Category Archives: Connections to Prior Knowledge

Basic Skills Older Students

A widespread problem at the secondary level is addressing basic skills deficiencies – gaps from elementary school. For example, I often encounter students in algebra 1 or even higher level math who cannot compute problems like 5÷2. Often the challenges arise from learned helplessness developed over time.

How do we address this in the time allotted to teach a full secondary level math course? We cannot devote class instruction time to teach division and decimals. If we simply allow calculator use we continue to reinforce the learned helplessness.

I offer a 2 part suggestion.

  1. Periodically use chunks of class time allocated for differentiation. I provide a manilla folder to each student (below left) with an individualized agenda (below right, which shows 3 s agendas with names redacted at the top). Students identified through assessment as having deficits in basic skills can be provided related instruction, as scheduled in their agenda. Other students can work on identified gaps in the current course or work on SAT problems or other enrichment type of activities.
  2. Provide instruction on basic skills that is meaningful and is also provided in a timely fashion. For example, I had an algebra 2 student who had to compute 5÷2 in a problem and immediately reached for his calculator. I stopped him and presented the following on the board (below). In a 30 second conversation he quickly computed 4 ÷ 2 and then 1 ÷ 2. He appeared to understand the answer and this was largely because it was in a context he intuitively understood. This also provided him immediate feedback on how to address his deficit (likely partially a learned behavior). The initial instruction in a differentiation setting would be similar.

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Introduction to Imaginary Numbers

The topic of imaginary numbers is one of the most abstract and therefore difficult math topics to teach in algebra. Here is how I introduce it to students (emphasis that this is only an introduction).

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I write 1, 2, 3… on the board (see photo above) and explain to the student “at some point in life you learned to count on your fingers, 1, 2, 3…” These are called the Natural numbers.

toddler counting

Then I explain, “later you were told that no cookies means ZERO cookies. Zero is a new type of number. We call 0, 1, 2, 3… the Whole numbers. You learned a new type of number.”

This continues, “A little later on you were told you could have half a cookie and so you learned about a new type of numbers called fractions.”

This continues with negatives. Then I explain that all these number types can be found on the number line. We call the set of all of these numbers Real Numbers.

I conclude with “Now we have a new type of numbers that are not found on the number line. These are called imaginary numbers. Just like before you had number types you had before and now you have a new one to learn.”

The point of this approach is to help the students understand that a new number set simply builds on previous number sets. Also, the students have encountered this situation before.

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Juggling Gaps and New Content

In math, many students with special needs fall behind. What results is a Catch-22 in programming and services. If the student is provided extra time to work on the gaps, he or she likely falls behind with current content. If the student is provided extra time to receive support for current topics, the gaps are not addressed

In both cases the extra support time can actually be counterproductive.

  • The focus on gaps likely results in the student working on different math topics which in effect means the student has TWO math classes – just what a student with math anxiety doesn’t need.
  • The focus on current topics means the student is trying to learn math topics for which he or she doesn’t have the prerequisite skills needed.

I recommend identifying the prerequisite skills for a current math topic and address ing these skills concurrently in math support or during the summer. For example, I used a Common Core coherence map (top photo below) to identify Common Core prerequisite standards for the standards a student faces in her upcoming school year. Then I listed these with each grade level standard (bottom photo below). The prerequisite skills can be identified using a task analysis approach as well. Screenshot 2018-06-12 at 6.03.52 AMScreenshot 2018-06-12 at 5.45.22 AM

This approach allows for a systematic approach to fill in gaps and to prioritize when they are to be addressed. When implemented effectively, the student can see the immediate benefit of the support time – it helps them in math class. Even better, the support teacher can match instruction and work with what is covered in math class.

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Congruent Triangles

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As Piaget highlighted, our brains make connections between new information and previous information (prior knowledge). I introduce the concept of congruent triangles by connecting it to prior knowledge of identical twins (photo above).

This connection is carried throughout the chapter. For example, to show triangles are congruent we look at parts of the triangle, just as we can look at shoe size, pants size and height of 2 people to determine if they are twins (see photo below).

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Function Notation for Algebra

Below is a video of a lesson I recorded on function notation using the Explain Everything app. The lesson starts by addressing the concept of function notation by connecting it to the use of the notation “Dr.” as in Dr. Nick of Simpson’s fame. The lesson builds on prior knowledge throughout with a focus on color coding and multiple representations.

This videos shows an instructional approach to teaching function notation and concepts in general and video lessons can be used for students who miss class or who need differentiation.

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Function Notation

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Function notation is challenging for many students yet we teachers overlook the reasons for the challenges. For example, students see the parentheses and rely on the rule they were taught previously, y(5) is “y times 5”. Because this is overlooked by teachers we often skim over the concept of notation and delve into the steps. This document is a means of presenting the concept and the use of function notation in a meaningful way. Feel free to use or revise and use the document as you wish.

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– symbol: negative vs subtraction

The use of the “-” symbol is challenging for many students. They don’t understand the difference between the use of the symbol in -3 vs 5 – 3. To address this I use a real life example of multiple uses of the same symbol (1st 2 photos below) then break down the “-” symbol (photo below at bottom). I suggest this be introduced immediately prior to the introduction of negative numbers.

negative vs subtractionNegative vs Subtraction B

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