I presented this paper in the context of creating school-wide math agreements, an idea I learned from the authors at last year`s NCTM conference. During a meeting with all the staff, I started a conversation about things we often say to children that are not as specific as you would like. It`s not a quick thing that happens to your students, but the quick answer is to teach them the value of space and how, if you multiply by ten, make the number ten times larger, thus moving the number to the next space value. I was so guilty of telling this rule to the children. This made multiplying by 10 so easy. In fact, it was so easy that my eldest son came home when he was in kindergarten and told me he could multiply by 10. He made some examples, and then I asked him how he knew. Well, for those of you who are members of the Build Math Minds site, I have a video on how I can help kids understand what happens when you multiply (or divide) by a power of ten and how it all relates to help them understand the value of the place. I will buildmathminds.com/45 link on the “View Notes” page. Still on the View Notes page, I`m going to link to a video training I did on the rules that violate the rules that expire, as well as the link to article 13 Rules that expire so you can read the 13 rules.
13 Expiring Rules from Karen S. Karp, Sarah B. Bush, and Barbara J. Dougherty changed my teaching forever. Not only do they share the 13 rules, but there is also a section on the expired mathematical language we use, and they give alternatives that can be used instead. This article is seriously a MUST-READ. To give you an overview of the article, I just want to share one of the rules with you today. How about sharing 2 and 1/2? Would the ratio be lower than the dividend and the divisor? When students first learn division, they focus on partially understanding that when you share a lot, you can`t have more than you started with. However, this rule expires when you start dividing integers and fractions/decimals and fractions by fractions or decimals by decimals. I found this article and it is worth sharing.
It describes common rules and vocabulary that teachers share and that elementary school students tend to generalize – tips and tricks that don`t promote conceptual understanding, rules that “expire” later in students` mathematical careers, or vocabulary that isn`t accurate. If you multiplied 3 x 1/3, would the product be larger than the factors? Many students learn that when you multiply, your product is always larger than your factors, but this rule only applies if you are working with positive integers. If fractions, decimal numbers, and negative numbers are introduced later, the rule is no longer true. We talk about a few examples, and then I distributed the article. They read, commented and discussed the article immediately and there. And finally, they wrote personal commitments about the things they would change in their practice. I asked them to include their promises in the feedback survey so I could follow up in the coming weeks. The mathematical conversations that emerged from this process were incredible. I hope you will try it! I was guilty of giving my students advice or rules that I thought would help them get an answer faster because in the past I thought my job was to help my students get answers.
Now I see my work as a teacher as helping students make sense, and many of these tips had no “meaning” behind them. It was really just rules to help my students get an answer. Next week`s preview: When choosing visuals and manipulations to use in and between school levels, choose wisely. Focus on those that allow connections in the different classes. A good example is KP Ten-Frame Tiles. They are so superior to Base Ten blocks! More information on this topic next week. But for now, check out Peggy`s latest article, which focuses on why KP Ten-frame tiles are superior to Ten base blocks from the get-go. Too often, teachers inadvertently teach math in too generalized and inaccurate ways. We teach tricks that promote nothing more than memorization. The result is that our students misunderstand the very ideas we are trying to illuminate. How many teachers have had to contradict their colleagues from previous years because what was being taught was so painfully limited? Over-generalization of generally accepted strategies, the use of imprecise vocabulary, and the use of tips and tricks that do not promote conceptual mathematical understanding can lead to misunderstandings later in students` mathematical careers.
I also used the fact that we say “multiplication makes bigger and division always makes you smaller” as an example of how we accidentally teach misunderstandings. This table shows some commonly used languages that “expire” and more appropriate alternatives. How many of these “rules” did you say in class? “1. If you multiply a number by ten, just add a zero to the end of the number. This “rule” is often taught when students learn to multiply an integer by ten. However, this Directive shall not apply where the decimal places are multiplied (e.g. 0,25 × 10 = 2,5, not 0,250). Although this statement may reflect a regular pattern that students identify with integers, it is not generalizable to other types of numbers.
Expiry date: Class 5 (5.NBT.2). “Well, it turned out that an older child on the bus told him the rule: you just add a zero. My husband thought it was great that our 5 year old knew about it and was able to solve these problems. Until I explain how it doesn`t work because it will multiply by decimal numbers. And of course, he`s probably asking the same thing that many of you are asking, which is, “What should we do instead of saying you`re just adding a zero?” Thirteen rules that expire: 1. If you multiply a number by ten, simply add a zero to the end of the number. 2. Use keywords to solve word problems. 3.
You can`t take a larger number from a smaller number. 4. Addition and multiplication make numbers larger. 5. Subtraction and division reduce numbers. 6. You always divide the largest number by the smallest number. 7.
Two negative points make it a positive one. 8. Multiply everything in parentheses by the number outside the parentheses. 9. Incorrect fractions should always be written as a mixed number. 10. The number you specify first when counting is always less than the number that comes next. I love this article and can`t wait to share it with my teachers at my school.
I was wondering if you had any suggestions for an engaging and challenging way to present this to my teachers. Thank you very much! To address vocabulary, I talked about using descriptions rather than terminology. For example, the term “flip-flops” is often used in first-grade classes. We discuss the idea that if children can say Tyrannosaurus rex and Stegosaurus and know exactly what they mean, they can say “commutative property” with understanding. “Flip-flop facts” may be a good first-class description of commutative property, but do not replace the actual term. We have all heard those statements. and I bet we all said them at some point. And every time we say this, we perpetuate misunderstandings that will be difficult, if not impossible, to teach in the years to come. While you`re at it, don`t forget to leave a review on iTunes.
I`d like to hear what you think about it and how we can make sure we`re offering you content you`ll really like. But another reason I don`t want to teach these tips and rules is that they don`t last. They are working for now. They work for the types of problems that students are working on right now, but when they move on to other types of problems, the rules no longer work. We have reviewed and revised the lists several times in a matter of months. All the lists have been displayed in our conference room so that the notes can see the lists of others and use them as stepping stones. I am working through these stages in my own school district. Won`t you join me on this journey to abolish the rules that expire by engaging in professional discourse and making school-wide agreements? Hey, have you already subscribed to the Build Math Minds podcast? If not, be sure to do it today because I don`t want you to miss episodes! Click here to subscribe to the podcast in iTunes.