Friend posted on Facebook about math and beer drinking. I was compelled to oblige him.

## Why Math Education is Important

**Tagged**beer, equation, exponential, natural light

Friend posted on Facebook about math and beer drinking. I was compelled to oblige him.

Announcing 2 workshops for educators working with students with special needs on math. These are designed to be hands on, with immediately implementation and can be delivered to schoolwide or district wide audiences.

- Identifying, writing, and monitoring progress for IEP math objectives that will support the entire year of math and that will allow all team members to track progress
- Math instructional strategies that are designed to address challenges specific to ADHD, ASD (autism), and LD (learning disabilities)

A common method to learn multiplication facts is through skip counting. In turn, this is a means of learning division facts (see next paragraph). The challenge for many students is they struggle to learn the skip count routines or cannot engage brute force memorization effectively (e.g., have a working memory deficit).

The challenge with multiplication by skip counting is keeping track of two sets of numbers while memorizing the order of the skip counting. That is another example of the rubbing belly and patting head phenomena in math where one extra task demand undermines the process.

A hack I use to scaffold this process to reduce the task demand during the learning process is to provide rows from a multiplication chart (below) for the facts of focus (3s and 4s in this example). The same approach can be used for division facts, e.g., in the image below right I have the student choose the row of the divisor (3) and then skip count to until reaching the dividend (12). The idea is the student has less task demands while learning the process and seeing the number pattern. This allows for more repetitions or rehearsal.

For students more severely impacted by a disability or who simply struggle with the patting head and rubbing belly of skip counting, the appropriate times table row can be provided for each problem to allow the student to circle (below). This allows for a hands on approach with even less task demand. You could also laminate the rows to make them reusable in lieu of several consumable ones requiring more paper. I like the consumable as I use that for data collection.

IXL.com is a site that provides online practice for math (and other topics). It has a hidden feature that allows for very effective differentiation. This can be highly useful in a general ed math class and in settings for special education services. This includes special ed settings with students working on a wide ranges of math topics, for algebra students who missed a lot of class or enter the course with major gaps, and for the general algebra population to meet the range of needs. IXL can be used before the lesson or after, for intervention.

By way of example, assume you have a student or students working on graphing a linear function using an XY table (image below). Using a task analysis approach, this topic can be broken up into smaller parts: completing an XY table, plotting points and drawing the line, interpreting what all of this means. I will focus on the first two in this post.

IXL has math content for preschool up to precalculus. For the topic of graphing (shown above) many of the steps are covered in earlier grades. For example, plotting points is covered in 3rd grade (level E), 4th grade (level F), and 6th grade (Level H). To prepare students for the graphing linear functions, they can be provided the plotting points assignments below to review or fill in gaps.

The tables used to graph are covered starting in 2nd grade (level D) and up through 6th grade (level H). These can also be assigned to review and fill in gaps.

When it is time to teach the lesson on graphing a linear function, IXL scaffolds all of the steps. For example, the image below in the top left keeps the rule simple. The top right image below shows that the students now have an equation in lieu of a “rule.” The bottom image below shows no table. All 3 focus on only positive values for x and y before getting into negatives.

The default setting on IXL is to show the actual grade level for each problem. I did not want my high school students know they were working on 3rd grade math so I made use of a feature on IXL to hide the grade levels (below), which is why you see Level D as opposed to Grade 2.

Here are excerpts from two handouts I use to help students understand how to write multiplication and rate word problems as math expressions. The image, below at top, shows a problem from the first handout I present. The students draw a single group represented by the rate expression (for elementary school word problems the term rate is not used). The image at the bottom is the same problem with scaffolding to write the multiplication problem. I find that students working on rates and slope in middle school, high school, and even in college struggle with this topic. I use this approach as part of a review of prerequisite skills before getting into rate and slope.

Here are some point sheets or check lists I have used to help shape behavior. This is a follow up to the post on classroom management.

Top left is a scaffolding I use to help students learn to solve math problems using multiplication (3rd grade). The situations are typically rate problems (e.g., 5 pumpkins per plant or $2 per slice of pizza) although the term “rate” is not used yet. The same concept of rate plays out in high school with slope of a line, applied to real life situations (top right).

