Category: Topics regarding sequences and prerequisites

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“All I Really Need to Know” in math is From Elementary and Middle School

A spin off to Fulghum’s book (below) is that by high school, students have been presented with almost all of the math they need if they are not pursuing college.

High school math, aside from some exceptions, is largely designed to prepare students for college and subsequent careers.

If your student is entering high school and does not have a postsecondary goal of college (2 or 4 year) then you can turn the main focus of the math education to topics covered before high school.

Some of the topics in geometry and statistics are applicable to real life and most of those would have been covered in the Statistics or Geometry and Measurement domains from previous grades. There are topics unique to high school math that are prerequisites for some vocations, e.g., trigonometry for surveying. Some applications of the high school math address real life, but the focus is on the math and not the applications.

The image below shows a breakdown of a sample of topics for life skills math and for two vocations. Here is a link to a PDF of the document shown above. You can see the math topics. Here are the links to the pages for the plumbing topics and the welding topics.

Related to this value of college education for certain job sectors. The director of the Office of Higher Education in Connecticut, Tim Larson, stated that many companies have proprietary software, programs, or procedures that they will teach new hires. The take away from this is that much of the actionable knowledge needed would not be covered in college. Many of the skills they are looking for are not academic in nature. The Wall Street Journal published two articles that speak to the change in requirements for some jobs, in which a college degree is no longer a requirement (“Rethinking the Need for College Degrees“, “Is this the end of college as we know it?”)

A college education (or at least the degree) provides incredible opportunities, but it is not needed for many students.

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Viability of a Grade Level in Math

When I am asked to consult or evaluate a student, often the student is years behind in math. As a result, I am often asked to determine the grade level of the student’s achievement. Regressing the math achievement to a single number is not viable. This post provides an explanation.

Common Scenario

Here is a common scenario. A school official reported out the grade level in math for a student. The 7th grade student tested at a 4th grade level. As a result, the student spent much of her 7th grade year working on 4th grade math. When I started working with her, I discovered that she was very capable of higher level math. Six months later, she was taking algebra 1.

The Math Spider Web

Unlike reading, math is not nearly as linear. It is more like a spider web of categories (called domains). For example, Geometry is not a prerequisite for Ratios and Proportions and Fractions is not a prerequisite for Expressions and Equations. Geometry and fractions may be included in problems associated with other domains but they are not foundational building blocks.

On the other hand, in reading, comprehension and decoding are essential in all grade levels. Unresolved trouble with decoding in 3rd grade causes major problems in 4th grade and beyond.

Reading Rockets

Grade Level

A student tests at a 3.2 in reading. This provides a clear picture of where the student is in the progression of reading ability. There are books written at that grade level.

If a student is reported to to test at a 3rd grade level in math, the student may have scored higher than 3rd grade in Geometry, at 3rd grade in measurement and data, and lower than 3rd grade in the other domains. True, in reading we have students who may decode at a high level and comprehend at a low level. That is more specific that sorting through 6 domains in math. Then consider that the comprehensive number of domains addressed by middle school increases to 11.

The Domains

The image below shows a breakdown of the Common Core of State Standards math domains. In a video, I use this graphic to unpack why it is more challenging to determine a single level of ability for math.

Addressing Grade Level Metric

If you are presented with a single grade level as an indicator of math ability, I recommend that you ask for a breakdown by category and how your student will be provided differentiation to address gaps. This is more appropriate than plowing through all of the math at a lower grade level.

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Proportional Relationship (with k) vs Linear Function (with m)

There have been interesting discussions on various Facebook Teacher pages about proportional relationships vs linear functions. This mirrors discussion about the constant of proportionality vs slope vs unit rates.

The difference between the proportional relationships and slope is context and the ratios. The ratio for the former addresses the variables themselves. The ratio for the slope addresses the change in the variables. This arises from context. To flesh this out let’s use the pay as a function of hours, with $15 an hour for the hourly rate.

