Author Archives: ctspedmathdude

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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Curriculum Based Assessments

Most testing for IEPs involves standardized testing. As I wrote in a previous post, this is important testing but is not sufficient. A major focus of special education is to make the general education accessible as possible. Hence, curriculum based testing is an important complement to the standardized based testing. For example, the KeyMath3 assessment will speak to problem solving or geometry but those are broad categories. If I am working with a 3rd or 4th grade student, I would be interested in the student’s level of mastery in computing the perimeter of a rectangle.

Also, math is very different than reading because math has a variety of categories of math, aka domains. A student testing at a 4th grade level in math does not reveal much information, as I explain in this previous post.

When I conduct evaluations or assessments, I go to the Common Core Standards and assess each with curriculum based problems, see below. The photo shows my planning document and then I transfer the problems to a student handout for the student to complete.

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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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The Process of Learning

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.

Here is a link to a video of me explaining this.

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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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Counting Out Value of Coins

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.

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Online Floating Rulers to Measure Length

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.

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Q&A Regarding Math and Online Learning

I will be fielding questions about math and online learning in real time. As a follow up, I will respond to questions through this blog post. If you did not catch the Instagram session and have questions, you can post them here through a comment. I will post replies on this post.

Below is a list of links to resources, e.g. online handouts, activities, which align with the discussion.

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Karen Linder (advocate and parent of a son with a disability) and I are collaborating on a series of Parent User Guide materials for parents. The goal is to scaffold parent efforts to use the IEP. We are presenting documents in smaller chunks to make it easier for parents to understand the steps they can take without being overwhelmed with an already challenging endeavor of advocating for your child’s needs.

Our first installment is part 1 in understanding IEP objectives (see screen shot of front page above). There is a document with templates to use and an accompanying video to provide additional support in how to use the templates. Please share feedback so we can improve our efforts to help parents.

Please feel free to reach out to us if you think we can be of service or if you have questions.

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