Math Differentiation – Educational Aspirations

Math Intervention for Enrichment

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This school year I’ve been given the opportunity to work with a select group of second grade math students. Since early October I’ve been seeing two groups of around 20 students for approximately 30 minutes twice a week. These 40 students were selected based on unit pre-assessment scores and teacher recommendations. The second grade students that I see tend to be in need of enrichment of the math skills that they’re exploring in class. This enrichment can take on many forms, but mainly I’ve been looking at have students develop a better understanding of numbers and patterns. I’ve been asked to expand on the unit being taught in class and report back progress that students have been making. The groups that I see are designed to be flexible and change depending on a particular math unit.

Here area few things I’ve observed as the year has unfolded:

1.) 30 minutes twice a week is a short time period. I’m all for packing in as much instruction as possible, but 30 minutes goes by very quickly. I’ve had to redesign many of my lessons to overlap the two days in a week. Retention can also be an issue with this. I spend each session with a bit of review and that has seemed to help.

2.) I’ve had to incorporate my own pre/post-assessment to show student growth. At first I thought this was extremely time consuming as students only have a small amount of time in my class and I want to make sure that the class time is being used appropriately. This year many of the classes in my school are using the same pre-test as the post-assessment. I’m using that model right now but it may change as the year progresses.

3.) I’m not able to meet with the second grade team every week so we decided to use Google Docs as a communication tool. My students’ pre/post assessment scores are located in the shared doc and can be assessed by any of the second grade teachers. I also attached a copy of the pre/post assessment to the document so teachers are aware of what topics I’m addressing.

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4.) I’ve been using effect size to show student growth. I learned about effect size in more detail after attending a Visible Learning conference over the summer. I feel like this has been a useful tool and has shown some insight into student gains in my class. This tool has also been important as it brings some finality to the units that I teach and can be used as one data point in transitioning students in/out of my class.

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5.) Student reflection is key. This year I’ve been giving students a copy of their pre-assessment stapled to their post-assessment. Students are then able to review their growth and ask questions. The focus is on student growth and not necessarily on point value or grade. Thankfully at second grade students aren’t used to traditional grades yet.

I’m looking forward to seeing how this enrichment opportunity develops over time and the positive impact it has on students.

Meaningful Math Practice

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Last week many of my students took a pre-assessment on an adaptive app. This particular app gave students questions in a certain math strand area and sent out a grade level equivalency score (GRE). Once students finished the pre-assessment they were given question at the GRE. If a student answered a question incorrectly they were sent to a help screen. The students were asked to watch a video about the concept. Some of the students watched the video while others made more attempts at finding a solution. Even after watching the video students still answered the question incorrectly. Every incorrect question asked student to watch a video and try the question again. Some of the students became frustrated and quit.

Most of the student were finding that the video wasn’t a helpful for math practice. This type of math exposure/practice wasn’t meaningful to the students. After observing this I started to analyze my school’s math practices. I started to question how many math exposures we truly give to students and how many of those opportunities are truly meaningful to students.

I find that students at my school are exposed to math in a variety of settings. Students are introduced to the idea of a particular math concept through a parent, teacher, nature, workbook, video, and many others. This experience is usually followed up with additional practice at some point. Students need to be given time to practice and apply what they’re learning. This often leads teachers to give students multiple exposures to specific math concepts. These exposures or practice opportunities give students time to experience math in different ways and through this I feel like students are able to comprehend/apply the math at a higher level.

Providing those multiple exposures is important. The form that the practice takes is just as important. While I’m in and out of different classrooms I find that the additional exposures sometimes take the forms below.

Worksheets

Although it may benefit some it’s not the only solution and I wouldn’t categorize this type of practice as extremely meaningful. Primarily, I find student math journals or worksheets used for math practice. I believe both of these have a role in practice but changing the exposure model has benefits and often those two mediums are used for homework. In my district student will at some point have to show an understanding of numbers on a worksheet. Generally these types of worksheets are found on unit assessments. I should also mention that digital worksheets fall into this category as well.

Activity/Projects

These are some of the more memorable experiences in class. Giving students a problem with multiple solutions can be refreshing and give insight to what students are thinking as they create a solution. This can also take the form of having students create projects with their peers.

Manipulatives

Taking out the pattern blocks can lead to some great learning opportunities. Fractions, base-ten blocks, algebra tiles, 3d Shapes, and many other manipulatives play a vital role in the classroom. Eventually these manipulatives take an abstract form on a worksheet/screen.

