Category Archives: Connections to Prior Knowledge

Math is a Language

The Gutenberg printing press was revolutionary because it provided a faster way to share words. In turn, these words and how they were structured were representations of ideas used to make sense of the world around us.

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.

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Information Processing Analogy – Big Picture

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.

Below is a model of information processing first introduced to me in a master’s course at UCONN.

Here is a summary of what is shown in the model.

  1. Our senses are bombarded by external stimuli: smells, images, sounds, textures and flavors.
  2. 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.
  3. 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.
  4. 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.

Effective instruction would address these challenges proactively. Here is a video regarding learning disabilities (LD) that summarizes the need in general for teachers to be highly responsive to student needs. Here is a great video that helps makes sense of what autism in terms of how stimuli can be received by those with autism (look for the street scene). Here is a video of a researcher explaining how ADHD responds to sensory input (he gets to a scenario that effectively encapsulates ADHD).

To address these challenges:

Ironically, this is likely a lot of information for your brain to process…

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Counting Money as a Game

The Allowance Game is retail game tailored to students who are learning to count money.

 

I revised it to make it more authentic and more functional.

  • I changed some money amounts to necessitate the use of pennies (see below)
  • I use real dollars and cents to provide more opportunities to handle real money
  • I differentiate by creating different task demands. For example
    • I was using the modified version with 2 clients
    • One was learning to count out dimes and pennies only, but could manage ONES simultaneously
    • This student would also be provided a coin chart I use to teach students how to count with coins
    • The other was practicing with ONES, and all coins up to a quarter
    • I collected data on a data sheet – 1 per student

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.

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Making Sense of Fractions – Regrouping with Mixed Numbers

It is easy to get caught up in the steps and rote memorization when working with fractions. The brain processes information more effectively when the information is meaningful. ADHD makes paying attention to rote memorization of steps even more challenging.

Below is an excerpt of work I completed with a middle school student who has ADHD. This was completed extemporaneously as intervention (you see his initial attempt was incorrect) but can be used as Universal Design in whole class instruction.

Here is a break down of how I helped the student after seeing his mistake in his initial attempt. First, I modeled the first mixed number as pizza pies.

Then I presented the problem in pizza terms. “You have 3 pies and 1 slice and you are going to give me 1 pie and 2 slices. Do you have enough slices?” <wait for response> “You don’t, so what can we do?” <wait for response> “We cut up one of the pies.” I have the student cut the pie into fourths.

I then make the connection with the mixed number and guide the student to taking away 1 pie and writing 4/4. This provides more concrete meaning for writing 1 as 4/4.

In turn, this provides meaning for the new mixed number and meaning for the subtraction of the whole numbers (pies) and the fractions (slices).

 

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WAR Card Game Modified for Height Comparision

Comparing numbers is very challenging for some students and likely speaks to a major gap in their number sense. It is also very challenging to address effectively with these students. The photo below shows an entry point into I have used with success in helping such a student.

Even for these struggling students, taller and shorter are likely prior knowledge that students understand intuitively. The people outline on the vertical number chart leverages this intuitive understanding to compare numbers.

  • I start by showing 2 different outlines and asking which is “taller” and stick use that term until the student gets the idea. This isolates the focus to comparing items.
  • Then I redo all the comparisons using the term “taller” and when the student makes a selection, 10 in this case, I reply “yes, 10 is MORE!” and have them repeat “more.”
  • Finally, I redo the comparisons asking which is more and for improper responses I ask which is taller then restate that the taller item is “more.”

For a change of pace, I created a revised version of the card game WAR by using these outlines (no face cards). The cards are 5″x7″ which I purchased on Amazon. Here is a link to the WORD document with the outlines I cut out and taped to the cards. After a student shows success with this revised game, I play regular WAR but use the people outlines for feedback to make a correction or as a prompt as necessary, with the intent to fade their use.

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Superhero Math

I was recruited to help a middle school student who is having a very rough time at this time in his life. It was shared with me that he likes Marvel superheroes and he is struggling with counting money and multiplication. Below are some ideas I presented for a test run and photos of the items I ordered for these suggested activities.

  • For multiplication

    • Put the heroes (or villains) in groups of 2 and have him count out 4 groups and compute. Use different groups and number per group. (IGNORE the numbers on the cards)

    • Get a group of 10 villain cards. Pretend heroes have to travel in groups of 2 and ask how many groups to get 10 heroes to fight the 10 villains. (IGNORE the numbers on the cards). Variations of this.

    • After gets the idea of groupings, focus on the number on the cards and show him two 5s and have him compute. Variations of this.

    • Play a game where he draws two cards and has to multiply the cards (start with very low numbers or maybe show him a 2 card and he has to pick another card to multiply by 2.

marvel playing cards

  • For Money

    • Tell him he earn money to buy these figures, one at a time – a monetary version of a token economy. Have him rank them by his favorite to least favorite and come up with a price for each with his favorite figures costing more. Start with the least favorite and make the price such that with a little practice he could count out the coins to pay for it. Maybe 17 cents with dimes and pennies. He has to count out the money correctly and independently to actually buy the item.

Avengers Figurines

  • Other options

    • If he needs work with addition you can play WAR in which 3 cards are played and each person adds to find the total. For subtraction do the same with 2 cards.

