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Month: September 2019

Patterns and Algebra

Visual patterns are a way to introduce and extend students’ understanding of algebra and functions. As such, there are a number of principles for helping students better understand algebra by examining visual patterns that have the same structure as their algebraic counterparts.

 

Principle #1: We can use the visual pattern to give more meaning to the algebraic structure.

What do you see changing in the pattern below? What stays the same? How is this pattern related to algebra?

3 squares built from smaller squares, with the diagonals painted black
A typical visual pattern

In the pattern above, children are likely to describe the squares as growing from left to right. They’ll notice that the diagonals of the squares are shaded and that the squares are two longer each time.

Each of these observations can be described in terms of equations based on the term number. For example, a child might notice that each square has 4 arms and 1 center square shaded, so you might annotate the diagram to show the arms and the center square.

3 growing squares with the diagonals from the center circled in red and the center square circled in blue.
Annotating for structure

Another child notices that the length of each arm is one more than the term number and that the total number of black squares is 4 times this length plus one additional square. Algebraically, this is represented with S = 4(n + 1) + 1. We color the parts of this equation to correspond with the visual to make the connections more clear. By doing this, we assign meaning to each part of the expression based on the visual.

 

Principle #2: We can use visual patterns to justify algebraic relationships.

Visual patterns can also be used to give meaning and to justify that a particular algebraic relationship is true, beyond what is possible to do with pure algebraic reasoning alone or a single visual example.

One might start by giving students the following image and asking them what they notice about the image.

Growing squares where each square is larger than the square before it by an odd number.
What looks mathematically important in this visual?

Collectively, students will notice that there are five squares, each square is larger than the square before it, each square is composed of smaller squares, each square has the square before it embedded in the lower left-hand corner, the number of white squares added on each time is odd, and a whole of other mathematical and non-mathematical observations.

The observation that each square is embedded in the next square and that the number of white squares added each time is an odd number can be written as follows.

1² = 0 + 1 = 0² + 1 = 1
2² = 1 + 3 = 1² + 3 = 4
3² = 4 + 5 = 2² + 5 = 9
4² = 9 + 7 = 3² + 7 = 16
5² = 16 + 9 = 4² + 9 = 25

By starting with the visual, students can reason inductively that “each square is just the square before it plus an odd number” and then this reasoning can be represented algebraically as n² = (n – 1)² + (2n + 1).

 

Principle #3: Visual patterns can be used to help students understand some of the language used in algebra.

A growing pattern of algebra tiles, showing 2 missing single units in each square.
Each square needs 2 more units to “complete it”

I did not learn during high school why “Completing the Square” was called Completing the Square. It wasn’t until I started teaching the idea using a visual to represent the square1 that the language made sense.

 

Principle #4: Visual patterns can be used to distinguish between different algebraic functions2.

Look at the two patterns below. How is each pattern changing as it increasing? How are these changes different between the different kinds of visuals?

One pattern growing by 2 each time and one pattern multiplying by 2 each time.
How are the sequences similar? How are they different?

By using patterns we can more easily contrast the difference between y = 2x and y = 2x which in written form are far more similar the corresponding visual sequences.

 

Further resources and inspiration:

 

On Misconceptions

Misconception: a view or opinion that is incorrect because [it is] based on faulty thinking or understanding1; a wrong or inaccurate idea or conception2.

These two definitions for misconceptions vary slightly, but the gist of the definitions are the same — there are some ways of thinking which do not match the world as we know it.

When we examine children thinking closely, we find that thinking often differs from our own. But this makes sense given that children have different experiences of the world than we do and often have not experienced the parts of the world that we have.

What should a teacher do about misconceptions? Should teachers try to prevent kids from having misconceptions? Should teachers label children who have misconceptions as wrong? Is there any harm in labelling children’s ideas as misconceptions? 3

It’s clear that some ways we use language to talk about children cause harm. If I consistently use the words “low” and “high”4 to describe my students, then the odds are greater that I also associate low and high expectations for these groups of students, which is correlated with student learning5. Here the language is harmful because it over simplifies the relationship between children and their background knowledge and results in students learning less than they would otherwise be capable of learning.6

The most problematic nature of the idea of misconceptions is that it frames how we respond to children’s ideas.

  1. A child writes 2 × 3 = 5 when they meant to write 2 × 3 = 6. Why did the child do this? Maybe they were overwhelmed with the task or tasks they were working on and defaulted to a previous relationship they know. It’s not a misconception per se, it’s something the child could probably find for themselves if asked to look at their work again.
  2. A child looks at the two angles below and concludes that angle A is larger because the rays are longer. This definition of larger is likely to be entirely consistent with every other experience of smaller and larger for this child. This child is attending to different properties of the geometry than the one intended by the author of the question.
    Two angles, A and B, one with longer rays but smaller angle, one with smaller rays but a larger angle.
    Is this a misconception? Or is this an entirely consistent worldview based on a different world than their teacher? Do we say to the child, “No, that’s wrong,” or do we value the thinking this child did and consider how to increase the size of their world?

My preference when working with children is to avoid over-simplistic words and phrases to describe their thinking. By default, the word misconception assumes a deficit view of children’s thinking7 and ignores the great thinking children did to come up with their ideas. What I prefer to the word misconception is language that describes more precisely the varied ways of thinking that children have. While it is the role of teachers to expand the world view of children and we need language to talk to colleagues about our role, the language we adopt frames the conversations we have.