Games

Games are exciting. Blending math concepts, games and a bit of competition can lead to learning opportunities. I find this especially evident when the teacher or student helps explain their mathematical thinking in the process.

Videos

Watching a brief video about a particular concept can be a great opportunity for students. Pausing and offering commentary or asking questions can help students delve deeper into a particular concept.

Class Discussions

Having a classroom discussion about a particular math concept can be powerful. Often these types of conversations can expand understanding of math concepts. Hearing other students’ experiences or strategies many benefit the class. It may also be helpful to document the class ideas and refer to the learning at a later time.

Reflections

Giving students opportunities to reflect on their learning can pay dividends throughout the school year. I find this to be especially beneficial as students look back at their progress to observe their own mathematical growth. The reflection can take place after any of the strategies shown above.

Math practice takes on many different forms. How do educators make it a meaningful experience for students?

Educanon and Formative Assessments

The second grade classrooms at my school reviewed subtraction strategies last week. Students were subtracting numbers on a number line and becoming more confident in using regrouping strategies. Based on pre-assessment results I felt like some of the students would benefit from additional enrichment. While talking with a few colleagues I revisited Educanon. I first heard of Educanon from Mary and I briefly used it last year. So I dusted off my username and password and logged into my account again.

A while back I created a subtraction video using Explain Everything. I turned off the microphone function (my dog was barking when I created this) and just used the pointer and drawing functions. The video was only around two minutes in length, but had 10 questions. I added a variety of questions, including multiple choice, fill in the blank and checklists. The last question asked the students to use a whiteboard to find the difference between two numbers.

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During class students were placed in two different stations. One station was Educanon and the other was working with base-ten blocks to subtract multidigit numbers. The stations rotated after approximately 10-15 minutes. All students logged in and finished the Educanon within the time period.

After class I was able to review the results. I felt like this data could be helpful for the teacher as well as the student.

The next day I printed out the student results and compiled a reflection sheet.

Each student’s answers were stapled to the back of the reflection sheet. As a class we reviewed each question together and students filled out their specific sheet. Out of all the categories on the sheet, I thought the “Key Vocabulary/Concept” section stood out. Students started to develop an understanding of what type of skills were being addressed from each question. This was also an opportunity to emphasize certain math vocabulary words. At the end of this reflection session, students circled their effort level on this formative assessment.

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I thought this was a beneficial activity for a second grade classroom. Students are also starting to think more about their own mathematical thinking and learning. I’ll be using the data and student reflection in preparation for more challenging subtraction concepts later in the year.

Using Genius Hour Ideas in Math Class

One of my goals last year was to incorporate more student content creation in my classroom. The journey was challenging but definitely worthwhile. Students created a variety of projects that helped showcase their math understanding. There were elements of choice in the projects and I felt like student engagement and curiosity bloomed.

This year I’m trying something different. I’ve always been impressed with the idea of using genius hour in the classroom. What intrigued me was the student choice and engagement piece. The idea of students owning their learning and being intrinsically motivated to participant in the learning process is important. Hearing stories from Paul and Joy inspired me to think of ways that I could apply a genius hour philosophy in an elementary math classroom.

I had a few discussions with colleague and kicked around a few ideas on how to get started. I started off with an informal wonder wall. Students started to generate questions that they would like to answer. I soon found out that this was a challenging task for the students. They weren’t used to this type of assignment. When asked to create a question for the wonder wall they had trouble. Many students asked what I wanted and were unsure of what questions to create. I showed the students the Google and Siri test. If the question that they came up with could immediately be Googleable or Siriable (words?) then they should probably find another question. This actually worked as the class used some horrible and decent examples. After a while and some modeling, students started to compile a few different questions.

I then placed the different math strands on the whiteboard: geometry, measurement & data, number & operations and algebra. The class then started to sort their questions into the different math strands.

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Students decided on what math strand to emphasize and documented it on their recording sheet. The class then discussed the math genius project flow chart.

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I wanted to give students a bigger picture of what’s going to happen over the next month or so. Since my classes only have about an hour to work on this project a week, it’ll probably take at least a month of sessions to finish. I really have no idea though. It could take a couple of months, but it depends on how the students progress.

After we reviewed the flow chart the class will be moving into the research portion. Students will use a variety of tools/resources to research their topic to find some sort of conclusion. The students will be using the sheet below to document their research.

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So far so good. Next week the classes will continue to research their questions and think about what type of presentation tool they’d like to use. This is definitely a journey and I’ll be documenting our progress through this blog.