    • You can play subtraction in which one person has superheroes and the other has villains. In order for a villain to win a villain card has to be higher than a hero card by 3 or more.

You can write an 11, 12 and 13 on the J, Q, K cards respectively. All the games can be presented though Direct Instruction – I do, we do, you do. The You do can be used as daily progress monitoring. If he needs prompting this can be recorded. This can be used for your progress reports. Attached is a data sheet I use for activities.

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Making Proportions Meaningful (and Therefore Accessible)

A student reported to our schools math lab where I reside. He had a handout on proportions shown in the photo below and stated that he didn’t know what to do.

2019-02-26 10.38.31

I find that in the vast majority of situations like this the student lacks the conceptual understanding of the topic. As is typically the case, I started my sessions with the student by focusing on something he more intuitively understood. Teens know money, phones, games, music and food.

In this case I started by showing him a photo on my phone shrunk then enlarged the photo and talked about how I could double the size of the photo. We talk about what doubling means then I show him a handout with the photo in two sizes (below).

I explained that the small photo was 3×2 inches and that I wanted to enlarge it. The bottom of the big photo is 6″ but I needed to figure out the height (vertical length) which is marked with an X.

I had him figure out the height (4). Then I explained that proportional means the shape is the same but bigger or smaller. In this case both the side and bottom were multiplied by 2. Then I showed him the “mathy” way of doing the problems. This progressed towards the handout he brought into math lab. By the end he was doing the proportions independently.

2019-02-26 10.37.27

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Velocity, Acceleration, Speeding Up and Slowing Down

DISCLAIMER: This is a very mathy, math geek post but it also has value in demonstrating instructional strategies and multiple representations.

We all understand speed intuitively. Velocity is speed with a direction. Negative in this case does not indicate a lower value but simply which way an object is traveling. Both cars below are traveling at equivalent speeds.

velocity vs speed

The velocity can be graphed (the red curve below). Where the graph is above the x-axis (positive) the car is traveling to the right. Below is negative which indicates the car is traveling to the left. The 2 points on the x-axis indicate 0 velocity meaning the car stops (no speed). (I will address the blue line at the end of this post as to not clutter the essence of what is shown here for the lay people who are not math geeks.)velocity graph with tangents

Below is an example of using instructional strategies to help make sense of the graph and of velocity, acceleration, speeding up and slowing down.

As stated previously, the points on the x-axis indicate 0 velocity – think STOP sign. As the car moves towards a stop sign it will slow down. When a car moves away from a stop sign it speeds up.

The concept and the graph analysis are challenging for many if not most students taking higher level math. This example shows how instructional strategies are not simply for students who struggle with math. Good instruction works for ALL students!

speeding up and slowing down

It is counterintuitive that when acceleration is negative the car can be speeding up. The rule of thumb is when acceleration and velocity share the same sign (+ or -) the object is speeding up. When the signs are different the object is slowing down. This rule is shown in the graph but the stop sign makes this more intuitive.

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Adding ones digits in 2 digit numbers with carrying

A major obstacle in math for many students with special needs is carrying in addition problems. Below is a task analysis approach.

First, I target the step of identifying the ONES and TENS place in the 2 digit sum in the ONES column (below it is 12). In a scaffolded handout I create a box to for the sum with the ONES and TENS separated. At first I give the sum and simply have the student carry the one.

sum of ones given.jpg

Then I have the student find the sum and write it in the box (14 below). Once mastered I have the student write the sum and carry the 1.

sum of ones given

They would have mastered adding single digit numbers before this lesson. I revert back to single digit numbers to allow them to get comfortable with writing the sum off to the side without the scaffolding. (In the example below I modeled this by writing 13.)

sum of ones with color no scaffolding

The last step is to add the TENS digits with the carried 1. I use Base 10 manipulatives to work through all the steps (larger space on the right is for the manipulatives) and have the student write out each step as it is completed with the manipulatives.

sum of ones with carrying with base 10 blocks first

Finally, the student attempts to add without the scaffolding. I continue with color but then fade it.

adding 2 digit numbers with carrying with color no scaffold

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The I in Instruction can be the same I in IEP and IDEA.

I am consistently surprised by the reliance on canned items for students who struggle. There are different reasons students struggle but we know that there are secondary characteristics and factors that inhibit effective information processing that can be addressed with some Individualization.

In a math intervention graduate course I teach at the University of Saint Joseph, my graduate students are matched up with a K-12 student with special needs. The graduate student implements instructional strategies learned from our course work. Below is the work of one of my grad students. From class work and our collaboration we developed the idea of using the fish and a pond as base 10 blocks for the student my grad student was helping. He likes fish and fish will get his attention. The grad student explained that if he has 10 fish the 10 fish go into a pond. In the photo below the student modeled 16 with a TEN (pond) and 6 ONES (fish).

2018-07-26 17.13.05.jpg

Similarly, another student likes Starbursts and that student’s respective grad student created Starburst packs to represent TENS and ONES (there are actually 12 pieces in a pack so we fudged a little).  The point is that it was intuitive and relevant for the student. The student understood opening a pack to get a Starburst piece.

2018-07-26 17.41.15

 

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