## Algebra 1: Concepts and Skills

I had an interesting discussion through a Facebook post recently regarding concepts vs skills. I want to share some information I have gathered regarding this topic. I do so, because there were a substantial number of teachers advocating for skill based learning. I hope to initiate some meaningful discussion.

Below left is a photo of an information processing model presented in a graduate level course on learning I took at UCONN. A key element I want to highlight is that information is more effectively processed if the information is meaningful. A theory behind this is Gestalt Theory in which the brain want to make information meaningful or organize it, e.g., the closure model in which our brains complete the triangle in the middle of the circle portions.

The meaning underlying math skills originates in the concepts. Below are the definitions for both, with the concepts being the “how or why” underlying the skills which are the “what to do” part.

I am not arguing that skills are unimportant or that rote practice is wrong. My position is that the concepts should drive the process. Here is a cartoon I think highlights the challenges with students having only skill based knowledge for topics that have important underlying concepts. I witnessed this first hand as a college adjunct instructor and found that a substantial number of students only understood slope by its formula. I also see a substantial number of students receiving special ed services who are taught at a skill level only to allow for progress. Often this is challenging for them when they have working memory or processing issues.

I will summarize in my own words an interpretation an article I read on the definition of Math, which stated there is no singular definition. The following was a theme that appeared to emerge. Math is a set of quantitative related ideas that can help explain the phenomena and the world. The mathematical symbols are used to represent these ideas. There are different ways to represent these ideas, e.g., we represent functions with tables, graphs, and equations. Formal proofs in Western Civilization are not the same a those in the East. Computer based proofs are not fully accepted by many math experts.

Technology has provided amazing ways to represent mathematical ideas. The most genius approach I have encountered is Dragonbox. The image below shows their initial representation of an equation through their algebra app. It develops the concept and the skills simultaneously.

Below is a list of some algebra 1 topics and some of the associated concepts. These are largely derived from math sources but include some massaging by me. I am happy to hear the working definitions of others.

## Learning Itself Has Changed

I teach a math methods class for special education teacher candidates at UCONN. During a lesson on an instructional strategy on making math meaningful, I experienced an epiphany of sorts. The students were partaking in a discovery lesson in which they rotated through using four different types of manipulatives (photo below, left). They would follow directions, take photos to document their work, and then the class would rotate to the next manipulative. Two were intuitive and easy to follow, one less so, and the integer chips (red and yellow below) were foreign to several students. This mirrored the energy and attention given to a computer based discovery lesson involving matching that I also conducted. Sandwiched in between was use of my beloved PowerPoint slides (below right) in which I shared key points about meaning making. At the start I shared that I would present for less than 10 minutes, which apparently was still too long for several of them.

My point is not to be critical but to share how I am stepping back to reflect on how this anecdote reflects a broader issue. Social media, not technology in general, appears to be changing how humans learn as the formative years of the younger generations are immersed in social media.

We all know at a visceral level from our Pandemic experience that the technology itself is not nearly enough to grasp the focus of our students at a level that intellectually engages them, as opposed to engaging them with activity that is often conflated with meaningful learning. (Most of us have experienced the phenomena of an activity that the students were actively completing but at the end did not appear to learn much.)

It occurs to me that the act of learning has profoundly changed because students have all the information in the world at their fingertips (article above lightly speaks to the depth of this). They learn about each other including what they had for breakfast or what they saw at the mall. They inform and teach each other about common interests and have a broader exposure to new ideas and interests. The article cited below speaks to “learning procedure…not restricted by time and space.” That statement alone speaks to how profoundly the younger generations learn and how they view the act of leaning.

I read a comment in a teacher Facebook group posted by a college instructor who was bemoaning the engagement and effort of his math classes. This led to a broader discussion about technology as a means of engaging the students. To me it appears to be #morethantechnology and I would like to learn from others about this topic.

NOTE: I don’t believe having students exchange Tweets and Snaps about curriculum topics is enough. People share and consume information in an on demand setting, including us older folks in our ancient Facebook groups and listservs. I think we have to make the information more engaging in a substantive fashion…but how?!

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## Postsecondary Goals and Objectives Document

If you find this helpful, maybe consider making a small donation to help build an accessible playscape for children with disabilities.

The images above show a postsecondary planning documented to help a family prepare for a postsecondary transition planning meeting with the IEP team. This student was likely to be placed in a day program and live in group housing. Below are some images showing a focus on employment (no specialized education or college) and on education.

Here is a link to a folder with these documents.

## Progress, but in What Direction!?

In the video, Charles Barkley has made great progress getting to Annapolis for the NCAA Basketball Tournament. Problem is, the tournament was in Indianapolis.

For obvious reasons, in special education we frequently discuss and recognize progress. As in the commercial, there can be a lot of progress, but in what direction? Is it moving the student towards postsecondary goals the family has established and will the student be prepared for postsecondary life?

