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24 Hudson Valley Parent ■ July 2014 A student or teacher with a profound un- derstanding of funda- mental mathematics could move readily and easily between different answers, like: 4×18 = 18+18+18+18 (be- cause by defi nition 4×18 is the sum of four 18s). 4×18 = 4×2×9 = 8×9 (because multi- plication is associative and we can regroup the factors). 4×18 = (4×10)+(4×8) (since four 18s is the same as four 10s and four 8s). 4×18 = (4×5)+(4×5)+(4×5)+(4×3) (since four 18s is the same as four 5s, four 5s, four 5s, and four 3s). All of the above methods and more are emphasized in the Com- mon Core. But gaining this fl uency takes time, so students don't com- plete their times tables until the end of 3rd grade and don't learn the standard algorithm for multiplica- tion until 4th or 5th grade. It's worth noting that even without the standard algorithm, 3rd grade students can still answer 4×18, through any of the approach- es outlined above (or any of a num- ber of others); in fact, this question is from a 3rd grade exercise. More signifi cantly, elementary arithmetic is not important because it allows you to calculate without a calculator, but because it prepares Dear Jeff, Are students learning less under the Common Core? Recently Louisiana posted sample 7th grade questions based on the new Common Core curriculum. A lot of people noted that this was the type of problem they solved in the 2nd or 3rd grade, raising questions whether students are learning less under the Common Core. To answer this question, we must answer another: What does it mean to learn mathematics? In Knowing and Teaching Ele- mentary Mathematics (1999), one of more infl uential studies on elemen- tary mathematics education, Liping Ma considers the question: Why do students in China routinely outper- form American students on interna- tional math exams? She found that while elementary mathematics teachers in China had considerably less formal schooling, they possessed "a profound under- standing of fundamental mathemat- ics" and conveyed this understand- ing to their students. For example, consider the ques- tion: "What is 4×18?" For decades, mathematics instruc- tion in the U.S. focused on computa- tional competency, whereby student learning in mathematics was mea- sured by whether they could apply a standard algorithm for multiplication: 18 ×4 72 However, this is just one of many possible methods to fi nding the same answer. you for algebra. The pre- vious curriculum did so poorly, so that every year, of the 700,000 students who took a college algebra course, more than half of them failed. Is this because algebra can only be learned by the most advanced students? Not at all. Research shows that students with a deep understanding of mathemat- ics — such as that developed by the Common Core — can learn algebra as early as 2nd grade. By the 3rd grade, students can solve simple algebra problems like 36÷k = 9, and word problems (the bane of most algebra students) pose no special diffi culty: Leanne needs 120 tiles for an art project. She has 56 tiles. If tiles are sold in boxes of 8, how many more boxes of tiles does Leanne need to buy? This leads to solving the equation 8x+56 = 120. So perhaps students are learn- ing less. But they will be able to do more. Jeff Suzuki teaches mathematics at Brooklyn College, and is one of the founders of the Mid-Hudson Val- ley Math Teachers Circle. Email your Common Core questions to editor@ excitingread.com. Common Core @ Home Only at hvparent.com: Dear Kiersten, How will my child's individual test score affect her personally? Answers from Common Core experts Part Five of a continuing series JEFF SUZUKI Mathematics