Christopher Danielson

Cheat Sheet / Updated 04-15-2022

As a parent, you’re most likely to encounter the Common Core State Standards for Mathematics (Common Core Standards for short) in the homework that your child brings home. The Common Core Standards are a set of statements about what students should know and be able to do at each grade level from kindergarten through high school. In states that have adopted the Common Core Standards (44 states and the District of Columbia), these standards replace each state’s pre-existing standards. In some states it represents a big change, whereas in other states, the changes are less significant. Understanding the Common Core Standards is important for being able to help your child in math in school. This Cheat Sheet gives you the lowdown on these standards.

Article / Updated 03-26-2016

A lot of misinformation is available about the Common Core Standards. These standards guide the math your child learns in school each year. In order to advocate for and to support your child, you need to be well informed. Here are some important facts that counter some of the common myths about the Common Core Standards. The Common Core standards are only for math and English language arts. The Common Core doesn’t have science, history, or sex education standards. You may have heard of a set of standards called the Next Generation Science Standards, but they aren’t part of the Common Core. Common Core includes the standard algorithms that you probably studied as a child. The old-fashioned way of solving addition and subtraction problems hasn’t gone away with Common Core. Students build up to those algorithms in the Common Core by exploring number patterns, by using the relationship between operations, and by representing their thinking with pictures and equations. Common Core classrooms value the ways that you probably think about numbers. You can probably solve a problem such as 1001 – 2 in your head. You probably do it without borrowing from the thousands place. Maybe you count backward or compare this problem to a similar one such as 1001 – 1. Students develop these ways of thinking and use them to solve problems in Common Core math classrooms. High schools may still offer precalculus and calculus courses. The Common Core standards outline three years’ worth of math courses for all high school students. Hence, it leaves a free year for precalculus for students who need it for their college majors. Schools can supplement this precalculus content across the other high school courses in order to prepare students for calculus courses in their senior year. Supplementing in this way requires creative program coordination, but the standards don’t prohibit it. Schools have had similar programs for many years because parents and students have increased the demand for Advanced Placement (AP) courses.

Article / Updated 03-26-2016

States, districts, schools, and teachers have invested a lot of time, money, and effort implementing the Common Core Math Standards — to say nothing of parents. Before the Common Core, each state wrote its own standards for math. Now, nearly all states have agreed on the mathematics students will study in school. Here’s what you need to know about the benefits of seeing through the process of adopting the Common Core Standards. Most of these benefits come from having a common set of standards — not common in the sense of ordinary, but in the sense of shared. Equitable access to mathematics: A topic you don’t study is one you don’t have an opportunity to learn. Not all students in US schools have similar opportunities to learn math. In many low-income schools — especially those in big cities, and those with large percentages of students of color — students are more likely to get stuck in endless cycles of review. Such students may not have an opportunity to learn any algebra, for example, because their curriculum constantly reviews fraction arithmetic. With common standards, schools agree to offer opportunities to learn everything at each grade level. Students who need to review fraction arithmetic will do so while they begin to study algebra, not instead of it. Robert Moses considered mathematics as a tool of liberation. Common standards insist that all schools offer this tool to all students. Consistent content for mobile students: The United States is a mobile society. Students frequently move between districts and even between states in the midst of the school year. This problem is especially noticeable in urban districts and in school districts that include large populations of agricultural or military workers. A common set of standards makes it much more likely that students who change classrooms midyear will be able to pick up where they left off. Higher quality standards: In many states, standards are written into law. Yet most legislators have no experience or knowledge of education. You can only imagine how this process can go wrong when 50 states write their own standards. The Common Core Standards take this process out of legislators’ hands. To be sure, states still control whether the Common Core Standards are the law of the land, but after a state has adopted these standards, the details of which math children study at which grade aren’t subject to the whims of the party in power. Improved textbook development: The Common Core Standards represent a rare opportunity to markedly improve the state of American textbook publishing. Most children in the United States study most of their school math from one or more textbook series published by a major publisher. These publishers have an incentive to create textbooks that can be widely adopted in many states. When states disagree about the grade level for the placement of a topic, publishers tend to include that topic at all grade levels, leading to incoherent and inefficient textbooks. Common standards across the majority of states should allow for more coherent textbook development. More meaningful professional development: Without common standards, teachers from different locales have difficulty discussing their work. Whether teachers connect at a conference, a training session, online, or at the coffee shop, common standards allow sixth grade teachers to talk to each other about the math they teach. Common standards facilitate sharing problems and solutions for student learning across district and state lines. Many high-achieving countries have systems in place for consistent and incremental improvement of education. The United States has no such system — at least in part because there is no common baseline on which to build. The result is that teachers see reform after reform sweep through the schools — out with the old and in with the new. Common standards are a necessary starting place for the kind of incremental improvement that supports teacher growth and development in the long term. Better prepared teachers: The mathematical preparation of teachers before Common Core relied on the whims of state legislatures. “What currently is the content of first grade math?” and “In what state will most of our graduates seek jobs?” were questions on the minds of teacher education faculty members before Common Core. As with K–12 textbooks, when answers to these questions vary, the content of the college textbooks students use needs to vary as well. Creating common agreements about what math teachers need to know will be much easier in an environment where nearly all states have agreed on what math students need to know and when they need to know it. Better testing: The federal No Child Left Behind Act (NCLB) of 2001 mandates year-by-year testing in grades 3–8. With or without the Common Core Standards, this high stakes testing is the current reality unless Congress alters or repeals NCLB. Before the Common Core Standards, each of the states developed their own tests, usually by contracting with a large publisher. The Common Core Standards present an opportunity to develop a smaller number of better tests. As with all opportunities, whether it’s successful remains to be seen. Better tracking of national trends: The National Assessment of Educational Progress (NAEP) is given periodically to fourth, eighth, and twelfth graders, and is often referred to with the nickname “The Nation’s Report Card.” NAEP is the main source of information available about national trends in student achievement. The development of the Common Core Standards and associated assessments means that there should soon be data about national trends in student achievement available that goes far beyond NAEP. For example, NAEP isn’t standards-aligned, so it’s not possible to use NAEP data to answer questions about the appropriateness of particular state standards.

