(This is the first post in a two-part series on this topic)
Susan Hobart asked this week’s “question-of-the-week":
What strategies help math facts stick besides the old “drill the skill” and, if someone is not proficient at addition facts, can learning multiplication help or confuse?
Today’s responses come from educators Anne Collins, Sue O’Connell, Alexandra Mattis and José Luis Vilson. Part Two will be published in a few days, and I’ll be including comments from readers there.
You might also be interested in a ten-minute conversation that I had with Anne and Sue on my BAM! Radio Show.
In addition, you might want to read previous posts on this blog about teaching math.
Response From Sue O’Connell
Sue O’Connell is the Director of Quality Teacher Development, providing on-site math professional development to schools and school districts across the country. She is the author of numerous books including Mastering the Basic Math Facts in Addition and Subtraction and Mastering the Basic Math Facts in Multiplication and Division (Heinemann 2011) and the newly-released Putting the Practices into Action: Implementing the Common Core Standards for Mathematical Practice K-8 (Heinemann, 2013). You can find all her books here:
What strategies help math facts stick besides the old “drill the skill”?
The old “drill the skill” strategy of learning math facts was based solely on memory. Any strategy that is based solely on memory has a weak foundation. A first step for helping math facts stick is for students to understand what they are being asked to memorize. Our goal would be for students to both understand and retain basic math facts. So, how might teachers do this?
- Provide opportunities for students to model math facts with counters, number lines, and area grids.
- Pose investigations to explore the patterns in math facts.
- Focus attention on properties that simplify the task (e.g., understanding the commutative property allows them to expand their repertoire of known facts - if they know 3×9, then they know 9×3).
- Have students talk about what they notice with connected facts (e.g., How do ×4 facts relate to ×8 facts? How do ×9 facts relate to ×10 facts? How does 20÷4 relate to 4×5?).
- Engage students in interactive math fact practice. Rather than flash cards, select engaging and varied tasks that give students lots of exposure to the fact sets.
The benefits of using these strategies? Students will understand patterns, properties and facts, develop greater number sense, and have a smoother path to retention of facts.
If someone is not proficient at addition facts, can learning multiplication help or confuse?
If proficiency is interpreted as being able to automatically call out the answer to addition facts, then it is not necessary for students to be proficient with addition facts to move on to multiplication facts. Some students may always have a difficult time memorizing facts, but to hold them back from exploring multiplication would be counterproductive. It would be more important for students to understand the operation of addition and be able to visualize the action of addition (combining, adding to) so they can make the connection to the operation of multiplication. While both involve finding a total, in multiplication students may be combining equal sets or increasing a set by twice or three times as much (multiplicative comparisons).
Response From Anne Collins
Anne Collins is the series editor of NCTM’s publication, Assessment Sampler: Tasks Aligned with Principles and Standards of School Mathematics and Using Classroom Assessment to Improve Student Learning. She is an elected member of the NCTM Board of Directors. Her series of five books titled Zeroing in on Numbers and Operations was published by Stenhouse in 2010:
In order for students to develop fact mastery, it is important that they: a) develop a strong understanding of number relationships, b) have developed efficient strategies for retrieving facts, and c) have ample time to practice those facts.
Students who understand that multiplication is about groups are likely to have an easier time learning their multiplication facts than those who are told that multiplication is a shortcut for addition. Students who have time to model multiplication using arrays--either with tiles or on graph paper--are better able to visualize the fact that multiplication is two-dimensional, with every product making a rectangle.
Students who represent multiplication in arrays are making an image that also reinforces the “groups of” strategy, which works successfully with most students. The strategy of knowing that “6 times 7 equals 5 groups of 7 plus one more 7" works effectively for those students who are able to add that additional group of 7. Notice that addition is necessary when using this strategy and that the efficiency of this method is lost if the student needs to count up from 35 to 42. Developing math fact understanding through appropriate problem situations helps students recognize the need to know their facts. It is important that students develop automaticity with both addition and multiplication facts because multiplication of factors with two or more digits will require both.
All students need to practice their newly developing skills so the facts “stick” and they develop automaticity with them. Multiplication War is a fun way for children to internalize their facts. In this game, two players start by equally sharing a deck of regular playing cards. Each player turns two cards faceup and calculates the product. The greater product wins the hand. Should there be a tie, the players play “war” by placing three cards facedown, upon which they place two cards faceup. The greater product for those final two cards wins the game. The game continues until one player is out of cards. Another computer/iPad game that involves reasoning as students are practicing their facts is Kakooma. In this game, students must determine the product of two factors shown in a square, pentagon, or hexagon, thereby adding a level of complexity to the game.
