Category Archives: Uncategorized

RTI – Response to Intervention

RTI Process

The photo above shows a model of the RTI (called SRBI in Connecticut) process. RTI is a systematic approach to addressing student academic needs. Here is a link to a video explaining the process and below is an outline of the process:

  1. Students are served in a classroom that provides high quality initial instruction. This includes the use of UDL, differentiation, formative assessment, instructional strategies to make content meaningful and concrete and to meet student needs in general. The general classroom is Tier I.
  2. Assessment is used to evaluate student progress AND the effectiveness of the instruction. If students are not understanding a math topic or unit (as demonstrated by data not observation) the student can be moved into Tier II which involves intensified focus of instruction and in a small group.
  3. Assessment is used again. If the student is not making sufficient progress despite changes in instruction the student can be moved into Tier III which involves maybe 1 on 1 or 1 teacher and 2 students. The level of intensity is ramped up further.

Here are a couple of key components:

  • The initial classroom includes an effort to meet individual needs.
  • Data is the key to decision-making. Assessment is objective.
  • The programming is evaluated using the data.
  • RTI is included in IDEA 2004.
  • Student placement at the different tiers is fluid. Students are moved into and out of tiers based on data.


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Snow Math

2017-03-14 10.03.23

Here’s a common word problem used for linear functions and equations (y=mx+b):

There are 6 inches of snow on the ground. Snow is falling at a rate of 2 inches per hour. Write a linear equation showing total snow as a function of time (in hours). The equation would be y=2x + 6.

Often the word problems like this are presented on a sheet of paper in isolation as an attempt to make the math relevant and to develop conceptual understanding. For students who have trouble with conceptual understanding, words on paper are likely too abstract or symbolic to allow applications like the one above to be meaningful.

The real life application is useful if presented more effectively. Here’s an approach to use the same scenario but in a more relevant and meaningful presentation. The photo above shows the current amount of snow – call it 6 inches. Students can be shown the photo to allow for a discussion about accumulation and for their estimates of the amount of snow shown. The photo below shows an excerpt from a storm warning. Showing this warning and a snow fall video can allow for a discussion about rate of snow fall and the purpose for storm warnings. Combined, this approach can lead into the above word problem.

2017-03-13 21.57.29 - Edited

Once the application is presented students can be asked to compute snow levels after 1 hour, 2 hours etc. Then they can be asked to determine how long it would take for the accumulation to reach 18 inches (the prediction for the day this post was composed). After computing the answers WITHOUT the equation the students can be shown how to use the equation – the “mathy way.”

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Corresponding Angles in Stain Glass


Found this (above) cool example of corresponding angles (see photo below for explanation). This window photo could be a nice introduction to this type of problem by printing it out on paper and having students match angles as the teacher shows the photo on the Smart Board or screen.


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One More or One Less Scaffolding


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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Application for Trigonometry


Making math meaningful and maybe interesting is a challenge. The photo above refers to a real life application for triangles and trigonometry (see photo below) that is found in a news story about Russian jets and a US destroyer. The jet was flying at an altitude of 100 yards and within 200 yards of the destroyer. Topics that could be addressed:

  • Altitude (and perpendicular)
  • Pythogorean Theorem
  • Trigonometry: e.g. find angle of elevation or depression
  • Vectors (include velocities)

A relevant, real life application is a method to make information meaningful. When talking about the altitude of a triangle (the up and down part shown in the photo below) the vocabulary term of altitude becomes more meaningful both in terms of context and with the visual below.


Here is the agenda I would follow to use this application as an activity.

  1. I would show the video (show on the webpage linked at bottom of handout) and explain what a destroyer and the jets are.
  2. Discuss the situation with Russia (age appropriate discussion)
  3. Show the picture and ask the students to draw a sketch.
  4. Review the sketch and refer to the parts of the triangle in real life terms, e.g. altitude.
  5. Task the students with a problem related to this problem – create your own, e.g. find the angle of elevation or use Pythagorean Theorem to find length of missing side.
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Evaluating the Effectiveness of Math Objectives


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.


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.


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 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.

Solve the word problem above using a calculator.

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Perseverance in Math

perseverance checklist

A major challenge for students is not content but how to “do math” which includes perseverance. The photo above shows a table that can be used to monitor progress on perseverance. It addresses two situations involving perseverance (see below).

The focus of perseverance in math is making an informed attempt when a path or next step is unclear (and does not necessarily result in a solution). Paths can be categorized as using a strategy, e.g. drawing a picture, or following an algorithm, e.g. steps to solve an equation. (See excerpt of CCSS Standards of Mathematical Practices below).

Perseverance in math involves two situations:

  • The initial entry point (strategy or algorithm) is not apparent but one is selected and implemented
  • An ongoing strategy or algorithm is determined to be insufficient and an alternative strategy or algorithm is selected and implemented

From the CCSS Standards of Mathematical Practice (bold font is my emphasis on the perseverance component)
1. Make sense of problems and persevere in solving them.

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

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Addressing the Concept of Addition Part 2

2016-02-24 10.26.09

In a previous post I presented an approach to teach and assess the concept of addition. This document shows all the steps I use including the one shown in the photo above.

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Starbucks would benefit from some math intervention

starbucks reward

Starbucks has changed their rewards program to “boost participation.” The chief strategy officer apparently could use some math intervention.

From USA Today (emphasis mine):

“Currently, customers with “gold” status have to visit 12 times to earn a free food or drink item. Under the new program, those customers will have to earn 125 stars to get a free reward.

Most customers spend around $5 with each visit, said Matt Ryan, global chief strategy officer. At that rate, a customer would still need to visit about 12.5 times, or spend a little more than $60, before earning a free reward.”

In other words, “most customers” will need to visit more often to get the same reward benefit.

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