Article / Updated 03-26-2016

One important change in the Common Core Math Standards is that students are expected to work through multi-digit computations by thinking about number relationships before they’re expected to follow standard algorithms. For parents who never had to think about their computations in school, this can make homework time a bit daunting. For example, a subtraction method like this one went around the Internet, with people expressing horror at how complicated the Common Core makes simple arithmetic. On the surface, this problem certainly does look complicated. Where does the 3 come from? What does 15 have to do with subtracting 12 from 32? Why not just do it the old-fashioned way? If you dig a little deeper, you can see that these good questions all have reasonable answers. Children learn to count by tens and fives in kindergarten and first grade, which means that multiples of five are familiar landmarks in the number system. The 3 doesn’t magically appear; instead it’s what you need to get to 15, a multiple of five. Then you can count on by fives. Some children may use the same sort of thinking and use 8 as their first number, which is what you need to get from 12 to 20. Adding 8 to 12 right away is another case of using a memorized fact strategically. (Here the related fact is that 2 + 8 = 10, so 12 + 8 = 20.) After the student is at 20, it’s another 10 to 30, and finally two more to get to 32. Other children may notice that 32 and 12 have the same units digit, so they may count 12, 22, 32 — two steps of ten. Don’t mistake the fact that multiple strategies are discussed in class for a mandate that all students master all of these strategies. Teachers aren’t trying to increase the number of things students need to remember. Rather, they’re exposing students to a number of correct ways of thinking so that students can recognize and build on their own ideas. The point of this kind of work is to help children develop addition and subtraction as related operations, not separate sets of facts to be learned. Understanding relationships among facts reduces the number of errors students make, the size of the errors they do make, and their reliance on calculators in the long run.

Article / Updated 03-26-2016

Place value is an important concept to know for Common Core math. The fact that it took thousands of years for humans to develop a place value number system is an important sign that place value is difficult for people to learn. The usual way of writing numbers is a place value number system. In other words, a limited set of symbols (called digits) builds numbers (0, 1, 2, 3, and so on up to 9) and you can write all numbers using these symbols. Most importantly, the values of these symbols change depending on where they appear in the number. In other words, their location (place) determines their worth (value). Another way to understand the difficult of learning place value is to put yourself in the place of a young child for a moment. Look at the following mathematical expressions and identify how they’re alike and how they’re different: 35, 3x, and In each case, one symbol (3) is placed next to another (5, x, and 1/2). The meaning of putting these symbols next to each other is different in each case. In the case of 3x, it means multiply. In the case of it means add. The case of 35 is by far the most complicated. Putting 3 and 5 next to each other doesn’t mean add 3 and 5, nor does it mean multiply these numbers. It means give 3 a new value and add 5 to that. Understanding the number 10 as ten ones and as one group of ten may seem basic, but the concept is difficult for many children. Similarly, one hour is the same as 60 minutes, but children may struggle to think about it both ways. The best way to become familiar with this sort of thinking is exposure. Talk with your child about things that come in groups; count the individual things and the groups. Go back and forth. Do it every day. It will become a natural way of seeing the world for you and your child, and it will help her with place value and later math as well.