Response From José Vilson
Mr. José Vilson is a math educator for a public middle school in the Inwood/Washington Heights neighborhood of New York City, as part of the NYC Department of Education. His book, This Is Not A Test: A New Narrative on Race, Class, and Education, was highlighted in a post on this blog last month. His blog was named one of the top 20 teacher blogs by Scholastic Inc., and part of the GOOD 100:
For one, I generally don’t believe in drill-the-skill, skill-and-drill, or any other synonymous tactic as a whole. I see the value in getting practice in working with a tool or technique before using it and trying to break the rules. Every artist copies another artist until they’ve mastered the techniques, then break out on their own.
Having said that, math shouldn’t be limited to a disconnected set of rules and jargon that doesn’t seem to mean much of anything. We should try to give everything we teach a context, or at least a story as to why we came up with the curriculum we did. For instance, in a sixth grade classroom, I was inclined to teach the Sieve of Eratosthenes. First, it was fun and I felt like it. Secondly, it was a rather dramatic way of introducing the idea of prime and composite numbers. The students felt less like they were doing tedious work and more like understanding how math was uncovered and why it matters.
For my English Language Learners, I also like to include cognates (and false cognates) to help children connect to the vocabulary. For example, when we started our percents unit, I always like to remind children of the root in “percent,” which is “cent.” I then prompted them to think about a word they know that akin to it, and in Spanish, they found “centavo,” meaning “penny.” Then I asked them to think why would a “centavo” be called a “penny.” From there, they connected that 100 pennies makes a dollar, so a penny is 1/100th of a dollar, or a percent. These types of words work perfectly to connect students with limited English language proficiency.
I also try to draw out as many diagrams and charts as possible. I tend to stay away from just writing steps and getting more into the actual thinking behind how we approached a problem. For instance, when solving multi-step equations, I have students think in terms of order of operations, but only when appropriate. Some students find that moving variables so they don’t get encumbered with negative numbers helps. As long as they follow some of the maxims, they can think through some of the solutions without going through a “process.”
Response From Alexandra Mattis
Alexandra Mattis is a mathematics teacher and special educator with over 10 years’ experience as a teacher, administrator, curriculum developer, tutor, and advocate in New York and Maryland. Through Your Math Rx: Clinical Mathematics Interventions, she offers tutoring, consulting, advocacy, and workshops focusing on students with special learning needs to families, schools, and professionals in Westchester County, NY and Fairfield County, CT. Follow her on Twitter @YourMathRx:
Our knee-jerk reaction is to drag out the flash cards and long problem sets. This is an easy, but ineffective, approach. Teachers, through constructivist-based curricula and the Common Core Standards, are now teaching arithmetic from multiple approaches. Parents may be wary of this, but it is all connected and adds layers of understanding to concepts so innate and “obvious” that they are difficult to explain.
The simple answer is that people learn information by using it. The more ways they use a concept, the more a child will internalize it. Strategies such as counting methods and manipulatives do help cement understanding, but real-life situations and games that require kids to solve a problem using arithmetic are the most authentic and effective.
Moving on is more a function of mindset than concepts. Arithmetic operations should naturally reinforce each other; learning multiplication can help solidify addition facts.
However, there are two potential roadblocks.
The first is a “mathematics learning disability,” which includes autism, nonverbal learning disability, visual- or auditory-processing disorder, and dyscalculia. If a child does not absorb the spirit and intent of a process, if a child lacks understanding of the meaning behind operations or numbers, they have a disability that requires a more intensive and personalized approach to mathematics.
The second is anxiety. This causes the child to see a long list of prerequisites they are not meeting, where they should see a web of interconnected concepts. An anxious child may not internalize the connections taught in class and may perceive each approach as being a separate tower of procedure and facts (this does not necessarily signal an undertrained or uncaring teacher). To avoid failure, they will need one-on-one help and at-home interaction with concepts.
For links to online (and real-life) activities, games, and advice, visit my site at YourMathRx.com.
In addition, for parents and teachers alike, the National Council of Teachers of Mathematics (NCTM) has some wonderful online lessons, activities, games,and support. There are also wonderful sites like math.com,coolmath.com, hotmath.com, and the Math Forum at Drexel that are full of games and advice. Apple support is incredibly kind and helpful if you want to check out the available apps for your family’s iPad (most are also available for Android tablets). Physical games are also wonderful to help families interact around math: The 24 Game, Equate, and Sumoku are classic, interactive games that are useful through high school, and companies like Learning Resources and Edupress make other fun, well-constructed games.
Thanks to Sue, Anne, José and Alexandra for their contributions!
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