Additional files: Source Sheet Check List

Estimating in the Elementary Classroom

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My school finished its ninth day of school yesterday. It’s been a journey as students are understanding class routines better. At this point in the year, students and teachers are starting to become more solid in their processes. Many of my students arrive to class at different times. Some students are at an elective or leave class a bit earlier/later than the rest of their peers. Regardless of the arrival time, when students enter the room they follow a flow chart. Students have their own folder and materials inside that are ready to go. I usually have some type of visual brainteaser for the week and a grade specific Scholastic math magazine. In the past I’ve used different types of math warm-up activities to start class.

This year I adapted my warm-up strategy. I wanted to individualize the type of responses within that warm-up time slot. After researching a few different tools, I decided to try Andrew Stadel’s Estimation180 this year. I think of Estimation180 as an opportunity for students to develop a stronger sense of numbers and practice estimation skills in the process. Initially, I thought that the site would be great for middle or high school students. I then found the below sheet and site that seemed helpful. This is one way in which student can document their thinking. The template also includes lessons that could link to Fawn’s Visual Patterns site.

This template inspired me to adapt the sheet to fit an elementary classroom. I changed the template a bit to work with a third grade math class. A few colleagues and I will be using this sheet early next week.

So now, students enter the classroom, pick up their folder and begin to work on their daily estimation challenge warm-up sheet. The estimation is displayed on the whiteboard. Students pick a high, low and exact estimate. I ask the students to prepare to tell me about the reasoning that they used to come to the concluding estimate. The class then completes the online portion of the site and submits a response. We then look at other responses and reasoning.

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After a brief discussion the result is revealed. Students write in the correct numbers and find the + / – . The entire activity takes about 5 – 10 minutes.

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I’m planning on using Estimation180 a few days a week and incorporate Visual Patterns for the rest of the days. The template also includes a few different reflection pieces. I feel like these activities provide students opportunities to produce a product and reflect on the results. At some point I’d like to add a journaling component to encourage more reflection and possible goal setting.

Math Acceleration or Enrichment

Acceleration v Enrichment — Acceleration vs Enrichment

A while back I was asked a question about student acceleration and differentiation. The question related to different types of acceleration opportunities for students that master math content before others. This question is often at the heart of differentiation for high achieving students. I thought awhile about the question and started to brainstorm what opportunities truly exist or if acceleration is needed in those circumstances. In an education setting acceleration is often associated with a curriculum that is moving faster or happening at a quicker pace than the norm.

In math at the elementary level, concepts are usually built upon one another and acceleration seems to be valued. Similar to a lattice fence, once one concept has been mastered, teachers often move the student to the next row/concept. The goal is to continually move students in an upwards trajectory towards the next concept on the ladder.

When acceleration is the focus, students are asked to master and then move to the next numerical concept. For example, If student A has mastered 2.0A.A.1 they automatically move to the next concept, 2.0A.B.2. Keep in mind that mastery is often defined by the author of the assessment. Mastery could be correctly answering a few abstract problems in a row or answering 90% of the answers correctly. In the author’s mind, the faster this process occurs over time the more the student learns. This isn’t always the case and the perceived notion of learning might not actually be occurring. This is especially prevalent with online adaptive software programs. This type of philosophy often facilitates minimal understanding and can lead to problems down the road. Also, students that are accelerated are often asked to answer questions more on an abstract level rather then explore mathematics constructively. Creating a personal level of mathematical understanding is valuable. Focusing in on only the abstract doesn’t always lead to a learning experience or a better understanding of math.

I believe acceleration has a place in the elementary classroom, but I don’t think that it should be the default. Honestly, I feel like accelerating is easier than providing opportunities for enrichment. Instead of acceleration why not emphasize enrichment for students that have already demonstrated mastery? I think the word enrichment gets caught up in buzzword land, so here’s a formal definition:

Miram-Webster defines enrichment as the process that improves the usefulness or quality of (something) by adding something to it.

Enriching math instruction doesn’t necessarily mean that students quickly move from one concept to another, but instead it may focus on practical application and problem solving. Developing strong problem solving skills enhances the usefulness of mathematics. I find that students benefit when given opportunities to enrich their understanding of mathematics. In addition, enrichment provides opportunities for students to practice relevant skills that become immediately useful. Logical thinking, abstract reasoning, and problem solving can all be part of the enrichment process. All of the skills that are practiced through enrichment activities can be used cumulatively throughout a math curriculum sequence. The picture below is just one example.

