Thanks so much for sharing your experiences with the lesson. You know your students best so if you gut is telling you to take a little longer on the lesson than I would listen to it. It’s a big lesson. It’s okay to spend more time.
It sounds like the students are struggling in the right ways. Working on abstract problems like these are a key skill in computer science. You might be able to make some progress if you give students some insight into the connection to the decimal number system and how number systems work in general which can be found at the end of that lesson.
A couple things to consider:
Have students to play around and actively rearrange the the combinations. The rules they make DO NOT have to fit the order that they originally wrote down.
It’s not essential that students make rules for getting to all 27 combinations, it’s more important that they grapple with the elements of a number system (common symbols and predictable rules)
Thanks. We ended up having to work it out as a class. We started lesson 7 and I think the instructions on the first part of the worksheet are confusing. But since we started lesson 7 on the same day day we finished lesson 6 they were able to follow the rules we created.
Based on what I’m seeing in my students so far, I think one group will struggle and the other will run with it. The struggling group is too ingrained in the “concrete world” and has trouble thinking abstractly - they need exact rules of how to do things and are having trouble understanding that they are the ones who are making up the rules… The other group doesn’t seem to have as many “rules” that they follow so I think they can do more abstract thinking.
I start the second day with a scavenger hunt on computer history which covers both hardware and software topics. This link to answer most questions is: http://lecture.eingang.org/index.html. It doesn’t like Internet Explorer; however. The reason I mention this is that the scavenger hunt provides a nice lead-in to number systems with its question about Shamanite Tradition.
Should I encourage the students to denote a representation for each combination? For example, should 3 circles represent ‘0’ and 2 circles with one triangle represent ‘1’? I am not sure what the protocol should look like to move from one combination to the next. I don’t see my students having any issue with discovering the different combination, I am just a bit confused on the rest of it. Thank you!
after your students have come up with all of the different combinations, encourage students to reorder the groups of three so there’s a pattern for getting from one group to the next. they should pick a pattern that gives them space to develop rules for recreating the list in order. these rules will be rules for counting in the number system your students have just created!
note that are LOTS of different types of rules students could come up with, and lots of different patterns they can looks for.
This activity was fantastic. Some key learnings from my class:
Emphasize that each number is only three shapes long - some students created longer patterns but still used place value to represent the numbers. I had students share these too and discussed why they had more values represented than
The word “Base” came out organically.
I found a good “testing” question was to ask students to represent a specific number and then asked them to show me a number higher or lower - especially to transfer over tens places (show me 18, then 19, then 20) to see if we could get the rhythm.
Overall fantastic - students ASKED to see eachothers because there were so many different ideas out there.
I would say 40% of my students could not grasp the system part of the activity. The students could grasp that there were 27 possible permutations, but not understand the method used to accumulate them. I think if I were to teach this again I would have students create all 27 first then work backwards to explain the process used to list of possible permutations.
I agree, students learn when they struggle through a process, before studying the number systems. I found the lessons 6-9 fit well together. Lesson 6 starts with students exploring the counting and representing numbers with symbols then they move on to creating their own numbering system. I feel the lesson is well written and I will use all the materials provided. I especially found the Teaching ,Tips and Tricks extremely helpful in showing how to teach the lesson . Being able to see the shapes being used and hearing the questions asked helps me to be better prepared to teach the lesson.
It might be easier to start with 1 circle and 1 square and ask how many permutations (2 or 2^1). Next, how may permutations with 2 circles and 2 squares (4 or 2^2). Kids figure these out real quick and can generalize it beyond the 3-shape, 3-length number of this exercise. They could readily, for instance, predict how many permutations would be possible for 4 shapes and a 5-length number (4^5).
Not to be picky, but in U1L6 Dot 2 I would change “If you just had a circle and a square, how many 3-shape permutations could you make?” to “If you had just circles and squares…”. My kids (and I) were confused the first time we read this about how one could make ANY 3-shape permutations from only 2 shapes. Sort of like fishes and loaves…
what sort of rules should we be coming up with. I’m teaching myself this course for next year and even though i’m a math teacher i’m struggling with this concept. Could you give me some example such as if i start with three circles what could be the logical next two permutations and why would these be appropriate?
There is not exactly a correct answer to this question. If you are a math teacher it might be easiest to think about the circle, triangle and square as representing a number system with 3 digits (0,1,2). In this case it probably seems logical to you that if we have 000 the next number is 001 and then 002. There isn’t a ordering thats as easy to see for circles triangles and squares so students have to come up with the ordering themselves. One group may say that the order is circle,triangle, then square another may say its circle, square and then triangle. In addition the order in which they change the circles could be different in that one group starts at the left end and the other starts at the right.
The big takeaway here is that we can represent numbers with these 3 shapes organized in different patterns and there are many ways that pattern could be made. That will allow you to discuss binary and hexidecimal with students later but early on here allows you to avoid numbers representing other numbers which can be super confusing for students.
