If you've been to Jewish kindergarten, you know the number 613. If you look at the pomegranates below, surely you doubt that each contains 613 seeds. I know I do.
This colleague's statement was tremendously helpful because it helped me ask myself a few key questions:
1) Why does Judaism like to assign numbers to rituals and historical events? (Apropos of Yom Kippur - the 100 Shofar blasts and Sisera's mother, for example.)
2) What are we supposed to do with these numbers? How literally are they to be taken? In the Sisera's mother example, I could make a compelling argument that if someone is weeping and mourning, it would be incredibly difficult if not downright impossible to separate and count all her cries.
3) What underlying messages about numbers and about intellectual values are we supposed to glean from the way the Torah deals with numbers?
Let's turn to our main example, the pomegranate. The short answer to the three above questions are as follows:
1) The pomegranate's seeds are said to number 613 so they can symbolize the Torah. (We take our counting of 613 mitzvot VERY seriously. That number is by no means an estimate.) We associate the pomegranate with the number 613 to show on Rosh Hashana a bridge between the physical and the spiritual world. The Torah, an intellectual concept, is reflected in the physical world, which God created and is a source of great religious potential. (Essentially, Torah and Madda are forever intertwined - and that's a good thing.)
2) I don't imagine anyone expects that every pomegranate has exactly 613 seeds. I confess, I was not sure when I embarked on this project for the first time whether the number was anywhere close. My first estimate of the number of seeds was between 100 and 200.
3) This question is by far the most central to our lesson. The fact that the number 613 is so important in Judaism is worth a look. 613 is not a likely choice. If 100 is a paradigmatic round number, 613 is its exact opposite (although not in mathematical terms!) being prime, and a special prime at that.
I'm going to argue to my students that the example of the pomegranate teaches us about the importance of precision, estimation, and attention to detail, among other things. The first two might seem to be mutually exclusive, but they are both essential components of mathematical study, and I'm going to to try to convince you (and my students) that they are important to Judaism, too.
We'll be doing this activity today, on the first day of school, in addition to the "meet the teacher" basics. Here's our lesson plan, based on my trial run at home with an almost 5-year-old:
1) Buy a pomegranate and a plastic tablecloth. Make sure you have 1 good knife on hand but keep it away from the kids.
2) Introduce the topic, and ask every kid to record an estimate of the number of seeds in the pomegranate. Slips of paper work for this, or if you use clickers or other mobile devices in your classroom, this is an ideal way to collect the data quickly. For more information about using estimation in your classroom, I recommend Andrew Stadel's Estimation180 website. What you can't allow the kids to do is make a wishy-washy estimate like I did above. 100-200 is really not one estimate - it's at least two. Tell the kids to pick one and stick with it. I decided not to share my estimate with my students - I didn't want them thinking I was pushing them in a particular direction.
3) A good estimate has multiple parts. Mr. Stadel likes to ask students, "What's too low?," "What's too high?," "What is your estimate?" and "What is your reasoning?" This all takes time, but is worth a quick discussion. If you're not using clickers, a think-pair-share is also a good venue for every kid to get a chance to speak.
4) Cut up the pomegranate. scoring the sides and breaking them apart by hand so you don't cut the seeds and get an even larger mess. With gentle hands, you won't have too much pomegranate juice to deal with. Lay out its seeds in orderly rows. Make sure the kids wash their hands first, and try not to let them drop seeds all over the floor. I'll talk about measurement error more in my Sukkot post (coming on 9/24), but in the meanwhile, we'll attempt to be precise by:
a) not dropping seeds
and
b) putting the seeds in clear rows of 10.
Counting to 200 or more would be difficult to do precisely, so groups of 10 allow us to just count rows.
Since this is the first day of school, some students will be setting up their binders, so the seed-counting will be done by those who finish quickly. The more kids, the faster it will happen.
I did the whole activity in 33 minutes, essentially alone because the 5-year-old opted out. For this to work in my classroom, it needs to take less than 20 minutes, so the more students helping, the faster it will go. I hope.
As a technical note, popping the seeds out is easy and fun, but putting them in rows is tricky on a plastic surface. Today's 6th grade session will hopefully be on a tablecloth that is paper on top and plastic on the bottom, to make the seeds easier to work with. You'll also note that the seeds don't photograph well on a shiny background. (If you're reading this post more than 36 hours after I post it, I'll probably have replaced the photos here with ones that don't look terrible.)
In conclusion, we note our final number and maybe even graph it on a number line or a histogram with everyone's estimates. Then we make the following points:
i) Judaism cares about numbers, quantities and math. Studying math can be a spiritual pursuit because God created, and generally works through, the natural world.
ii) Precision matters. The rabbis care very much that there are 613 laws. They will double up laws and leave some out to arrive at 613. In our math studies, we apply this by taking care in our work and learning how to precisely answer challenging questions.
iii) Estimation is a useful skill.
The first pomegranate contained 592 seeds. My largest estimate was 3 times too small. Wowza! Estimation is a skill and if you don't practice it, you won't develop it. I always tell my students - especially the strong ones - you should know before calculating whether your answer will be closer to 40 or 400. In this case, I flunked that test. (I could have gone back and edited my original estimate but honestly, being that far off really drives home my point.)
Without multiple trials, I can't calculate variance, standard deviation or perform any other statistical analysis to predict whether other pomegranates will be the same. I encourage you to share your pomegranate counts with me using the form linked below if you do this activity so that I can build a data set over time.
I can tell you that 592 seeds in the pomegranate means that it was missing less than 4% from the traditional estimate. That's incredibly close. Not only is my childhood not ruined, I am impressed yet again with the mathematical insight of our Sages - a theme which will feature prominently on this blog.
Shana tova! Add your pomegranate data here.
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Update: My 6th graders did the activity in about 20 minutes. I scored the pomegranate and divided into sections and they did the rest. They counted 610 seeds. It was a nice way to start the year.
You'll note that they grouped their tens randomly instead of in grids. Most heartening was the way almost everyone was able to participate.
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