Category Archives: Scaffolding

Graphing Linear Functions

Screenshot 2017-12-22 at 11.52.09 AM

Graphing linear functions and the underlying concept are challenging for many students. The video below shows a scaffolded approach to teaching how to graph. This approach also addresses the concept of the graph as a visual representation of all possible solutions (see photo above). Students often do not realize that the line is actually comprised of an infinite set of points which represent all the solutions. Here is a link to the document used in the video.

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1 to 1 Correspondence Scaffolded

IMG_20171221_080306139 - Edited

I have encountered several students who struggle with 1 to 1 correspondence with the educators struggling to figure out how to teach this to these students who continue to struggle. This post reveals an approach I used with a student.

I broke down the task using a formal task analysis approach. This approach involves identifying the different individual steps and to address these steps in isolation. Here is the sequence I use and suggest.

  1. Conduct a pretest using a task analysis pretest data sheet I created for this topic. I do not use any scaffolding and prompt the student to count out the objects (in this case decks of cards) and to do so independently. I prompt the student after they show they cannot complete a step which allows the student to attempt the next step. (Think of teaching a student to get dressed and he cannot put his socks on. You help him with the socks then ask him to put on his shoes.)
  2. I then focus on the movement of the objects. I provide scaffolding for start and stop piles (see mats with track photos above). The student is asked to move the cards one at a time without counting.
  3. The student must learn the “rules of the game” which includes how to place the items in the stop pile. Students may be confused about placement, e.g. one student ran out of room while placing the decks in a straight line and I had to demonstrate that it was OK to place them on different spots on the mat. Once the student demonstrates mastery of moving the items we move on to the next step. IMG_20171221_080726834 - Edited
  4. We then focus on counting in isolation. The card decks are labeled with numbers (photo below) and the student does not move anything but simply reads the numbers. (More on these numbers in a later step.) More numbers can be added as necessary.IMG_20171221_080604425 - Edited
  5. The next step (photo below) is to have the student read the number on each card. I have a stack of decks of cars on the start pile with the numbers facing down. I show the student the number of the deck that I am moving to the stop pile and the student reads off the number. I place the used deck face down to hide the number. This activity forces the students to focus on each item as he reads the number. One student kept counting ahead to the next number and I prompted him to return his focus to the current number. This is the crucial step as it focuses on the 1 item 1 number aspect of counting. IMG_20171221_080526960 - Edited
  6. The next step is to have the student move the decks from the start pile to the stop pile and to read each number while doing so. I turn each deck face up as a prompt for the student to move and read.
  7. The student then is prompted to select the cards on his own and read (the cards can be in a pile in order by number).
  8. Eventually 1 then 2 then 3 decks have the number missing which adds an extra task demand for the student – identify the next number as he is moving the item.
  9. Finally the items do not have any numbers and the student counts, with the mats eventually be faded.

Note: this is especially effective for students with ADHD because it helps to focus and organize their task demand for the activity of counting.

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Kahoot Game

Screenshot 2017-12-21 at 12.31.22 PM

A Kahoot is an online and app quiz game that allows students to answer questions using a personal device (e.g. simulated phone in photo above). The teacher can create the questions (e.g. example question I created in photo above).

My approach is to use a Kahoot to scaffold learning. In this post I use plotting points as an example.

  • I start with simple questions, e.g. identify the letter and number coordinates for the dog and chick below. Notice in the top photo below that I provide the actual coordinates in question 1 (“for the dog C4”) as a scaffold to show the students what to do.
  • Then I show numeric coordinates for a point, but only with positive numbers.
  • Eventually I present problems that address all 4 quadrants and ask questions about the parts of the coordinate plane (photo bottom one, below).
  • Notice that the questions have times (in seconds). This indicates the time allotted to answer each question (teacher sets this). For students with special needs I print a hard copy to allow them more time to read the question. If necessary, they can respond by circling the answer on the handout.

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Elapsed Time Scaffolded

Elapsed time scaffolded

The photo above shows a scaffolded handout to break down elapsed time for a student. The problem is divided into 3 parts: time from 10:50 to 11:00, time from 3:15, time from 11:00 to 3:00 (see photo below). This is based on how we may compute elapsed time by focusing on minutes then on hours. Notice the 3 clocks (in photo above) with no hands which allows the student to engage the clocks by having to determine and show how many minutes passed, e.g. 10:50 to 11:00.

The final answer would be 4 hours and 25 minutes.

