required to make sure students know the material.

Second, when trying to implement change for the improvement of math, the greatest thing you can do as a teacher is be a reflective practitioner.  When I worked with teachers, I often gave the suggestion to include a reflection piece on their lesson plans and at the end of each day or at the end of the week, jot a few notes describing how the week went.  Was the activity a complete disaster? Did I spend too much time lecturing?  Did I provide enough opportunities for students to work cooperatively?  How can I tweak this activity so that students are more engaged?  And so on and so on.

It is not uncommon to refer to what was done in previous years to plan out the current school year.  What is useful is to see what went well, what didn’t go so well, and what could be done better.  By making it a priority to reflect, you will begin to make those adjustments to your lessons that will improve the quality of your math instruction.

Lastly, a key priority for me is to always build a relationship with my students.  When I served as a math instructional coach to middle and high school math teachers, I reminded them every year of the importance of building a relationship with their students.  Your students need to know that you care about them and that they can trust you.  When we relate to our students and they begin to genuinely care about us beyond simply having respect for us, they want to make us look good.

So if that means, putting forth a little more effort in class, being just a little more attentive when you are speaking, trying a little harder on that test, they will do it.  When they reach that point of caring, they will understand that what you do is reflected through how they academically perform and when they measure up, you shine.

Summing it All Up
In identifying some key priorities for implementing change to improve math outcomes, I suggest being more flexible, practice being a reflective practitioner, and strive for better relationships.

What are some other key priorities you see are necessary for improving math outcomes?

Converting Fractions to Percentages

By Beth Richardson, RME High School Mathematics Coordinator

As a high school math teacher, I taught a wide range of students from ESL Algebra 1 and regular Geometry to Pre-AP Algebra 2. The resounding similarity I saw between all of my students was that, for some reason, students cringe when they see rational numbers. They feel like rational numbers automatically make the problem “hard”. I was amazed that by high school, students were still struggling with something as simple as converting from a fraction to a percent. Perhaps this is because, as teachers, we sometimes teach our students shortcuts that leave out the logic behind the scenes of the procedures they learn.

The IES Practice Guide, which is supported by research evidence, recommends that teachers “'help students understand why procedures for computations with fractions make sense’ and ‘develop students’ conceptual understanding of strategies for solving ratio, rate, and proportion problems before exposing them to cross-multiplication…’ (Siegler et al., 2010).”

Some common shortcuts teachers use are changing the fraction to a decimal then multiplying by 100 or changing the fraction to a decimal then moving the decimal to the right twice.

Examples:

Neither method above leads students to a percentage as the final answer, unless the student “remembers” to tag it on at the end. Units are crucial when converting in any context. In order for students to understand why they must multiply by 100% rather than 100 when converting from fraction to percent, units must be used properly.

Instead, students should be taught to set up proportional relationships, including units, between the fraction and unknown out of 100%. It is important that students understand that when the units of the numerator and denominator are the same, they cancel and the fraction is unit-less.

Example:
25 students went on a field trip and 5 wore a hat. What percentage of the students wore a hat?

20% of the students wore a hat on the field trip.

Through the process above, students see why they are multiplying by 100% and why the units in their answer must be a percentage. Also, students can use number sense to reason that x must be a percentage between 5 and 100.

Summing It All Up
Fellow teachers: it’s not safe to assume that our students understand why they are doing a particular procedure, even if it is one they “should” have mastered several grade levels ago. If we take a little more time to illustrate examples with labeled units and explanation, we will hopefully catch any previous misconceptions our students have and steer them on the right path towards math success.

Now it’s your turn. Share with us common misconceptions, similar to what we described above, that you’ve found in your classroom!

Resources:
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.

Siegler, R., Carpenter, T., Fennel, F., Geary, D., Lewis, J., Okamoto, Y., Thompson, L., & Wray, J. (2010) Developing effective fractions instruction for kindergarten through 8th grade: A practice guide. Washington, DC: National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education. Retrieved from http://ies.ed.gov/ncee/wwc/publications/practiceguides/