When I work with IEP teams or with families, I look to establish the postsecondary goals and evaluate progress accordingly. Often, I am called in to help when a student is in middle or high school and is years behind in math. With only a few years remaining with support through IDEA, it is crucial that progress be evaluated in terms of a long range plan that gets the student ready for postsecondary life and the student is ready BEFORE leaving high school (or a transition program). In other words, the student arrives in Indianapolis and is there before the game starts.

## College Preparation and Retests

In a recent Facebook group for high school math teachers, an interesting discussion arose about retests and how to respond to students who have a low score on a given test. A common perspective that I shared for a time is that we should eschew retests as we are preparing students for college. In this post I offer counter points to this argument, and attempt to unpack the college preparation issue. The focus is on math.

It appears that colleges universally have placement tests. Below are excerpts from universities in Florida, Texas, California, Washington, Nebraska, and Connecticut, along with a sprinkling of 2 year schools from some of these states. If a student makes an A in precalculus in high school, that does not qualify the student to take calculus or even to retake precalculus! A test score is required, whether it is the SAT or the college’s placement test. Indirectly, this appears to show that the colleges are not taking high school results at face value. Whether the student took a retest or not, or didn’t need one he or she still must prove content mastery to some degree.

Then consider what is happening at colleges. Dr. S.A. Miller of Hamilton College wrote an essay about grades in college (excerpt below left). In it, she cites Dr. Stuart Rojstaczer‘s article in the Christian Science Monitor about grade inflation in college. The focus here is not on how the grades are inflated, e.g., with a retest, but the evidence shows grades skewed towards the higher end and has increasingly become more pronounced over the past few decades. The rigor and accountability cited in the argument against retests is not what it appears.

Given the need to demonstrate content mastery through placement tests, and college academic expectations that are not quite what they seem, it makes sense to focus on mastery of content. This is illustrated by what is happening in Connecticut, a state that ranked 3rd in US News and World Report state K-12 education rankings and 2nd in Wallet Hub’s rankings of state school systems.

In a Connecticut State Board of Education report, over 40% of students entering a state community college or state university (not including UCONN) needed to take a remedial course.

If this is happening in Connecticut, it may shed light on what is happening in other states. It certainly appears that there is something not working in terms of content mastery in math. I am not attempting to place blame and certainly do not suggest that the problem is a lack of retests. Another issue is self-help study skills and how well students are playing their part of the learning process. A report of a survey conducted by Manchester Community College in Connecticut included the most common reason people struggle in their classes. The 2nd most cited response, by 60% of faculty and 61% of students, was that the students do not know how to study effectively.

It appears the issue of retests is more an issue of why is there a need for retests to be considered and the implications of not filling in the content knowledge gaps. I will conclude with a bit of irony. UCONN is ranked as the 23rd best public university. Students are allowed 3 attempts on the math placement test.

## Inclusion vs Proximity

Some educators and parents of students with special needs are unclear about what is meant by the term inclusion. Some think it is having the student with a disability in the same location as “nondisabled peers.” Some think it involves doing the same exact tasks or academic work.

Sesame Street figured this question out years ago. The girl in the red shirt in the video below (video set to start with her) was experiencing inclusion, not because she was next to the other kids. She was not jumping rope but was most certainly included and appeared to love it! (Note: “inclusion” is not defined in IDEA, so formally this issue would be one of least restrictive environment.)

Below is a genius representation of inclusion (not my idea).

It appears that inclusion is sometimes viewed as a dichotomous choice. For example, I observed the student in a school who was the most severely impacted by a disability sitting in a grade level history class during a lesson communism. This was an effort to provide inclusion but was he was experiencing proximity.

Below is an example of inclusion for a student with autism in an algebra 1 class. Below left is a typical math problem. To the right is one I created for the student with autism. It was designed to help him understand the concept of matching inputs and outputs without using a lot of the math terminology. In his case, the focus in math was on concepts.

## Counting Money at the Store

The way a student counts money in school on a school desk or table (top photo) is the way he or she will attempt count at the register as seen in the 2nd photo in which the student pulled all bills from his wallet then counted, with some bills folded. (Bonus if you can identify the woman in the photo!!!)

In the top photo (below) I had the student pull bills out from his wallet, with the bills unfolded and in order in his wallet (you can see he pulled a \$20 bill first). In the next photo you can see that he is counting out the bills from the wallet as he did in practice.

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I am asked by individuals, sometimes out of the blue, to provide insight about math for special education topics. I make an ongoing effort to share what I have and continuing to learn to help students. What is a math topic you want more information about? I will respond to general or even specific situations.

Enter your request in a comment.

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## “They Will Never Need This Math”

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.
• I was called in by a district to help a 10th grader who was not grasping the basic math or pre-algebra that was presented for months. He was showing significant task avoidance. The postsecondary education goal was for him to attend a community college. I started algebra work with him immediately and he was grasping it.
• 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 a vocation, cover the math needed for that vocation. For example, I worked out a long range plan for 7th grader whose mother shared that may work in an auto repair setting. The math needed for that vocation is measurement so the plan focused on measurement and life skills/consumer math.
• 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.

Students should be presented the math they NEED.