That Day We Added It All Up 那一天,我们聚在一起学数学

As teachers, we continue to learn new approaches to explore mathematics and are faced with new opportunities to provide the best learning experiences for our students. We are constantly participating in different professional learning to keep our teaching practices in line with the most up-to-date research on how students best learn. We collaborate constantly with fellow teachers, administrators, and instructional coaches to learn from each other, and to make sure our classrooms are true learning environments.

作为老师,我们一直在学习对数学进行探索的新方法,同时还面临着向我们的学生提供最佳学习体验的机会。我们不断参加各种职业培训,以在给学生提供最好的学习体验这件事上让我们的教学实践跟上最新的研究成果。我们不断与其他老师、行政管理人员以及指导教练合作,相互学习,来保证我们的课堂是真正适合学习的环境。

I am also a parent of twins in elementary. As parents, we want to be informed and involved in our child’s education, so we can help at home. We engage with our kids asking questions about what they learned at school at every dinner conversation. We want to be a positive influence on our kid’s emotional and academic life.

正是因为这些原因,作为上海美国学校浦西校区的小学部数学指导教练,作为正在读四年级的双胞胎的家长,我开始参与到我们的家长社区中来。在过去两年的时间里,我们举办了多场“数学交流会”,介绍了我们在上海美国学校教授和学习数学的方法。自第一场活动吸引了90位家长以来,这个活动的反响非常好。

These first meetings were about sharing with our parent community that recalling facts, procedures, and formulas is merely arithmetic, and while it seems to “work” for elementary and middle school, it does not prepare students to solve real mathematical problems, which is the main goal of mathematics: to solve “messy” or “unfamiliar” complex problems.  Mathematics is about making connections and solving problems by making sense of numbers, by using numbers flexibly, by using concrete and visual representations to build understanding. When students truly understand a concept and they can visualize it, use a model or representation, and can explain it, then they are ready to make a connection to a new math concept where they can apply their knowledge accordingly to solve the new “messy” problem.

在这些活动中,我们让学生家长认识到,记住数学事实、程序和公式仅仅是停留在算术层面,尽管这似乎对于小学和中学来说“奏效”,但这并不能让学生解决真正的数学问题,解决“棘手的”或“不熟悉的”复杂问题才是数学教学的主要目的。数学就是要对数字进行理解,以灵活的方式使用数字,使用有形和直观的方法帮助理解,从而建立联系,解决问题。当学生真正理解一个概念,能够对其进行可视化呈现,使用模型或表征,并对其进行解释的时候,他们就能够与全新的数学概念建立联系,运用他们学到的知识解决新出现的“棘手”问题。

Parents continued to want to learn more about this conceptual approach to learning mathematics, so we invited them to participate in four sessions called “Math in March.”

家长们希望进一步了解数学学习的概念化方法,于是我们邀请他们参加了四场名为“三月数学”的活动。

We started by showing the progression of addition and subtraction strategies our students use to build understanding in solving problems. Parents were engaged using a variety of CONCRETE manipulatives, to make sense of place value when adding and subtracting without the need to use any rules or tricks such as “carry” or “borrow”. They were able to use place value strategies to understand how to make tens, or how to “unbundle” a ten when subtracting. They continued to use PICTORIAL representations such as tape diagrams or number lines to model the problems. They understood the difference between modeling and fake modeling (which is just drawing an answer found by an algorithm). Using visual representations is how we really MODEL a problem, then we can choose other strategies to help us solve.

首先我们向家长们展示学生们用来解决问题的加减策略。家长们使用一系列“有形”教具在加减运算过程中对位值进行理解,无需使用“进位”、“借位”等任何规则或技巧。他们能够使用位值策略理解如何加到十,或者是在进行减法运算时如何对十进行“拆分”。随后,他们使用尺图、数轴等“图形”表示对问题进行建模。他们理解了真建模和假建模(只是用算法求出答案)之间的差别。我们使用直观的方法对问题进行建模,随后我们可以选择其他策略帮助我们解决问题。

We moved onto ABSTRACT strategies and then expanded into multiplication and division, and in later sessions into fractions. (Fractions were a hit with the parent community. They were so excited to use fraction tiles, unifix cubes, number lines, area models, and tape diagrams to understand how to add, subtract, multiply and divide fractions conceptually!)

