Money is intuitive for many students, even when the underlying math is not. For example, I often find that students who do not understand well the concept of Base 10 place value do understand $10 and $1 bills. With this in mind, I created a virtual scaffolded handout that builds on student intuitive understanding of the bills through the use of $10 and $1 bills to represent regrouping. Here is a video showing how I use it.
In the photo below, at the top, a $10 bill was borrowed into the ones column. The reason is that $7 needed to be paid (subtracted) but there were only five $1 bills. In the photo below, bottom, the $10 bill was converted into ten $1 bills. On the left side of the handout, the writing on the numbers shows the “mathy” way to write out the borrowing.
Once the student begins work with only the numbers, the $10s and $1s can be referenced when discussing the TENS and ONES places of the numbers. This will allow the student to make a connection between the numbers and their intuitive, concrete representation of the concept.
Often education and special education focuses solely on content. In turn, the content may focus only on steps and facts to memorize as opposed to ideas and concepts.
A challenge for many students during k-12 education then in post-secondary life is being an independent, self-sufficient learner. The adults supporting them often focus on short term success at the expense of long term success in terms of independence.
I propose shaping the independent learning process early and often. An activity I use is completing jigsaw puzzles.
With guidance, completing a puzzle can activate 3 processes of learning: critical thinking, mindfulness, and perseverance. By having a strategy of identifying the side pieces of the puzzle, the student is analyzing pieces which is critical thinking. Paying attention to the shapes of pieces in mindfulness. Continuing to try different pieces when pieces don’t fit is an act of perseverance. Start with fewer pieces and focus on the process, then use increasingly more pieces of the same puzzle before moving on to another puzzle.
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.
Counting out the total value for a set of coins can be very challenging for students who struggle with money.
Here is a video showing how to use a WORD document table a former colleague and I created to support students in a life skills class. The video shows how this handout is used but does so with a virtual version the can be completed on the computer (see image below). Here is a link to the virtual document.
To help students learn how to measure with a ruler, I focus on minimizing the number of tic marks on the ruler at first. The image below shows an excerpt from a WORD document with a halves ruler that I use and an instructional strategy. It also contains a quarters and an eighths ruler that students can slide around the WORD document as shown above and in a video explaining this artifact and how I created it.
This is useful for distance learning as well as in class. Here is a link to the WORD document with the rulers shown in the video.
I use the following method as a entry point for double digit numbers.
The photo below shows 2 packs of Popsicle sticks counted as 10 each, followed by single sticks counted as 1 each. The student counts on from 20, with the use of the scaffolded handout (photo at bottom). The handout focuses only on counting on from 20 and shows a photo of 2 of the bundles of sticks. Similar handouts involve counting on from 10 or from 30 etc.
By engaging in the actual counting, the student learns the 10s by doing. This would be followed by counting on from each 10 without the handout.
The use of Popsicle sticks is useful for 2 reasons. First, a bundle of items like shown below is more concrete than the rods for Base 10 blocks. Second, pulling packs of sticks apart of bundling 10 sticks together is an act that is concrete for students and ties into their prior knowledge regarding the grouping of objects (e,g. pack of gum).
The graphic organizer below is used to show the student the steps for addition. It also addresses the concept of addition (which I have addressed previously) as an act of pulling “together” two parts to form a whole.
The student is prompted to move the first part (set of coins in this case) to the number chart. This can be completed with 1 to 1 correspondence or with subitizing (identifying the number of items without counting). Then, the student is prompted to move the second part while counting on, e.g. 7, 8 etc. (as opposed to starting from the left and counting from the first coin: 1, 2 etc.). The chart scaffolds the counting on and allows the student to see the total as a magnitude.
It is important to first teach the students the “rules of the game”, i.e. how to use the graphic organizer. To do this have the student simply move the first part to the number chart then the second part. The student can also be prompted to state the addition problem (written at the top). When the student is fluent with these steps the counting on can be implemented.
The next step would be to replace the coins with the symbolic representation, 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.
The students loved playing the game, it was engaging so they practiced the counting out money, I was able to collect data, and I was able to differentiate. When I co-taught a Consumer Math course, I would assign a para (instructional assistant) to facilitate the game with a couple students and to collect data.