When our 3rd child was born, we decided to buy a used Honda Odyssey as 3 young kids were not fitting into a sedan. Being the stats geek I am (master’s in statistics at the University of South Carolina – total geek) I collected mileage and price data for all the used Odysseys for sale on dealer sites throughout South Carolina. I then created a the scatterplot shown below. I went to a dealer, showed an agent my graph, and he immediately exclaimed “Where did you get that? We create graphs like that every week!”
It was this experience that led me to the idea of using used car data to introduce linear functions. Shopping for a used car has proven to be a relevant, real life activity the students enjoy.
Used car shopping to collect data on 10 used cars of a single make and model.
Creating a scatterplot for price vs mileage of the used car of choice.
Creating a line of best fit (regression line) to model the data.
Creating a linear bi-variate equation (regression equation) to model the data.
The activity is presented on a WORD document (feel free to revise). It shows screenshots to walk student through the Carmax website (subject to Carmax revising their website). The screenshots make it easy for the student to navigate, which increases independence. (NOTE: there is an ample number of Youtube videos on using Google Sheets for this activity.)
The end product looks like this. Note the importance of using 1000s of miles as the slope is more meaningful, -$140.64 per thousand miles, as opposed to 14 cents per mile. I would start with the scatterplot alone to unpack the variables, the relationship between the variables, and the ordered pairs. Then the line and equation can be introduced to show a meaningful use of the line and the equation. The y-intercept has meaning with “0 miles” equating to a new car (I do not explain that new cars have miles already accumulated until we unpack the math).
I have frequently encountered the presentation of absolute value as a positive value or opposite. This is part of the repertoire of memory devices we (certainly I have in the past) use as a short cut to learning how to do the steps for a problem. The meaning of the absolute value of a number is it’s distance from 0 (below).
Below is an image of a Do Now or Initiation handout I use to introduce absolute value. From the start I focus like a laser on the meaning of distance for absolute value. I start with a situation that may be prior knowledge for them. Then take a step towards the mathy part with the numbers and slowly make my way to the symbol.
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.
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 104 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.
As a parent of a child with a disability and as a math educator, I am repeatedly struck by the fact that a group of adults (educators and professionals) convene to discuss and plan how to help a child. A great deal of time, resources, and money is concentrated on that child. Awesome! Unfortunately, in math education I frequently encounter situations in which this collective energy is concentrated on math that is more about boxes to check than engaging the student in math that he or she will need in post-secondary life.
IDEA enumerates the purpose of special education, with the transition goals aligned with employment, living skills, and future education that are desired for each individual student. This is explicit and aligns with the goal most teachers likely have, to make a difference in the lives of their students.
Despite this, when I am called in to help with math programming for a student I often find the math being presented to the student is not aligned with the post-secondary goals and often appear to the result of following the general ed curriculum, by default. Here are some examples.
I co-taught an algebra 1 class with a student impacted by autism to the point that he needed a paraprofessional guiding him through the daily work. He worked in isolation with the para and struggled with the basic elements of the course. It was not until his junior year that he was moved to a consumer math class.
A senior was in a consumer math course I taught. The course was for students who could not access the general curriculum, yet her transition goal for education was to attend a community college. This setting likely require a math course (that did not have consumer math topics) and a placement test.
Over 25 years of teaching math I have periodically heard educators minimize the struggles of students with math with the rationalization “they will never need this math.” My response is to ask why “then we are presenting this math to them?!”
So what math do they need? Here is a list of blog posts that address this question. In short, here is what I share with IEP teams, educators, parents, and special ed teacher candidates I teach.
If the goal is a career that involves a 4 year degree, then boxes must be checked. The student will have to have the math courses needed to get into the college and to prepare for the math in his or her major. This is the “mathy math” that will be on a college placement test as well.
For a 2 year degree at a school with open admissions, the focus of the high school math can be narrowed to the math course required (if any) and on the placement test. Typically, this would involve a focus on algebra. For the aforementioned 10th grader, we did not cover geometry and prioritized the algebra topics to cover.
For another middle school student whose goal was to have a job and to be as independent as possible. He loved sports and his mother said he would love to work in a sports related store. For him I recommended data and statistics (not the mathy type but meaningful and applied stats and data) to help him make sense of and discuss sports stats. This was complemented by a recommendation for consumer math.
Several special ed teachers identified solving multi-step equations as the most challenging math topic to teach in middle school math. Here is my approach to teaching multi-step equations like 3m + 4m + 1 = 15. .
First, I use a task analysis approach to break down the math topic like we cut up a hotdog for a baby in a high chair. MOST of the steps involved are prior knowledge or prerequisites skills. I present these in a Do Now (warm up, bell ringer, initiation) – see image below. This allows me to fill in the gaps and to lay the foundation for the lesson. The prerequisite skills include simplifying expressions and solving 2 step equations. I also present meaning for the equation with a relevant real life problem that is modeled by this equation. By attempting the walkathon problem without the “mathy” approach, the students will more likely understand the equation and why they add 3m and 4m.
After reviewing the Do Now I use Graspable Math, which is a free online application that allows users to enter their own expressions and equations. These can be manually simplified and solved by moving parts around. Here is a tutorial on how to do this. This allows them to manually work with the simplifying and the equation before working on the handout, in a concrete-representational-abstract approach.
This is followed by a scaffolded handout with the use of color coding. I have student work on the first step in isolation as that is the new step (the other steps are prior knowledge and were addressed in the Do Now). This avoids all the work on the other steps that can result in sensory overload and allows me to address mistakes in the new content immediately.