接着我们介绍了“抽象”的方法以及乘除法和分数运算。(分数深受家长喜爱。他们非常兴奋地使用分数瓦片、方块积木、数轴、面积模型和尺图理解如何从概念上对分数进行加减乘除!)

The constant sounds of “ahhhhh” and “ohhhhh” were just fabulous when parents were shown a visual representation of what ½ x ¼ looks like! Parents were able to see past the formula and tricks and able to see in a new, clearer way what one half of one fourth really looks like, and why the product is one eighth. The room was loud, Chinese and English language, iPhone flashes, unifix cubes everywhere! Our parents were doing mathematics!

当看到 ½ × ¼ 的直观表示时,家长们发出了一阵阵的惊叹声。家长们能够透过公式和技巧以更加清晰的全新方式看到四分之一的二分之一到底长什么样以及乘积为什么是八分之一。家长们议论纷纷,有说中文的,也有说英文的。大家纷纷拿出手机拍照,到处都是方块积木。看哪,家长们在学数学!

Annie Barnard, mother of 5th grade and kindergarten students, said of the experience, “Most of us were impressed with simple questions we had been taught in the past and knew how to do, but never got to understand why. You taught us to visualize the math. Seeing the answer make sense in front of our eyes really gave us a good vibe for the first time, and got us wanting to learn math again. We’re now helping each other understand how to solve math problems without worrying about not being able to remember the right way – because there isn’t one way to solve math, there are many ways.”

作为五年级学生和幼儿园学生的妈妈,Annie Barnard表示:“让我们大多数人印象深刻的是那些我们过去学过并且知道如何解答但从不知道其中缘由的简单问题。我们在用直观的方法学习数学。亲眼看到答案呈现在眼前真可谓让我们眼前一亮,也让大家升起了再次学习数学渴望。我们现在能够相互帮助,理解如何解答数学问题,而不用担心记不住正确方法,因为解答数学问题并不只有一种方法,而是有很多方法。”

What can parents do at home to support their child to feel successful in mathematics?

家长们在家里能够做些什么,从而让他们的孩子在数学上获得成就感呢?

*If you find a mistake in their homework, instead of saying, “That is the wrong answer, do it again.” Try saying, “guide me through this problem, how did you get that answer?” Most likely, when explaining their strategy, they will catch their own mistake.

*如果你在他们的家庭作业中发现了错误,不要说“这个答案不对,重做一遍”,而是要说“给我讲讲这个问题,你是如何得到这个答案的”。大多数情况下,他们在解释做题策略时,就会发现自己的错误。

*When doing homework, ask: “Can you think of a different way to solve that?”

*在做家庭作业时,你可以说:“你能不能想出解这道题的另一种方法?”

*Play with numbers, for example: find all the different ways to make 24, or all the different ways to solve 16 x 9.

*玩数字游戏,例如:找到得到24的所有方法或解答16 × 9的所有方法。

*Solve math puzzles.

*解答数学难题。

*Play math games to practice fluency, fun games, not flashcards!

*通过玩数学游戏做到熟练掌握,要玩有趣的游戏,而不是数字卡片!

Across both campuses, at SAS we use a balanced approach to teaching and learning mathematics. Teachers and students not only focus on concepts and procedures, they focus on the use of visual representations to model real-world situations, they problem solve, they communicate and reason in their thinking, and make connections to their future learning.

在上海美国学校的两个校区,我们使用一种两者兼顾的方法开展数学教学。老师和学生不仅专注于概念和程序,还专注于使用直观方法来对现实生活情形进行建模,解决问题,沟通并推理他们的想法,与未来学习建立联系。 

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