Students often need to have a foundational understanding of mathematics to be successful at the middle and high school levels. Logical thinking and abstract reasoning skills tend to contribute to the background knowledge for algebra and geometry concepts. Problem solving is a skill that’s used throughout school and life. Enrichment opportunities encourage students to use the math learned and apply it to practical situations. It also enables students to solve problems using trial and error and find multiple solutions. Perseverance skills are also practiced during math enrichment opportunities. Instead of completely emphasizing the upward trajectory of concepts, students that experience enrichment opportunities develop skills laterally and may cement a more solid mathematical foundation in the process. It may also enable students to see mathematics in a new light, not just a lattice of concepts placed in chronological order. Feel free to review Mathwire, NRichMaths and Andrew Stadel’s Math Acts, for a few different examples of how to incorporate math enrichment opportunities.

There isn’t really one right answer to the question found at the beginning of this post. The solution includes a possible combination of acceleration and enrichment, but immediately leaping to acceleration might not be the best option.

How do you use math enrichment in the classroom?

National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington, DC: Authors

Enrichment. 2014. In Merriam-Webster.com.Retrieved July 21, 2014, from http://www.merriam-webster.com/dictionary/enrichment

photo credit: Filter Forge via photopin cc

Introducing Fractions

Today’s second grade math lesson included an introduction to fractions. In the past I’ve introduced our fraction unit with pie manipulatives. They work great, but I was looking for a more hands-on lesson that motivated as well as provided opportunities for enrichment. While thinking about how I could make my fraction lesson more engaging, I decided to research a few different options. Specifically, I wanted to find a way to incorporate a lesson with multiple answers. I thought what’s healthy (district wellness plan), easy to peel, has different sectional pieces, and is relatively easy to clean up? I ended up deciding on purchasing a bag of clementines. I also put together this sheet for the activity.

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The lesson went well and it’s definitely one that I’ll keep in the repertoire. Students estimated the amount of slices, identified fractional pieces, found the numerator/denominator, turned the fraction into a mixed number and finally ate the clementine. A few pictures are below.

How do you introduce fractions?

Web-Based Math Differentiation in Elementary Schools

Different Learning Paths for Different Students

It’s apparent that student achievement data, in many different forms (formative, standardized, norm-referenced, common assessments, etc.) is becoming increasingly valued by administrators and teachers alike. Teacher PLC teams analyze this data to become more aware of strengths/concerns and differentiate their instruction accordingly. Instead of whole group instruction, teachers are beginning, or already using guided groups to meet the diverse academic needs of their classroom.

Once needs are identified, teachers put together plans to address the needs in the classroom. Generally, teachers utilize guided reading/math groups, small groups, resource specialists, to meet the needs of individual students, whether the needs are remedial or for enrichment purposes. One of the goals is to meet the need with some type of teacher support or intervention; although this is not always possible with time constraints and limited staffing. Have time to be able to individualize instruction is vital for any teacher. At times, time and staffing limit the amount of differentiation that can occur. Teachers continue to look for ways to supplement their instruction for differentiation in and outside of their classroom. Which leads me to this question …

What free online tools can be used to supplement math differentiation in and outside of the classroom?

Note: All of the tools below are aligned to the Common Core Standards and can be accessed at school or home. I’m not suggesting that these tools replace school interventions, but they may be helpful if used appropriately. Click the pictures to enlarge.

MobyMax

MobyMax is an adaptive online curriculum provider that creates individual education plans for students. Student take a pre-assessment that seems to be fairly accurate (at least in my opinion). The pre-assessment determines where to start instruction and helps students practice skills that they haven’t yet mastered. Student data is collected on every lesson and problem that is completed, so progress monitoring is quite painless. Mobymax also has an app for easy access.

Scootpad

Scootpad is another adaptive curriculum provider that enables teachers to assign specific CCSS concepts to individual students. Teachers determine the mastery level and they are able to keep track of individual student progress. As of right now, there aren’t any lessons associated with the questions.

TenMarks

TenMarks is used to introduce or reinforce teaching in the classroom. Students are able to review online lessons and are asked questions related to the topic. Teachers are able to track student progress over time with TenMarks.

XtraMath

Xtramath is designed to help students improve their math computation fluency. This isn’t a program that’s for everyone. I’ve found that students who need practice with multiplication/division tables benefit from this web-based intervention. The program is very user-friendly and has a progress monitoring component which seems beneficial.