Article / Updated 03-26-2016

You will want to make sure that your child is familiar with comparing logarithms for Common Core math. In addition to comparing numbers with ratio and unit rate, you can actually compare numbers a third way — with logarithms. A logarithm is basically an exponent. In the equation 10x = 100, writing log10(100) is how you solve for x; log is short for logarithm (in this case, x = 2). Comparing numbers with logarithms is a high school and college-level topic, but people use logarithms instinctively. As an example that logarithms are instinctive — the distance between 1 and 1,000 doesn’t feel all that different from the distance between 1,000 and 1,000,000 for most people. Try this: Draw a number line with 1 at one end and 1,000,000 at the other end. Now put 1,000 on that number line. Most people end up putting 1,000 a bit to left of center on the number line. If this is true for you, it’s because your brain reacts to the logarithms of large numbers, not to their actual values. According to the values of these numbers, 1,000 should be half of the way along the line — so small a fraction of the distance that it’s difficult to see the space between 1 and 1,000.

Article / Updated 03-26-2016

An important way to think about functions for Common Core math is as relationships between variables. If you think of a function as a relationship, you can keep an eye out for useful features. These features include whether (and where) a function is increasing or decreasing, whether a function is linear or not, and so on. You can read a graph from left to right. If the graph rises as you read from left to right, it means that the y-values are increasing. If the graph falls as you read from left to right, the y-values are decreasing. If the graph is horizontal, the y-values are constant. A function can be increasing for all x-values (such as the line y = 2x + 2), or it can be increasing in some places and decreasing in other places (such as y = x2 + 2). In eighth grade, students may simply describe the change they see in graphs of functions, and often these functions don’t have symbolic form. This graph describes Jim’s distance from home as a function of the number of hours since he woke up on Tuesday. Tell the story of Jim’s day, accounting for all of the information you see in the graph. Be sure to include reasons for the increasing, decreasing, and constant values you see. For example, the graph is constant at zero for the first part of Jim’s day because he woke up at home and left for work an hour and a half later. The graph increases to show his commute to work, where he stayed until lunch. He walked to lunch in the middle of the day (notice that the increase in distance from home shows a lesser rate than during his drive to work). He drove home at the end of the day before leaving on a trip. He spent that night at a hotel 100 miles from home.

Article / Updated 03-26-2016

In addition to the content standards that state what students need to learn at each grade level, the Common Core Standards for Mathematical Practice describe how students should approach their mathematical work and what kinds of tasks teachers and curriculum should present to students. You can use the following list to keep track of the eight Common Core Standards for Mathematical Practice. They are as follows: Making sense of problems and persevering in solving them Reasoning abstractly and quantitatively Constructing viable arguments and critiquing the reasoning of others Modeling with mathematics Using appropriate tools strategically Attending to precision Looking for and making use of structure Looking for and expressing regularity in repeated reasoning

Article / Updated 03-26-2016

Homework is a hot topic in the transition to Common Core Standards. Homework assignments that ask students to think in new ways can be intimidating to parents. When something comes home that looks unfamiliar to you, don’t panic. Homework is just a way of giving students additional time to think about the things that they’re learning — what teachers call time on task. Your child’s teacher assigns homework with the best interests of your child and of the class in mind. When your child needs help with homework, remember that the homework is your child’s responsibility, not yours. You obviously want to help your child, but not do the homework for her. Above all, don’t let the homework add extra stress and take over your home life. Start by asking your child to explain the problem she is working on and to explain that day’s lesson from class. Then, you can support your child’s learning by asking her questions such as these: How did you do this in class? Can I see your work from class today? How do you know that? (Ask this question of both right and wrong answers.) What do know how to do very well on this assignment? What do you not know how to do very well on this assignment? How can you get help at school outside of your regular math time when you need it? Use your child’s answers to these questions to guide your next move. Help her see how the day’s lesson can help her solve the problem she is working on, tell how you think about these problems, and ask follow-up questions.

Article / Updated 03-26-2016

In Common Core math, third graders use a number line to see how fractions relate to each other. A number line is useful for showing the relative location of fractions, as well as their distance from each other. You may remember a number line running across the front wall of an elementary classroom when you were in school, with a point for zero, a point to the right of that marked 1, then 2 and so on, with an arrow on the end to suggest that the line and the numbers continue forever. The two most important numbers on a number line are 0 and 1. After you mark them on the number line, every other number has a defined location. Correspondingly, when third graders use the number line to study fractions, they mark 0 and 1 and then partition the interval between these numbers into same‐sized pieces. For example, the number line in the figure shows the interval between 0 and 1 partitioned into fifths. The mark closest to 0 is labeled one‐fifth, which is a unit fraction. The remaining marks are labeled as multiples of one‐fifth. Students think about three‐fifths as being the distance from zero which ends at the third mark to the right of 0 when the interval is cut into fifths. Fifths on the number line. Third graders basically show numbers on a number line in two ways: They show a fraction as a location on the number line as the figure demonstrates. More importantly, they also think of a fraction as a distance on the number line. One-fifth isn't just the tick mark labeled it's also the distance between adjacent pairs of tick marks on this number line.