Tag Archives: solve

Concepts vs Skills – Need Both

In general math is taught by focusing on the steps. Conduct a Google search for solving equations and you will see the steps presented (below). You need a video to help your student understand solving and you typically get a presenter standing at the board talking through the examples. (I’ve posted on my approach to solving equations.)

When the math is taught through the skill approach the student may be able to follow the steps but often does not understand why the steps work (below). The brain wants information to be meaningful in order to process and store it effectively.

calvin hobbs toast

To help flesh this situation out consider the definitions of concept and skills (below). Concept: An idea of what something is or how it works – WHY. Skill: Ability” to execute or perform “tasks” – DOING.

definition conceptdefinition skill

Here is how the concept first approach can play out. One consultation I provided involved an intelligent 10th grader who was perpetually stuck in the basic skills cycle of math (the notion that a student can’t move on without a foundation of basic skills). He was working on worksheet after worksheet on order of operations. I explained down and monthly payments then posed a situation shown at the top of the photo below. I prompted him to figure out the answer on his own. He originally forgot to pay the down-payment but then self-corrected. Then I showed him the “mathy” way of doing the problem. This allowed him to connect the steps in solving with the steps he understood intuitively, e.g. pay the $1,000 down payment first which is why the 1000 is subtracted first. Based on my evaluation the team immediately changed the focus of this math services to support algebra as they realized he was indeed capable of doing higher level math.

solving equation with conceptual understanding first

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Introduction to Equations – (Meaning Making)

This is a meaning making approach to introducing equations. I will walk through the parts shown in the photo in the space below this photo. (A revised edition of this handout will be used in a video on this topic.)

intro to equations

First I explain the difference between an expression (no =) and an equation (has =). An equation is two expressions set equal to each other (21 is an expression).

intro to equations definition equation

I then develop the idea of a balanced equation and will refer to both sides of the see saw as a prelude to both sides of the equation. I also focus on the same number of people on both sides as necessary for balance.intro to equations balanced vs unbalanced

At this point I am ready to talk about an unknown. Here is the explanation I use with the photo shown below.

  • I start with the seesaw at the top. The box has some guys in it but we don’t know how many.
  • We do notice the seesaw is balanced so both sides are equal.
  • This means there must be 2 guys in the box.
  • I follow by prompting the students to figure out how many guys are in the box(es) in the bottom two seesaws.
  • Finally, I explain that the number of guys in the box is the solution because it makes the seesaw balanced.

intro to equations definition solution

There are multiple instructional strategies in play.

  • Connection to student prior knowledge – they intuitively understand a seesaw. This lays the foundation for the parts of an equation and the concept of equality.
  • Visual representation that can be recalled while discussing the symbolic representation, e.g. x + 1 = 3
  • Meaning making which allows for more effective storage and recall of information.
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Evaluating the Effectiveness of Math Objectives

iep-math-objective-improve

The math objectives present in the photos on this post were written for former students of mine. These types of objectives are ineffective and ubiquitous. When I have sat in IEP meetings the majority of the time I am the only person who is capable of evaluating IEP math objectives. This post provides some guidance for others to evaluate these objectives.

In the photo above the objective has 3 major flaws.

  • “Using tools” is ambiguous. A 4 year old can use a calculator even if he does not know what he is doing.
  • “Problem solving skills” is a broad term that needs to be defined.
  • “Improve” can mean a student increases a success rate from 0% to 1%. That speaks for itself.

iep-math-objective-create-visual-aids

In the objective above there are 2 major problems.

  • “Multi-step word problems” is very broad. If a student shows she can solve a problem that requires addition and then subtraction but no multiplication or division is this mastery?
  • Often the accommodations are built into the objective and therefore the assessment. I have repeatedly had educators tell me this is what the student needs. That is valid if there is no intention for the student to do the work independently but often that point is overlooked.

iep-math-objective-demonstrate-understanding

The objective above is similar to the previous example. The examples in the objective include “solving” and “graphing.” Is the student supposed to demonstrate mastery in all the different types of algebra concepts? Or, if he can solve equations is the objective mastered?

How can caregivers evaluate these objectives? 

The language of an effective objective can be used, almost verbatim, as  problem. For example

  • Objective: Billy will add fractions with like denominators.
  • Problem: Add the following fractions (with like denominators).

Have the person writing the objectives provide an example problem that can be used to assess mastery of the objective. If the problem includes additional information or language beyond what is written in the objective then the objective is ineffective. For example:

  • In objective 1 above (the first one) the objective is to use tools to improve problem solving skills.
  • Below is a possible problem (from LearnZillion.com) that could be used. I would ask the author of the objective to explain what problem solving skills should be demonstrated and to explain what constitutes improvement. Neither of these terms is explicitly stated in this problem. It is very likely that valid responses to these questions is not possible and hence the objective needs to be revised.
story-problem-2-step

Solve the word problem above using a calculator.

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