There are two types of assessment based on its goals or use. One is what I call assessment in the service of teaching. The second is what I call assessment in the service of learning. Assessment in the service of teaching refers to the use of assessment information to improve teaching while assessment in the service of learning refers to the use of assessment information in the form of feedback to keep the learners to the task of learning. This post is about assessment in the service of learning. Continue reading “Six ways to give feedback to students to keep them in the task of learning”
Tag: lesson study
Math Teachers at Play blog carnival #50
When I sign-up to host the May edition of Math Teachers At Play blog carnival organized by Denise of Let’s Play Math blog, I didn’t know it will be its 50th edition. Wasn’t I lucky? It’s a milestone for MTAP. Kudos to the organizer and supporters of MTAP. But I got one little problem. It is a tradition in math blog carnivals to always starts with saying something mathematically significant about the n in its nth edition! Oh dear. Things I associate with the number 50 are mostly non-mathematical like golden anniversaries!
Wikipedia to the rescue:
Fifty is the smallest number that is the sum of two non-zero square numbers in two distinct ways: 50 = 12 + 72 and 50 = 52 + 52. It is also the sum of three squares, 50 = 32 + 42 + 52.
And I didn’t know that until I hosted this carnival! I’m a teacher I have to ask: “So what’s the next bigger number to 50 that is the sum of two non-zero square numbers in two distinct ways?”; “What are other numbers that can be expressed as a sum of the squares of consecutive numbers?”; “What about those numbers that can be expressed as sum of cubes?”;… There is always something to investigate in math. One of the major objectives of school math is to get students into this thinking habit without us telling them to do so but I’m digressing from my topic now. Let’s get to the great posts submitted for this edition.
1 – How many bricks are in this building? Says its author Paul Murray: This is an activity I’ve used for years and recently wrote up for a class. It integrates many problem-solving methods, multiplication, addition, and place value concepts, estimation, and organization of data. It also takes the students outside with a clearly defined task to accomplish.
2 – Wolfram Alpha. Says it author Coleen Young: This page is from the student version of my blog and has several slideshows showing the syntax for WolframAlpha including a fun show at the end on the sillier questions one can ask! I started this student version because they can just be given the link. One of my former students emailed me recently to tell me how much she was using WolframAlpha at university.
3 – New intuitive ways of learning math by Mohamed Usama. Says Mohamed, “I am a student and I love game programming. CREVO is just my virtual startup where I publish all my ideas and other news. Math Operations is a game that won local game development competition. That time, I developed this game in Flash. It was just a 48 hour competition but still idea was executed well. At the time when I was receiving my prize I announced that soon, I’ll publish it for all Android devices and here it is. I finally developed this game for all Android & Amazon Kindle Fire devices. Designed graphics (SD & HD) for tablet as well. Last week I published my new version 1.5 and its available on Google Play (Amazon is still reviewing it). I hope you people will love it. I need high support because I really want to make games for kids, education sector is what my target is.
4 – Guess my rule says its author John Golden is a story of an algebra lesson based on a simple, common social game.
5 – You Want School Reform? Brace Yourself…. submitted by Matt Wilson. Writes Matt in the post “Anybody building a house needs to start by building a foundation, but our system is teaching foundation building without ever teaching anyone what a house actually is…”
6 – Missing Angles says the author of Five Triangles is a non-trivial math problem for middle school students requires some actual thinking.
7 – An elegant solution: An algebra problem from 1798 by Dan Pearcy. Says Dan, I stumbled across this great little problem on John Cook’s blog (The Endeavour) during the weekend. The reasons that it’s so great are two-fold: (1) Most people think they’ve solved it when they have four solutions from their equation when in fact they have not considered that the equation could be written in four different ways. (2) The solutions are so elegant. Possibly because they are all based around the golden ratio.
8 – More on Microsoft Equation Editor says John Chase is a follow-up and more in-depth discussion of http://mrchasemath.wordpress.com/2012/03/15/microsoft-office-equation-editor/.
9 – Sidewalk Math: Functions. No name was supplied but its from a blog called “The Map is Not the Territory”.
10 – 9 TED talks to get your teens excited about math shared by Caroline Mukisa. A great collection.
11 – Thinking (and teaching) like a mathematician. Says Denise, “Being ‘good at math’ means much more than being able to work with numbers. It means making connections, thinking creatively, seeing familiar things in new ways, asking “Why?” and “What if?” and “Are you sure?” If we want to teach real mathematics, we teachers need to learn to think like mathematicians. We need to see math as a mental game, playing with ideas.”
12 – Tiger’s Mum presents Geometry: 2D and 3D posted at The Tiger Chronicle.
