The effort to provide intervention to fill in gaps is challenging for different reasons. One reason is the effort to balance support for current content while filling in gaps. This post shows an example of how to fill in gaps while working through the current topic.

Overview

Various rubrics used to assess teacher instruction includes an effort to build on or connect to prior knowledge. If the student has gaps with prior knowledge, the lesson becomes less accessible for students with the gaps. Previously, I addressed how to support both current content and fill in gaps. The idea is to systematically fill in gaps by addressing prerequisite skills as they arise in new lessons.

Example

The handout out below shows an example of how this can play out. The first page is used as a do now for the content presented on page 2. If you are teaching a student how to solve 1 step equations and are moving into integers, page 1 is a a means of supporting the new content while filling in possible gaps. The first image shows the student will need to evaluate -13 – 3 as part of the solving in the lesson. This can be addressed in the do now, as shown in the 2nd image, page on the right. (Notice all the problems on page 1 are steps to solve on page 2 problems.) This is useful for students with special needs and for differentiation.

Students start work on solving equations in 6th grade. They work on it in 7th and 8th and into algebra classes in high school and college. Despite this, many struggle at all levels with solving. I have witnessed and heard about this at each level. This post addresses one possible reason, which is the students are not grasping the concepts. This may be due to the way equations are presented. This post unpacks the possible reason and presents a scaffolded handout to unpack the concepts and vocabulary.

Add-on Steps We Teach

The image below shows a earlier scaffolded handout I had used for years. It includes two elements that in my experience are common: referring to the “sides” of the equation, and the vertical orientation of the step with inverses.

I think these are “add-ons” that we teachers incorporate into math. I assume there are advantages and disadvantages for each. In this case, the disadvantages as I see them are a follows.

The term side may be somewhat ambiguous. We use it in the contexts of which side are you on, stand off to the side, side of a building, there are two sides to every story etc. We know this because most of us probably have experienced something like what is shown in the image below left.

The vertical orientation (image above “-1”) does not produce an expression that they have seen before. We are changing the rules of the game and then wonder why they are confused -see another in image below right. (I have used my share of add-ons.)

Explanation of the Handout

I am attempting to wean myself off of the use of add-ons. This handout The handout scaffolds both mental steps and written steps. In addition to the skill based steps, they focus on concepts and vocabulary. Here is a YouTube video showing how it works, shown on a Jamboard accessible using this link. (You have to make a copy to access it.)

The first 5 steps focus on mental steps which are identifying components of the equations using vocabulary. The steps prompt them to write what their thinking is. The example can be presented in “I Do” fashion with think alouds.

The solving steps guide a horizontal orientation. This allows students to focus on the expressions as a whole and not in split-level fashion.

Because the expressions with steps included are horizontally oriented, it is easier for the students to see the expressions as a whole and then simplify them.

The 0 is written and addressed with an additional step to highlight the identity.

I considered adding another step for the students to indicate that the final equation shows the solution.

Blank Templates

The second sheet has blank templates. One possibility is that you could assign students problems and they can complete the first 4 on here.

Solving the variable on both sides is the Cerberus of 1 variable linear equations. It has multiple steps, simplifying expressions, and eliminating a variable expression. The later is a new step, added on to all the other steps. This post describes a scaffolded handout to guide students through the mental steps and the written steps.

Overview

The process starts with mental steps of identifying the two variable terms. This directs students to focus on identifying that the equation has a variable on both sides which in turn leads them to understand the algorithm they will follow. The circling focuses attention on the operations. Then the students choose which variable term to eliminate and identify the inverse operation. The written steps are then guided.

Choosing the Variable Term

As I did in the scaffolded handout for solving a 2-step equation, I have students solve the equation two different ways. This time by eliminating each variable term respectively. This allows them to see for themselves which term may provide the path of lesser resistance.

Solving 2-step Equations is an escalation in terms of task demand. Starting in elementary school, they are exposed to math sentences in which there is a single operation. Now they are asked to choose a number to eliminate. This post provides details about a scaffolded handout. It addresses the choice of number to eliminate and how one choice may provide the path of least resistance.

The Steps

Similar to the handout for solving 1-step equations, this scaffolded handout engages the students with the mental steps in addition to the steps they write out when they show their work. A key concept is identifying that there are two numbers to eliminate and then eliminating them, one at a time.

Choosing the Number

In lieu of teaching the method of doing PEMDAS or order of operations in reverse, I focus on presenting both the respective steps for eliminating either number in the expression with the variable. I believe it strengthens their understanding of the steps for solving and it shows them a reason to choose one route over the other – but they are free to choose! Here is a FB Reel and Youtube video addressing this.

Many students struggle with solving equations. Many struggle with simplifying expressions. Putting these together becomes an algebraic version or Orthrus (brother to the more famous 3-headed dog, Cerberus). This post shares a scaffolded handout to guide students through the process while making sense of each step.

Explanation of Handout

The first two are mental steps. By addressing them explicitly with students providing a written evidence of their thinking, the mental steps can be observed and assessed. The operations are circle to be proactive in addressing common misconceptions. This can invoke a discussion on the meaning of the operations negative and subtraction. Once the expression is simplified, prior knowledge of 2-step equations kicks in. Here is a link to a previous handout.

I previously posted scaffolded handouts for solving 1 step linear equations. I added a couple steps to create a new handout. In it I flesh out ALL of the steps, including identifying the inverse, the number to eliminate, simplifying the expressions to get the identity (0 or 1) and then simplifying again to eliminate the identity (see images below).

Here is an introduction to solving equations using a Jamboard (see photo at very bottom for how to make a copy). A seesaw is used to unpack the concept of an equation as two sides that are equivalent. The box is used to unpack the concept of a variable representing an unknown number (or oranges in this context). The form of a solution for an equation is established, with students revising to create other solutions. Here is a YouTube video and a FB Reel showing how it works.

Students are then provided a couple slides to match seesaw representations with actual equations. The matching provides a scaffold to support the connection between representations.

Then the seesaw and box representation is used to unpack the steps. Students are provided the steps as written directions on how to model the given solving steps with the seesaw.

The students are then provided the equation and seesaw representation, along with the solving steps provided as moveable pieces. Students slide pieces and move seesaw objects to make the connection between the two. Here is a link to a post with an updated version of this handout.

Finally, students are given the equation and tasked with completing all of the steps including the initial set up. This slide can be copied with new equations entered for additional practice (including having the variable on the right or writing the number before the variable (e.g., 5+m=8).

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.