These types of problems start in 3rd grade (below, top left), play out in 6th grade (below, top right), into 8th grade (below, middle), and into high school algebra and statistics (below, bottom). I referenced this connection previously regarding word problems and dominoes. This highlights how crucial it is that strategically selected gaps in a student’s math education be addressed in context of long range planning.

This post was inspired by some posts on teacher Facebook pages by new teachers asking for ideas. Classroom management is a common, if not the most common, issue that arises among teacher candidates and new teacher. It is challenging to some degree for almost all teachers. As a long time teacher trainer for the states of Connecticut and South Carolina, I had compiled presentation ideas about classroom management I used to share with the candidates I was helping. I hope this post can serve as a cheat sheet for new teachers.

First, I identify 3 stages of classroom management:

- Prevention – actions taken to avoid common undesired behaviors, and shaping desired behaviors
- Intervention – actions taken to address problems as soon as they arise
- Remediation – actions taken to address full blown problems.

The use of the image below left was inspired by a teacher candidate who lamented that during a school observation of a class there were no behavioral issues that would allow her to see classroom management in action. I explained that in fact she saw effective classroom management because of a preventative process implemented. Similarly, there are a couple retail chain stores I have observed with parallel issues with some process, such as how used dishes are cleared out or how bathrooms are maintained (at least the men’s rooms).

I will focus on prevention, which involves being proactive as opposed to being reactive. Like the man waiting for the leaf to fall, some teachers fall into the trap of waiting for a problem to occur. There are two books I recommend that help with being proactive: The First Days of School and Every Minute Counts.

Being proactive involves helping students understand what they are to do. This involves creating positive “Norms!” which is short for what normally happens in class. These can be positive or negative (often assumed to be positive). An example of a negative norm could be students standing at the door the last couple minutes waiting to leave. To establish positive norms, a teacher can set expectations and procedures for various situations in daily class functioning: how to enter and leave the classroom, ask a question, sharpen a pencil, work through problems in classwork etc. For example, when asking a question, are students allowed to blurt out answers or do they raise their hand and wait to be called upon?

Norms will NOT be established by simply posting rules or by verbally explaining our beloved syllabi. Posting rules (bottom left) is akin to a speed limit sign. Many if not most cars routinely drive over the speed limit on an interstate. The drivers do not respect the sign because they know there is an unwritten actual limit that one must cross before getting a ticket. Similarly, students know that often classroom rules are malleable as well.

Here is an example. Years ago, during the first day of class for my freshmen, one student was barely doing any work, despite my prompts. I pulled him aside to ask why he wasn’t working. He replied, “If I do all of my work now, you will expect me to do all my work all year.” He understood that often classroom expectations are conditional.

This Garfield cartoon (below) speaks to such expectations. Most students will toe the line once they understand what the line is. Again, not by what is posted but the teacher’s actions, day in and day out. In other words, their behaviors are shaped.

The image, bottom left, shows one of my classes at the start of a class during a first week. I was projecting a daily point sheet I use to provide them immediate feedback on how they are meeting expectations. The first expectation is that they are to work on the Do Now within 2 minutes of the bell. I have that part enlarged on the projection in the photo. To help shape behavior, I use a “Gotcha” ticket in which I am catching students doing something right. I do this to provide feedback on my 3 classroom expectations by writing a short blurb about what they did RIGHT. Every Monday I would start class by collecting the tickets I gave them and conduct a lottery with a handful of small prizes from Dollar Tree type stores. Some students would not turn in their tickets because the written praise was reward enough!

An organized classroom can support establishing norms. The images below are examples from my classrooms. (Elementary school teachers are routinely doing this!)

- Left: I color code each course (e.g., green for Algebra 2). Each class has a file box in which each student has a folder in which I return papers and share individual information as necessary. (e.g., I researched some content on welding for a student interested in that vocation.) The white paper next to some file boxes is the Do Now for the day.
- Middle: Each student is assigned a number and I use a shoe caddie to keep track of their respective calculators.
- Right: This is a learning wall with notes for the current topics. When students would ask a question about a problem, I would direct them to find the matching problem on the wall and explain the first step (8th grade is coded red and 7th grade was coded blue.)
- Bottom: On the right is a board with the daily objective and do now for each class – again, color coded. The posted photos of leaders like MLK Jr. are representations of my expectations: Responsible, Respectful, Resourceful as seen in the Gotcha Ticket. They learn these expectations by our daily activities and feedback – shaping.