13 – Another Proof of the Sum of the First n Positive Integers and The Mathematics of the Poles shared by Guillermo Bautista. The first shows a geometric proof and the second post is a discussion on the connection among poles of the earth, the latitudes and longitudes, and the polar coordinates.
14 – Planning and Analyzing Mathematics Lessons in Lesson Study by Erlina Ronda (that’s me). This is a powerpoint presentation for researching lessons with your colleagues. Lesson study is schools-based teacher-led professional development model.
15 – The nature of math vs the nature of school math. This is my top post this month. Everybody is concerned about the great divide between math and math education.
The next MTap Carnival will be hosted in Math Mama Writes.
Math Knowledge for Teaching Addition
This post is the second in the series of post about the Math Knowledge for Teaching (MKT) where I present task/lesson that teachers and interested readers of this blog can discuss. The first is about Tangents to Curves, a Year 12 lesson. This second post is for young learners.
The task
How many small cubes make up this shape?
This is a pretty simple task. Any Grade 1 pupil will have no difficulty giving the correct answer. All they need to do is to count the cubes. Yesterday, in one my workshop with teachers about lesson study, we viewed a Japanese lesson using the same task but was used in such away that children will learn not just counting.
The lesson
Before this lesson the class already learned that putting together concept and the symbol + and =.
The pupils were given small cubes to play with on their tables. After a minute, the 2x2x2 cube was shown on the TV screen and the teacher asked the class to predict how many small cubes make-up the shape. Some used their cubes to make a similar shape without the teacher encouragement to do so. The cubes were only there to help those who might have trouble imagining the bigger cube were some parts are not shown. The pupils counted the visible cubes one-by-one and then those not seen in the drawing (a drawing of the cube is posted on the board). But, the teacher was not just after the answer 8, he was after the learners’ counting strategy. So he asked: Can you use the + sign to show us your counting strategy? Some of the students answers were: 4+4 = 8, 2+2+2+2 = 8, 6+1+1=8. But, the teacher was not only after this, he wanted the class to realize that this number expressions may have come from a different way of looking at the cube. He started with those who wrote 4+4 to show the class how this counting was done. There were two different strategies: halving the cube vertically and the other horizontally which the students demonstrated using the cubes. All throughout the teacher was asking the class, “Can you follow the thinking? “Do you have a different idea?” “Who has another idea?”
After the summarizing the different ideas of the pupils in the first task, the teacher gave the second task:
What is your idea for counting the small cubes in this shape? Show your idea in numbers and symbols.
The shape was projected on the TV screen as the teacher rotated the shapes. The pupils came-up with different combinations of visible and not visible cubes like 7+3 = 10, 4+6 = 10, etc. They were invited to explain these expressions and their thinking using the drawing on the board. The teacher did not have any difficulty getting the answer he wanted from the pupils: “We already know that this shape (the big cube) is 8 so we just add 2 (8+2 = 10).
Questions for Teachers Discussion/Reflection:
- What about numbers will the pupils learn in the lesson?
- What is the role of technology and visuals in this lesson?
- What about mathematics is given emphasis in the lesson?
- What mathematics teaching and learning principles underpin the design of the lesson?
Remember this quote from George Polya: What the teacher says in the classroom is not unimportant, but what the students think is a thousand times more important.
For further reading:
Patterns in the tables of integers
Mathematics is said to be the science of patterns. Activities that involve pattern searching is a great way to engage students in mathematical thinking. Here are some of my favorites for teaching positive and negative integers. If you are wondering why most of my posts are about integers it’s because I’m doing a Lesson Study with a group of Mathematics I (Year 7) teachers about this topic. Last week we concluded the first cycle of our research lessons on teaching subtraction of integers.
The first task students need to do with the tables is to list 3-5 observations. From there you can start asking the ‘whys’ for each observation. Sample questions are provided for each table below:
1. Adding integers
Sample questions for discussion:
a) Under what conditions will the sum be positive? negative? zero?
b) Why are there the same numbers in a diagonal?
c) How come that the sum is increasing from left to right, from bottom to top?
2. Taking away integers
Sample questions for discussion:
a) Under what conditions will the difference be positive? negative? zero?
b) Why are there the same number in a diagonal?
c) How come that for each row/column, the difference is decreasing?
3. Multiplying integers
Sample question for discussion:
a) Under what conditions will the sum be positive? negative? zero?
4. Dividing Integers
Sample question for discussion:
a) Under what conditions will the difference be positive? negative? zero?
b) Does dividing integers still results to an integer? What do we call these new numbers?
Feel free to share your ideas/questions for discussion.
You may also want to share other math concepts that students can learn with these tables.