By Amy Hand, Head of Middle School and Ilsa Dohmen, Director of Teaching & Learning
Quick: do the following subtraction problems without a calculator!
1. 76 – 59
2. 1000 – 384
3. 812 – 604
4. 1023 – 529
So began our parent learning event on Wednesday, December 11, which focused on Hillbrook’s JK – 8 mathematics program. We asked parents to engage in this quick arithmetic task and then contrasted their strategies with those that our students learn to illustrate an important point: the way that we learned to do math is quite different from the way our kids are currently learning to do math. While the former, “traditional” approach usually emphasized procedures and encouraged the use of a single standard algorithm for each computational skill learned, today’s teachers emphasize conceptual understanding and number sense, teaching young mathematicians a range of strategies intended to develop fluency with numerical reasoning. For instance, the first problem can be solved by “counting up”: 59 + 1 = 60, and 60 + 16 = 76, so 59 + 17 = 76; this means that 76 – 59 = 17.
The second problem can be answered by exploiting constant differences: the distance between 1000 and 384 on the number line must be the same as the distance between 999 and 383 since we shifted each number one unit to the left. We can find the answer to that simpler problem (999 – 383) instead: 616. Students may use “decomposing”—using place value to think of a number in parts—to answer the third problem, perhaps by reasoning that 800 – 600 = 200 and 12 – 4 = 8, so the difference is 200 + 8, or 208. Last but not least, we can always rely on the good old standard algorithm we all learned in school for #4: stack the numbers and use regrouping (which was called “borrowing” when I was in middle school) to get the answer of 494.
Of course, any of the above strategies can be used for any of the problems—which, in a way, is the point: when you learn to think flexibly and creatively about numbers and their relationship to each other, you can use your critical thinking skills and number sense to select the strategy that is most efficient (or just easiest for you!) for any given problem.
This key idea exemplifies Hillbrook’s instructional approach to mathematics, which aims to balance the development of procedural fluency (i.e. getting the answer to skill-based problems quickly and accurately, which is what many people associate with being “good at math”) with the development of number sense, critical thinking, and problem-solving capacity. By attending to these four domains, we hope that students become not only better mathematicians, but better thinkers.
Instructional Coach Autumn Vavosa then shared some of the substantive outcomes of Hillbrook’s math audit completed in Spring of 2017: our Math Philosophy Statement, our math learning outcomes by grade, and our math process skills (all of which are articulated in the presentation’s slide deck). Middle School Math and Science Lead Clara Ngo elaborated on the importance of the process skills: these are the capacities that describe how students engage with, think about, and communicate their understanding of mathematics to others. Prioritizing these process skills alongside our content skills is a critical feature of our program and one that allows our students to develop thinking skills that transcend the study of mathematics.
In order to see how these guiding principles about math instruction and math learning play out in our classrooms day to day, parents were then presented with two math problems (you can check them out in the linked slide deck!), one from our Lower School curriculum and one from Middle School. We asked parents to engage with these problems in the same way that we ask our students to; this meant that they started with independent think time, then collaborated with peers while LS teachers and MS math faculty facilitated their work, and finally created a poster showing their reasoning and solution to the problem. We concluded with a structure frequently used in math classrooms: the gallery walk, in which people read and review others’ work and post their own feedback and questions in response using post-its.
Afterward, attendees reflected on how it felt to do this math together in this way. One parent shared, “figuring out the problem to start, on my own, I had to manage my stress. Then talking to people really helped me. But with these problems, I was still stressed. Bar modeling helped me a lot–those were more visual and that helped it make sense.” As Head of Middle School Amy Hand noted, “It’s not like following a recipe. Real math is not completing a set of steps to get the predetermined answer. It’s more like filling in a crossword. You don’t go all the way through the clues in order. You probably start with some clues that are easiest for you to solve the puzzle, then go back through to see what else you can figure out from there.”
Another parent commented that “our group generated a lot of great ideas but having never worked together before, none of us was really listening well. This made me think about how challenging it is for the teachers to help students hear each other, so they can benefit from the ideas.”
Parents also asked great questions, including how the school balances skill practice and memorization that is helpful in math with doing richer problem-solving and expressed a common interest in knowing more about what is happening in their child’s math class so they can continue the conversation at home; one noted, “I read to my kids and we talk about stories, I’m building routines for also doing math at night before bed.”
Another theme in the questions concerned how our program allows for differentiation among learners within a class. Math teachers noted that the ideal problems to use in class share certain features: they are open-ended, which means the structure of the problem doesn’t immediately point to a particular path for solving it; they are “low floor/high ceiling,” meaning that they are accessible to all learners while being rich enough to offer ample opportunity for highly competent or curious learners to explore; they can be solved using multiple pathways, which allows students to learn from each other and make connections between strategies; and they are adaptable so that a teacher can create a simpler or more complex related problem in depending on individual needs. Amy shared, “The great thing about using problems to motivate math learning, versus skills to motivate math learning, is that there are always natural extensions. For example, I could say to a child, ‘Okay, you used x to represent the amount of candy each kid had at the start of this problem. Can you set up a different equation in which x represents how much candy was left over? And do you get the same solution when you solve it?’”
We were grateful for the opportunity to share our math program with parents, to joyfully engage in the “doing” of mathematics together, and to hear directly from parents what their questions and concerns are regarding math learning. If you were unable to attend this event, we encourage you to take a look at the slide deck that was shared at Wednesday’s event. You can also view the Math Overview Handouts by grade level, and these three articles we shared:
- Think you’re bad at math? You may suffer from ‘math trauma’
- All The Mathematical Methods I Learned In My University Math Degree Became Obsolete In My Lifetime
- Memorizers are the lowest achievers and other Common Core math surprises
To continue the conversation, Hillbrook families can join us Monday, January 27, at 8:30 AM on campus for our “Reaching Beyond Ourselves to Make a Difference” parent learning event. Future dates and topics can be found on our Parent Education Map.