As a complement to a set of classroom norms for effective classroom management is engaging instruction. Math teachers are often guilty of standing in front of the class going through multiple examples. To many students, this sounds like the teachers in the Peanuts cartoons, “wah waaah wah ….” Oral communication of information is far less effective than student centered learning. If students are not engaged by your instruction they will find something else to engage them – often an undesired behavior.

It is very challenging to differentiate while standing at the front of the room. Engage students with hands on work and walk around to differentiate. Parents do this with their children. Below my son is having a snack and watching a fire truck parade on YouTube and I was free to attend to other matters, like helping his brother.

Also, make the initial instruction accessible to all students using a Universal Design for Learning (UDL). The cartoon explains this approach. I use scaffolded handouts, color coding, manipulatives, meaning making strategies, and multiple representations. I refer to this as reverse differentiation. In lieu of waiting to provide specialized instructional strategies after an issue arises, I provide the strategies to all in the initial instruction and allow them to opt out, e.g., one student would repeatedly tell me, “Mr. E, you know I don’t use highlighters.” and he didn’t need them. An example is a lesson on perimeter and area (below) I taught to a group of elementary students (image below). I started the lesson with them building a rectangular pen for their animals and then they counted the number of pieces to determine perimeter. The formula was the LAST thing I showed them.

Finally, we can be responsive to student needs. In the image below, the female circled in red was having serious family issues. She could not focus so I offered her an alternative of playing math games on the computer instead. Often, I hear teachers warn that other students will complain about different treatment. I have almost never encountered this (see all the students on task in the photo). If the students see a teacher differentiating and attempting to meet a variety of needs for all students, they are very accepting.

In a previous post I asked readers to identify a math topic that they wanted help unpacking. Scientific Notation was cited. Here is my approach to unpacking this topic.

My first step in presenting a new topic is meaning making. For scientific notation, the underlying idea is NOTATION – “*the act, process, method, or an instance of representing by a system or set of marks, signs, figures, or characters*.” We can represent numbers in different ways, one of which is scientific notation. This is useful to represent very large or very small numbers (as happens in science). It is useful because in lieu of writing out a bunch digits, the power of 10 can be used as a shortcut. In the image above you see that 4.5 x 10^{4} has two parts, the decimal and the 10s.

Before I get into these big or small numbers, I address the concept of notation because that word is in the topic. To introduce a concept, I typically start with a related topic that is relevant for students. In this case it is money. To mirror the two parts of scientific notation, I list the bills and how many of each. In the left image below, I show both parts and pair combinations that are the same value (a single $10 bill and ten $1 bills). I then show how I can convert a single $10 bill by dividing by 10 and then multiplying the number of bills by 10 (middle image). This previews the steps used in scientific notation. Then (right image) I show the same approach for dollars and cents (which previews decimals). Note: to help flesh out the dollars and cents I would first use the linked Jamboard.

The image below left keeps the concept of money, but the images are faded. The students are still working with money and how many but now with numbers only. The middle image introduces decimals, but the same steps are used (divide by 10 and multiply by 10).

Finally, the matched pairs shown in the previous handout pages (images above) are presented with an explanation of the parts of scientific notation (below left). I explain the idea of scientific notation as a special way to write numbers, list the two parts, and then I show examples by circling the ones in each pair (bottom left) that fit the criteria. Then they identify numbers that are written in scientific notation (below right).

Following this introduction lesson, I would explain the applications (linked above) and go into more detail on how to rewrite the number in scientific notation.

I am asked by individuals, sometimes out of the blue, to provide insight about math for special education topics. I make an ongoing effort to share what I have and continuing to learn to help students. What is a math topic you want more information about? I will respond to general or even specific situations.

Enter your request in a comment.

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