This handout can have the equations removed and be used as a blank template to follow. In turn this would be followed with regular solving worksheets.
Unit rate (e.g., hamburger meat on sale for $2.39 per pound or you make $13 per hour) is an incredibly important topic in middle and high school. First, unit rates and unit costs are common in life. Second, in the Common Core State Standards math categories you can see that Ratios and Proportions (which includes unit rate) are a 6th and 7th grade topic and are then replaced by Functions in 8th grade. Below is a photo showing a graph of a function you can see that the slope in an application is a unit rate.
The unit rate is also conceptually challenging whether it is in a function or is a unit cost at the store. This is a major sticking point for many students in special ed who have fallen behind. To address this, I used the approach below.
First, I present a pack of items the student likes (4 pack of Muscle Milk for this student). Use a Jamboard to show a 4 pack and the price of the 4 pack (photo on left). Then I “pull out” the 4 individual bottles and divide the $8 among the bottles to show $2 for each bottle. Finally, I have the student shop for packs of items at a grocery store or Amazon and compute the price for 1 item using a mildly scaffolded handout.
I Follow the same steps for ounces or pounds but show how 4 oz is divided into single ounces (in lieu of a pack divided into single items). Then the student shops for items that can easily be divided to get a unit cost.
Math is a language with words and other symbols that also makes sense of the world around us. We consume and know more math than we realize or allow ourselves credit for.
When buying the latest iteration of an iPhone, we may call forth algebra. How much will you pay if you buy an iPhone for $1000 and pay $80 a month for service? Well, that depends on how many months you will use this iteration before moving on to the next iPhone. The number of months is unknown so algebra gives us a symbol to represent this unknown number of months, x (or n or whichever letter you want).
Just as there is formal and informal English (or other language), we can engage algebra formally or informally. You don’t need to write an equation such as y = 1000 + 80x to figure out how much you will pay. You can do this informally, compute 80 times 10 months + 1000 on the calculator. Then try 80 times 12 months etc.
Math provides us a means of organizing and communicating ideas that involve quantities like the total cost for buying an iPhone.
The difficulty in learning math is that it is often taught out of context, like a secret code. In contrast, a major emphasis in reading is comprehension through meaning, such as activating prior knowledge (see below).
In fact, math absolutely can and, in my view, should be taught by activating prior knowledge. My approach is to work from where the student is and move towards the “mathy” way of doing a problem.
Without meaning, students are mindlessly following steps, not closer to making sense of the aspects of the world that involve numbers.
Effective instruction is effective because it addresses the key elements of how the brain processes information. I want to share an analogy to help adults (parents and educators) fully appreciate this.
Our senses are bombarded by external stimuli: smells, images, sounds, textures and flavors.
We have a filter that allows only some of these stimuli in. We focus on the ones that are most interesting or relevant to us.
Our working memory works to make sense of the stimuli and to package it for storage. Our working memory is like a computer, if there is too much going on, working memory will buffer.
The information will be stored in long term memory.
Some will be dropped off in some random location and our brain will forget the location (like losing our keys)
Some will be stored in a file cabinet in a drawer with other information just like it. This information is easier to find.
Here is the analogy. You are driving down the street, like the one shown below.
There is a lot of visual stimuli. The priority is for you to pay attention to the arrows for the lanes, the red light and the cars in front of you. You have to process your intended direction and choose the lane.
There is other stimuli that you filter out because it is not pertinent to your task: a car parked off to the right, the herbie curbies (trash bins), the little white arrows at the bottom of the photo. There is extraneous info you may allow to pass through your filter because it catches your eye: the ladder on the right or the cloud formation in the middle.
Maybe you are anxious because you are running late or had a bad experience that you are mulling over. This is using up band width in your working memory. Maybe you are a relatively new driver and simple driving tasks eat up the bandwidth as well.
For students with a disability that impacts processing or attention, the task demands described above are even more challenging. A student with ADHD has a filter that is less effective. A student with autism (a rule follower type) may not understand social settings such as a driver that will run a red light that just turned red. A student with visual processing issues may struggle with picking out the turn arrows.
There are numerous hidden tasks that we undertake while at the grocery store. We process them so quickly or subconsciously that we are not aware of these steps.
As a result, we may overlook these steps while educating students on life skills such as grocery shopping. Subsequently, these steps may not be part of the programming or teaching at school and therefore generalization is left for another day. Yet, the purpose of IDEA is, in essence, preparing students for life, including “independent living.”
Step 1 is to administer a baseline pretest during which we start with no prompting to determine if the student performs each task and how well each is performed. As necessary, prompting is provided and respective documentation is entered into the table (to indicate prompting as opposed to independent completion). For example, I worked with a client who understood the meaning of the shopping list but started off for the first item without a basket or cart. I engaged him with a discussion about how he would carry the items. At one point I had him hold 7 grapefruits and it became apparent to him that he needed a cart. (I documented this in the document.)
Other issues that arose were parking the cart in the middle of the aisle, finding the appropriate section of the store but struggling to navigate the section for the item (e.g. at one point I prompted him to read the signs over the freezer doors), and mishandling the money when prompted to pay by the cashier announcing the total amount to pay.
Step 2 is to identify a task or sequence of tasks to practice in isolation based on the results of the pretest. For example, this could involve walking to a section of the store and prompting the student to find an item. Data collection would involve several trials of simply finding the item without addressing any other steps of the task analysis.
Step 3 would be to chain multiple steps together, but not the entire task analysis yet. For example, having the student find the appropriate section and then finding the item in the section.
Eventually, a post-test can be administered to assess the entire sequence to identify progress and areas needing more attention.