Category Archives: instructional strategies

Learning Math – The Patting Head and Rubbing Belly Phenomena

In education, math especially, there exist a learning situation I call the patting head and rubbing belly phenomena. In this phenomena students are presented a math problem that consists of several steps they know how to do and then maybe one or two additional steps that are new. Adding the additional step is like adding the task of patting your head while you rub you belly. The additional math step seems so simple, but attempting it simultaneously with an additional task can make the entire effort exceedingly challenging. A related scenario is generalization to different settings, but that is different. This is true for all types of math, whether it is the general curriculum or life skills/consumer math.

I have written about how we cut up a hotdog for a baby in a highchair and that we could do the same for math topics using a task analysis and chunking approach. Related to this, I recommended that support class be used not to backfill gaps but to address prerequisite skills for upcoming or current math topics covered in a general ed math class.

This phenomena plays out in life skills math or consumer math in a stealthy manner because the steps or tasks seem so simple. For example, many of us have worked with a child or student who was learning to count money. When learning about a nickel or a quarter, the coin name and value are easily identified. Once both are introduced, many students confuse the two and may even freeze while attempting the work with the coins.

There is an ABA based process for addressing this using a task analysis and chaining in which steps are worked on in isolation before connecting (chaining) the steps together (and not all of them at once until the end). One related strategy to help implement this approach is through scaffolded handouts in which the steps are enumerated and the structure of the handout isolates the tasks. I have used this approach for 1 to 1 correspondence up to AP Statistics (see below).

When working out a draft of an IEP, I suggest having the task analysis and chaining explicitly identified in the accommodations page and ask for an example of what this looks like (using an example math topic).

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Introduction to Unit Cost (Unit Rate)

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.

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Addressing Multiplication as a Gap

There is a delineated sequence for teaching multiplication over the years, including repeated addition, set modeling, arrays, single digit etc (below). It exists to build conceptual understanding of the multiplication facts that are at some point memorized by many students. When I work with students who are a more than a year behind in the sequence for multiplication, I find that programming for these students to help them catch up sometimes involves shortcuts such as a reliance on rehearsal or resorting to use of the multiplication table in isolation. I am not against use of the table or narrowing the focus, but am promoting a more comprehensive approach.

Here is a sequence, on a Jamboard, I used for a recent student who was struggling for a long time with multiplication (explanation of each step shown below images). The student was interested in Minecraft so I used Minecraft items such as stone bricks and a wagon. I would spend as much time on each step, as necessary.

  • Count out the total number of stone bricks. This allows an assessment of how the student counts: by 3s or individually. If individually, I would prompt the student to count by 3s.
  • Add 3 + 3
  • Show a short video on the wagon (this adds interest and gives the students a bit of a break)
  • Present the bricks in 2 groups of 3, in context of 2 wagons with 3 bricks each.
  • Present the same problem as a multiplication problem but with the image for one of the factors in lieu of two numbers.
  • Use the multiplication table to skip count.
  • Present additional multiplication problems for independent attempts. The student completed both problems independently, without the table. For him this was a major success.

The follow up to this would be to assess his ability to do higher groups of 3s and groups of other numbers. For some students, I work on mastery of individual numbers before moving on. This builds confidence and allows for fluency in the process of skip counting out to the appropriate number. NOTE: I don’t worry about rote memorization of the facts but of fluency in the process of skip counting out the answers.

For students who are older, I sometimes recommend that the student be presented problems with visuals but then use a calculator to compute. This can develop conceptual understanding and also address the working memory and other related issues that undermine learning math facts.

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Introduction to 2 Variable Inequalities

Previously, I shared how I use a Google Jamboard to introduce 1 variable linear equalities with a focus on conceptual understanding. I use the same approach for the 2 variable version (example problem below).

A conceptual gap that typically arises is the students do not understand what the shading represents. This is what I am addressing from the start using a Jamboard. First, the focus is on understanding the inequality and identifying a single point that works (below).

The next step is for students to determine more points that are solutions for the inequality, with no equal to part. (below).

The equal to part is addressed separately (below).

The equal to and the greater parts previously addressed are combined together.

The inequality is will be expanded to include an operation (+ 2) with a focus on the equal to part first.

The greater than with no equal to is addressed.

Then the equal to and greater than are addressed sequential. The equal to results in dots in a straight line and in lieu of plotting all the points, a line is drawn (building on the intro to 1 variable inequalities). This is followed by the greater than part and shading in lieu of plotting all of the dots above. THIS is where they gain an understanding of what the aforementioned shading is.

Finally, the dashed line is addressed by showing, as was done with the 1 variable inequalities, that there is a cutoff point that is not part of the solution set so in lieu of plotting a bunch of open circles, a dashed line is drawn.

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Life Skills Math – Not So Easy

As I wrote previously, shopping is dense with math tasks as are grocery stores. Here are some division situations that are sneaky challenging and require a student to know when and why to divide before even reaching for the calculator. I will use these to help illustrate the fact that life skills math is not simply counting money or using a calculator to add up prices. There is a great deal of problem solving and thinking skills that need to be developed.