  • If we focus on the fact that to compute the total pay, we multiply by the hours worked by 15 we have a proportional relationship and the $15 an hour is a constant of proportionality.
  • If we focus on the fact that every increase of 1 hour results in an increase of $15 in our total pay, we have a linear function and the $15 an hour is a rate of change (or slope of the line). Because of the context, we have different constants (k vs m).

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CCSS MATH Coherence Map – 3rd Grade Groups of Objects to HS Slope

A hidden treasure is the Common Core of State Standards Math Coherence Map. It is an interactive flow chart that shows connections between the various standards.

  • If you are teaching math, you can see the connections between what you are teaching, what was taught previously, and how you are preparing students for their future math education.
  • If you are a special education teacher, you can see the progression of prerequisite skills. If you write IEP objectives for grade level standards, you can address the prerequisite standards and progress made through these prerequisites can show progress towards mastery of the IEP objectives.

In this post, I show the progression from 1st grade standard on the = sign and 2nd grade standard of repeated addition all the way to interpreting slope in high school math.

After clicking “Get Started” you will narrow your focus to the grade, the cluster, and then the math domain.

The flow chart shows connections between a selected standard and others, including prerequisites. In this case 8.F.B.4 – 8th grade content that is a prerequisite for the high school math work. Click on the 8.F.B.4 standard and it pops up (below right).

In turn, the 8th grade standard is connected to a 7th grade prerequisite regarding ratios and proportions.

Notice that the 7th grade standard, 7.RP.A.2, appears to be a dead end (bottom left). I picked up the path by clicking on Grade seen at the top left of the screenshot and made my back to that standard and the connections to prerequisites appeared. (Same happened in 3rd grade shown further down in this post.)

The path continues from ratio and proportions in 7th grade to unit rate in 6th grade, multiplication word problems and multiplication in elementary school.

I want to emphasize that students are working on unit rate and slope problems in ELEMENTARY SCHOOL! 3.OA.A.1 below addresses groups of objects model for multiplication and 4.OA.A.2 addresses word problems involving multiplication.

I was recently working with a student entering middle school on multiplication word problems. To unpack the word problems and the concept of multiplication in context, I had her draw (photo below) groups and groups of objects to help her identify the unit rate (although we don’t use that term yet). This work will impact her math education through the high school math and even into college (slope has been a common gap for the college students in the math courses I taught at various colleges and universities).

This approach I used with the student could be used for high school students, especially those with special needs.

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Math Intervention is Packing a Suitcase

In working with students with special needs on math programming and services, a common and important issue is that the student is behind and there is a tension between math intervention to fill gaps and addressing ongoing grade level content.

Unpacking the situation

There is no single grade level for math, as is the case for reading. Math progression is more like a web, not a line. For example, if a student can do 5th grade geometry but only 3rd grade level fractions, do we average out the grade level math to be 4th grade? (No.) Do we identify the student as working at a 3rd grade level? (No.) 5th grade level? (No.)

Like a suitcase, there is a capacity to the daily time a student has for school services. I often encounter situations in which the services recommended involve the student working on grade level content and catching up on the gaps during support time. If the student has only been learning 75% of the math content each year, he or she needs that support time to help learn the new content to get closer to 100%. There is too much being stuffed into the suitcase. Something has to give.

The focus of the services and programming often shifts away from post-secondary plans, with a focus on the short term. Like the situation facing the man in the image below, there are long term implications.

Recommendations

There are two recommendations I make in regards to addressing the gaps, without overstuffng the suitcase.

The IRIS Center is part of the Peabody College of Vanderbilt University.
  • Use triage to shift focus to the priority topics. For example, the parents of a student in 7th grade but working on math from lower grade levels wanted to pursue a math track that would allow the student to go to community college. I mapped out a long range plan (image below) that focuses on algebra as that is the type of math most likely encountered in a math requirement. Here is another plan which was to prepare a student to possibly work in a field related to cars.
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