Elapsed Time Scaffolded and completed

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One More or One Less Scaffolding

more-less-screen-for-hundreds-table

The photo above shows a screen for a hundreds table was shared by one of my students in a Math for Children graduate course. She found it on Pinterest for use a class presentation. I love this idea and came up with some revisions I think can make it more effective. It seems to me that this screen may be too busy with 4 different numbers showing. Additionally, the view of the other numbers outside the screen could be distracting.

Below are a couple revisions I would suggest.

one-less-hundreds-table-screen one-more-hundreds-table-screen

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Making Slope Less Complicated

slope-graph-real-life-application

Slope is the rate of change associated with a line. This is a challenging topic especially when presented in the context of a real life application like the one shown in the photo. The graphed function has different sections each with a respective slope.

One aspect of slope problems that is challenging is the different contexts of the numbers:

  • The yellow numbers represent time
  • The orange numbers represent altitude
  • The pink numbers represent the slopes of the lines (the one on the far right is missing a negative)

Before having students find or compute slope I present the problem as shown in the photo above and discuss the meaning of the different numbers. What I find is that students get the different numbers confused and teachers often overlook this challenge. This approach is part of a task analysis approach in which the math topic is broken into smaller, manageable parts for the student to consume. Once the different types of numbers are established for the students we can focus on actually computing and interpreting the slope.

This instructional strategy is useful for all grade levels and all math topics.

 

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Introduction to Slope

which-jobSlope is one of the the most important topics in algebra and is often understood by students at a superficial level. I suggest introducing slope first by drawing upon prior knowledge and making the concept relevant (see photo above).  This includes presenting the topic using multiple representations: the original real life situation, rates (see photo above) and tables, visuals,  and hands on cutouts (see photos below).

10-dollars-per-hour-graphA key aspect of slope is that it represents a relationship between 2 variables. Color coding (red for hours, green for pay) can be used to highlight the 2 variables and how they interact –  see photo above and below.5-dollars-per-hour-graph

The photo below can be used either in initial instruction, especially for co-taught classes, or as an intervention for students who needs a more concrete representation of a rate (CRA). The clocks (representing hours) and bills can be left in the table for or cut out.cut-outs-hours-bills

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Address Challenges at Performance Points

performance-point

In the photo above you see a contrast between how children learn and how educators often teach necessary skills. Children learn to ride a bike by actually performing the target skills. This is a performance point – the setting in which the child actually performs. In school students are often taught necessary skills in isolation, away from the performance points. Imagine teaching a child to ride a bike by having him sit at a desk while the parent points out all the steps for riding a bike.

Often accommodations and supports are provided in isolation or out of context. Students with autism have lunch buddies in a contrived setting with an educator leading conversation. Students with ADHD have a weekly time to organize their notebooks. Students who have trouble functioning in a general ed classroom may be pulled out as a result.

Below are a couple of examples of how support can be provided at the points of performance. The photo below shows a checklist I used for a students with autism in my algebra class. They would follow the checklist and self-evaluate by checking off each step as it was completed. They were learning how to perform necessary skills at the point of performance.

checklist-in-class

Another overlooked point of performance is in organizing a notebook. Students should organize a notebook while IN CLASS and on a DAILY basis. I use the rubric below to help support students with this task.

checklist-for-notebook

Dr. Russell Barkley, an expert on ADHD, talks about performance points for students with ADHD in his book and in his ADHD Report. This focus at the “points of performance” can and should apply to any student with a disability (and students in general).

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Analyzing a Graph

analyze-graph-using-color

Students can hit a road block at the steps that appear to be very simple. For example, in the problem below the students are prompted to find the highest point on the graph. Many think the graph refers to the entire coordinate plane and they pick 5 as the high point. It is the highest point on the y-axis but not the graph. I introduce the problem by highlighting the actual graph in pink and explain that this highlighted line is what is meant by the graph.

Analyze Graph Using Color Coordinate Plane Only - Edited.jpg

The use of color also helps students distinguish between the x and y axes and what the variables x and y represent in the context of the problem (# minutes and # kilometers in this problem) – see photo above. This problem also involves plugging in a # for x (blue) IN the function (red). In the photo below you see how I use color to help emphasize this.

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Modified Multiplication Table – Area Model Included

Modified Multiplication Table Using Area Model

The idea is that the student will have to count squares. By doing so the student is more engaged (or less passive) in determining the product and has to engage the visual representation.

Here is a link to the document.

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