For example, if a student has $60 to spend on gifts for her 3 teachers the student needs to understand that she can spend up to $20 per teacher (before even talking about taxes).

An entry point for division can involve a dividing situation the students intuitively understand, e.g., sharing food. Start with 2 friends sharing 8 Buffalo wings evenly (below).

This can lead into the 3 teachers sharing the $60 evenly (below). In turn, this can be followed by the online shopping shown above.

This approach can be used to develop an understanding of unit cost (cited in the shopping is dense post). Start with a pack of items to allow the students to see the cost for a single item before getting into unit cost by ounces, for example.

I have had success with teaching these division related concepts using sheer repetition as much of our learning is experiential learning. Using a Google Jamboard as shown in the photos allows for the repetition.

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Introduction to Inequalities

For students with special needs, the teacher speaking “math” to students sounds like the teacher from the Peanuts cartoons.

This is apparently the case when students are learning about inequalities such as x < 4 because I have seen many high school and college students struggle with this topic. The challenge is that teachers are often focused on the math symbols and steps as opposed to the math concepts. In contrast, below are Google Jamboard slides I use (you can make a copy and edit) to introduce the concept of inequalities.

First, I start with a topic of interest and possibly prior knowledge for the students (age to get a drivers license – below). I present the idea of an inequality in context before I show any symbols. In this case, students identify ages that “work”.

Then I introduce the symbol (below). In this case, I include equal to for the inequality (x > 16 vs x > 16). The students plot the same points then we discuss that there are many other ages that work. These ages are called solutions. We put a closed circle on all of the solutions. Then discuss that ages are not exactly whole numbers so we can plot points on all the decimals. Then we discuss that the solutions keep going to the right so we keep drawing dots to the right. There are so many dots we draw a “line” instead of all the dots.

Then we do the same steps for a situation in which the number listed (52 in the case below) is NOT a solution. The students put dots closer and closer to the number but cannot put a closed circle on 52 as a solution (top photo below). Then we present the symbols and talk about the number as a cutoff point that we get really close to but cannot touch. Therefore we use an open circle to show the number is NOT a solution.

This introduction can be followed by problems on a handout, ideally with context then without.

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Shopping is Dense with Math Tasks

I recently worked with a student on an online grocery shopping activity – finding ingredients for mac and cheese. We had the ingredients listed in a column on a Google Doc (allows both of us to edit the doc simultaneously) and then he cropped and pasted a photo of each ingredient (see photo below). The goal was for him to identify the total he need and the total cost in planning for actual shopping or to continue with the online shopping. Note: he wasn’t actually buying anything at this point but this was a step in preparing him to do so.

This activity is dense with math tasks and shopping related tasks. The math tasks include the following:

  • Identify the price (vs quantity of the item or unit price).
  • Interpret the quantity for the ingredient.
  • Identify the units (oz and cups)
  • Convert units
  • Compare amount in box with amount needed.
  • Determine how much more is needed, if any.
  • Compare choices before selecting the item, (Barilla Pasta vs another brand).

To convert units, the “mathy” approach can be used or the student may simply use an app. For this student we chose an online unit converter (see below). This is more complicated that it appears. The student must choose the units and the order (in this case convert cups to ounces or vise versa), distinguish between imperial and US cups, understand that you enter the quantity (the search results in 1 US ounce appearing by default), and then interpret the decimal (keep in mind the ingredient quantities are in fractions).

Life skills math is more complex and challenging that parents and educators may realize. As a result, the planning for developing these skills should begin much sooner rather than later – not to mention the actual logistical tasks of shopping, e.g. finding an item in the grocery store.

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Critical Thinking

Often we adults engage students with closed-ended questions and then consider this as having a conversation with the student. I witnessed this first hand in a high school consumer math course I co-taught. The adults sat with the students the first day after December break for a conversation about their break. The questions were consisted of and were similar to the following. “Did you have fun?” “Did you eat a lot?” For some, like my son, this is appropriate. For many others, we are offering low hanging fruit that does little to move them forward.

Ask open-ended questions that prompt the student to engage in critical thinking such as analyzing and evaluating – below, courtesy of Jessica Shabatura. Work this into IEPs and 504 to have teachers implement this. For example, I asked the students what they liked about break. Then I asked why they liked it. Here is an example of me questioning my son, who does not have a disability, when he was maybe 4.

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Secondary Characteristics: A Performance Factor

For students with a disability, performance does not align with ability.

In my view, there are 3 different categories of performance factors: the disability, gaps in achievement, and secondary characteristics. (Percents are contrived to provide a visual representation.)

To address these secondary characteristics, which manifest as a set of behaviors, I suggest a focus on shaping with a token board.

Here is a video explaining this.

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Using $10 and $1 Bills to Represent Regrouping in Addition or Subtraction

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.

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